Research Article On the Cross Correlation Properties of MIMO...

Research ArticleOn the Cross Correlation Properties of MIMO WidebandChannels under Nonisotropic Propagation Conditions

Ana Maria Pistea1 and Hamidreza Saligheh Rad2

1Military Technical Academy Bucharest 050141 Bucharest Romania2Tehran University of Medical Sciences Tehran 16656 Iran

Correspondence should be addressed to Hamidreza Saligheh Rad hamidsalighehgmailcom

Received 20 August 2014 Accepted 12 December 2014

Academic Editor Xinyi Tang

Copyright copy 2015 A M Pistea and H Saligheh Rad This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Wideband (WB) and ultrawideband (UWB) systems combined with multiple-input multiple-output (MIMO) technology increaseboth the systems performances and the complexity of the channel models required to evaluate their capabilities Because inreal scenarios waves propagate nonisotropically the accuracy of the channel model is increased if nonisotropic propagation isconsidered The channel bandwidth is the key term in the evaluation process of these systems because large bandwidths introducefrequency selectivity a unique phenomenon of WB and UWB systems with more complexity in the latter case This is due to thefact that unlike WB technology in which the propagating signal is the only affected parameter by the frequency selectivity in theUWB case this phenomenon also affects the antenna propagation pattern (APP) In this paper we developed a novel channelmodelbased on the statistical analysis of two-dimensional cross correlation functions (CCFs) ofWBUWBMIMOnonisotropic channelsA mathematical solution to assess the frequency selective behavior of the UWB APP is also presented The CCF reveals how thepower spectral density (PSD) of the channel is influenced by bandwidth nonisotropic propagation and APP

1 Introduction

The design of high quality wideband (WB) and ultrawide-band (UWB) multiple-input multiple-output (MIMO) wire-less systems requires the development of two-dimensional(2D) space-time-frequency (STF) channel models along withthe evolution from narrowband (NB) toWBUWB transmis-sionsThis evolution from NB toWBUWBMIMO channelsbrings new and more complex propagation phenomenainvolving the following open questions

(1) What are the main propagation phenomena thatappear in a 2D propagation scenario when the chan-nel bandwidth increases

(2) In a 2D scenario how does the influence of theantenna propagation pattern (APP) on NB channelsdiffer from the influence of the APP on WBUWBchannels characteristics

(3) Is a 2D channel model developed for the study of WBchannels adequate for the analysis of UWB channels

An exhaustive approach to characterize WBUWB MIMOchannels and to answer the questions enumerated aboveis to statistically analyze the behavior of their space-time-frequency (STF) channel transfer functions (CTFs) Wepropose a model for both WB and UWB MIMO chan-nels based on two-dimensional (2D) STF cross correlationfunction (CCF) between the CTFs of two subchannels ofa multicarrier (orthogonal frequency division multiplexingOFDM) channel A subchannel represents the connectionbetween two antennas one at the base station (BS) and theother at the mobile station (MS) which transmitreceive thesignal at specific time and frequency

The calculation of the CCF for WBUWB MIMO chan-nels has attracted the attention of several researchers forexample [1ndash5]Ma andPatzold extended anNBMIMOchan-nelmodel to a version adequate forWBchannels analysisThemain improvementmade to extend the applicability of theNBmodelwas to add the elementswhich introduce the frequencyselectivity caused by the propagation channel This modelpresents the 2D CCF derived only for isotropic propagation

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 904153 13 pageshttpdxdoiorg1011552015904153

2 International Journal of Antennas and Propagation

[1] Patzold and Hogstad developed a WB MIMO channelmodel with temporal spatial and frequency correlationproperties This model presents the 2D CCF derived forboth isotropic and nonisotropic propagation assuming aspecific geometry of scatterers around receiver [2] Saleh andValenzuela (S-V) developed one of the first UWB channelmodels which was proposed to reproduce the multipatheffect of indoor environments Using S-V results Zou et aldetermined a CCF expression of multiband OFDM-UWBdiscrete-time base-band channel impulse response [3 4]Abdi and Kaveh propose an ST-CCF for a MIMO frequencynonselective Rician fading channel assuming a one ring ofscatterers model around the MS [5] This model is as anexample of an outdoor environment where the von-Misesprobability density function (PDF) is suggested to model thenonisotropic propagation environment around the user

As a summary our literature review shows that thegeneral approach is to develop channel models for WBtransmissions separated from the channel models for UWBcommunications Another trend is to include the effects ofthe propagation environment (waves clustering frequencyselectivity) and to ignore the frequency selective behaviorof the antenna patterns Because UWB systems were mainlydeveloped for indoor applications the research in the fieldof UWB outdoor channel modeling was not a priority [6ndash8] From the literature review it was also observed that mostof the MIMO-CCF models are based on a specific geometryfor scatterers distribution around MS to characterize thenonisotropic propagation

Lately UWB technology in combination with MIMOsystems finds more and more interest in areas such aslocation tracking application for outdoor emergency services[9 10] location tracking and sensor applications for mobileoutdoor users [11] and medical [12] and electronic warfareapplications [13] Even more some research results [14ndash16]proved that a joint radar and wireless communication systemwould constitute a unique platform for future intelligentenvironmental sensing and ad hoc communication networksin terms of both spectrum efficiency and cost effectivenessThis evolution regarding the use of UWB systems in outdoorenvironments as radar or communication system has led tothe formulation of specific requirements aiming at investi-gating the influence of outdoor propagation on UWBmobilesystems [17 18]

The model proposed in this paper describes the CCFbetween two subchannels of an outdoor WBUWB MIMOchannel employing directional and omnidirectional antennasand in the presence of nonisotropic wave propagation in2D space Instead of assuming a specific geometry forscatterers in space (models based on specific geometries ofscatterers in the space are just able to predict the behaviorof that particular propagation scenario) we used specificmathematical relationships between the physical parametersof the wireless channel along with appropriate assumptionson their PDFs to derive a channel model based on the CCFof outdoor WB and UWB MIMO channels This model isan extension of a 2D NB channel model [19] The evolutionthrough systems with (ultra)wide bandwidths implies newpropagation effects that make the NB model inappropriate

for WBUWB transmissions In order to extend the NBchannel model to a version adequate for WBUWB channelsanalysis the mathematical structure of the model needs tobe updated with the elements which characterize the newpropagation phenomena (represented by channel frequencyselectivity waves clustering) and with the elements whichdefine the performances of the new systems (represented byoperating bandwidth frequency selectivity of the antenna)We represent the nonisotropic scatterers by the Fourier seriesexpansion (FSE) of the PDF of the propagating directionsWe also consider the effect of directional and omnidirec-tional antenna element patterns by the FSE of the APPsThe expression of the CCF turns out to be a compositionof linear expansions of Bessel functions of the first kindFourier analysis of the derived CCF is used to determinethe power spectral density (PSD) of WB and UWB channelsin a stationary scenario The mathematical set-up of thestationary scenario transforms the MIMO channel into amultiple-input single-output (MISO) channel to be analyzedas a special case The contribution in this 2D-CCF model isfourfold

(1) It includes the main propagation phenomena thatappear in 2D WBUWB MIMO outdoor scenariosand permits the evaluation of combined effects ofchannel bandwidth nonisotropic scattering APP(omnidirectional or directional) and the arrayinterelement distance on the CCF of WB and UWBMIMO channels

(2) This model can be used for the analysis of both WBand UWB propagation channels The key to apply amodel for the analysis of both types of channels is touse the appropriate expressions for APPs Since WBAPP is not frequency selective the relative bandwidthbeing a small fraction of the central frequency andthe UWB APP is frequency selective and differentantenna patterns expressions for these two types ofsystems should be used In this paper we proposemathematical expression for UWB APP

(3) It accurately reflects the influence of theMS direction(and speed) on the spectral characteristics of WB andUWB channels

(4) It gives mathematical expressions to evaluate thechannel power spectrum of WB and UWB wirelesschannels

In [20] a three-dimensional (3D) version of this modelwas presented The difference between 2D and 3D versionsof the proposed model consists of APP and nonisotropicpropagation representations In the case of the 2D modelthe APP and the propagation environment are representeddepending on azimuth angle spread (AAS) The 3D versionof the model represents these two elements as a functionof both AAS and elevation angle spread (EAS) From theobtained results it was observed that the 2Dmodel provides agood representation of how the frequency selectivity affectsthe signal spectrum A 2D model for most propagationenvironments is more eligible when we get closer to theground because a propagation environment behaves similar

International Journal of Antennas and Propagation 3

to a 2D environment Another advantage is that such modelrepresents the propagation environment fairly accuratelywith minimum complexity and difficulty The model pre-sented in this paper is an extension of the conference paperpresented in [21] There are significant differences betweenthe results presented in [21] and the content of the currentwork where both the CCF and the PSD are derived andevaluated forWBUWBMIMO channels while [21] describesonly the PSD of WBUWB systems in a stationary case forMISO channels The latter paper is mostly about PSD thederivation process of the CCF (which is the root of the PSD)was not presented and the model cannot be extended tobe used in other scenarios (eg indoor propagation MIMOchannels) This statement is also valid for [20] which doesnot contain the derivation process of the three-dimensionalCCF In the present work are provided details regardingthe calculation of the moment generating functions (MGFs-Φ

120591119904) of the delay profile (120591 119879) and details regarding the

calculation of the number of Fourier series coefficients (FSCs)for different PDFs of the AAS necessary to approximateLaplace and von-Mises distributions and how they are furtherinvolved in assessing the CCF Since in [20 21] CCF was notderived and graphically represented this connection betweenFSCs APP distribution of the propagation environment andCCF behavior cannot be established

The rest of this paper is organized as follows in Section 2the notations and the parameters included in the 2DWBUWB channels model are described in Section 3 theWBUWB CCF expression is derived and is numericallyevaluated In Section 4 the Fourier analysis of the CCFwhichresults in the PSD of theWB andUWB channels is presentedin Section 5 conclusions are presented

2 Two-Dimensional WidebandUWB MIMOModel Description

In this section we describe the propagation scenario thenotations used throughout this paper and the elements(accompanied by theirmathematical expressions) introducedin the structure of the CTF valid for WB and UWB channelsWe also emphasize the differences which exist between WBand UWB channel models and how these differences areincluded in our model

By definition a channel is consideredWB if the fractionalbandwidth (119861

119891) (the ratio of bandwidth at minus10 dB points to

center frequency) is 1 lt 119861119891le 20 while UWB channels

are characterized by 20 lt 119861119891le 200 [22ndash24] For both

WB and UWB scenarios we consider a moving MS witha constant speed vector 119881 (msec) and a fixed BS in a 2Dnonisotropic propagation environment The resulting CTF isdetermined by the 119901th transmitting antenna element at theBS side the propagation environment and the119898th receivingantenna element at the MS CTF description is based ontwo categories of elements (C1) elements assigned to thecommunication system which are the system bandwidth andAPPs at the BS and at the MS and (C2) elements assignedto the wireless channel represented by nonisotropic prop-agation frequency selectivity and clustering phenomenon

Both categories of elements enumerated above changed theirmathematical representation once with the evolution fromNB to WB and UWB channels as it is presented in thefollowing lines

(C1) The APPs of the 119901th and119898th antenna of the BS andthe MS array 119866119861

119901(Θ

119861

120596) 119866119872

119898(Θ

119872

120596) give the response ofthe antenna in terms of the propagation directions Θ andfrequency 120596 These functions implicitly include the effectof mutual coupling caused by the neighboring antenna ele-ments Are all periodic functions of 2120587 where Θ119861- and Θ

119872-are unity vectors which represent a propagation direction(DOD orDOA) at BS orMS respectivelyWe assume that theDOA (to the receiver) and the DOD (from the transmitter)are independent Therefore we represent them by their FSCsas follows

119866 (Θ 120596) =

infin

sum

119896=minusinfin

G119896119890119895119896Θ

G119896=

1

2120587int

120587

minus120587

119866 (Θ 120596) 119890minus119895119896Θ

119889Θ

(1)

For NB channels it is assumed that the response of theantenna does not change significantly over the bandwidthsince the relative bandwidth is a small fraction of the centralfrequency [19] This assumption is also true for WB channelsbut proved to be false once with the evolution throughultrawide bandwidths UWB antenna patterns are frequencyselective and this characteristic should be considered inthe design process of UWB channel models The radiationpattern depicts the relative strength distribution of the trans-mitted or received power by the antenna In [25 page 19] it issuggested that in order to correctly evaluate the received andthe transmitted power of UWB signals the radiation patternshould be determined across the entire frequency spectrumThis requirement is the consequence of the frequency selec-tivity phenomenonThus according to themethod suggestedin [25] depending on the signal bandwidth we propose twoapproaches for 2D APP calculation as follows

(a) for WB signals APP is calculated depending on thecentral angular frequency 120596

(b) for UWB signals we calculate APP depending on thecentral frequency and by integrating 119866(Θ 120596) acrossall the frequencies of the transmitted signal

Table 1 presents the APPs of two WBUWB antennasThe helical (directional) and rectangular (omnidirectional)antennas are often used for antenna arrays and (ultra)widebandwidth applications [22]

119891119867 119891

119871are the upper and the lower frequencies of the

UWB channel bandwidth the parameter ℎ is proportionalwith the size of the antenna and 119866

0is the real and positive

constant antenna gain that varies for each antenna(C2) The nonisotropic propagation is characterized by

the nonuniform distribution of the waveforms around theMS and the BS in space In order to describe a nonisotropicchannel the PDF of the propagating directions known asazimuth angular spread (AAS) can be used In the proposedmodel the PDF of the propagation directions is denoted by119891119861

(Θ119861

) at the BS and119891119872

(Θ119872

) at theMS One candidate for


Table 1 2D antenna radiation patterns

Antenna type APP 119866(Θ 120596) forallΘ isin [minus120587 120587)

Helical antennaWB 119895119866

0sdot120596

2119888sdot ℎ sdot sinΘ

UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660sdot120596

2119888sdot ℎ sdot sinΘ119889120596

Rectangular antennaWB 119895119866

0

sin(((1205962119888) sdot ℎ sdot sinΘ))((1205962119888) sdot ℎ sdot sinΘ)

UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660


119889120596

Table 2 Nonisotropic AAS and corresponding Fourier seriescoefficients

PDF 119891Θ(Θ) forallΘ isin [minus120587 120587)F

119896

Laplace119891Θ(Θ) =

119890|((minusradic2Θ)120590)|

radic2120590

F119896=119890minus((120587(radic2+119895119896120590))120590)

(119890((2radic2120587)120590)

minus 1198901198952119896120587

)

2120587(119895radic2119896120590 minus 2)

Von-Mises119891Θ(Θ) =

119890|119899lowastcos(Θminus120583)|

21205871198690(119899)

F119896=119869119896(119899)

1198690(119899)

the PDF of the nonisotropic AAS is Laplace distribution [26]Another distribution which characterizes the nonisotropicenvironment is the von-Mises PDF [27] Since these PDFsare periodic functions with period 2120587 in Table 2 we representthem by the corresponding FSC

119891Θ(Θ) =

+infin

sum

119896=minusinfin

F119896119890119895119896Θ

F119896=

1

2120587int

120587

minus120587

119891Θ(Θ) 119890

minus119895119896Θ

119889Θ

(2)

Von-Mises PDF is strongly influenced by parameter 119899which determines the order of the channel nonisotropyIn other words 119899 controls the width of DOA of scatteredcomponents This parameter can take values in the range 119899 isin

[0infin) When 119899 rarr infin 119891Θ119872(Θ) = 120575(Θ minus 120583) the propagation

environment is considered extremely nonisotropic-scatteredconcentrated at Θ = 120583 where 120583 isin [minus120587 120587) is the meanDOA at the MS For large 119899 say 3 le 119899 le 20 we havea typical nonisotropic environment [27] When FSCs aredetermined this parameter appears in the argument of theBessel functions represented by the modified Bessel functionof the first kind 119869

119896(sdot) and the zero-order modified Bessel

function 1198690(sdot)

The isotropic propagation environment can be mathe-matically represented in a similar way using the followingexpression of the FSC F

119896= (12120587)120575

119896 Laplace and

von-Mises distributions are specific to WBUWB MIMOchannels

The complexity of the nonisotropic propagation environ-ment is increased by two phenomena which appear only inthe case of WB and UWB channels

Clustering Phenomenon NB systems receive most of themultipath components (MPCs) within the symbol duration

In these circumstances the receiver acts as if there was a sin-gle multipath component Due to the better time resolutionof WB and UWB systems the multipath components tend toarrive in cluster rather than in a continuum as it is commonfor NB channels The authors of [28] define a cluster as anaccumulation ofMPCswith similar time of arrivals and angleof arrivals As a consequence of this phenomenon the CTF ofWBUWBchannels is the summation of the dominant 119868pathsand 119871 clusters In the case of NB channels the CTF appearedto be only the summation of 119868 paths

Thepropagation delay of the 119894th pathwithin the 119897th clusterbetween the 119901th (at the BS) and 119898th (at the MS) antennaelements 120591

119901119898119894119897is decomposed into three components one

component which represents the delay depending on thedistance between BS and MS and other two componentswhich vary as a function of the local coordinates of BS andMS

120591119901119898119894119897

= 120591119894119897minus (120591

119861

119901119894119897+ 120591

119872

119898119894119897)

120591119861

119901119894119897≜

119886119861

119901Θ

119861

119894119897

119888 120591

119872

119898119894119897≜119886119872

119898Θ

119872

119894119897

119888

(3)

where (sdot)119879 represents the transpose operator 120591119894119897is the delay

between local coordinates 119874119861 or 119874119872 and 120591119861119901119894119897

120591119872119898119894119897

representthe propagation delays from antenna 119886119861

119901to 119886119872

119898located in their

corresponding coordinates119874119861 or119874119872 119879119897is the cluster arrival

rate and is considered to be constant in timeΘ119861

119894119897andΘ119872

119894119897were

previously defined

Frequency Selectivity Phenomenon This refers to the differ-ent attenuation that the signal subbands undergo In ourmodel the frequency selectivity of the radio channel ischaracterized by two components These components arerepresented by APP (valid only for UWB channels) andby the term (120596

119887119908120596)

120578 The parameter 120596119887119908

represents thesignal bandwidth 120596

119887119908= 120596

119867minus 120596

119871 120596

119871and 120596

119867are the

lower and the upper angular frequencies 120578 depends on thegeometric configuration of the objects which produce thesignalrsquos diffraction The following values can be assignedto 120578 minus1 (diffraction by corner or tip) 05 (diffraction byaxial cylinder face) and 1 (diffraction by broadside of acylinder) [29] When modeling NB channels it was adequateto define the path gain depending on the path time-delay[19] This is not sufficient for WBUWB MIMO channels


where the frequency selectivity phenomenon influences thegain of the channel In the ray based propagation modelswhich can be applied to signals transmitted at high frequencyranges like WB and UWB signals it can be assumed thatone propagation path has DOA and time of arrival that doesnot depend on frequency but has a frequency dependentcomplex path gain In our model the path gain of the 119894thpath within the 119897th cluster which propagates from the 119901thantenna to the 119898th antenna is expressed as the extension ofFriisrsquo transmission formula 119892

119901119898119894119897= 12120596120591

119894119897

When putting together the elements described in (C1)and (C2) the CTF of WB and UWB channels results in thefollowing expression

ℎ119901119898

(119905 120596)

= (120596119887119908

120596)

120578 119871

sum

119897=1

119868

sum

119894=1

119866119861

119901(Θ

119861

119894119897 120596)

times 119866119872

119898(Θ

119872

119894119897 120596) 119892

119901119898119894119897119890119895(120601119894119897minus120596119894119897119905minus120596120591119894119897minus120596119879119897)

(4)

The resulting CTF is represented by the double sum-mation of clusters 119897 and of the propagation waveforms 119894over a maximum number of clusters 119871 and paths 119868 Withineach cluster (path) the signal reaches the receiver with aresponse described by the PDFs of some random variablesThese random variables are phase delay DOD and DOA Inthe resulting CTF each 119897 cluster and implicitly each 119894 waveare associated with a path attenuation gain 119892

119901119898119894119897 a path

phase shift 120601119894119897 and a time-varying delay 120591

119901119898119894119897(119905) ≜ 120591

119901119898119894119897+

(119905119888)119881119879

Θ119872

119894119897TheDoppler shift of the 119894th receivedwavewithin

the 119897th cluster is represented by 120596119894119897

= (120596119888)119881119879

Θ119872

119894119897where

119881 and 119888 are the MS velocity vector and the speed of lightrespectively

3 Two-Dimensional Cross CorrelationFunction of WBUWB MIMO Channels

The CCF expression of the CTFs ℎ119901119898(1199051 120596

1) and ℎ

119902119899(1199052 120596

2)

of two arbitrary subchannels of a MIMO channel is the resultof the following definition

119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2) ≜ 119864 [ℎ

119901119898(119905

1 120596

1) ℎ

lowast

119902119899(119905

2 120596

2)] (5)

In the CCF expression there are three dimensions spacetwo pairs of antenna elements (119901119898 119902 119899) time (119905

1 119905

2)

and central frequencies (1205961 120596

2) According to these three

dimensions we call it STF-CCF The expectation operationis performed over all introduced random variables In thepresence of enough number of multipaths by invoking thecentral limit theorem the CTF can be considered a Gaussianrandom process Therefore the above second-order statisticsfully characterize statistical behavior of the channel By

replacing (4) with (5) the CCF results in the followingexpression

119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

times

119871

sum

119897=1

119868

sum

119894=1

119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119872

119898(Θ

119872

11989411198971

1205961)

times 11989211990111989811989411198971

times 11989211990211989911989421198972

times 119890(minus11989512059611198791199011198981198971

minus119895120596112059111990111989811989411198971(1199051))]

times 119864 [119866119861lowast

119902(Θ

119861

11989421198972

1205962)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895(1206011198941 1198971

minus1206011198942 1198972)

times 119890(minus11989512059621198791199021198991198972

minus119895120596212059111990211989911989421198972(1199052))]

(6)

Regrouping dependent and independent random vari-ables and using the elements described in (C1) and (C2) weobtain

119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times

119871

sum

119897=1

119868

sum

119894=1

119864 [(1205911198942119897212059111989411198971)minus1

times 119890119895((120596212059111989421198972

minus120596112059111989411198971)+(12059621198791198972

minus12059611198791198971))

]

times 119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

]

times 119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119861lowast

119902(Θ

119861

11989421198972

1205962)

times 119890119895((1205961119888)119886

119861119879

119901Θ119861

11989411198971minus(1205962119888)119886

119861119879

119902Θ119861

11989421198972)

]

times 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

(7)

Equations (8)divide(14) present the calculation methodologyof the expectations in (7)We present the calculationmethod-ology at the MS side but one should note that the samecalculation procedure is valid for the BS sideThe calculationof this equation is performed for two cases

Case 1 Thepaths are considered similar and this is equivalentto 119894

1= 119894

2= 119894 119897

1= 119897

2= 119897 some of the random variables in this

expression become identical and by using the Fourier series


expansions for both APPs and AASs we are able to calculatethese expectations either at the BS or at the MS

Case 2 The paths are considered to be dissimilar This isequivalent to 119894

1= 119894

2 119897

1= 119897

2 we assume that the DOD or

DOA PDFs of different propagation waves are independentof each other that is Θ119872

11989411198971

and Θ119872

11989421198972

(or Θ119861

11989411198971

and Θ119861

11989421198972

) areindependent

(i) The calculation of the first expectation in (7) is carriedout as follows

119864 [(1205911198942119897212059111989411198971)minus1

119890119895(120596212059111989421198972

minus120596112059111989411198971)

times 119890(12059611198791198981199011198971

minus12059621198791198981199011198972)

]

=

Φ(minus1)

120591(119895 (120596

2minus 120596

1))

timesΦ119879(119895 (120596

1minus 120596

2)) Case 1

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)

timesΦ119879(minus119895120596

2)Φ

119879(119895120596

1) Case 2

(8)

The elements denoted by Φ120591119879(119904) represent the moment gen-

erating functions (MGFs) of the delay profile (120591 119879) evaluatedat the difference between two frequencies Φ

120591119879(119904) suggests

that channel similarity depends on frequency differences andnot on absolute frequencies The value of this parametertends to decay when the frequency differences increase Thedefinition of the MGF of a random variable 119909 with the PDF119891119883(119909) is defined as follows [30 page 415] Φ

119883(119904) = 119864[119890

119895119904119883

] =

intinfin

minusinfin

119890119895119904120585

119891119883(120585)119889120585 The probability densities of the absolute

times of arrival of clusters and paths used to calculate theMGFs are represented by [31]

119901119879119897(119879) =

Λ119871+1

119879119871

119890minusΛ119879

119871 119901

120591119894119897(120591) =

120582119868

120591119868minus1

119890minus120582120591

(119868 minus 1) (9)

where Λ and 120582 are the cluster arrival rate and ray arrival rateThe parameter 1Λ is typically in the range of 10ndash50 ns while1120582 shows wide variations from 05 ns in non-line-of-sight(NLOS) situations to more than 5 ns in LOS situations [32]

(ii) The calculation of the second expectation in (7) isperformed as follows

Considering the case of the planar wave propagation wetake into account the phase contribution of surrounding scat-terers by a random phase change parameter 120601

119894119897sim 119880[minus120587 120587)

Since the path phase shifts 120601119894119897 appear in (4) in the form of

119890119895120601119894119897 the correlations of 119890119895120601119894119897 over different paths and clusters119864[119890

119895(1206011198941 1198971minus1206011198942 1198972

)

] have impact on the channel characteristicsWe assume that 119864[119890119895(1206011198941 1198971minus1206011198942 1198972 )] can take the following values

119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

] = 1 Case 11198962

Case 2(10)

(iii) The calculation of the third expectation in (7) is basedon the following methodology

Case 1 119864 [119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

119894119897minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

119894119897)

]

= int

120587

minus120587

119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890(119895119888)(119889

119872

119898119899)119879Θ119872

119894119897 119891Θ119872 (Θ

119872

119894119897) 119889Θ

119872

119894119897

= int

120587

minus120587

infin

sum

119896=minusinfin

(G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

times 119890119895119896Θ119872

119894119897+119895(|119889119872

119898119899| cos(Θ119872

119894119897minusang119889119872

119898119899)119888)

119889Θ119872

119894119897

= 2120587

infin

sum

119896=infin

119895119896

119890119895119896ang119889119872

119898119899

times (G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898119899

10038161003816100381610038161003816

119888)

(11)

whereG119872

119898119896G119872

119899119896 andF119872

119896are the FSCs of the APPs (defined

in (1)) and the AAS (defined in (2)) in the correspondingcoordinates respectively 119869

119896(119906) ≜ (119895

minus119896

120587) int120587

0

119890119895(119896120585+119906 cos 120585)

119889120585 isthe 119896th-order Bessel functionotimes is the linear convolution and| sdot | is the Euclidean norm

Thedefinition of the linear convolution can be formulatedas follows [33 page 155] given two discrete-time signals 119909

119899

and 119910119899 the linear convolution between them is defined as

follows 119911119899≜ 119909

119899otimes 119910

119899= sum

infin

119896=infin119909119896119910119899minus119896

The vectors and 119889

119872

119898119899and 119889

119861

119901119902are the separation vectors

between two antenna elements at the MS (119898 119899) and at the BS(119901 119902) Large distances often result in less STF correlation asthe Bessel functions asymptotically decrease The norms of119889119872

119898119899and 119889

119861

119901119902represent shifted distances (between 120596

1119886119861

119901and

1205962119886119861

119902) at the BS and (between 120596

2(1199052119881 + 119886

119872

119899) and 120596

1(1199051119881 +

119886119872

119898)) at the MS Parameters 119889(sdot)

(sdotsdot)contain space time and

frequency separations between ℎ119901119898(1199051 120596

1) and ℎ

119902119899(1199052 120596

2)

These two vectors illustrate theimpact of the antennasrsquo loca-tion (spatial correlation) of thecarrier frequencies (frequencycorrelation) of the time indices and of the mobile speed onthe CCF at BS and MS The value of the 119889119872

119898119899and 119889119861

119901119902can be

determined based on the following equations

119889119872

119898≜ 120596

1(119886

119872

119898minus 119905

1119881) 119889

119872

119899≜ 120596

2(119886

119872

119899minus 119905

2119881)

119889119872

119898119899≜ (120596

21199052minus 120596

11199051) 119881 + (120596

1119886119872

119898minus 120596

2119886119872

119899)

119889119861

119901≜ 120596

1119889119861

119901 119889

119861

119902≜ 120596

2119889119861

119902 119889

119861

119901119902≜ 120596

1119886119861

119901minus 120596

2119886119861

119902

(12)


Similar to the calculation in the first case for the second casewe have (in MS)

Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)

We formulate the CCF including both cases using the deriva-tions presented above

119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)

We evaluate the CCF presented in (14) for the scenario 1198941=

1198942= 119894 119897

1= 119897

2= 119897 and 1198962 = 0 The final expression of the CCF

obtained in this scenario is presented in

119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40

Laplace distribution

Von-Mises distributionn = 10

n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk

Figure 1 Fourier series coefficients for two AAS PDFs to approx-imate Laplace and von-Mises distributions (with orders of non-isotropy 119899 = 3 10) determined to correspond to the real PDFs forthe nonisotropic propagation environment

TheCCF appeared to be the convolution of two categoriesof FSCs as follows

(i) F(sdot)

119896are the 119896th FSCs of PDFs of the nonisotropic

AAS at both MS and BS sides This means that theCCF is represented as the same linear combination ofFSCs associatedwith these PDFsThis allows accuratemodeling for various 2D wireless propagation envi-ronments In ourmodel we determined the necessarynumber of FSCs by calculating integral (2) for each ofthe two PDFs that describe the nonisotropic propa-gation environment In Figure 1 the FSCs of Laplaceand von-Mises PDFs (at the MS) are compared Forthe von-Mises distribution FSCs are presented whenthe propagation environment has two different ordersof nonisotropy

(ii) G(sdot)

(sdot119896)are the 119896th FSCs of 2D WBUWB APPs at

both BS and MS sides used to investigate the impactof directional and omnidirectional antennas At thispoint we differentiate the WB APP which is not fre-quency selective from the frequency selective UWBAPPs

31 CCF Numerical Evaluation Low correlation betweenreceived signals is a necessary condition for good MIMOperformances This low correlation is achieved when eachantenna provides a unique weighting for each 119897 cluster with119868 multipath components In order to achieve space diversitywith MIMO systems the antennas need to be as compact aspossible but also to be able to recover signals with dissimilarmultipath fading An optimal solutionmay be a low thresholdfor correlation rather than zero correlation between antennasLow correlation is often considered when CCF lt 07 [25]Correlation values are especially significant in applicationswhere the evaluation of the channel capacity is necessary


0

1

2

01

23

40

1

Antenna spacing (cm)

CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times

WB channel Laplace distribution

UWB channel Laplace distribution

WB channel von-Mises distribution

UWB channel von-Mises distribution

Figure 2 CCF corresponding to WB and UWB signals with omnidirectional APP Laplace and von-Mises PDFs of the propagationenvironment with antenna spacing 1205822 = 119888120587120596

119867and frequency offsets Δ119891 = 119891

2minus 119891

1

Figures 2 and 3 depict the effect of the distribution of thenonisotropic propagation environment theAPP (directionalomnidirectional) the antenna spacing and the carrier fre-quency offset Δ119891 = 119891

2minus 119891

1(where 119891

1is constant at 5GHz

for WB channels and at 31 GHz for UWB channels) on thenormalized CCF|119877(119905

1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the

reported figures the unit for the antenna spacing is half ofthe highest frequency of the signal bandwidth 1205822 = 119888120587120596

119867

where 120596119867= 2120587119891

119867

In [22 page 26] measurement results for spatial correla-tion of UWBMIMOchannels with omnidirectional antennasare presented The measured data show that the spatialcorrelation is a monotonously decreasing function of theantenna distance A complete characterization of MIMOUWB channels correlations is provided in [34] It was statedthat for about 4 cm antenna spacing the correlation functionfollows a pattern of the first kind zeroth-order Bessel functionwith distance It was also found that a value of 2 cm forantenna spacing is sufficient for CCF le 05 In [35ndash37] it wasstated that the CCF values are higher for directional antennasthan for omnidirectional antennas This is because theirbeamwidth limits the effective angular spread and their ability

to capture multipath signals from all directions The resultsregarding the spatial correlation presented in Figures 2 and3 are consistent with the CCFrsquos behavior described in thementioned references In the reported figures the influenceof the frequency correlation is also evaluated It can beobserved how CCF decreases as Δ119891 increases This decreaseresults not only from the Bessel functions (as the centralfrequencies appear in the Bessel operands) but also from theterm produced by theMGF of the delay profile As the figuresshow the CCF is maximum when Δ119891 = 0 and when thereis no separation between the antennas of the MIMO systemCCF also decreases whenWB APP is replaced by UWB APPThis is because UWB signals offer a higher degree of diversityowing to their abundant multipaths In Figure 1 it can beobserved that the number of FSCs necessary to approximateLaplace PDF is larger than the number of FSCs necessaryto approximate von-Mises distribution This indicates thata propagation environment described by Laplace PDF ischaracterized by a higher order of nonisotropy which pro-vides better diversity and lower correlation between MIMOsubchannelsThe graphical representations of CCF show thatthe correlation decreases faster when the environment has


0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)123Antenna s

15times





Figure 3 CCF corresponding toWB and UWB signals with directional APP Laplace and von-Mises PDFs of the propagation environmentwith antenna spacing 1205822 = 119888120587120596


2minus 119891

1

Laplacian distribution for both WB and UWB channels In[38] the influence of parameters like antenna element spac-ings environment parameters and scattering density on thespatial correlation properties of multilink MIMO channelswas presented Based on different propagation phenomenalike those previously mentioned new key technical issues indeveloping and realizing multiinput multioutput technologywere presented in [39]

4 Fourier Analysis of the 2D STF-CCF ofWBUWB MIMO Channels

The derived CCF can be used to analyze the power spectraldensity (PSD) of WB and UWB MIMO channels PSDgives the distribution of the signal power among variousfrequencies and also reveals the presence or the absence ofrepetitive patterns and correlation sequences in the signalprocess These structural patterns are useful in a wide rangeof applications like data forecasting signal detection andcoding pattern recognition and radar and decision-makingsystems The analyzed PSD corresponds to a multiple-inputsingle-output (MISO) channel which is a particular case ofMIMO channel This analysis corresponds to the stationary

scenario when 1205961= 120596

2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast

1minus119896(120596) otimesF

119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)

Using the Fourier transform of 119869119896(119906) which is given by

119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896

119879119896(Λ)) radic1 minus Λ2 |Λ| lt 1

0 |Λ| ≧ 1

(18)where 119865[sdot] denotes the Fourier transform and 119879

119896(Λ) ≜

cos[119896 cosminus1(Λ)] is the 119896th-order Chebyshev polynomial


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)

(b) Rectangular antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)


Uniform distributionVon-Mises distribution n = 10

(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)

(d) Rectangular antenna

Figure 4 PSD of WB channels MS moves in the positive direction of the 119909-axis ((a) (b)) and 119910-axis ((c) (d)) two antenna types employedat the MS nonisotropic propagation (Laplacian or von-Mises distributed) and isotropic environment (uniformly distributed)

function of the first kind [[16] page 486] the Fouriertransform of the CCF derived for stationary case versus thetime-difference index Δ119905 ≜ 119905

2minus 119905

1 results in the following

equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861

119896and Λ is a frequency

variable in the interval minus(120596119888)|119881| lt Λ lt (120596119888)|119881|Note that 119877

11990111199021(Λ 120596) = 0 for all |Λ| ⩾ (120596119888)|119881| We denote

119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)

2119879119896(sdot) is theChebyshev

polynomials which form a complete orthogonal set on theinterval minus1 ⩽ 119906 lt 1 with respect to the weighting function1radic1 minus 1199062

Therefore any bandlimited CCF (in the intervalminus(120596119888)|119881| ⩽ Λ ⩽ (120596119888)|119881|) can be expanded in terms ofChebyshev polynomials as shown in (19)The PSD 119877119872

(Λ) isthe last term in (19) and shows the channel variations causedaround or by the MS speed and direction and the impact of


104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)


Figure 5 PSD of UWB channels MSmoves in the positive direction of the 119909-axis ((a) (b)) and 119910-axis ((c) (d)) two antenna types employedat the MS nonisotropic propagation (Laplacian or von-Mises distributed) and isotropic environment (uniformly distributed)

the nonisotropic environment of the channel bandwidthand of the APP

119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)

In Figures 4 and 5 the PSD of WB and UWB signalsis depicted The presented results illustrate (20) with theinfluence of the following elements

(i) The channel bandwidth forWB channels was equal to200MHzwith the central frequency119891 = 25GHz ForUWB channels the results correspond to a bandwidthdelimited by 31 divide 106GHz

(ii) APPs specific toWB and UWB systems employed atthe MS side (directional and omnidirectional anten-nas)

(iii) nonisotropic propagation environment around theMS (Laplace and von-Mises PDFs)

(iv) direction of the MS speed the positive 119909-axis or 119910-axis direction

From the comparative analysis of PSD obtained forWB and UWB channels relevant conclusions can be drawnregarding its distribution in the interval minus(120596119888)|119881| lt Λ lt

(120596119888)|119881| as follows

(i) ThePSDfluctuations aremore pronounced in the caseof UWB channels compared to WB channels This is


the effect of the frequency selectivity phenomenonwhose effects increase with signal frequency andbandwidth The noticeable increase of frequencyselectivity that occurs in the case of UWB chan-nels clearly differentiated them from other channelswith narrower bandwidth This characteristic may behelpful in the identification process of the channelrsquostype for its bandwidth from its PSD only In Figures4(a) 4(b) 5(a) and 5(b) it can be observed thatMS movement in the positive direction of the 119909-axis produces a PSD with larger values at positiveΛ than at negative Λ This asymmetry of the PSDis also determined by the Doppler spectrum whichconcentrates towards positive frequency axis

(ii) The PSD shape is asymmetrical (Figures 4(a) 4(b)5(a) and 5(b)) or symmetrical (Figures 4(c) 4(d)5(c) and 5(d)) as a consequence of the interactionbetween the beam of the antenna pattern the direc-tion of the MS speed and the distribution of thepropagation directions around the MS Figures 4(c)4(d) 5(c) and 5(d) depict the shape of the PSDwhichis symmetrically distributed around Λ = 0 becausethe PDF which characterizes the nonisotropic propa-gation and the APPs are symmetrical aroundΘ119872

= 0

and are perpendicular towards the speed directionFor both WB and UWB channels the maximumDoppler shift is (120596119888)|119881| (ie 119877119872

(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)

The obtained results are consistent with those proposedin [19 37] In [19] the PSD of a narrowband channel innonisotropic propagation environment is similar to the PSDshape we obtained forWB channels Between our results andthose presented in [38] there are similarities regarding the U-shape of the PSD but there are also differences determinedby parameters specific to channels with large bandwidthslike frequency selectivity higher central frequency and APPstypically used for this type of systems Comparing the PSDshape obtained for WB channels with the PSD obtainedfor UWB channels and even with the PSD obtained fornarrowband channels in [19] we can conclude that thechannel bandwidth has a great influence on the channelpower spectrumWith the increase of signal bandwidth largervariations can be observed over (because of the increasedfrequency selectivity) the PSD of UWB channels

5 Conclusion

In this paper we proposed an outdoor channel model forWBUWB MIMO systems with moving receiver basedon the mathematical expression of a 2D STF-CCF Theimpact of channel bandwidth nonisotropic propagation andomnidirectional and directional antennas on the CCF wasevaluated The increase of the channel bandwidth (fromWBto UWB) generated the necessity to develop two approachesfor the APPs calculation The derived CCF appeared to be acombination of linear series expansion of Bessel functionsThe coefficients of the CCF expansion are represented bythe convolution of the Fourier coefficients of the APPs and

the AASs specific to WB and UWB channels The Fouriertransform of the CCF in a stationary scenario showedthat the PSD diverges from the U-shaped function thatis ClarkeJake model This deviation depends on the AASthe employed antennas and the MS speed direction Theincrease of the channel bandwidth also generated a morefrequency selective PSD for UWB channels compared to thePSD of WB channels This information can be useful forapplications of signal detection and recognition since theincreased frequency selectivitymakes the identification of thechannelrsquos type for its bandwidth from its PSD only possible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper has been financially supported within the Projectentitled ldquoHorizon 2020 Doctoral and Postdoctoral StudiesmdashPromoting the National Interest through Excellence Com-petitiveness andResponsibility in the Field of Romanian Fun-damental and Applied Scientific Researchrdquo Contract num-ber POSDRU15915S140106 This project is cofinancedby the European Social Fund through Sectoral OperationalProgramme for Human Resources Development 2007ndash2013ldquoInvesting in Peoplerdquo

References

[1] Y Ma and M Patzold ldquoWideband two-ring MIMO channelmodels for mobile-to-mobile communicationsrdquo in Proceedingsof 10th International Symposium on Wireless Personal Commu-nications pp 380ndash384 Jaipur India December 2007

[2] KWitrisal andM Pausini ldquoStatistical analysis of UWB channelcorrelation functionsrdquo IEEE Transactions on Vehicular Technol-ogy vol 57 no 3 pp 1359ndash1373 2008

[3] N Gvozdenovic andM Eric ldquoLocalization of users inmultiuserMB OFDM UWB systems based on TDOA principlerdquo inProceedings of the 19th Telecommunications Forum (TELFORrsquo11) pp 326ndash329 Belgrad Serbia November 2011

[4] Q Zou A Tarighat and A H Sayed ldquoPerformance analysis ofmultiband-OFDM UWB communications with application torange improvementrdquo IEEE Transactions on Vehicular Technol-ogy vol 56 no 6 pp 3864ndash3878 2007

[5] A Abdi and M Kaveh ldquoA space-time correlation model formultielement antenna systems inmobile fading channelsrdquo IEEEJournal on Selected Areas in Communications vol 20 no 3 pp550ndash560 2002

[6] T Santos J Karedal P Almers F Tufvesson and A F MolischldquoModeling the ultra-wideband outdoor channel measurementsand parameter extraction methodrdquo IEEE Transactions onWire-less Communications vol 9 no 1 pp 282ndash290 2010

[7] M Di Renzo F Graziosi R Minutolo M Montanari and FSantucci ldquoThe ultra-wide bandwidth outdoor channel frommeasurement campaign to statistical modellingrdquo Mobile Net-works and Applications vol 11 no 4 pp 451ndash467 2006

[8] C F Souza and J C R Dal Bello ldquoUWB signals transmissionin outdoor environments for emergency communicationsrdquo


in Proceedings of the 11th IEEE International Conference onComputational Science and Engineering Workshops pp 343ndash348 San-Paolo Italy July 2008

[9] Y Wang X Yu Y Zhang et al ldquoUsing wavelet entropy todistinguish between humans and dogs detected by UWB radarrdquoProgress in Electromagnetics Research vol 139 pp 335ndash352 2013

[10] W Zhou J T Wang H W Chen and X Li ldquoSignal modelandmoving target detection based onMIMOsynthetic apertureradarrdquo Progress in Electromagnetics Research vol 131 pp 311ndash329 2012

[11] A EWaadt SWang C Kocks et al ldquoPositioning inmultibandOFDM UWB utilizing received signal strengthrdquo in Proceedingsof the 7th Workshop on Positioning Navigation and Communi-cation (WPNC rsquo10) pp 308ndash312 March 2010

[12] W Z Li Z Li H Lv et al ldquoA new method for non-line-of-sight vital sign monitoring based on developed adaptive lineenhancer using low centre frequency UWB radarrdquo Progress inElectromagnetics Research vol 133 pp 535ndash554 2013

[13] I C Vizitiu Fundamentals of Electronic Warfare Matrix RomPublishing House 2011

[14] K Mizui M Uchida and M Nakagawa ldquoVehicle-to-vehiclecommunication and ranging system using spread spectrumtechnique (Proposal of Boomerang Transmission System)rdquo inProceedings of the 43rd IEEE Vehicular Technology Conferencepp 335ndash338 Secaucus NJ USA May 1993

[15] C Sturm and W Wiesbeck ldquoWaveform design and signalprocessing aspects for fusion of wireless communications andradar sensingrdquo Proceedings of the IEEE vol 99 no 7 pp 1236ndash1259 2011

[16] B J Donnet and I D Longstaff ldquoCombiningMIMO radar withOFDM communicationsrdquo in Proceedings of the 3rd EuropeanRadar Conference (EuRAD rsquo03) pp 37ndash40 September 2006

[17] CEPT Report 34 Report B from CEPT to European Commis-sion in response to the Mandate 4 on Ultra-Wideband (UWB)2009

[18] European Communications Office ldquoSpecific UWB applicationsin the bands 34ndash48GHz and 6ndash85GHz location trackingapplications for emergency services (LAES) location trackingaplications 2 (LT2) and location tracking and sensors applica-tions for automotive and transportation environments (LTA)rdquoECC Report 170 2011

[19] H S Rad and S Gazor ldquoThe impact of non-isotropic scatteringand directional antennas on MIMO multicarrier mobile com-munication channelsrdquo IEEE Transactions on Communicationsvol 56 no 4 pp 642ndash652 2008

[20] A M Pistea and H Saligheh Rad ldquoThree dimensional space-time-frequency description ofWB andUWBMIMO channelsrdquoIEEE Communications Letters vol 16 no 2 pp 205ndash207 2012

[21] AM Pistea T PaladeAMoldovan andH S Rad ldquoThe impactof antenna directivity and channel bandwidth on the powerspectral density of wideband and UWB MISO channelsrdquo inProceedings of the 7th International Conference on Wireless andMobile Communications (ICWMC rsquo11) pp 36ndash41 LuxembourgCity Luxembourg June 2011

[22] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons 2nd edition 1996

[23] M Ghavami L B Michael and R Kohno Ultra WidebandSignals and Systems in Communication Engineering JohnWileyamp Sons 2004

[24] A FMolisch K BalakrishnanD Cassioli et al ldquoIEEE 802154achannel modelmdashfinal reportrdquo Technical Report DocumentIEEE 80215-04-0662-02-004a 2005

[25] T Kaiser and F Zheng Ultra-Wideband Systems with MIMOJohn Wiley amp Sons 2010

[26] A Abdi J A Barger and M Kaveh ldquoA parametric modelfor the distribution of the angle of arrival and the associatedcorrelation function and power spectrum at themobile stationrdquoIEEE Transactions on Vehicular Technology vol 51 no 3 pp425ndash434 2002

[27] J Ren andRG Vaughan ldquoSpaced antenna design in directionalscenarios using the vonmises distributionrdquo in Proceedings of the70th IEEE Vehicular Technology Conference (VTC rsquo09) pp 1ndash5Anchorage Alaska USA September 2009

[28] Q Spencer M Rice B Jeffs and M Jensen ldquoA statisticalmodel for angle of arrival in indoor multipath propagationrdquo inProceedings of the 47th IEEE Vehicular Technology Conferencevol 3 pp 1415ndash1419 Phoenix Ariz USA May 1997

[29] R C Qiu and I-T Lu ldquoWideband wireless multipath channelmodeling with path frequency dependencerdquo in IEEE Interna-tional Conference on Communications (ICC rsquo96) ConferenceRecord Converging Technologies for Tomorrowrsquos Applicationsvol 1 pp 277ndash281 Dallas Tex USA June 1996

[30] J P M de Sa Applied Statistics Using SPSS STATISTICAMATLAB and R Springer 2003

[31] J Wallace Characterization and identification of ultrawidebandradio propagation channels [PhD thesis] Faculty of the Gradu-ate School University of Southern California 2005

[32] A F Molisch ldquoUltrawideband propagation channels-theorymeasurement and modelingrdquo IEEE Transactions on VehicularTechnology vol 54 no 5 pp 1528ndash1545 2005

[33] M C Jeruchim P Balaban and K S Shanmugan Simulationof Communication Systems Applications of CommunicationsTheory Plenum Press New York NY USA 1992

[34] W Q Malik ldquoSpatial correlation in ultrawideband channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp604ndash610 2008

[35] D Gesbert H Bolcskei D A Gore and A J Paulraj ldquoOutdoorMIMOwireless channels models and performance predictionrdquoIEEE Transactions on Communications vol 50 no 12 pp 1926ndash1934 2002

[36] P Jin N A Goodman and K L Melde ldquoExploiting directionalantennas for reduced-dimension space-time RAKE receivingrdquoIEEE Transactions on Vehicular Technology vol 57 no 6 pp3880ndash3885 2008

[37] C R Anderson ldquoLow-antenna ultra wideband spatial cor-relation analysis in a forest environmentrdquo in Proceedings ofthe Vehicular Technology Conference pp 1ndash5 Barcelona Spain2009

[38] X Cheng C-X Wang H Wang et al ldquoCooperative MIMOchannelmodeling andmulti-link spatial correlation propertiesrdquoIEEE Journal on Selected Areas in Communications vol 30 no2 pp 388ndash396 2012

[39] X Cheng B Yu L Yang et al ldquoCommunicating in the realworld 3D MIMOrdquo IEEE Wireless Communications Magazinevol 21 no 4 pp 136ndash144 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



[1] Patzold and Hogstad developed a WB MIMO channelmodel with temporal spatial and frequency correlationproperties This model presents the 2D CCF derived forboth isotropic and nonisotropic propagation assuming aspecific geometry of scatterers around receiver [2] Saleh andValenzuela (S-V) developed one of the first UWB channelmodels which was proposed to reproduce the multipatheffect of indoor environments Using S-V results Zou et aldetermined a CCF expression of multiband OFDM-UWBdiscrete-time base-band channel impulse response [3 4]Abdi and Kaveh propose an ST-CCF for a MIMO frequencynonselective Rician fading channel assuming a one ring ofscatterers model around the MS [5] This model is as anexample of an outdoor environment where the von-Misesprobability density function (PDF) is suggested to model thenonisotropic propagation environment around the user

As a summary our literature review shows that thegeneral approach is to develop channel models for WBtransmissions separated from the channel models for UWBcommunications Another trend is to include the effects ofthe propagation environment (waves clustering frequencyselectivity) and to ignore the frequency selective behaviorof the antenna patterns Because UWB systems were mainlydeveloped for indoor applications the research in the fieldof UWB outdoor channel modeling was not a priority [6ndash8] From the literature review it was also observed that mostof the MIMO-CCF models are based on a specific geometryfor scatterers distribution around MS to characterize thenonisotropic propagation

Lately UWB technology in combination with MIMOsystems finds more and more interest in areas such aslocation tracking application for outdoor emergency services[9 10] location tracking and sensor applications for mobileoutdoor users [11] and medical [12] and electronic warfareapplications [13] Even more some research results [14ndash16]proved that a joint radar and wireless communication systemwould constitute a unique platform for future intelligentenvironmental sensing and ad hoc communication networksin terms of both spectrum efficiency and cost effectivenessThis evolution regarding the use of UWB systems in outdoorenvironments as radar or communication system has led tothe formulation of specific requirements aiming at investi-gating the influence of outdoor propagation on UWBmobilesystems [17 18]

The model proposed in this paper describes the CCFbetween two subchannels of an outdoor WBUWB MIMOchannel employing directional and omnidirectional antennasand in the presence of nonisotropic wave propagation in2D space Instead of assuming a specific geometry forscatterers in space (models based on specific geometries ofscatterers in the space are just able to predict the behaviorof that particular propagation scenario) we used specificmathematical relationships between the physical parametersof the wireless channel along with appropriate assumptionson their PDFs to derive a channel model based on the CCFof outdoor WB and UWB MIMO channels This model isan extension of a 2D NB channel model [19] The evolutionthrough systems with (ultra)wide bandwidths implies newpropagation effects that make the NB model inappropriate

for WBUWB transmissions In order to extend the NBchannel model to a version adequate for WBUWB channelsanalysis the mathematical structure of the model needs tobe updated with the elements which characterize the newpropagation phenomena (represented by channel frequencyselectivity waves clustering) and with the elements whichdefine the performances of the new systems (represented byoperating bandwidth frequency selectivity of the antenna)We represent the nonisotropic scatterers by the Fourier seriesexpansion (FSE) of the PDF of the propagating directionsWe also consider the effect of directional and omnidirec-tional antenna element patterns by the FSE of the APPsThe expression of the CCF turns out to be a compositionof linear expansions of Bessel functions of the first kindFourier analysis of the derived CCF is used to determinethe power spectral density (PSD) of WB and UWB channelsin a stationary scenario The mathematical set-up of thestationary scenario transforms the MIMO channel into amultiple-input single-output (MISO) channel to be analyzedas a special case The contribution in this 2D-CCF model isfourfold

(1) It includes the main propagation phenomena thatappear in 2D WBUWB MIMO outdoor scenariosand permits the evaluation of combined effects ofchannel bandwidth nonisotropic scattering APP(omnidirectional or directional) and the arrayinterelement distance on the CCF of WB and UWBMIMO channels

(2) This model can be used for the analysis of both WBand UWB propagation channels The key to apply amodel for the analysis of both types of channels is touse the appropriate expressions for APPs Since WBAPP is not frequency selective the relative bandwidthbeing a small fraction of the central frequency andthe UWB APP is frequency selective and differentantenna patterns expressions for these two types ofsystems should be used In this paper we proposemathematical expression for UWB APP

(3) It accurately reflects the influence of theMS direction(and speed) on the spectral characteristics of WB andUWB channels

(4) It gives mathematical expressions to evaluate thechannel power spectrum of WB and UWB wirelesschannels

In [20] a three-dimensional (3D) version of this modelwas presented The difference between 2D and 3D versionsof the proposed model consists of APP and nonisotropicpropagation representations In the case of the 2D modelthe APP and the propagation environment are representeddepending on azimuth angle spread (AAS) The 3D versionof the model represents these two elements as a functionof both AAS and elevation angle spread (EAS) From theobtained results it was observed that the 2Dmodel provides agood representation of how the frequency selectivity affectsthe signal spectrum A 2D model for most propagationenvironments is more eligible when we get closer to theground because a propagation environment behaves similar















119901(Θ

119861

120596) 119866119872

119898(Θ

119872



119866 (Θ 120596) =

infin

sum

119896=minusinfin

G119896119890119895119896Θ

G119896=

1

2120587int

120587

minus120587

119866 (Θ 120596) 119890minus119895119896Θ

119889Θ

(1)





119891119867 119891






(Θ119861

) at the BS and119891119872

(Θ119872






0sdot120596


UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660sdot120596

2119888sdot ℎ sdot sinΘ119889120596


0


UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660


119889120596



119896


119890|((minusradic2Θ)120590)|

radic2120590

F119896=119890minus((120587(radic2+119895119896120590))120590)

(119890((2radic2120587)120590)

minus 1198901198952119896120587

)

2120587(119895radic2119896120590 minus 2)



21205871198690(119899)

F119896=119869119896(119899)

1198690(119899)


119891Θ(Θ) =

+infin

sum

119896=minusinfin

F119896119890119895119896Θ

F119896=

1

2120587int

120587

minus120587

119891Θ(Θ) 119890

minus119895119896Θ

119889Θ

(2)







119896= (12120587)120575

119896 Laplace and








120591119901119898119894119897

= 120591119894119897minus (120591

119861

119901119894119897+ 120591

119872

119898119894119897)

120591119861

119901119894119897≜

119886119861

119901Θ

119861

119894119897

119888 120591

119872

119898119894119897≜119886119872

119898Θ

119872

119894119897

119888

(3)



120591119872119898119894119897


119901to 119886119872




119894119897andΘ119872

119894119897were

previously defined


119887119908120596)

120578 The parameter 120596119887119908


119887119908= 120596

119867minus 120596

119871 120596

119871and 120596

119867are the




119901119898119894119897= 12120596120591

119894119897


ℎ119901119898

(119905 120596)

= (120596119887119908

120596)

120578 119871

sum

119897=1

119868

sum

119894=1

119866119861

119901(Θ

119861

119894119897 120596)

times 119866119872

119898(Θ

119872

119894119897 120596) 119892


(4)


119901119898119894119897 a path


119901119898119894119897(119905) ≜ 120591

119901119898119894119897+

(119905119888)119881119879

Θ119872



= (120596119888)119881119879

Θ119872

119894119897where




1) and ℎ

119902119899(1199052 120596

2)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2) ≜ 119864 [ℎ

119901119898(119905

1 120596

1) ℎ

lowast

119902119899(119905

2 120596

2)] (5)


1 119905

2)





119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

times

119871

sum

119897=1

119868

sum

119894=1

119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119872

119898(Θ

119872

11989411198971

1205961)

times 11989211990111989811989411198971

times 11989211990211989911989421198972

times 119890(minus11989512059611198791199011198981198971

minus119895120596112059111990111989811989411198971(1199051))]

times 119864 [119866119861lowast

119902(Θ

119861

11989421198972

1205962)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895(1206011198941 1198971

minus1206011198942 1198972)

times 119890(minus11989512059621198791199021198991198972

minus119895120596212059111990211989911989421198972(1199052))]

(6)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times

119871

sum

119897=1

119868

sum

119894=1

119864 [(1205911198942119897212059111989411198971)minus1

times 119890119895((120596212059111989421198972

minus120596112059111989411198971)+(12059621198791198972

minus12059611198791198971))

]

times 119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

]

times 119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119861lowast

119902(Θ

119861

11989421198972

1205962)

times 119890119895((1205961119888)119886

119861119879

119901Θ119861

11989411198971minus(1205962119888)119886

119861119879

119902Θ119861

11989421198972)

]

times 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

(7)



1= 119894

2= 119894 119897

1= 119897






1= 119894

2 119897

1= 119897



11989411198971

and Θ119872

11989421198972

(or Θ119861

11989411198971

and Θ119861

11989421198972

) areindependent


119864 [(1205911198942119897212059111989411198971)minus1

119890119895(120596212059111989421198972

minus120596112059111989411198971)

times 119890(12059611198791198981199011198971

minus12059621198791198981199011198972)

]

=

Φ(minus1)

120591(119895 (120596

2minus 120596

1))

timesΦ119879(119895 (120596

1minus 120596

2)) Case 1

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)

timesΦ119879(minus119895120596

2)Φ

119879(119895120596

1) Case 2

(8)



120591119879(119904) suggests


119883(119904) = 119864[119890

119895119904119883

] =

intinfin

minusinfin

119890119895119904120585



119901119879119897(119879) =

Λ119871+1

119879119871

119890minusΛ119879

119871 119901

120591119894119897(120591) =

120582119868

120591119868minus1

119890minus120582120591

(119868 minus 1) (9)




119894119897sim 119880[minus120587 120587)



119895(1206011198941 1198971minus1206011198942 1198972

)


119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

] = 1 Case 11198962

Case 2(10)


Case 1 119864 [119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

119894119897minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

119894119897)

]

= int

120587

minus120587

119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890(119895119888)(119889

119872

119898119899)119879Θ119872

119894119897 119891Θ119872 (Θ

119872

119894119897) 119889Θ

119872

119894119897

= int

120587

minus120587

infin

sum

119896=minusinfin

(G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

times 119890119895119896Θ119872

119894119897+119895(|119889119872

119898119899| cos(Θ119872

119894119897minusang119889119872

119898119899)119888)

119889Θ119872

119894119897

= 2120587

infin

sum

119896=infin

119895119896

119890119895119896ang119889119872

119898119899

times (G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898119899

10038161003816100381610038161003816

119888)

(11)

whereG119872

119898119896G119872

119899119896 andF119872



119896(119906) ≜ (119895

minus119896

120587) int120587

0

119890119895(119896120585+119906 cos 120585)



119899


follows 119911119899≜ 119909

119899otimes 119910

119899= sum

infin

119896=infin119909119896119910119899minus119896


119872

119898119899and 119889

119861



119898119899and 119889

119861


1119886119861

119901and

1205962119886119861


2(1199052119881 + 119886

119872

119899) and 120596

1(1199051119881 +

119886119872




1) and ℎ

119902119899(1199052 120596

2)


119898119899and 119889119861

119901119902can be


119889119872

119898≜ 120596

1(119886

119872

119898minus 119905

1119881) 119889

119872

119899≜ 120596

2(119886

119872

119899minus 119905

2119881)

119889119872

119898119899≜ (120596

21199052minus 120596

11199051) 119881 + (120596

1119886119872

119898minus 120596

2119886119872

119899)

119889119861

119901≜ 120596

1119889119861

119901 119889

119861

119902≜ 120596

2119889119861

119902 119889

119861

119901119902≜ 120596

1119886119861

119901minus 120596

2119886119861

119902

(12)



Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)


1198942= 119894 119897

1= 119897



119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40



n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk



(i) F(sdot)



(ii) G(sdot)





0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























119901(Θ

119861

120596) 119866119872

119898(Θ

119872



119866 (Θ 120596) =

infin

sum

119896=minusinfin

G119896119890119895119896Θ

G119896=

1

2120587int

120587

minus120587

119866 (Θ 120596) 119890minus119895119896Θ

119889Θ

(1)





119891119867 119891






(Θ119861

) at the BS and119891119872

(Θ119872






0sdot120596


UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660sdot120596

2119888sdot ℎ sdot sinΘ119889120596


0


UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660


119889120596



119896


119890|((minusradic2Θ)120590)|

radic2120590

F119896=119890minus((120587(radic2+119895119896120590))120590)

(119890((2radic2120587)120590)

minus 1198901198952119896120587

)

2120587(119895radic2119896120590 minus 2)



21205871198690(119899)

F119896=119869119896(119899)

1198690(119899)


119891Θ(Θ) =

+infin

sum

119896=minusinfin

F119896119890119895119896Θ

F119896=

1

2120587int

120587

minus120587

119891Θ(Θ) 119890

minus119895119896Θ

119889Θ

(2)







119896= (12120587)120575

119896 Laplace and








120591119901119898119894119897

= 120591119894119897minus (120591

119861

119901119894119897+ 120591

119872

119898119894119897)

120591119861

119901119894119897≜

119886119861

119901Θ

119861

119894119897

119888 120591

119872

119898119894119897≜119886119872

119898Θ

119872

119894119897

119888

(3)



120591119872119898119894119897


119901to 119886119872




119894119897andΘ119872

119894119897were

previously defined


119887119908120596)

120578 The parameter 120596119887119908


119887119908= 120596

119867minus 120596

119871 120596

119871and 120596

119867are the




119901119898119894119897= 12120596120591

119894119897


ℎ119901119898

(119905 120596)

= (120596119887119908

120596)

120578 119871

sum

119897=1

119868

sum

119894=1

119866119861

119901(Θ

119861

119894119897 120596)

times 119866119872

119898(Θ

119872

119894119897 120596) 119892


(4)


119901119898119894119897 a path


119901119898119894119897(119905) ≜ 120591

119901119898119894119897+

(119905119888)119881119879

Θ119872



= (120596119888)119881119879

Θ119872

119894119897where




1) and ℎ

119902119899(1199052 120596

2)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2) ≜ 119864 [ℎ

119901119898(119905

1 120596

1) ℎ

lowast

119902119899(119905

2 120596

2)] (5)


1 119905

2)





119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

times

119871

sum

119897=1

119868

sum

119894=1

119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119872

119898(Θ

119872

11989411198971

1205961)

times 11989211990111989811989411198971

times 11989211990211989911989421198972

times 119890(minus11989512059611198791199011198981198971

minus119895120596112059111990111989811989411198971(1199051))]

times 119864 [119866119861lowast

119902(Θ

119861

11989421198972

1205962)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895(1206011198941 1198971

minus1206011198942 1198972)

times 119890(minus11989512059621198791199021198991198972

minus119895120596212059111990211989911989421198972(1199052))]

(6)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times

119871

sum

119897=1

119868

sum

119894=1

119864 [(1205911198942119897212059111989411198971)minus1

times 119890119895((120596212059111989421198972

minus120596112059111989411198971)+(12059621198791198972

minus12059611198791198971))

]

times 119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

]

times 119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119861lowast

119902(Θ

119861

11989421198972

1205962)

times 119890119895((1205961119888)119886

119861119879

119901Θ119861

11989411198971minus(1205962119888)119886

119861119879

119902Θ119861

11989421198972)

]

times 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

(7)



1= 119894

2= 119894 119897

1= 119897






1= 119894

2 119897

1= 119897



11989411198971

and Θ119872

11989421198972

(or Θ119861

11989411198971

and Θ119861

11989421198972

) areindependent


119864 [(1205911198942119897212059111989411198971)minus1

119890119895(120596212059111989421198972

minus120596112059111989411198971)

times 119890(12059611198791198981199011198971

minus12059621198791198981199011198972)

]

=

Φ(minus1)

120591(119895 (120596

2minus 120596

1))

timesΦ119879(119895 (120596

1minus 120596

2)) Case 1

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)

timesΦ119879(minus119895120596

2)Φ

119879(119895120596

1) Case 2

(8)



120591119879(119904) suggests


119883(119904) = 119864[119890

119895119904119883

] =

intinfin

minusinfin

119890119895119904120585



119901119879119897(119879) =

Λ119871+1

119879119871

119890minusΛ119879

119871 119901

120591119894119897(120591) =

120582119868

120591119868minus1

119890minus120582120591

(119868 minus 1) (9)




119894119897sim 119880[minus120587 120587)



119895(1206011198941 1198971minus1206011198942 1198972

)


119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

] = 1 Case 11198962

Case 2(10)


Case 1 119864 [119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

119894119897minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

119894119897)

]

= int

120587

minus120587

119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890(119895119888)(119889

119872

119898119899)119879Θ119872

119894119897 119891Θ119872 (Θ

119872

119894119897) 119889Θ

119872

119894119897

= int

120587

minus120587

infin

sum

119896=minusinfin

(G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

times 119890119895119896Θ119872

119894119897+119895(|119889119872

119898119899| cos(Θ119872

119894119897minusang119889119872

119898119899)119888)

119889Θ119872

119894119897

= 2120587

infin

sum

119896=infin

119895119896

119890119895119896ang119889119872

119898119899

times (G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898119899

10038161003816100381610038161003816

119888)

(11)

whereG119872

119898119896G119872

119899119896 andF119872



119896(119906) ≜ (119895

minus119896

120587) int120587

0

119890119895(119896120585+119906 cos 120585)



119899


follows 119911119899≜ 119909

119899otimes 119910

119899= sum

infin

119896=infin119909119896119910119899minus119896


119872

119898119899and 119889

119861



119898119899and 119889

119861


1119886119861

119901and

1205962119886119861


2(1199052119881 + 119886

119872

119899) and 120596

1(1199051119881 +

119886119872




1) and ℎ

119902119899(1199052 120596

2)


119898119899and 119889119861

119901119902can be


119889119872

119898≜ 120596

1(119886

119872

119898minus 119905

1119881) 119889

119872

119899≜ 120596

2(119886

119872

119899minus 119905

2119881)

119889119872

119898119899≜ (120596

21199052minus 120596

11199051) 119881 + (120596

1119886119872

119898minus 120596

2119886119872

119899)

119889119861

119901≜ 120596

1119889119861

119901 119889

119861

119902≜ 120596

2119889119861

119902 119889

119861

119901119902≜ 120596

1119886119861

119901minus 120596

2119886119861

119902

(12)



Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)


1198942= 119894 119897

1= 119897



119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40



n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk



(i) F(sdot)



(ii) G(sdot)





0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0sdot120596


UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660sdot120596

2119888sdot ℎ sdot sinΘ119889120596


0


UWB 1

(119891119867minus 119891

119871)int

119891119867

119891119871

1198951198660


119889120596



119896


119890|((minusradic2Θ)120590)|

radic2120590

F119896=119890minus((120587(radic2+119895119896120590))120590)

(119890((2radic2120587)120590)

minus 1198901198952119896120587

)

2120587(119895radic2119896120590 minus 2)



21205871198690(119899)

F119896=119869119896(119899)

1198690(119899)


119891Θ(Θ) =

+infin

sum

119896=minusinfin

F119896119890119895119896Θ

F119896=

1

2120587int

120587

minus120587

119891Θ(Θ) 119890

minus119895119896Θ

119889Θ

(2)







119896= (12120587)120575

119896 Laplace and








120591119901119898119894119897

= 120591119894119897minus (120591

119861

119901119894119897+ 120591

119872

119898119894119897)

120591119861

119901119894119897≜

119886119861

119901Θ

119861

119894119897

119888 120591

119872

119898119894119897≜119886119872

119898Θ

119872

119894119897

119888

(3)



120591119872119898119894119897


119901to 119886119872




119894119897andΘ119872

119894119897were

previously defined


119887119908120596)

120578 The parameter 120596119887119908


119887119908= 120596

119867minus 120596

119871 120596

119871and 120596

119867are the




119901119898119894119897= 12120596120591

119894119897


ℎ119901119898

(119905 120596)

= (120596119887119908

120596)

120578 119871

sum

119897=1

119868

sum

119894=1

119866119861

119901(Θ

119861

119894119897 120596)

times 119866119872

119898(Θ

119872

119894119897 120596) 119892


(4)


119901119898119894119897 a path


119901119898119894119897(119905) ≜ 120591

119901119898119894119897+

(119905119888)119881119879

Θ119872



= (120596119888)119881119879

Θ119872

119894119897where




1) and ℎ

119902119899(1199052 120596

2)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2) ≜ 119864 [ℎ

119901119898(119905

1 120596

1) ℎ

lowast

119902119899(119905

2 120596

2)] (5)


1 119905

2)





119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

times

119871

sum

119897=1

119868

sum

119894=1

119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119872

119898(Θ

119872

11989411198971

1205961)

times 11989211990111989811989411198971

times 11989211990211989911989421198972

times 119890(minus11989512059611198791199011198981198971

minus119895120596112059111990111989811989411198971(1199051))]

times 119864 [119866119861lowast

119902(Θ

119861

11989421198972

1205962)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895(1206011198941 1198971

minus1206011198942 1198972)

times 119890(minus11989512059621198791199021198991198972

minus119895120596212059111990211989911989421198972(1199052))]

(6)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times

119871

sum

119897=1

119868

sum

119894=1

119864 [(1205911198942119897212059111989411198971)minus1

times 119890119895((120596212059111989421198972

minus120596112059111989411198971)+(12059621198791198972

minus12059611198791198971))

]

times 119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

]

times 119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119861lowast

119902(Θ

119861

11989421198972

1205962)

times 119890119895((1205961119888)119886

119861119879

119901Θ119861

11989411198971minus(1205962119888)119886

119861119879

119902Θ119861

11989421198972)

]

times 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

(7)



1= 119894

2= 119894 119897

1= 119897






1= 119894

2 119897

1= 119897



11989411198971

and Θ119872

11989421198972

(or Θ119861

11989411198971

and Θ119861

11989421198972

) areindependent


119864 [(1205911198942119897212059111989411198971)minus1

119890119895(120596212059111989421198972

minus120596112059111989411198971)

times 119890(12059611198791198981199011198971

minus12059621198791198981199011198972)

]

=

Φ(minus1)

120591(119895 (120596

2minus 120596

1))

timesΦ119879(119895 (120596

1minus 120596

2)) Case 1

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)

timesΦ119879(minus119895120596

2)Φ

119879(119895120596

1) Case 2

(8)



120591119879(119904) suggests


119883(119904) = 119864[119890

119895119904119883

] =

intinfin

minusinfin

119890119895119904120585



119901119879119897(119879) =

Λ119871+1

119879119871

119890minusΛ119879

119871 119901

120591119894119897(120591) =

120582119868

120591119868minus1

119890minus120582120591

(119868 minus 1) (9)




119894119897sim 119880[minus120587 120587)



119895(1206011198941 1198971minus1206011198942 1198972

)


119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

] = 1 Case 11198962

Case 2(10)


Case 1 119864 [119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

119894119897minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

119894119897)

]

= int

120587

minus120587

119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890(119895119888)(119889

119872

119898119899)119879Θ119872

119894119897 119891Θ119872 (Θ

119872

119894119897) 119889Θ

119872

119894119897

= int

120587

minus120587

infin

sum

119896=minusinfin

(G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

times 119890119895119896Θ119872

119894119897+119895(|119889119872

119898119899| cos(Θ119872

119894119897minusang119889119872

119898119899)119888)

119889Θ119872

119894119897

= 2120587

infin

sum

119896=infin

119895119896

119890119895119896ang119889119872

119898119899

times (G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898119899

10038161003816100381610038161003816

119888)

(11)

whereG119872

119898119896G119872

119899119896 andF119872



119896(119906) ≜ (119895

minus119896

120587) int120587

0

119890119895(119896120585+119906 cos 120585)



119899


follows 119911119899≜ 119909

119899otimes 119910

119899= sum

infin

119896=infin119909119896119910119899minus119896


119872

119898119899and 119889

119861



119898119899and 119889

119861


1119886119861

119901and

1205962119886119861


2(1199052119881 + 119886

119872

119899) and 120596

1(1199051119881 +

119886119872




1) and ℎ

119902119899(1199052 120596

2)


119898119899and 119889119861

119901119902can be


119889119872

119898≜ 120596

1(119886

119872

119898minus 119905

1119881) 119889

119872

119899≜ 120596

2(119886

119872

119899minus 119905

2119881)

119889119872

119898119899≜ (120596

21199052minus 120596

11199051) 119881 + (120596

1119886119872

119898minus 120596

2119886119872

119899)

119889119861

119901≜ 120596

1119889119861

119901 119889

119861

119902≜ 120596

2119889119861

119902 119889

119861

119901119902≜ 120596

1119886119861

119901minus 120596

2119886119861

119902

(12)



Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)


1198942= 119894 119897

1= 119897



119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40



n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk



(i) F(sdot)



(ii) G(sdot)





0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901119898119894119897= 12120596120591

119894119897


ℎ119901119898

(119905 120596)

= (120596119887119908

120596)

120578 119871

sum

119897=1

119868

sum

119894=1

119866119861

119901(Θ

119861

119894119897 120596)

times 119866119872

119898(Θ

119872

119894119897 120596) 119892


(4)


119901119898119894119897 a path


119901119898119894119897(119905) ≜ 120591

119901119898119894119897+

(119905119888)119881119879

Θ119872



= (120596119888)119881119879

Θ119872

119894119897where




1) and ℎ

119902119899(1199052 120596

2)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2) ≜ 119864 [ℎ

119901119898(119905

1 120596

1) ℎ

lowast

119902119899(119905

2 120596

2)] (5)


1 119905

2)





119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

times

119871

sum

119897=1

119868

sum

119894=1

119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119872

119898(Θ

119872

11989411198971

1205961)

times 11989211990111989811989411198971

times 11989211990211989911989421198972

times 119890(minus11989512059611198791199011198981198971

minus119895120596112059111990111989811989411198971(1199051))]

times 119864 [119866119861lowast

119902(Θ

119861

11989421198972

1205962)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895(1206011198941 1198971

minus1206011198942 1198972)

times 119890(minus11989512059621198791199021198991198972

minus119895120596212059111990211989911989421198972(1199052))]

(6)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times

119871

sum

119897=1

119868

sum

119894=1

119864 [(1205911198942119897212059111989411198971)minus1

times 119890119895((120596212059111989421198972

minus120596112059111989411198971)+(12059621198791198972

minus12059611198791198971))

]

times 119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

]

times 119864 [119866119861

119901(Θ

119861

11989411198971

1205961)119866

119861lowast

119902(Θ

119861

11989421198972

1205962)

times 119890119895((1205961119888)119886

119861119879

119901Θ119861

11989411198971minus(1205962119888)119886

119861119879

119902Θ119861

11989421198972)

]

times 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

(7)



1= 119894

2= 119894 119897

1= 119897






1= 119894

2 119897

1= 119897



11989411198971

and Θ119872

11989421198972

(or Θ119861

11989411198971

and Θ119861

11989421198972

) areindependent


119864 [(1205911198942119897212059111989411198971)minus1

119890119895(120596212059111989421198972

minus120596112059111989411198971)

times 119890(12059611198791198981199011198971

minus12059621198791198981199011198972)

]

=

Φ(minus1)

120591(119895 (120596

2minus 120596

1))

timesΦ119879(119895 (120596

1minus 120596

2)) Case 1

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)

timesΦ119879(minus119895120596

2)Φ

119879(119895120596

1) Case 2

(8)



120591119879(119904) suggests


119883(119904) = 119864[119890

119895119904119883

] =

intinfin

minusinfin

119890119895119904120585



119901119879119897(119879) =

Λ119871+1

119879119871

119890minusΛ119879

119871 119901

120591119894119897(120591) =

120582119868

120591119868minus1

119890minus120582120591

(119868 minus 1) (9)




119894119897sim 119880[minus120587 120587)



119895(1206011198941 1198971minus1206011198942 1198972

)


119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

] = 1 Case 11198962

Case 2(10)


Case 1 119864 [119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

119894119897minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

119894119897)

]

= int

120587

minus120587

119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890(119895119888)(119889

119872

119898119899)119879Θ119872

119894119897 119891Θ119872 (Θ

119872

119894119897) 119889Θ

119872

119894119897

= int

120587

minus120587

infin

sum

119896=minusinfin

(G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

times 119890119895119896Θ119872

119894119897+119895(|119889119872

119898119899| cos(Θ119872

119894119897minusang119889119872

119898119899)119888)

119889Θ119872

119894119897

= 2120587

infin

sum

119896=infin

119895119896

119890119895119896ang119889119872

119898119899

times (G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898119899

10038161003816100381610038161003816

119888)

(11)

whereG119872

119898119896G119872

119899119896 andF119872



119896(119906) ≜ (119895

minus119896

120587) int120587

0

119890119895(119896120585+119906 cos 120585)



119899


follows 119911119899≜ 119909

119899otimes 119910

119899= sum

infin

119896=infin119909119896119910119899minus119896


119872

119898119899and 119889

119861



119898119899and 119889

119861


1119886119861

119901and

1205962119886119861


2(1199052119881 + 119886

119872

119899) and 120596

1(1199051119881 +

119886119872




1) and ℎ

119902119899(1199052 120596

2)


119898119899and 119889119861

119901119902can be


119889119872

119898≜ 120596

1(119886

119872

119898minus 119905

1119881) 119889

119872

119899≜ 120596

2(119886

119872

119899minus 119905

2119881)

119889119872

119898119899≜ (120596

21199052minus 120596

11199051) 119881 + (120596

1119886119872

119898minus 120596

2119886119872

119899)

119889119861

119901≜ 120596

1119889119861

119901 119889

119861

119902≜ 120596

2119889119861

119902 119889

119861

119901119902≜ 120596

1119886119861

119901minus 120596

2119886119861

119902

(12)



Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)


1198942= 119894 119897

1= 119897



119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40



n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk



(i) F(sdot)



(ii) G(sdot)





0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1= 119894

2 119897

1= 119897



11989411198971

and Θ119872

11989421198972

(or Θ119861

11989411198971

and Θ119861

11989421198972

) areindependent


119864 [(1205911198942119897212059111989411198971)minus1

119890119895(120596212059111989421198972

minus120596112059111989411198971)

times 119890(12059611198791198981199011198971

minus12059621198791198981199011198972)

]

=

Φ(minus1)

120591(119895 (120596

2minus 120596

1))

timesΦ119879(119895 (120596

1minus 120596

2)) Case 1

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)

timesΦ119879(minus119895120596

2)Φ

119879(119895120596

1) Case 2

(8)



120591119879(119904) suggests


119883(119904) = 119864[119890

119895119904119883

] =

intinfin

minusinfin

119890119895119904120585



119901119879119897(119879) =

Λ119871+1

119879119871

119890minusΛ119879

119871 119901

120591119894119897(120591) =

120582119868

120591119868minus1

119890minus120582120591

(119868 minus 1) (9)




119894119897sim 119880[minus120587 120587)



119895(1206011198941 1198971minus1206011198942 1198972

)


119864 [119890119895(1206011198941 1198971

minus1206011198942 1198972)

] = 1 Case 11198962

Case 2(10)


Case 1 119864 [119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

119894119897minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

119894119897)

]

= int

120587

minus120587

119866119872

119898(Θ

119872

119894119897 120596

1)119866

119872lowast

119899(Θ

119872

119894119897 120596

2)

times 119890(119895119888)(119889

119872

119898119899)119879Θ119872

119894119897 119891Θ119872 (Θ

119872

119894119897) 119889Θ

119872

119894119897

= int

120587

minus120587

infin

sum

119896=minusinfin

(G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

times 119890119895119896Θ119872

119894119897+119895(|119889119872

119898119899| cos(Θ119872

119894119897minusang119889119872

119898119899)119888)

119889Θ119872

119894119897

= 2120587

infin

sum

119896=infin

119895119896

119890119895119896ang119889119872

119898119899

times (G119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898119899

10038161003816100381610038161003816

119888)

(11)

whereG119872

119898119896G119872

119899119896 andF119872



119896(119906) ≜ (119895

minus119896

120587) int120587

0

119890119895(119896120585+119906 cos 120585)



119899


follows 119911119899≜ 119909

119899otimes 119910

119899= sum

infin

119896=infin119909119896119910119899minus119896


119872

119898119899and 119889

119861



119898119899and 119889

119861


1119886119861

119901and

1205962119886119861


2(1199052119881 + 119886

119872

119899) and 120596

1(1199051119881 +

119886119872




1) and ℎ

119902119899(1199052 120596

2)


119898119899and 119889119861

119901119902can be


119889119872

119898≜ 120596

1(119886

119872

119898minus 119905

1119881) 119889

119872

119899≜ 120596

2(119886

119872

119899minus 119905

2119881)

119889119872

119898119899≜ (120596

21199052minus 120596

11199051) 119881 + (120596

1119886119872

119898minus 120596

2119886119872

119899)

119889119861

119901≜ 120596

1119889119861

119901 119889

119861

119902≜ 120596

2119889119861

119902 119889

119861

119901119902≜ 120596

1119886119861

119901minus 120596

2119886119861

119902

(12)



Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)


1198942= 119894 119897

1= 119897



119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40



n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk



(i) F(sdot)



(ii) G(sdot)





0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Case 2 119864 [119866119872

119898(Θ

119872

11989411198971

1205961)119866

119872lowast

119899(Θ

119872

11989421198972

1205962)

times 119890119895((1205961119888)(119886

119872

119898minus1198811199051)

119879Θ119872

11989411198971minus(1205962119888)(119886

119872

119899minus1198811199052)

119879Θ119872

11989421198972)

]

= (21205872

)

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119898 (G119872

119898119896(120596

1) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119898

10038161003816100381610038161003816

119888)]

times [

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889119872

119899 (G119872

119899minus119896(120596

2) otimesF

119872

119896) 119869

119896(

10038161003816100381610038161003816119889119872

119899

10038161003816100381610038161003816

119888)]

(13)


119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

1minus 120596

2))Φ

119879(119895 (120596

2minus 120596

1))

times W (119889119861

119901119902H

119861

119896) timesW (119889

119872

119898119899H

119872

119896)

+ 1198962

Φ(minus1)

120591(119895120596

2)Φ

(minus1)

120591(minus119895120596

1)Φ

119879(minus119895120596

2)Φ

119879(119895120596

1)

times W (119889119861

119901G

119861

119901119896(120596

1) otimesF

119861

119896)

timesW (119889119861

119902G

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898G

119872

119898119896(120596

1) otimesF

119872

119896)

times W (119889119872

119899G

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(14)


1198942= 119894 119897

1= 119897



119877119901119898119902119899

(1199051 119905

2 120596

1 120596

2)

=(120596

11988711990811205961198871199082

)120578

(12059611205962)120578

(412059611205962)

times Φminus(1)

120591(119895 (120596

2minus 120596

1))Φ

119879(119895 (120596

2minus 120596

1))

timesW (119889119861

119901119902G

119861

119901119896(120596

1) otimesG

119861lowast

119902minus119896(120596

2) otimesF

119861

119896)

timesW (119889119872

119898119899G

119872

119898119896(120596

1) otimesG

119872lowast

119899minus119896(120596

2) otimesF

119872

119896)

(15)

where

W (119889H119896) ≜ 2120587

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119889

H119896(120596) 119869

119896(|119889|

119888)

H119896= G

119896(120596

1) otimesG

lowast

minus119896(120596

2) otimesF

119896

(16)

0 5 10 15 20 25 30 35 40



n = 3

100

10minus1

10minus2

10minus3

10minus4

k

ℱk



(i) F(sdot)



(ii) G(sdot)





0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)12

3Antenna15 times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15 times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

05

15times108

02

04

06

08

Δf (MHz)1

23Antenna

15times

005

115

2

0

2

40

02

04

06

08

1


CCF

times108

Δf (Hz)1

152Antenna s times







2minus 119891

1


2minus 119891

1(where 119891



1 119905

2 120596

1 120596

2)|2

|119877(1199051 119905

1 120596

1 120596

1)|2 In the


119867

where 120596119867= 2120587119891

119867




0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)12

3Antenna sp15

times

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)

0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108

Δf (MHz)0

1

2

01

23

40

1


CCF

08

06

04

02

05

15times108


15times







2minus 119891

1





2= 120596 and 119898 = 119899 = 1 Considering

ang119889119872

11= ang119881 + ang(119905

2minus 119905

1) we get

11987711990111199021

(1199051 119905

2 120596 120596)

= 1205871205962120578

119887119908

21205962120578+2W (119889

119861

119901119902H

119861

119896)

times

infin

sum

119896=minusinfin

119895119896

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896)

times 119869119896(120596 (119905

2minus 119905

1) |119881|

119888)

(17)


119869119896(Λ) ≜ 119865 [119869

119896(119906)] = int

infin

minusinfin

119890minus119895Λ120585

119869119896(120585) 119889120585

= (2 (minus119895)

119896


0 |Λ| ≧ 1


119896(Λ) ≜



102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5


RM(Λ

)

(cΛ)(|V|120596)


102

101

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6


RM(Λ

)



(cΛ)(|V|120596)

(c) Helical antenna

102

101

100

10minus1

10minus2

10minus3

10minus4


RM(Λ

)



(cΛ)(|V|120596)




2minus 119905


equation11987711990111199021

(Λ 120596)

≜ int

infin

minusinfin

119890minus119895ΛΔ119905

11987711199011119902

(1199051 119905

2 120596 120596) 119889Δ119905

=1205962120578

119887119908

21205962120578+2W (119889

119872

119902119901H

119861

119896)120587119888

|119881|

times

infin

sum

119896=minusinfin

[(119890119895119896ang119881

G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896]

(19)

where H119861

119896≜ G119861

119901119896(120596) otimes G119861lowast

119902minus119896(120596) otimesF119861




119862119896= (119879

119896(119888Λ|119881|120596))radic1 minus (119888Λ|119881|120596)






104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










104

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)

(a) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8

10minus10


RM(Λ

)

(cΛ)(|V|120596)


102

100

10minus2

10minus4

10minus6

10minus8




RM(Λ

)

(cΛ)(|V|120596)

(c) Helical antenna

102

100

10minus2

10minus4

10minus6

10minus8


RM(Λ

)



(cΛ)(|V|120596)




119877119872

(Λ) ≜

infin

sum

119896=minusinfin

119890119895119896ang119881

(G119872

1119896(120596) otimesG

119872lowast


119872

119896) 119862

119896 (20)







(120596119888)|119881| as follows





= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












= 0


(Λ) = 0 if |Λ| ⩾

(120596119888)|119881|)


5 Conclusion





Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article On the Cross Correlation Properties of MIMO...

Documents

Transcript of Research Article On the Cross Correlation Properties of MIMO...