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Abstract

Multipath fading is a common phenomenon in wireless signal transmission. When a

signal is transmitted over a radio channel, it is subject to reflection, refraction and

diffraction. The communication environment changes quickly and thus introduces morecomplexities and uncertainties to the channel response. This simulator offers a better

understanding of this phenomenon. In order to observe the effects of multipath fadingchannel on the transmitted signal, a whole digital communication system simulator was

developed. Three kinds of digital communication systems: baseband transmission via

additive white Gaussian noise (AWGN) channel, passband transmission via single

AWGN channel, and passband transmission via multipath fading channel, are simulated.


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Contents

Introduction

Baseband transmission via AWGN channel

Signal sourceSampling

Waveform encoding and decoding

Pulse Code Modulation (PCM)

Simple DPCM

General DPCM

Delta ModulationError control coding

Block codesConvolutional codes

Line codingBaseband filtering

Eye pattern

Optimum receiver in AWGN channel

Lowpass filtering

Passband transmission via single AWGN channel

M-ASKQASKM-FSK

M-PSK

Passband transmission via multipath fading channel

Time variant multipath fading channel

Characterization of multipath fading channel

Channel model for multipath fading channel

Signal design for multipath fading channel

RAKE demodulator

Simulation results

Reference

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22.1

2.2

2.32.3.12.3.2

2.3.3

2.3.42.4

2.4.1

2.4.22.5

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3.13.2

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1. Introduction

This report describes a simulator for digital communication systems. Three kinds of

digital communication systems: baseband transmission via additive white Gaussian noise(AWGN) channel, passband transmission via single AWGN channel, and passband

transmission via multipath fading channel, are simulated, as shown in Figure 1. Passbandtransmission via multipath fading channel is emphasized.

Multipath fading is a common phenomenon in wireless signal transmission. When a

signal is transmitted over a radio channel, it is subject to reflection, refraction and

diffraction. Especially in the urban and suburban areas where cellular phones are most

often used, the communication environment changes quickly and thus introduces morecomplexities and uncertainties to the channel response. This simulator offers a better

understanding of this phenomenon.

In order to observe the effects of multipath fading channel on the transmitted signal, awhole digital communication system simulator was developed. For simplicity, complex

techniques, such as multiplexing and spread spectrum communication, were not

simulated. Analyses of the performances of the systems are not given. Section 2 briefly

describes baseband transmission via AWGN channel. Section 3 briefly presents passband

transmission via single AWGN channel. Passband transmission via multipath fading

channel is described in detail in Section 4.

2. Baseband transmission via AWGN channel

Baseband transmission is usually used in wired communication system, such as wired

telephone and wired local area network(LAN). Figure 2 shows the structure of a digital

Fig 1 Contents of the simulator


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communication system with baseband transmission, which consists of signal source,

sampler, waveform encoder, error control encoder, code transformer, baseband filter,AWGN channel, optimum receiver, inverse code transformer, error control decoder, and

lowpass filter. The following subsections describe the above blocks. Users can display

the required figures at different steps in Figure 2 by clicking the small rectangles, whosecolor stands for a display format specified in the legend at the bottom of Figure 2.

2.1 Signal source

A signal source is used to transform information such as voice into electric signal. In

this simulator, a sine waveform is generated with the amplitude, frequency, and initial

phase specified by the user. Figure 3 shows the pre-filtered signal to be transmitted with

amplitude 3V, frequency 1000Hz, and initial phase 45. After some processing, thissignal will be transmitted through an AWGN channel, and be recovered by the receiver.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

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Original signal

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Sampled s ignal

Fig 2 Baseband transmission via AWGN channel

Fig 3 Original signal Fig 4 Sampled signal


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2.2 Sampling

Figure 3 is an analog signal. To be transmitted via a digital communication system,

the signal must be discretized both in time and amplitude. A sampler is to discretize theanalog signal in the time domain. According to the Nyquist sampling theorem, in order to

restore the signal at the receiving site, the sampling frequency should be at least twice ofthat of the maximum signal frequency. Figure 4 shows a sampled signal (discrete signal)

with the sampling frequency 50 times of the signal frequency.

2.3 Waveform encoding and decoding

Waveform encoding converts a source signal into a digital code using a quantizationmethod. The waveform coded signal is represented by a set of integers {1, 2, , N},where N is finite. Waveform decoding recovers the original information signal sequence

using the waveform-coded signal. This simulator supports four waveform-coding

schemes.

2.3.1 Pulse Code Modulation (PCM)

PCM is the simplest and oldest waveform-coding scheme. PCM is a process that

assigns a signal value to inputs that are within a specified range. Inputs that fall in a

different range of values are assigned a different signal value. The input signal is in effect

digitized by scalar quantization, as show in Figure 5. If the quantizer performs uniformquantization, the PCM is called uniform PCM, as Figure 5.

As long as the statistics of the input signal are close to the uniform distribution,

uniform PCM works fine. However, in coding of certain signals such as speech, the input

distribution is far from being uniformly distributed. For a speech waveform, in particular,

there exists a higher probability for smaller amplitudes and lower probability for large

amplitudes. Therefore, it makes sense to design a quantizer with more quantization

regions at lower amplitudes and less quantization regions at larger amplitudes. The

resulting quantizer will be a non-uniform quantizer having quantization regions ofvarious sizes.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

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Signal after waveform decoding

Fig 5 PCM with uniform quantization


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The usual method for performing non-uniform quantization is to first pass the

samples through a nonlinear element that compresses the large amplitudes (reducesdynamic range of the signal) and then perform a uniform quantization on the output. At

the receiving end, the inverse (expansion) of this nonlinear operation is applied to obtain

the sampled value. This technique is called companding (compressing-expanding). Thereare two types of companders that are widely used for speech coding: A-law compander

and -law compander. Figures 6 and 7 are the characteristics of A-law compander and -law compander, respectively. Figures 8 and 9 are the recovered signal using A-law

compander and -law compander, respectively. Except for the discussion of thewaveform schemes in this subsection, the following results are based on PCM with

uniform quantization.

2.3.2 Simple DPCM

When a band-limited random process is sampled at the Nyquist rate or faster, the

sampled values are usually correlated random variables. This means that the previoussamples give some information about the next sample, and this information can be

employed to improve the performance of the PCM system. In the simplest form of

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Fig 8 PCM with A-law companding Fig 9 PCM with A-law companding

Fig 6 A-law compander characteristics Fig 7 -law compander characteristics


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differential pulse-code modulation (DPCM), the difference between two adjacent

samples is quantized. Because two adjacent samples are highly correlated, theirdifference has small variations and, therefore, to achieve a certain level of performance

fewer bits are required to quantize it.

Figure 10 shows a block diagram of this simple DPCM scheme. As seen in this

figure, the input to the quantizer is not simply 1 nn XX but rather 1 nn YX . 1 nY is

closely related to Xn-1, and the above choice has the advantage that accumulation of

quantization noise is prevented. Figure 11 shows the recovered signal using simple 4-bit

DPCM system.

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x 10-3

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(a) Encoder (b) Decoder

Fig 10 A simple DPCM scheme

Fig 11 Simple DPCM, 4 bits

(a) Encoder (b) Decoder

Fig 12 A general DPCM scheme


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2.3.3 General DPCM

In a more complex version of DPCM, instead of using only the previous sample, the

last p samples are used to predict the value of the next sample. Then the differencebetween the sample Xn and its predicted value is quantized. Usually a linear predictorof

the form = p

i iniXa

1is employed, and the coefficients of the predictor ai are chosen to

minimize the mean-squared error between the sampleXn and its predicted value.

Figure 12 shows a block diagram of a general DPCM system. Because we are using a

p-step predictor, we are using more information in predictingXn and, therefore, the range

of variations ofYn will be less. This in turn means that even lower bit rates are possiblehere. Differential PCM systems find wide applications in speech and image compression.

Figure 13 shows the recovered signal using second-order and 4-bit DPCM.

2.3.4 Delta Modulation

Delta modulation (M) is a simplified version of the simple DPCM scheme. In Mthe quantizer is a 1-bit (2-level) quantizer with magnitudes . Figure 14 shows therecovered signal using M. From Figure 14, we see that when the input signal changesdramatically, the quantized signal cant follow the input signal. This phenomenon is

called overloading. Many schemes have been developed to solve this problem.

2.4 Error control coding

Error-control coding techniques are used to detect and/or correct errors that occur in the

message transmission in a digital communication system. The transmitting side of the

error-control coding adds redundant bits or symbols to the original information signal

sequence. The receiving side of the error-control coding uses these redundant bits of

symbols to detect and/or correct the errors that occurred during transmission. The

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Fig 13 General DPCM,p=2, 4 bitsFig 14 M


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transmission coding process is known as encoding, and the receiving coding process is

known as decoding.

2.4.1 Block codes

There are two major classes in error-control code: block and convolutional. In block

coding, successive blocks of K information (message) symbols are formed. The codingalgorithm then transforms each block into a codeword consisting ofN symbols where

N>K. This structure is called an (N, K) code. The ratio K/Nis called the code rate. A key

point is that each codeword is formed independently from other codewords. This

simulator supports three block-coding schemes: Hamming code, cyclic code, and BCHcode.

Figure 15 shows the corrupted signal via an AWGN with variance 0.02 without error-

control coding. Figures 16 through 18 are the signals via the same channel in Figure 15using Hamming code, cyclic code, and BCH code, respectively, withN=7, K=4. In Figure

15, the bit error rate is 0.015. The bit error rates in Figures 16 through 18 are 0s.

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Fig 15 Without error-control coding Fig 16 Hamming code,N=7, K=4

Fig 17 Cyclic code,N=7, K=4 Fig 18 BCH code, N=7, K=4


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2.4.2 Convolutional codes

Convolutional codes differ from block codes in that there are no independent

codewords. The encoding process can be envisioned as a sliding window, Mblock wide,

which moves over the sequence of information symbols in steps of K symbols. M iscalled the constraint length of the code. With each step of the sliding window, the

encoding process generatesNsymbols based on theMKsymbols visible in the window.A convolutional code so constructed is called an (N, K,M) code. Convolutional codes are

commonly used in applications that require relatively good performance with low

implementation cost.

The Viterbi method is used for decoding the convolutional codes. The Viterbi

algorithm is a maximum likelihood(ML) decoding procedure that takes advantage of thefact that a convolutional encoder is afinite state machine. The criterion used for decision

-making is the metric for soft decision decoding and the Hamming distance for hard

decision coding. Details on the Viterbi method can be found in most advanced

communication books.

Figure 19 shows the recovered signal via the same channel as Figure 15 with

convolutional coding. The bit error rate is 0.

2.5 Line coding

Binary 1s and 0s, such as in PCM signaling, may be represented in various serial-bitsignaling formats called line codes. The output of the error-control encoder in this

simulator is unipolar (1s and 0s). Since the direct current component of unipolar signal is

not zero, it is not proper to transmit unipolar signals. Instead, bipolar signaling (1s and

1s) schemes are employed. This simulator provides four bipolar signaling schemes:

bipolar non-return-to-zero signaling (NRZ2), bipolar non-return-to-zero inverse

signaling (NRZI2), bipolar return-to-zero signaling (RZ2), and Millercoding, as shown

in Figure 20. Details of the above signaling schemes can be found in most

communication books. The figures in the following section use Miller code.

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Fig 19 Convolutional coding, (3,2,2) code


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2.6 Baseband filtering

Rectangular waveforms in Figure 20 have high frequency components. If the channel

bandwidth is limited, rectangular waveforms are not appropriate for this kind of channels.

In this simulator, a raise cosine filteris employed to make the waveform smooth. Figures

3.1 3.2 3.3 3.4 3.5 3.6 3.7

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Signal after baseband filtering

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Signal after baseband filtering

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Fig 20 Line codes

(a) Original (b) NRZ2 (c) NRZI2 (d) RZ2 (e) Miller

Fig 21 Raise cosine filtered signal

for RZ2 signaling

Fig 22 Raise cosine filtered signal

for Miller coding


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Eye pattern of the signal after baseband filtering

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Eye pattern of the s ignal corrupted by the AW GN

21 and 22 show the raise cosine filtered signals form RZ2 signaling and Miller codes,

respectively.

2.7 Eye pattern

An eye-pattern plot is a simple and convenient tool to study the effects of inter-

symbol interference (ISI) and other channel impairments for digital transmission. The

received signal is plotted against time. When the x-axis time limit is reached, the signalgoes back to the beginning of the time point. Thus the plots overlay each other. In the

optimal condition, the decision pointis at the widest eye opening point. Figures 23 and

24 are the eye patterns of the output of the baseband filtering and the signal corrupted by

the AWGN channel, whose variance is 0.01.

2.8 Optimum receiver in AWGN channel

It is convenient to subdivide the receiver into two parts, the signal demodulator and

the detector. The function of the signal demodulator is to convert the received waveform

r(t) into an N-dimensional vector r=(r1, r2, , rN), where N is the dimension of thetransmitted signal waveforms. The function of the detector is to decide which of the M

possible signal waveforms was transmitted based on observation of the rector r.

The signal and the noise are expanded into a series of linearly weighted orthonormal

basis functions {n(t)}. Two realizations of the signal demodulator are simulated. One isbased on the use of signal correlators, named correlation-type demodulator, as shown in

Figure 25. The second is based on the use ofmatched filters (MF), named matched-filter-type demodulator, as shown in Figure 26. In this simulator, the signal is one-dimensional,

and the realizations of above two demodulators are the same.

The maximum a posteriori probability (MAP) criterion is used to maximize the

probability of a correct decision and, hence, minimizes the probability of error. This

Fig 23 Eye pattern of thebaseband filtered signal

Fig 24 Eye pattern of signalcorrupted by the AWGN channel


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3.1 3.2 3.3 3.4 3.5 3.6 3.7

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Signal corrupted by the AW GN

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Output of the optimum receiver

detector is optimum when the signal has no memory. When the transmitted signal has

memory, the optimum detector is a maximum-likelihood (ML) sequence detector, whichbases its decisions on observation of a sequence of received signals over successive

signal intervals. Although this simulator does not provide ML sequence detector, it has

been shown that a detector based on the MAP criterion and one that is based on the MLcriterion make the same decisions as long as the a priori probabilities are all equal. Figure

27 is the signal, corresponding to that in Figure 22, corrupted by the AWGN channel.Figure 28 is the decisions made by the optimum receiver.

2.9 Lowpass filtering

In order to recover the analog signal from the output of the waveform decoder, a

lowpass filter is needed. This simulator provides a butterworth filterwith the order anddigital cutoff frequency specified by the user. Figure 29 shows the lowpass filtered signal,

Fig 27 Corrupted signal of Figure 22Fig 28 Output of the optimum

receiver for Figure 25

Fig 25 Correlation-type demodulator Fig 26 Matched-filter-type demodulator


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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

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Restored s ignal

Original signal

Restored s ignal

with order of 4 and cutoff frequency of 0.1, compared with the original analog signal. It

is obvious that there is a delay due to the infinite impulse response(IIR) filter.

3. Passband transmission via single AWGN channel

Wireless communication systems must use passband transmission. Figure 30 shows

the structure of a digital communication system with passband transmission via single

AWGN channel, which consists of signal source, sampler, waveform encoder, errorcontrol encoder, digital modulation (digital mapping and analog modulation), AWGN

channel, digital demodulation (analog demodulation and digital demapping), error control

decoder, and lowpass filter. The following subsections describe the above blocks that did

not show up in Section 2, that is, digital modulation and demodulation. Users can display

the required figures at different steps in Figure 30 by clicking the small rectangles, whosecolor stands for a display format specified in the legend at the bottom of Figure 30.

Fig 29 Lowpass filtered signal compared with the original signal

Fig 30 Passband transmission via single AWGN channel


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This section discusses passband digital modulation and demodulation methods.

Digital modulation includes two parts: digital to analog mapping and analog modulation,as shown in Figure 31. Digital to analog mapping techniques convert the received digital

data into analog signals. Analog modulation then modulates the mapped signals.

Typically, digital signals form a finite set of symbols. For example, in binarytransmission it is possible to consider pairs of binary numbers as symbols. In this case,

the symbol set is {00, 01, 10, 11}, making four distinct symbols. A digitalmapping algorithm must have at least four mapping points to map the symbols uniquely.

The number of points in the signal set is called the M-ary number, also known as

alphabet size. The arrangement of the signal set in the signal space, the space that digital

mapping algorithm uses to map the mapping points, is called the constellation.

In general, digital demodulation is the inverse of the modulation process. Amodulated signal is demodulated and then demapped from an analog signal to a digital

signal. Figure 32 shows how digital demodulation divides into an analog demodulationand an analog to digital mapping. This demodulation structure applies to all

demodulation methods except for M-FSK and P-FSK, which use correlation techniques.

3.1 M-ASK

M-ary amplitude shift keying (ASK) modulation includes two parts: M-ASK mapping

followed by analog amplitude modulation (AM). M-ASK mapping is a one-dimensional

coding process that maps the input digital symbols into real numbers in the interval [-x,

x], where x is the specified maximum number. The input symbols are integers in therange [0, M-1], where M is the M-ary number.

Three AM methods are supported in this simulator: double-sideband with

transmission carrier amplitude modulation (AM-DSB-TC), double-sideband carrier

Fig 31 Digital modulation

Fig 32 Digital demodulation


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suppressed amplitude modulation (AM-DSB-SC), and single-sideband amplitude

modulation (AM-SSB). Details on the AM methods can be found in references.

Figure 33 shows the constellation of 4-ASK. Figures 34 through 36 are the spectrums

of the modulated signals using AM-DSB-TC, AM-DSB-SC, and AM-SSB, respectively.

3.2 QASK

The M-ary quadrature amplitude shift keying M-QASK method is the most

commonly used digital modulation method in communication systems. In general, an M-

QASK process takes the input digital symbol and maps it into two independent

components: in-phase and quadrature. An analog quadrature amplitude modulation

(QAM) is used to modulate the in-phase and quadrature signals. On the receiving side,the signal is demodulated into in-phase and quadrature signals. A demapping process

recovers the message signal by using the in-phase and quadrature signals.

There are many mapping schemes available to map the input symbol to the in-phaseand quadrature components. This simulator supports square constellation, as shown in

Figure 37.

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ASK constellation

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Fig 33 4-ASK constellation Fig 34 Spectrum using AM-DSB-TC

Fig 35 Spectrum using AM-DSB-SC Fig 36 Spectrum using AM-SSB


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QASK Constallation

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Baseband spectrum for FSK (Hz)

FSK constellation

3.3 M-FSK

M-ary frequency shift keying (FSK) modulation modulates a digital signal bychanging the frequency of the output signal depending on the value of the input signal.

The M-FSK modulation divides into two parts: mapping and analog modulation. The

mapping process maps the input symbol into the value of the frequency shift from the

carrier frequency, and the analog modulation is analog frequency modulation (FM). If the

carrier frequency is Fc, and the tone space f (the frequency separation between twoconsecutive frequencies in the modulated signal), then the frequency range of amodulated signal is in the range [Fc, Fc+(M-1)f]. Figure 38 shows the constellation of4-FSK.

The demodulation process of the M-FSK uses a length-M vector signal where the

frequency of the ith

element in the vector signal matches the modulated signal when the

input symbol is i. The demodulation process computes the correlation values between the

signal array and the received signal and, after calculating the maximum correlation value,

decides what symbol was most likely transmitted. There are two different methods to

compute the correlation value: the coherent and noncoherent method, as shown in Figures

39 and 40, respectively. Using the coherent method, you must know the phaseinformation of the modulated signal from the receiving side where phase-locked loops areused. The noncoherent method does not require phase information; it recovers the phase

of the modulated signal during the demodulation. However, the noncoherentdemodulation method is more computationally complex than the coherent demodulation

method.

3.4 M-PSK

The M-ary phase shift keying (PSK) modulation modulates a signal by changing the

phase values in the modulated output signal. The M-PSK modulation divides into two

parts: mapping and analogphase modulation (PM). M-PSK distinguishes between the

Fig 37 4-ary QASK square constellation Fig 38 4-ary FSK constellation


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digital messages by setting different initial phase shifts in the modulation. The digital

input signals to a M-PSK modulator are in the range [0, M-1]. The phase shift for input

digit i is 2i/M. The structure of the demodulator for M-PSK is similar to that of M-FSK.The demodulation process calculates the correlation value between the input signal and a

vector of carrier frequency sinusoidal signal. Each sinusoidal signal in the vector has its

phase set to a possible result from the signal set. Figure 41 shows the constellation of 4-

PSK. Figure 42 is part of the modulated signal using 4-PSK. We can see phase change in

Figure 42.

4. Passband transmission via multipath fading channel

This section discusses the most important contexts of this report. Figure 43 shows the

ASK/PSK Constel lat ion0 0.2 0.4 0.6 0.8 1 1.2

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Fig 39 Coherent demodulation of M-FSK Fig 40 Noncoherent demodulation of M-FSK

Fig 41 Constellation of 4-PSK Fig 42 Modulated signal using 4-PSK


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structure of a digital communication system with passband transmission via multipath

fading channel, which consists of signal source, sampler, waveform encoder, errorcontrol encoder, digital modulation (digital mapping and analog modulation), AWGN

channel, filter bank demodulator, error control decoder, and lowpass filter. The following

subsections discuss the multipath fading channels. Users can display the required figuresat different steps in Figure 43 by clicking the small rectangles, whose color stands for a

display format specified in the legend at the bottom of Figure 43.

4.1 Time variant multipath fading channel

Fundamentally, mobile radio communication channels are time varying, multipath

fading channels. In a radio communication system, there are many paths for a signal totravel from a transmitter to a receiver. Sometimes there is a direct path where the signal

travels without being obstructed. In most cases, components of the signal are reflected by

the ground and objects between the transmitter and the receiver such as buildings,

Fig 43 Digital transmission via multipath fading channel

Fig 44 Multipath fading channel


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vehicles, and hills or refracted by different atmospheric layers. Theses components travel

in different paths and merge at the receiver. Figure 44 illustrates this phenomenon. Eachpath has a different physical length. Thus, signals on each path suffer different

transmission delays due to the finite propagation velocity. The superposition of these

signals at the receiver results in destructive of constructive interference, depending on therelative delays involved. The fact that the environment changes as time passes leads to

signal variation. This is called time variant. Signals are also influenced by the motion of aterminal. A short distance movement can cause an apparent change in the propagation

paths and in turn the strength of the received signals.

4.2 Characterization of multipath fading channel

Both the propagation delays and the attenuation factors are time-variant as a result of

changes in the structure of the medium. The received bandpass signal may be expressedin the form

( ) ( ) ( )( ) =n

nnttsttx (1)

where s(t) is the transmitted signal, ( )tn

is the attenuation factor for the signal received

on the nth

path and ( )tn

is the propagation delay for the nth path. s(t) can be expressed as

( ) ( )[ ]tfjl

cetsts2

Re= (2)

where sl(t) is the equivalent lowpass transmitted signal. Substitute (2) into (1) yields the

result

( ) ( ) ( ) ( )( )

= tfj

n

nl

tfj

ncnc ettsettx

22Re (3)

It is apparent from (3) that the equivalent lowpass received signal is

( ) ( ) ( ) ( )( ) = n

nl

tfj

nl ttsettrnc 2 (4)

It follows that the equivalent lowpass channel is described by the time-variant impulse

response

( ) ( ) ( ) ( )( ) = n

n

tfj

n tettcnc 2; (5)

( )tc ; represents the response of the channel at time tdue to an impulse applied at time t-.

When there are a large number of signal propagation paths, the central limit theorem

can be applied. Thus ( )tc ; can be modeled as a complex-valued Gaussian random

process. The envelope ( )tc ; at any instant tisRayleigh-distributed, as shown in Figure

45. In this case the channel is said to be aRayleigh fading channel. In the event that thereare fixed scatterers or signal reflectors in the medium, in addition to randomly moving


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1 1.2 1.40

50 0

1000

1500

2000

2500

3000

3500

4000

4500

5000

scatterers, ( )tc ; can no longer be modeled as having zero mean. In this case, the

envelope ( )tc ; has aRice distribution, as shown in Figure 46, and the channel is said to

be aRicean fading channel .

Assuming that c(;t) is wide-sense-stationary (WSS), we can define the

autocorrelation function ofc(;t) as

( ) ( ) ( )[ ]ttctcEtc

+= ;;2

1;, 21

*

21 (6)

where * means conjugation. In most radio transmission media, the attenuation and phase

shift of the channel associated with path delay 1 is uncorrelated with the attenuation and

phase shift associated with path delay 2. This is usually called uncorrelated scattering.Under this assumption, we have

( ) ( ) ( )21121 ;;, = tt cc (7)

( ) ( )0;cc

= is simply the average power output of the channel as a function of the

time delay . For this reason, c() is called the multipath intensity profile or the delay

power spectrum of the channel. The range of values of over which c() is essentiallynonzero is called the multipath spreador delay spreadof the channel and is denoted by

Tm.

The time-variant transfer functionC(f;t) can be defined as the Fourier transform of

c(;t). That is,

( ) ( )

= detctfC fj2;; (8)

If c(;t) is modeled as a complex-valued zero-mean Gaussian random process in the tvariable, it follows that C(f;t) also has the same statistics. Under the assumption that thechannel is WSS, the autocorrelation function of C(f;t) is the Fourier transform of the

multipath intensity profile, i.e.,

( ) ( ) ( ){ }ttfftf cCC == ;;,; 21

(9)

Fig 45 Rayleigh distribution Fig 46 Ricean distribution


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where f=f2-f1. C(f;t) is called the spaced-frequency, spaced-time correlation function

of the channel. Let t=0, then( ) ( ){ }

cCf

= (10)

As a result of the Fourier transform relationship between C(f) and c(), the reciprocalof the multipath spread is a measure of the coherent bandwidth (f)c of the channel. Thatis,

( )m

cT

f1

(11)

If (f)c is small in comparison to the bandwidth of the transmitted signal, the channel is

said to befrequency-selective. In this case, the signal is severely distorted by the channel.On the other hand, if (f)c is large in comparison with the bandwidth of the transmittedsignal, the channel is said to befrequency-nonselective .

Define the Fourier transform ofC(f;t) with respective to the variable tto be thefunction SC(f;). With fset to zero, the relation becomes

( ) ( )

= tdetS tjCC 2;0 (12)

The function SC() is apower spectrum that gives the signal intensity as a function of the

Doppler frequency . Hence, we call SC() the Doppler spectrum of the channel. The

range of values ofover which SC() is essentially nonzero is called the Doppler spreadBdof the channel. Due to the Fourier transform relationship between SC() and C(t), thereciprocal ofBd is a measure of the coherence time (t)c of the channel. That is,

( )d

cB

t1

= (13)

If the coherence time is larger than the symbol period, the channel is said to be a slow-

fading channel. On the other hand, if the coherence time is smaller than the symbol

period, the channel is afast fading channel.

The scattering function of the channel is defined as

( ) ( )

= fdtdeetfS fjtjC 22;; (14)

The relationships among the functions C(f;t), c(;t), C(f;), and s(;) aresummarized in Figure 47.

4.3 Channel model for multipath fading channel

We may view the channel response in (5) as the sum of a number of vectors

(phasors), each of which has a time-variant amplitude n(t) and phase n(t). In general, it

takes large dynamic changes in the physical medium to cause a large change in {n(t)}.On the other hand, the phase {n(t)} change by 2or more radians with relatively small


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changes of the medium characteristics. {n(t)} is expected to be in an unpredictable

(random) manner. Thus c(;t) can be modeled as a complex-valued Gaussian randomprocess using the central limit theorem.

This simulator simulates a slow-fading frequency-selective channel. A general model

for a time-variant slow-fading frequency-selective channel is illustrated in Figure 48. The

channel model consists of a tapped delay line with uniformly spaced taps. The tap

spacing between adjacent taps is 1/W, where W is the bandwidth of the signal transmitted

through the channel. The tap coefficients, denoted as ( ) ( ) ( ){ }tjnn nettc= , are usually

modeled as complex-valued Gaussian random processes that are mutually uncorrelated.

The length of the delay line corresponds to the multipath spread. That is,

W

LT

m= (15)

whereL represents the maximum number of possible multipath signal components. Using

the simulator, the user can specify the means and variances of the possible channels oruse the default values.

Fig 47 Relationships among the channel correlation functions and power spectra


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4.4 Signal design for multipath fading channel

There are lots of carrier modulation methods that are available for digital and analog

modulation, as discussed in Section 3. However, signal fading generally causes large

fluctuations in the received signal amplitude, and consequently it is extremely difficult to

discriminate among multiple amplitude levels in the received signal. For this reason,

ASK and QASK are generally avoided for digital communication through fading

channels. In this simulator, users can choose 2-FSK (coherent), 2-FSK/noncoherent, or 2-

PSK as the digital modulation method. ASK, QASK, and the corresponding analogmodulations are disabled in Figure 43. For simplicity, spread spectrum and ISI are not

discussed in this report.

4.5 RAKE demodulator

Observing Figure 48, it is apparent that the tapped delay line model with statistically

independent tap weights provides us with L replicas of the same transmitted signal at the

receiver. Hence, a receiver that processes the received signal in an optimum manner willachieve the performance of an equivalentL

th-order diversity communications system.

This simulator employs binary signaling over the multipath-fading channel. We have

two equal-energy signals sl1(t) and sl2(t). The received signal is expressed as

( ) ( ) ( ) ( )tzWktstctrL

k

likl += =1

( ) ( ) 2,1,0, =+= iTttztvi

(16)

where z(t) is a complex-valued zero-mean AWGN process. Assume that the channel

tap weights are known. Then the optimum receiver consists of two filters matched to v1(t)

Fig 48 Tapped delay line model of frequency-selective channel


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and v2(t), followed by samplers and a decision circuit that selects the signal

corresponding to the largest output. An equivalent optimum receiver employs cross

correlation instead of matched filtering. In either case, the decision variables for coherentdetection of the binary signals can be expressed as

( ) ( ) 2,1,Re0

* =

= mdttvtrU

T

mlm(17)

Figure 49 illustrates the operations involved in the computation of the decisionvariables. In this realization of the optimum receiver, the two reference signals are

delayed and correlated with the received signal rl(t).

4.6 Simulation results

Figure 50 shows part of the transmitted signal and the corresponding received signal

through the multipath fading channel (without AWGN), whose means and variances of

the tap coefficients are default values shown in Figure 43. Figure 51 shows part of theoriginal binary data and the output of the RAKE demodulator. The bit error rate of this

example is 0.

Fig 49 Optimum demodulator for wideband binary signals


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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-6

-1.5

-1

-0.5

0

0.5

1

1.5

Input and Output of the m ult ipath fading channel

Input of the multipath channel

Output of the mult ipath channel

55 60 65 70 75 80 85 90 95 100-1

0

1

2

Original digital signal

55 60 65 70 75 80 85 90 95 100-1

0

1

2

Output of the RAKE demodulater

Reference:

1. J. G. Proakis, Digital Communications, McGraw Hill, 1995;

2. L. W. Couch, Digital and Analog Communication Systems, Prentice Hall,1997;

3. Z. Y. Shen, et al, Principals of Communication Systems, Xidian Univ. Press,1997;

4. J. H. Schiller, Mobile Communications, Addison-wesley, 2000;

5. X. M. Kong, A Simulator for Time-varying Multipath Fading Channels, Master

thesis, 2000.

Fig 50 Input and output of the

multipath-fading channel

Fig 51 Original binary data and the

output of the RAKE demodulator

Digital Communication via Multipath Fading Channel

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