ARTICLE IN PRESS

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Signal Processing

Signal Processing 90 (2010) 2276–2287

0165-16

doi:10.1

$ Thi

983335| E

E-m

journal homepage: www.elsevier.com/locate/sigpro

Sampling theorems for Doppler-stretched wide-band signals$

Antonio Napolitano|

Universit �a di Napoli ‘‘Parthenope’’, Dipartimento per le Tecnologie, I-80143 Napoli, Italy

a r t i c l e i n f o

Article history:

Received 30 September 2009

Received in revised form

15 February 2010

Accepted 15 February 2010Available online 19 February 2010

Keywords:

Doppler effect

Cyclostationarity

Sampling

Spectrally correlated signals

84/$ - see front matter & 2010 Elsevier B.V. A

016/j.sigpro.2010.02.016

s work is partially supported by the NATO

.

URASIP member.

ail address: antonio.napolitano@uniparthenop

a b s t r a c t

The paper deals with Doppler-stretched wide-band signals occurring in mobile

communications and in radar/sonar problems in the presence of moving targets in

the case of constant relative radial speeds. Lo �eve bifrequency spectrum and cross-

spectrum are used to characterize in the spectral domain the received signal and to

jointly characterize the transmitted and received signals. It is shown that, even if both

the transmitted and received signals are singularly almost-cyclostationary (ACS), they

are not jointly ACS but, rather, jointly spectrally correlated. The problem of uniformly

sampling these nonstationary signals is addressed. It is shown that, unlike the case of

the wide-sense stationary signals, several kind of aliasing effects occur for the various

spectral (cross-)statistical functions used to characterize these signals. Lower bounds on

the sampling frequency are derived to avoid these aliasing effects.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

In mobile communications and in radar/sonar applica-tions, the Doppler effect due to constant relative radial speedbetween transmitter and receiver and/or a moving targetproduces a time-scale factor in the (real-valued) band-passreceived signal with respect to the transmitted one. Ittraduces into a frequency shift of the carrier frequency and atime-scale factor in the complex envelope of the receivedsignal. If the product between bandwidth and data-recordlength cannot be considered much smaller than the ratiobetween medium propagation speed and radial speed, theso called ‘‘narrow-band approximation’’ does not hold andthe time-scale factor in the complex envelope cannot beconsidered unitary [17, pp. 339–340]. In such a case, thereceived signal is said to be a Doppler-stretched version ofthe transmitted one and wide-band correlation techniquesmust be used for synchronization [13], channel identifica-tion [2], and source localization purposes [19–21].

In this paper, transmitted and received signals arecharacterized singularly and jointly in the spectral domain

ll rights reserved.

Grant ICS.NUKR.CLG

e.it

by the Lo�eve bifrequency spectrum [8]. It allows toproperly describe in the frequency domain second-ordernonstationary signals [3,8,9,14] and is suitable to treat in acommon framework auto- and cross-spectral functions oftransmitted and received signals in the case of constantrelative radial speed. Assuming an ACS transmitted signal,even if the Doppler channel introducing a time-scale factoris linear not almost-periodically time-variant, the receivedsignal is shown to be in turn ACS, but with different cyclicfeatures. Thus, both transmitted and received signalsexhibit Lo�eve bifrequency spectrum with spectral massesconcentrated on lines with unit slope, but with differentspectral densities and axis intercepts. In contrast, thetransmitted and received signals are not jointly ACS.Their joint nonstationarity can be characterized by resort-ing to the model of the spectrally correlated (SC) signalsintroduced in [9]. Specifically, the transmitted andreceived signals are jointly SC with Lo�eve bifrequencycross-spectrum having spectral masses concentrated onlines with slope equal to the reciprocal of the time-scalefactor introduced by the Doppler channel. Even in thecase of time-scale factor s very close to 1, as it happensfor communications signals, the difference js�1j can besignificant in channel modeling [15], synchronization [13],and in the problem of spectral density estimation anddetermination of the spectral density support [1,6,7,9,10].

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Furthermore, in sonar and acoustic aircraft applications,the value of s is significantly different from 1 and influ-ences the shape of the spectral densities and theirbandwidths [19–21].

SC signals have Lo�eve bifrequency spectrum withspectral masses concentrated on a countable set of supportcurves in the bifrequency plane. Thus the class of the SCsignals extends that of the ACS signals which are obtainedas special case when the support curves are lines with unitslope. Another class of nonstationary signals that extendsthat of the ACS signals is the class of the generalizedalmost-cyclostationary (GACS) signals [11,12] which is auseful model in the case of transmitted ACS signal, narrow-band condition, and constant relative acceleration betweentransmitter and receiver. The class of the ACS signals turnsout to be the intersection between the classes of the GACSand SC signals [9,11].

In the following, when it does not create ambiguity, thegeneric expression ‘‘spectral statistical function’’ will beused to denote either Lo�eve bifrequency (cross-)spectrumor its density, which in the special case of ACS signals iscoincident with the cyclic spectrum.

In the paper, sampling theorems are derived for spectralfunctions characterizing the almost-cyclostationarity ofthe received sampled signal and the joint nonstationarityof the transmitted and received sampled signals. It isshown that for ACS and SC signals, unlike the case ofwide-sense stationary (WSS) signals, the following threeconditions are not equivalent: (1) replicas in the Lo�evebifrequency (cross-)spectrum of the discrete-time signaldo not overlap; (2) the main replica falls in the principalbifrequency domain [�1/2,1/2]� [�1/2,1/2]; (3) for spec-tral densities of continuous- and discrete-time signals themapping n¼ f=fs holds in the whole frequency domainn 2 ½�1=2;1=2�, where f and n are the frequency variablesof the continuous- and discrete-time signals, respectively,and fs is the sampling frequency. Considering one condi-tion instead of the other in digital implementations ofsignal processing algorithms can lead to noncompletealiasing-effect removal.

In the case of a transmitted ACS signal, samplingtheorems are derived and sufficient conditions areprovided to satisfy the above-mentioned conditions withreference to Lo�eve bifrequency spectrum and cross-spectrum and their densities. Furthermore, samplingtheorems for time-domain statistical functions are alsopresented. The derived sufficient conditions allow to avoidseveral kind of aliasing effects in the expressions that linkspectral statistical functions of the sampled Doppler-stretched received signal to those of the sampledtransmitted signal. These relations are useful in thediscrete-time implementation of cyclostationarity-basedDoppler-channel parameter-estimation algorithms as thatin [2] and of wide-band source localization methods asthose proposed in [20,21].

The paper is organized as follows. In Section 2, theanalytical model for continuous-time Doppler-stretchedwide-band signals is introduced and for these signalssecond-order spectral characterization is provided bythe Lo�eve bifrequency (cross-)spectrum. In Section 3,the problem of sampling Doppler-stretched signals is

addressed and sampling theorems are provided in bothfrequency (Section 3.2) and time (Section 3.3) domains.Numerical results are reported in Section 4 to illustrate thetheoretical results. Conclusions are drawn in Section 5.

2. Continuous-time Doppler-stretched signal

2.1. Signal model

Let us consider a communications or radar/sonarsystem where

zTXðtÞ ¼ RefxaðtÞej2pf0tg ð1Þ

is the (continuous-time) transmitted signal with complexenvelope xa(t) and carrier frequency f0 and Ref�g denotesreal part. In the presence of relative motion betweentransmitter and receiver and/or a surrounding scatterer ora target characterized by constant relative radial speed v

(within the observation time interval), the transmittedsignal experiences a linearly time-varying delay. Conse-quently, the complex envelope ya(t) of the received signal

zRXðtÞ ¼ RefyaðtÞej2pf0tg ð2Þ

is the Doppler-stretched signal [17, pp. 339–340]

yaðtÞ ¼ Aejjxaðst�daÞej2pfat ð3Þ

with Fourier transform

Yaðf Þ ¼ Aejj 1

jsjXa

f�fa

s

� �e�j2pðf�faÞ da=s ð4Þ

where Xa(f) is the Fourier transform of xa(t). In (3) and (4),A is the scaling amplitude, j the phase, da the delay, fa thefrequency shift, and s the time-scale factor introduced bythe Doppler channel. In the case of nonrelativistic Dopplereffect, for stationary transmitter and moving receiverfa=�(v/c) f0 and s=(c�v)/c, whereas for moving transmit-ter and stationary receiver fa=�(v/(c+v)) f0 and s=c/(c+v),where c is the medium propagation speed. For a mono-static stationary radar in the presence of moving target,fa=�(2v/c)f0 and s=(c�v)/(c+v) [19].

If xa(t) has bandwidth B and the data-record length is T,under the so-called narrow-band condition

BT5c=v ð5Þ

the time-scale factor s in the received complex envelopecan be considered unitary [17, pp. 339–340] and theDoppler effect can be modeled as a simple frequency shiftof the complex envelope of the transmitted signal.

Several situations are encountered in practice wherethe time-scale factor cannot be considered unitary. In [9],it is shown that in problems of spectral density estimationof direct-sequence spread-spectrum (DSSS) signalsadopted in code-division multiple access (CDMA) systems,in the case of 512 chip per symbol and v=100 km h�1, thescaling factor s can be considered unitary only if thenumber of processed symbols does not exceed fewthousands. In space communications, disregarding thepresence of the scaling factor in the complex envelopedoes not allow synchronization [13]. In aircraft acoustics,in the case of moving transmitter with v=800 km h�1 andstationary receiver, it results sC0:6. In sonar applications

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the scaling factor s is significantly different from 1 even formoderate radial speeds [19]. In all these cases, the narrow-band condition given in (5) is not fulfilled and wide-bandsignal processing techniques need to be adopted [20,21].

2.2. Second-order characterization of the transmitted signal

In the following, the transmitted signal is modeled assecond-order ACS since this model is appropriate foralmost all modulated signals adopted in communications,radar, sonar, and telemetry [4]. ACS signals have second-order moments that are almost-periodic functions of time:

EfxaðtþtÞxð�Þa ðtÞg ¼Xa2A

Raxaxð�ÞaðtÞej2pat ð6Þ

where ð�Þ denotes an optional complex conjugation andthe cycle frequencies a range in the countable set A(depending on ð�Þ) containing possibly incommensurateelements. In [16,18], it is shown that both autocorrelationfunction EfxaðtþtÞx�aðtÞg and conjugate correlation functionEfxaðtþtÞxaðtÞg are necessary for a complete second-orderwide-sense characterization of the complex-valued signalxa(t). The coefficients Ra

xaxð�Þa

ðtÞ of the (generalized) Fourierseries expansion in (6) are referred to as (conjugate) cyclicautocorrelation functions. ACS signals are characterized inthe frequency domain by Lo�eve bifrequency spectrum [8]with support contained in lines with slope 71:

EfXaðf1ÞXð�Þa ðf2Þg ¼

Xa2A

Saxaxð�Þaðf1Þdðf2�ð�Þða�f1ÞÞ ð7Þ

where the Fourier transform Xa(f) of xa(t) is assumed toexists with probability 1 (at least) in the sense ofdistributions [22] (see also [9] for a link with theharmonizability property). In (7), dð�Þ denotes Dirac delta,(�) is an optional minus sign to be considered only if ð�Þ ispresent, and the (conjugate) cyclic spectra Sa

xaxð�Þa

ðf1Þ are theFourier transforms of the (conjugate) cyclic autocorrela-tion functions Ra

xaxð�Þa

ðtÞ. From (7) it follows that for ACSsignals correlation exists between spectral componentsthat are separated by quantities equal to the cyclefrequencies (that belong to the countable set A). Incontrast, for wide-sense stationary signals, distinct spec-tral components are uncorrelated.

If xa(t) is strictly bandlimited, i.e., the power spectrumis such that S0

xax�aðf Þ ¼ 0 for jf j4B, then the support of the

(conjugate) cyclic spectrum is such that [4, Eqs. (3.100)and (3.111), and Fig. 1]:

suppfSaxaxð�Þaðf ÞgDfða;f Þ 2 R�R : jf jrB; ja�f jrBg ð8aÞ

Dfða;f Þ 2 R�R : jf jrB; jajr2Bg ð8bÞ

Therefore, Saxaxð�Þa

ðf Þ ¼ 0 for jf j4B; jaj42B. Moreover, ac-

counting for (8a) and the fact that each term of the sum in

(7) is nonzero only for f2 ¼ ð�Þða�f1Þ, it results that

suppfSaxaxð�Þaðf1Þdðf2�ð�Þða�f1ÞÞgDfðf1;f2Þ 2 R�R

: jf1jrB; ja�f1jrB; f2 ¼ ð�Þða�f1Þg ð9aÞ

¼ fðf1;f2Þ 2 R�R : jf1jrB; jf2jrB; f2 ¼ ð�Þða�f1Þg ð9bÞ

2.3. Second-order characterization of the Doppler-stretched

received signal

In this section, the second-order spectral characteriza-tion of the Doppler-stretched received signal ya(t) isprovided under the assumption that the transmitted signalxa(t) is ACS. Moreover, in the case of xa(t) strictly band-limited, supports of the terms of the Lo�eve bifrequencyspectrum of ya(t) are derived.

Let xa(t) be ACS. Accounting for (4) and (7), the Lo�evebifrequency spectrum of ya(t) is given by

EfYaðf1ÞYð�Þa ðf2Þg ¼ A2ejkj 1

jsje�j2pðf1�faÞda=se�ð�Þj2pðf2�faÞda=s

�Xa2A

Saxaxð�Þa

f1�fa

s

� �dðf2þð�Þf1�½ð�Þsaþkfa�Þ

ð10Þ

where k9ð1þð�Þ1Þ, that is, k¼ 0 if (�)=� and k¼ 2 if(�)= +. From (10), it follows that the Lo�eve bifrequencyspectrum of ya(t) has support contained in lines withslopes 71. That is, the Doppler-stretched signal ya(t) isACS and, accordingly with [2], its cyclic spectra can berelated to those of the transmitted signal xa(t) by therelationship

Sayayð�Þaðf1Þ ¼ A2ejkje�j2pða�kfaÞda=s 1

jsjSða�kfaÞ=s

xaxð�Þa

f1�fa

s

� �ð11Þ

Therefore, due to the Doppler channel, cycle frequencies ofxa are scaled by s and conjugate cycle frequencies of xa arescaled by s and shifted by 2fa. Moreover, according to (9a),the support of the generic term of the sum over a in (10) issuch that

supp Saxaxð�Þa

f1�fa

s

� �d

f2�fa

s�ð�Þ a� f1�fa

s

� �� �� �

D ðf1; f2Þ 2 R�R :f1�fa

s

��������rB;

f1�fa

s�a

��������rB;

�

f2�fa

s¼ ð�Þ a� f1�fa

s

� ��¼ ðf1;f2Þ 2 R�R :

f1�fa

s

��������rB;

�

f2�fa

s

��������rB; f2 ¼ ð�Þsaþkfa�ð�Þf1

�Dfðf1; f2Þ 2 R�R

: jf1jr jfajþBjsj; jf2jr jfajþBjsj; f2 ¼ ð�Þsaþkfa�ð�Þf1g

ð12Þ

Note that the ACS nature of ya(t) is not an obvious result.In fact, relation (3) describes a transformation of the inputsignal xa(t) into the output signal ya(t) which is linear time-variant but not almost-periodically time-variant. Only inthis last case it is known that ACS signals are transformedinto ACS signals [4].

2.4. Joint second-order characterization of the transmitted

and received signals

By using (4) and (7), the Lo�eve bifrequency cross-spectrum of ya(t) and xa(t) can be expressed as

EfYaðf1ÞXð�Þa ðf2Þg ¼ Aejj 1

jsje�j2pðf1�faÞ da=s

Xa2A

Saxaxð�Þa

f1�fa

s

� �

�d f2�ð�Þ a� f1�fa

s

� �� �ð13Þ

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Thus, when sa1, the Lo�eve bifrequency cross-spectrumhas support contained in lines with slopes different from71. That is, even if both signals xa(t) and ya(t) are(singularly) ACS, they are not jointly ACS. The jointnonstationary behavior of xa(t) and ya(t) can be describedresorting to the class of the SC signals introduced in [9].In fact, jointly SC signals have Lo�eve bifrequency cross-spectrum with spectral masses concentrated on acountable set of curves in the bifrequency plane [9].Consequently, from (13) it follows that ya(t) and xa(t) arejointly SC with support curves constituted by lines withslopes 71/s. Assuming the transmitted signal xa(t) strictlyband-limited with bandwidth B, according to (9a) thesupport of the generic term of the sum over a in (13) issuch that

supp Saxaxð�Þa

f1�fa

s

� �d f2�ð�Þ a� f1�fa

s

� �� �� �

D ðf1;f2Þ 2 R�R : jf1jr jfajþBjsj; jf2jrB;�

f2 ¼ ð�Þ a� f1�fa

s

� ��ð14Þ

By taking the double inverse Fourier transform of bothsides of (13) leads to

Efyaðt1Þxð�Þa ðt2Þg ¼ Aejjej2pfat1

Xa2A

ej2pat2 Raxaxð�Þaðst1�t2�daÞ ð15Þ

Accordingly with the frequency-domain result, fort1 ¼ tþt, t2=t, the cross-correlation functionEfyaðtþtÞxð�Þa ðtÞg is not an almost-periodic function of t,that is, ya(t) and xa(t) are not jointly ACS.

3. Discrete-time Doppler-stretched signal

In this section, sampling theorems on spectral statis-tical functions of ACS transmitted signals and receivedDoppler-stretched signals are reported. The theoreticalresults of this section are illustrated by a numericalexperiment in Section 4. A reference to the correspondingresults of the numerical experiment is given as commentfor each theorem.

3.1. Sampling ACS signals

In this section, results for cyclic statistics of sampledACS signals [5] are reviewed and refined within theframework of the Lo�eve bifrequency spectrum.

Let xðnÞ9xaðtÞjt ¼ nTsbe the discrete-time signal

obtained by uniformly sampling with period Ts=1/fs thecontinuous-time signal xa(t). Its Fourier transform XðnÞ islinked to Xa(f) by

XðnÞ ¼Xn2Z

xðnÞe�j2pnn ¼1

Ts

Xk2Z

Xaððn�kÞfsÞ ð16Þ

If xa(t) is ACS, by using (7) and (16), the followingaliasing formula for the Lo�eve bifrequency spectrum ofa sampled ACS signal is obtained

EfXðn1ÞXð�Þðn2Þg ¼

1

Ts

Xn12Z

Xn22Z

Xa2A

Saxaxð�Þaððn1�n1ÞfsÞdððn2�n2Þ

�ð�Þða=fs�ðn1�n1ÞÞÞ ð17Þ

where, in its derivation, the scaling property of the Diracdelta fsdðnfsÞ ¼ dðnÞ is used [22, Section 1.7]. From (17),it follows that x(n) is a discrete-time ACS signal. In fact, itsLo�eve bifrequency spectrum has support contained in lineswith slope 71 in the bifrequency plane ðn1; n2Þ and can bewritten as

EfXðn1ÞXð�Þðn2Þg ¼

X~a2 ~A

~S~axxð�Þ ðn1Þ

Xn2Z

d n2�n�ð�Þð ~a�n1Þð Þ ð18Þ

In (18), the countable set ~A is linked to A by therelationships

~A ¼[a2Af ~a 2 ð�1=2;1=2� : ~a ¼ ða=fsÞmod 1g ð19Þ

A¼[~a2 ~A

[p2Z

fa 2 R : a¼ ~afs�pf sg ð20Þ

with mod 1 denoting the modulo 1 operation with valuesin (�1/2,1/2] and accordingly with [5], the (conjugate)cyclic spectrum of x(n) is given by

~S~axxð�Þ ðnÞ ¼

1

Ts

Xp2Z

Xq2Z

Sa�pf s

xaxð�Þa

ðf�qf sÞjf ¼ nfs ;a ¼ ~afsð21Þ

The following theorem provides sufficient conditions toavoid aliasing in the Lo�eve bifrequency spectrum of x(n).Moreover, known results from [5] are obtained in adifferent way which is suitable to be generalized in thenext section to the case of Doppler-stretched signals.

Theorem 1 (ACS signals: sampling theorem for Lo�eve

bifrequency spectrum). Let xa(t) be ACS and band-limited

with bandwidth B. For fsZ4B it results

EfXðn1ÞXð�Þðn2Þg ¼

1

Ts

Xa2A

Saxaxð�Þaðn1fsÞdðn2�ð�Þða=fs�n1ÞÞ;

jn1jr1

2; jn2jr

1

2ð22Þ

Moreover,

EfXðn1ÞXð�Þðn2Þg ¼ 0 for 1=4r jn1jr1=2 and 1=4r jn2jr1=2

ð23Þ

Proof. Replicas in the aliasing formula (17) are separatedby 1 in both n1 and n2 variables. Thus, from (9b) it followsthat fsZ2B is a sufficient condition such that replicas donot overlap. Such condition, however, does not assure thatthe mappings f1 ¼ n1fs and a¼ ~afs in

~S~axxð�Þ ðn1Þ ¼

1

TsSa

xaxð�Þaðf1Þjf1 ¼ n1 fs ;a ¼ ~afs

ð24Þ

hold 8 ~a 2 ½�1=2;1=2� and 8n1 2 ½�1=2;1=2�. For example,for ð�Þ present and a¼�B, equality (24) holds only for n1 2

½�1=2;0� and not for n1 2 ½0;1=2�. In fact, for n1 2 ½0;1=2�,the density of the replica with n1=0, n2=1 in (17) is alsopresent in the right-hand side of (24). In contrast,condition fsZ4B assures in (9b) with fi ¼ nifs, i=1,2,substituted into, that jn1jr1=4, jn2jr1=4, ja=fsj ¼

jn1þð�Þn2jr jn1jþjn2jr1=2 and, consequently, that themappings f1 ¼ n1fs and a¼ ~afs in (24) hold 8 ~a 2 ½�1=2;1=2�and 8n1 2 ½�1=2;1=2�. Moreover, (22) holds. Finally, notethat the effect of sampling at twice the Nyquist rate leadsalso to (23). &

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3.2. Sampling Doppler-stretched signals: frequency domain

In this section, sampling theorems are proved for thespectral functions characterizing the uniformly sampledDoppler-stretched received signal and jointly characterizingthe sampled transmitted and received signals. In addition,sufficient conditions are derived to avoid aliasing in therelationships linking spectral functions of sampled trans-mitted and received signals in order to make these rela-tionships formally analogous to their continuous-timecounterparts (10), (11), and (13) which are obtained startingfrom the continuous-time physical model (3)–(4) for thepropagation channel. This formal analogy allows, for exam-ple, to a straightforward discrete-time implementation of theDoppler-channel parameter estimation algorithm proposedin continuous-time in [2], where, starting from (11), twominimum mean-square estimation (MMSE) procedures areproposed to blindly estimate the parameters A, j, da, fa, and s.

Let

yðnÞ9yaðtÞjt ¼ nTs¼ AejjxaðsnTs�daÞe

j2pfanTs ð25Þ

be the discrete-time signal obtained by uniformlysampling with period Ts the Doppler-stretched signal (3).Define the normalized frequency shift n9faTs and thenormalized delay d9da=Ts, where d is possibly noninteger.Accounting for (4), (7), and (16), the following aliasingformula for the Lo�eve bifrequency spectrum of y(n) isobtained when xa(t) is an ACS signal:

EfYðn1ÞYð�Þðn2Þg ¼

1

TsA2ejkj 1

jsje�j2p½n1þð�Þn2�kn� d=s

�Xn12Z

Xn22Z

Xa2A

Saxaxð�Þa

ðn1�n1�nÞfs

s

� �

�ej2pðn1þð�Þn2Þ d=s

�d n2�n2�n�ð�Þ safs�ðn1�n1�nÞ

� �� �

ð26Þ

By comparing (26) and (17), it follows that the discrete-time signal y(n) is ACS since its Lo�eve bifrequencyspectrum has support contained in lines with slope 71in the bifrequency plane ðn1; n2Þ. In addition, if xa(t) isband-limited with bandwidth B, accounting for (9b), thesupport of the replica with n1=n2=0 in the aliasing formula(26) is a scaled version of that in (12) and is reported herefor future reference

supp Saxaxð�Þa

n1�ns

fs

� �dðn2�ð�Þðsa=fs�n1þknÞÞ

� �

D ðn1; n2Þ 2 R�R : jn1�njrB

fsjsj; jn2�njr

B

fsjsj;

�

afs¼n1�kn

sþð�Þn2

�ð27aÞ

D ðn1; n2Þ 2 R�R : jn1jr jnjþB

fsjsj; jn2jr jnjþ

B

fsjsj;

�

afs¼n1�kn

sþð�Þn2

�ð27bÞ

Starting from (26) to (27b), the following results can beproved, which provide sufficient conditions to avoid aliasingin the Lo�eve bifrequency spectrum of y(n) and its density.

Theorem 2 (Doppler-stretched signal: sampling theorem

for Lo�eve bifrequency spectrum). Let xa(t) be ACS and band-

limited with bandwidth B. For fsZ2ðBjsjþjfajÞ it results that

EfYðn1ÞYð�Þðn2Þg ¼

1

TsA2ejkj 1

jsje�j2pðn1þð�Þn2�knÞ d=s

�Xa2A

Saxaxð�Þa

n1�ns

fs

� �d n2�nð

�ð�Þafs

s�ðn1�nÞ� ��

; jn1jp1

2; jn2jp

1

2

ð28Þ

which is coincident with its continuous-time counterpart (10)but for the amplitude scale factor 1/Ts and the scaling factor fs

in both frequency variables.

Proof. Replicas in (26) are separated by 1 in both n1 and n2

variables. Thus, from (27a) it follows that fsZ2Bjsj is asufficient condition such that replicas do not overlap.Note that, even if replicas do not overlap, the mappingni ¼ fi=fs, i=1,2, does not link (10) and (26) forni 2 ½�1=2;1=2�. A sufficient condition to assure such amapping or, equivalently, that replica (n1, n2) in (26) lies inthe square ðn1; n2Þ 2 ðn1�1=2;n1þ1=2Þ� ðn2�1=2;n2þ1=2Þ,according to (27b) is jnjþBjsj=fsr1=2, that is,fsZ2ðBjsjþjfajÞ. &

An application of Theorem 2 is illustrated in Section 4 bythe results reported in Figs. 1 and 2.

Theorem 3 (Doppler-stretched signal: sampling theorem

for the density of Lo�eve Bifrequency Spectrum). Let xa(t)be ACS and band-limited with bandwidth B. For

fsZ2ð2BjsjþjfajÞ, the density of Lo�eve bifrequency spectrum

of y(n) is given by

~SðkÞ

yyð�Þ ðn1Þ ¼1

jsjA2ejkje�j2pða=fsÞ d=s 1

TsSa

xaxð�Þa

n1�ns

fs

� �;

jn1jr1=2; n2 ¼ ð�Þafs

s�n1þð�Þkn� �

8a

ð29Þ

where k is an index in one-to-one correspondence with a 2 A.In (29), for (�)=� it results n1�n2 ¼ as=fs and for (�)= + it

results n1þn2 ¼ as=fsþ2n.

Proof. According to (27a), the mapping n1 ¼ f1=fs for n1 2

½�1=2;1=2� between the densities in (26) with n1=n2=0and those in (10) is assured provided that the followingimplication holds 8a:

n1 ¼ n7B

fsjsj ) jn2jr1� jnjþ B

fsjsj

� �ð30Þ

where

n2 ¼ ð�Þafs

s�n1þð�Þkn� �

ð31Þ

From (31) with n1 ¼ n7Bjsj=fs substituted into we have

jn2j ¼afs

s�n1þð�Þkn����

����¼ afs

s�n8 B

fsjsjþð�Þkn

��������rsup

a2A

afs

��������jsj

þB

fsjsjþjð�Þkn�nj ¼ 2B

fsjsjþ

B

fsjsjþjfaj

fsð32Þ

ARTICLE IN PRESS

Fig. 1. (Continuous-time) Doppler-stretched signal yaðtÞ ¼ xaðstÞej2pfat , where xa(t) is a PAM signal with Nyquist pulse. (a) Magnitude and (b) support of

the bifrequency spectral correlation density of ya(t), as a function of f1/fr and f2/fr.

A. Napolitano / Signal Processing 90 (2010) 2276–2287 2281

Thus, inequality in (30) is satisfied provided thatð3B=fsÞjsjþðjfaj=fsÞr1�ððjfaj=fsÞþðB=fsÞjsjÞ that is,fsZ2ð2BjsjþjfajÞ. &

The sufficient condition on the sampling frequency inTheorem 3 can be made more stringent by requiring that notonly the mapping between spectral frequencies n1 ¼ f1=fs

holds for n1 2 ½�1=2;1=2� but also that the mapping betweencycle frequencies ~a ¼ a=fs holds for ~a 2 ½�1=2;1=2�. This newcondition assures the formal analogy between formulas interms of cyclic spectra for continuous- and discrete-timecases. It can be obtained starting from the aliasing formulafor the discrete-time (conjugate) cyclic spectrum of y(n)which can be derived accounting for (11) and (21):

~S~ayyð�Þ ðnÞ ¼

1

Ts

Xp2Z

Xq2Z

Sa�pf s

yayð�Þa

ðf�qf sÞ

�����f ¼ nfs ;a ¼ ~afs

¼ A2ejkje�j2pð ~a�knÞ d=s 1

jsj

1

TsXp2Z

Xq2Z

S½ð~a�kn�pÞfs �=s

xaxð�Þa

n�ns

fs�qfs

s

� �ej2pp d=s ð33Þ

It is worth to observe that the formal analogy betweencyclic spectra of the discrete-time sampled signals andtheir continuous-time counterparts is not straightforwardand should be carefully stated, as clarified by the followingconsideration. From (21) it follows that

~Sð ~a�knÞ=s

xxð�Þn�n

s

� �¼

1

Ts

Xp2Z

Xq2Z

S½ð~a�knÞ=s�p�fs

xaxð�Þa

n�ns

fs�qf s

� �ð34Þ

Even if the function in the left-hand-side (lhs) of (34) canbe seen as the discrete-time counterpart of the cyclicspectrum in the right-hand-side (rhs) of (11), bothfunctions in the rhs and lhs of (34) are periodic in n and~a with period s and, hence, are not (conjugate) cyclic

spectra of discrete-time signals. The discrete-time counter-part of (11) is provided by the following result.

Theorem 4 (Doppler-stretched signal: sampling theorem for

the (conjugate) cyclic spectrum). Let xa(t) be ACS and band-

limited with bandwidth B. For fsZ4ðBjsjþjfajÞ it results that

~S~ayyð�Þ ðnÞ ¼ A2ejkje�j2pð ~a�knÞ d=s 1

jsj~Sð ~a�knÞ=s

xxð�Þn�n

s

� �;

jnjr 1

4; j ~ajr 1

2ð35Þ

which is formally analogous to its continuous-time counter-

part (11).

Proof. By setting a¼ ð ~a�knÞfs=s and f ¼ ðn�nÞfs=s into(8a), we have

supp Sð~a�knÞfs=s

xaxð�Þa

n�ns

fs

� �� �D ð ~a; nÞ 2 R�R : jn�njr B

fsjsj;

�

j ~a�knj�jn�njr B

fsjsj

�D ð ~a;nÞ 2 R�R : jnjr jnjþ B

fsjsj;

�

j ~ajrkjnjþ2B

fsjsj

�D ð ~a;nÞ 2 R�R : jnjr 1

4; j ~ajr 1

2

� �

ð36Þ

where the last inclusion relationship holds provided thatinequality fsZ4ðBjsjþjfajÞ is satisfied.

Replicas in (33) are separated by 1 in both ~a and ndomains. Consequently, condition fsZ4ðBjsjþjfajÞ assures

that only replica with p=0 and q=0 gives nonzero

contribution in the principal domain ð ~a; nÞ 2 ½�1=2;1=2��

½�1=2;1=2�. Then, accounting for the fact that only replicas

with p=q=0 are coincident in (33) and (34), Eq. (35)

immediately follows. &

An application of Theorems 3 and 4 is illustrated inSection 4 by the results reported in Fig. 3.

ARTICLE IN PRESS

Fig. 2. (Continuous-time) Doppler-stretched signal yaðtÞ ¼ xaðstÞej2pfa t , where xa(t) is a PAM signal with Nyquist pulse. Bifrequency spectral correlation density of

the discrete-time signal yðnÞ ¼ yaðtÞjt ¼ nTs, as a function of n1 and n2. (a) Magnitude and (b) support for fs ¼ 2jsjB0 . (c) Magnitude and (d) support for fs ¼ 4jsjB0 .

A. Napolitano / Signal Processing 90 (2010) 2276–22872282

The spectral cross-characterization of y(n) and x(n)requires to prove sampling theorems in the case of signalsthat are not jointly ACS but, rather, jointly SC. The spectralsupports of the densities of the Lo�eve bifrequency cross-spectrum of (jointly) SC signals in general do not behave asthose of ACS signals. Therefore, results found in the ACScase cannot be extended to cross-statistics.

Accounting for (4), (7), and (16), the following aliasingformula for the Lo�eve bifrequency cross-spectrum of y(n)and x(n) can be derived when xa(t) is ACS:

EfYðn1ÞXð�Þðn2Þg ¼

1

TsAejj 1

jsje�j2pðn1�nÞ d=s

�Xn12Z

Xn22Z

Xa2A

Saxaxð�Þa

n1�ns

fs�n1fs

s

� �

�d ðn2�n2Þ�ð�Þ a=fs�n1�n�n1

s

� �� �

�ej2pn1 d=s ð37Þ

From (37), it follows that the discrete-time signals x(n)and y(n) are jointly SC with Lo�eve bifrequency cross-spectrum having support contained in lines with slope71/s in the bifrequency plane ðn1; n2Þ.

If xa(t) is band-limited with bandwidth B, accounting for(9b), the support of the replica with n1=n2=0 and fixed a in

ARTICLE IN PRESS

Fig. 3. (Continuous-time) Doppler-stretched signal yaðtÞ ¼ xaðstÞej2pfa t , where xa(t) is a PAM signal with Nyquist pulse. Slice of the magnitude of the

spectral correlation density of the continuous-time signal ya(t) along the support line f2 ¼ f1�s=Tp (thin line) and of the rescaled ðn1 ¼ f1=fsÞ spectral

correlation density of the discrete-time signal yðnÞ ¼ yaðtÞjt ¼ nTs, along n2 ¼ n1�sðTs=TpÞ (thick line) as a function of f1/fr. (a) fs ¼ 2jsjB0 and (b) fs ¼ 4jsjB0 .

A. Napolitano / Signal Processing 90 (2010) 2276–2287 2283

the aliasing formula (37) is given by

supp Saxaxð�Þa

n1�ns

fs

� �d n2�ð�Þ a=fs�

n1�ns

� �� �� �

D ðn1; n2Þ 2 R�R : jn1�njrB

fsjsj; jn2jr

B

fs;

�

afs¼

n1�nsþð�Þn2

� ��ð38aÞ

D ðn1; n2Þ 2 R�R : jn1j�jnjrB

fsjsj; jn2jr

B

fs;

�

afs¼

n1�nsþð�Þn2

� ��: ð38bÞ

Eqs. (38a) and (38b) allow to establish the followingresults.

Theorem 5 (Sampling theorem for Lo�eve bifrequency cross-

spectrum). Let xa(t) be ACS and band-limited with bandwidth

B. For

fsZmaxf2ðBjsjþjfajÞ;2Bg ð39Þ

it results that

EfYðn1ÞXð�Þðn2Þg ¼

1

TsAejj 1

jsje�j2pðn1�nÞ d=s

Xa2A

Saxaxð�Þa

n1�ns

fs

� �

�d n2�ð�Þafs�n1�n

s

� �� �;

jn1jp1

2; jn2jp

1

2ð40Þ

which is formally analogous to its continuous-time counter-

part (13).

Proof. Replicas in the aliasing formula (37) are separatedby 1 in both n1 and n2 variables. Thus, from the supportbound (38a) it follows that Bjsj=fsr1=2 and B=fsr1=2

that is,

fsZmaxf2Bjsj;2Bg ð41Þ

is a sufficient condition such that replicas do not overlap.Moreover, from (38b), it follows that a sufficient conditionto obtain that the replica (n1, n2) lies in the square ðn1; n2Þ 2

ðn1�1=2;n1þ1=2Þ � ðn2�1=2;n2þ1=2Þ is jnjþBjsj=fsr1=2and B=fsr1=2, that is, (39). &

An application of Theorem 5 is illustrated in Section 4 bythe results reported in Figs. 4 and 5.

Theorem 6 (Sampling theorem for the density of Lo�eve

bifrequency cross-spectrum). Let xa(t) be ACS and band-

limited with bandwidth B. For

fsZmaxf2ðBjsjþjfajÞ;4Bg ð42Þ

it results that the density of the Lo�eve bifrequency cross-

spectrum (37) along the support line n2 ¼ ð�Þða=fs�ðn1�nÞ=sÞ

is given by

~SðkÞ

yxð�Þ ðn1Þ ¼1

jsjAejje�j2pðn1�nÞ d=s 1

TsSa

xaxð�Þa

n1�ns

fs

� �; jn1jr1=2

ð43Þ

where k is an index in one-to-one correspondence with a 2 A.

Proof. Due to (38a), the mapping n1 ¼ f1=fs1 for n1 2

½�1=2;1=2� between the densities with n1=n2=0 in (37)and those in (13) is assured provided that the followingimplication holds 8a:

n1 ¼ n7B

fsjsj ) jn2jr1�

B

fsð44Þ

where

n2 ¼ ð�Þafs�n1�n

s

� �ð45Þ

ARTICLE IN PRESS

Fig. 4. (Continuous-time) PAM signal xa(t) with Nyquist pulse. (a) Magnitude and (b) support of the bifrequency spectral cross-correlation density of

yaðtÞ ¼ xaðstÞej2pfa t and xa(t), as a function of f1/fr and f2/fr.

A. Napolitano / Signal Processing 90 (2010) 2276–22872284

From (45) with n1 ¼ n7Bjsj=fs substituted into we have

jn2j ¼afs�n1�n

s

��������r a

fs

��������þ n1�n

s

��������rsup

a2A

afs

��������þ n7Bjsj=fs�n

s

��������

¼2B

fsþ

B

fs¼

3B

fsð46Þ

Thus, inequality in (44) is satisfied provided that3B=fsr1�B=fs that is, fsZ4B. Then, accounting for (39),we obtain (42). &

An application of Theorem 6 is illustrated in Section 4 bythe results reported in Fig. 6.

3.3. Sampling Doppler-stretched signals: time domain

In this section, time-domain counterparts of samplingtheorems stated in the frequency domain in Section 3.2 arederived.

By inverse Fourier transforming both sides of (11) givesthe following expression for the (conjugate) cyclic auto-correlation function of the Doppler-stretched signal ya(t):

Rayayð�ÞaðtÞ ¼ A2ejkjej2pfate�j2pða�kfaÞ da=sRða�kfaÞ=s

xaxð�Þa

ðstÞ ð47Þ

Then, accounting for the time-domain counterpart of (21)(see [5]), the following aliasing formula for the (conjugate)cyclic autocorrelation function of y(n) can be proved:

~R~ayyð�Þ ðmÞ ¼

Xp2Z

Rð~a�pÞfs

yayð�Þa

ðmTsÞ ¼ A2ejkjej2pnme�j2p½ ~a�kn � d=s

�Xp2Z

Rð~a�kn�pÞfs

xasxð�Þas

ðmTsÞej2pp d=s ð48Þ

In (48), xasðtÞ9xaðstÞ and, hence, according to (47) (with

A=1, j¼ 0, fa=0, and da=0) it results Raxasxð�Þas

ðtÞ ¼ Ra=s

xaxð�Þa

ðstÞ.

From (48), the following result can be derived.

Theorem 7 (Sampling theorem for the (conjugate) cyclic

autocorrelation function). Let xa(t) be ACS and band-limited

with bandwidth B. For

fsZ2Bjsj ð49Þ

it results that

~R~ayyð�Þ ðmÞ ¼ A2ejkjej2pnme�j2p½ ~a�kn � d=sRð

~a�knÞfs=s

xaxð�Þa

ðsmTsÞ ð50aÞ

¼ A2ejkjej2pnme�j2p½ ~a�kn � d=s ~R~a�knxsxð�ÞsðmÞ; j ~ajr1=2

ð50bÞ

with xsðnÞ9xaðstÞjt ¼ nTs. Eq. (50b) is the unaliased sampled

version of (47).

Proof. From (48) (with A=1, j¼ 0, n ¼ 0, and d=0) itfollows that

~R~axsxð�ÞsðmÞ ¼

Xp2Z

Rð~a�pÞfs

xasxð�Þas

ðmTsÞ ð51Þ

Thus, we have that if d=s 2 Z or if Raxasxð�Þas

ðtÞ ¼ 0 for jaj4 fs=2,

then the sum in the rhs of (48) equals the rhs of (51) and,

hence, ~R~axsxð�ÞsðmÞ for j ~ajr1=2. A sufficient condition to

assure Raxasxð�Þas

ðtÞ ¼ 0 for jaj4 fs=2 is that the bandwidth of

xas(t) is less than fs/2. That is, Bjsjr fs=2, which isequivalent to (49). &

An expression for ~R~ayyð�Þ ðmÞ alternative to (50a) and

(50b) can be obtained starting from (35). In fact, forfsZ4ðBjsjþjfajÞ it follows that the (conjugate) cyclic auto-correlation function of y(n) can be expressed as

~R~ayyð�Þ ðmÞ ¼

Z 1=2

�1=2

~S~ayyð�Þ ðnÞe

j2pnm dn¼ A2ejkje�j2pð ~a�knÞ d=s

ARTICLE IN PRESS

Fig. 5. (Continuous-time) PAM signal xa(t) with Nyquist pulse and its Doppler-stretched version yaðtÞ ¼ xaðstÞej2pfat . Bifrequency spectral cross-correlation

density of the discrete-time signals yðnÞ9yaðtÞjt ¼ nTsand xðnÞ ¼ xaðtÞjt ¼ nTs

, as a function of n1 and n2. (a) Magnitude and (b) support for fs ¼ 2B0 . (c)

Magnitude and (d) support for fs ¼ 4B0 .

A. Napolitano / Signal Processing 90 (2010) 2276–2287 2285

�1

jsj

Z 1=4

�1=4

~Sð ~a�knÞ=s

xxð�Þn�n

s

� �ej2pnm dn; j ~ajr 1

2

ð52Þ

Finally, the cross characterization of y(n) and x(n) isimmediately obtained by taking in (15) t1=(n+m)Ts andt2=nTs and accounting for (20) and (19):

EfyðnþmÞxð�ÞðnÞg ¼ Aejjej2pnðnþmÞX~a2 ~A

ej2p ~an

Xp2Z

Rð~a�pÞfs

xaxð�Þa

ððsmþðs�1Þn�dÞTsÞ ð53Þ

where (sm +(s�1)n �d) in general is noninteger.

4. Numerical results

In this section, a numerical experiment is conductedaimed at illustrating the theoretical results of Section 3.2.These results, which are formulated in terms of Lo�eve bifreq-uency spectra, can be equivalently expressed in terms ofbifrequency spectral cross-correlation densities [9] by repla-cing Dirac with Kronecker deltas. For example, (13) becomes

Syax�a ðf1; f2Þ ¼ Aejj 1

jsje�j2pðf1�faÞ da=s

Xa2A

Saxaxð�Þa

f1�fa

s

� �

�d½f2�ð�Þða�ðf1�faÞ=sÞ� ð54Þ

ARTICLE IN PRESS

Fig. 6. (Continuous-time) PAM signal xa(t) with Nyquist pulse and its Doppler-stretched version yaðtÞ ¼ xaðstÞej2pfat . Slice of the magnitude of the spectral

cross-correlation density of the continuous-time signals ya(t) and xa(t) along the support line f2 ¼ ðf1�faÞ=s�1=Tp (thin line) and of the rescaled ðn1 ¼ f1=fsÞ

spectral cross-correlation density of the discrete-time signals yðnÞ ¼ yaðtÞjt ¼ nTsand xðnÞ ¼ xaðtÞjt ¼ nTs

, along n2 ¼ ðn1�nÞ=s�ðTs=TpÞ (thick line) as a function

of f1/fr. (a) fs ¼ 2B0 and (b) fs ¼ 4B0 .

A. Napolitano / Signal Processing 90 (2010) 2276–22872286

where dg denotes Kronecker delta, that is, dg ¼ 1 for g¼ 0and dg ¼ 0 for ga0.

In the experiment, a pulse-amplitude-modulated (PAM)signal xa(t) is considered with stationary white modulatingsequence, Nyquist-shaped pulse with excess bandwidth Z,and symbol period Tp. It is second-order cyclostationarywith period Tp and strictly bandlimited with bandwidthB¼ ð1þZÞ=ð2TpÞ which can be slightly overestimated byB0 ¼ 1=Tp. Thus, it has three cycle frequencies ah ¼ h=Tp, h 2

f0;71g [4]. In the experiment, it is assumed Z¼ 0:85 andTp=4/fr, where fr is a fixed reference frequency. TheDoppler-stretched signal ya(t) in (3) has A=1, j¼ 0, da=0,fa=0.075 fr, and s=0.75.

In Fig. 1, (a) magnitude and (b) support of thebifrequency spectral correlation density Syay�a ðf1; f2Þ arereported as functions of f1/fr and f2/fr. In Fig. 2, thebifrequency spectral correlation density Syy� ðn1; n2Þ ofthe discrete-time signal yðnÞ ¼ yaðtÞjt ¼ nTs

, is representedas a function of n1 and n2 for two values of the samplingfrequency. Specifically, (a) magnitude and (b) support forfs ¼ 2jsjB0 and (c) magnitude and (d) support for fs ¼ 4jsjB0

are reported. According to Theorem 2, fs ¼ 2jsjB0Z2jsjB is asufficient condition to assure non overlapping replicas inthe Lo�eve bifrequency spectrum of ya(t). However, only themore stringent condition fsZ2ðBjsjþjfajÞ can assure that(28) holds. For the considered numerical values, fs ¼ 4jsjB0

is such that the sufficient conditions on fs of Theorems 3and 4 are both satisfied. In Fig. 3, the slice of the magnitudeof the spectral correlation density of the continuous-timesignal ya(t) along the support line f2 ¼ f1�s=Tp (thin line)and of the rescaled ðn1 ¼ f1=fsÞ spectral correlation densityof the discrete-time signal yðnÞ ¼ yaðtÞjt ¼ nTs

alongn2 ¼ n1�sðTs=TpÞ (thick line) as a function of f1/fr arereported (a) for fs ¼ 2jsjB0 and (b) for fs ¼ 4jsjB0. Accordingwith the results of Theorems 3 and 4, only in case (b) thereare no aliasing replicas and the spectral correlation density

of ya(t) is coincident with that rescaled of y(n) consideredin the main frequency domain.

Let us consider now the bifrequency spectral cross-correlation density Syax�a ðf1; f2Þ between the jointly SCsignals ya (t) and xa (t). Accounting for (54) we have

Syax�a ðf1; f2Þ ¼1

jsj

X1

h ¼ �1

Sahxax�a

f1�fa

s

� �df2�ðf1�faÞ=sþah

ð55Þ

where Sahxax�aðf1Þ denote the cyclic spectra of the PAM signal

xa(t).In Fig. 4, (a) magnitude and (b) support of the spectral

cross-correlation density (55) of the continuous-timesignals ya(t) and xa(t) are reported as functions of f1/fr

and f2/fr.In Fig. 5, the bifrequency spectral cross-correlation

density Syx� ðn1; n2Þ of the discrete-time signalsyðnÞ9yaðtÞjt ¼ nTs

and xðnÞ ¼ xaðtÞjt ¼ nTsis represented as a

function of n1 and n2 for two values of the samplingfrequency. Specifically, (a) magnitude and (b) support forfs ¼ 2B0 and (c) magnitude and (d) support for fs ¼ 4B0 arereported. Since B0ZBZBjsjþjfaj, due to Theorem 5,condition fs ¼ 2B0 assures nonoverlapping replicas in theLo�eve bifrequency cross-spectrum, and, hence, in itsbifrequency density Syx� ðn1; n2Þ. However, such a conditiondoes not assure, for each support line, the mapping in thewhole principal frequency domain between spectraldensities of continuous- and discrete-time signals. In fact,in Fig. 6, the slice of the magnitude of the spectral cross-correlation density of the continuous-time signals ya(t) andxa(t) along the support line f2 ¼ ðf1�faÞ=s�1=Ttp (thin line)and of the rescaled ðn1 ¼ f1=fsÞ spectral cross-correlationdensity of the discrete-time signals yðnÞ ¼ yaðtÞjt ¼ nTs

andxðnÞ ¼ xaðtÞjt ¼ nTs

along n2 ¼ ðn1�nÞ=s�ðTs=TpÞ (thick line) asa function of f1=fr is reported (a) for fs ¼ 2B0 and (b) forfs ¼ 4B0. According to Theorem 6 and the numerical values

ARTICLE IN PRESS

A. Napolitano / Signal Processing 90 (2010) 2276–2287 2287

of s and fa, only condition fs ¼ 4B0 assures the lack ofaliasing replica in the considered cross-spectral densityfunction. In such a case, the spectral cross-correlationdensity of the continuous-time signals and that rescaled ofthe discrete-time signals are coincident in the principalfrequency domain.

5. Conclusion

In this paper, the Lo�eve bifrequency spectrum is usedas framework to describe in the spectral domain the non-stationarity of a Doppler-stretched signal when the trans-mitted signal is ACS. It is shown that even if the Dopplerchannel is linear not almost-periodically time-variant, thereceived signal is still ACS. Moreover, the transmitted andreceived signals are jointly SC with spectral masses in thebifrequency plane concentrated on lines with slope equalto the reciprocal of the time-scale factor introduced bythe Doppler channel. Sampling theorems are proved forspectral statistical functions characterizing the Doppler-stretched received signal and jointly characterizing thetransmitted and received signals. Sufficient conditions arefound in terms of sampling frequency fs, transmitted signalbandwidth B, frequency-shift fa, and time-scale factor s. Foran ACS signals with bandwidth B, the known conditionfsZ4B [5] to avoid aliasing in both cycle and spectralfrequency domains for (conjugate) cyclic spectra is foundagain within the framework of the Lo�eve bifrequencyspectrum (Theorem 1). It is shown that for the ACSDoppler-stretched signal, fsZ2Bjsj and fsZ2ðBjsjþjfajÞ

assure, in the Lo�eve bifrequency spectrum of the sampledsignal, no overlap of replicas and main replica in theprincipal frequency domain, respectively (Theorem 2);fsZ2ð2BjsjþjfajÞ guarantees the mapping between spectraldensities of the continuous- and discrete-time signals inthe whole principal frequency domain (Theorem 3);fsZ4ðBjsjþjfajÞ assures the main replica of the (conjugate)cyclic spectrum in the principal spectral and cyclefrequency domain (Theorem 4). It is also shown that forthe jointly SC transmitted and received Doppler-stretchedsignals, fsZmaxf2Bjsj;2Bg and fsZmaxf2ðBjsjþjfajÞ;2Bg

assure, in the Lo�eve bifrequency cross-spectrum, no overlapof replicas and main replica in the principal frequencydomain, respectively (Theorem 5); fsZmaxf2ðBjsjþjfajÞ;4Bg

provides the mapping between cross-spectral densities ofthe continuous- and discrete-time signals in the wholeprincipal frequency domain (Theorem 6). The conditionfsZ4Bjsj is shown to be sufficient to avoid aliasing in thecycle-frequency domain for the (conjugate) cyclic auto-correlation function (Theorem 7). It is worth to observe thatin communications applications the several obtainedbounds on fs can be reduced to the two bounds fsZ2B

and fsZ4B since jsjC1 and jfaj5B can be generallyassumed. Such conditions, however, do not hold in sonarand acoustic aircraft applications. The derived lowerbounds for the sampling frequency are such that relation-ships in continuous-time linking spectral statistical func-tions of transmitted ACS and received Doppler-stretched

signals are formally analogous to their discrete-timecounterparts. Thus, the MMSE procedure described in [2]can be straightforwardly implemented in discrete-time bysubstituting cyclic spectra of continuous-time signals withcyclic spectra of their sampled versions and integrals withsums. Numerical results have been presented to illustratethe theoretical results.

References

[1] J.C. Allen, S.L. Hobbs, Detecting target motion by frequency-planesmoothing, in: Proceedings of the Twenty-Sixth Annual AsilomarConference on Signals, Systems, and Computers, Pacific Grove, CA,October 1992.

[2] V. De Angelis, L. Izzo, A. Napolitano, M. Tanda, Cyclostationarity-based parameter estimation of wide-band signals in mobile com-munications, in: Proceedings of 13th IEEE Statistical Signal Proces-sing Workshop (SSP’05), Bordeaux, France, July 17–20, 2005.

[3] J. Dmochowski, J. Benesty, S. Affes, On spatial aliasing in microphonearrays, IEEE Transactions on Signal Processing 57 (4) (April 2009)1383–1395.

[4] W.A. Gardner, A. Napolitano, L. Paura, Cyclostationarity: half acentury of research, Signal Processing 86 (4) (April 2006) 639–697.

[5] L. Izzo, A. Napolitano, Higher-order cyclostationarity properties ofsampled time-series, Signal Processing 54 (November 1996)303–307.

[6] Q. Jin, K.M. Wong, Z.Q. Luo, The estimation of time delay and Dopplerstretch of wideband signals, IEEE Transactions on Signal Processing43 (April 1995) 904–916.

[7] K.-S. Lii, M. Rosenblatt, Spectral analysis for harmonizable processes,The Annals of Statistics 30 (1) (2002) 258–297.

[8] M. Lo�eve, Probability Theory, Van Nostrand, Princeton, NJ, 1963.[9] A. Napolitano, Uncertainty in measurements on spectrally correlated

stochastic processes, IEEE Transactions on Information Theory 49(September 2003) 2172–2191.

[10] A. Napolitano, Mean-square consistent estimation of the spectralcorrelation density for spectrally correlated stochastic processes, in:Proceedings of IEEE International Conference on Acoustics, Speech,and Signal Processing (ICASSP 2007), Honolulu, Hawaii, USA, April16–20, 2007.

[11] A. Napolitano, Estimation of second-order cross-moments of gen-eralized almost-cyclostationary processes, IEEE Transactions onInformation Theory 53 (6) (June 2007) 2204–2228.

[12] A. Napolitano, Discrete-time estimation of second-order statistics ofgeneralized almost-cyclostationary processes, IEEE Transactions onSignal Processing 57 (5) (May 2009) 1670–1688.

[13] J. Oberg, Titan calling, IEEE Spectrum 41 (October 2004) 28–33.[14] T.A. Øigard, L.L. Scharf, A. Hanssen, Spectral correlations of fractional

Brownian motion, Physical Review E E74 (2006) 031114.[15] R.A. Schlotz, Problems in modeling UWB channels, in: Proceedings of

the Thirty-Sixth Asilomar Conference on Signals, Systems, andComputers, Pacific Grove, CA, 3–6 November 2002.

[16] P.J. Schreier, L.L. Scharf, Second-order analysis of improper complexrandom vectors and processes, IEEE Transactions on Signal Proces-sing 51 (3) (March 2003) 714–725.

[17] H.L. Van Trees, Detection, Estimation, and Modulation Theory, PartIII, John Wiley & Sons, Inc., New York, 1971.

[18] P. Wahlberg, P.J. Schreier, Spectral relations for multidimensionalcomplex improper stationary and (almost) cyclostationary pro-cesses, IEEE Transactions on Information Theory 54 (4) (April 2008)1670–1682.

[19] L.G. Weiss, Wavelets and wide-band correlation processing, IEEESignal Processing Magazine 11 (1) (January 1994) 13–32.

[20] L.G. Weiss, R.K. Joung, L.H. Sibul, Wideband processing of acousticsignals using wavelets transforms. Part I. Theory, Journal of theAcoustical Society of America 96 (2) (August 1994) 850–856 (pt. 1).

[21] L.G. Weiss, Wideband processing of acoustic signals using waveletstransforms. Part II. Efficient implementation and examples, Journalof the Acoustical Society of America 96 (2) (August 1994) 857–866(pt. 1).

[22] A.H. Zemanian, Distribution Theory and Transform Analysis, Dover,New York, 1987.