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KEEE494: 2nd Semester 2009 Lab 4

Lab 4: Quadrature Amplitude Modulation

1 ModulationA quadrature amplitude-modulated (QAM) signal employs two quadrature carriers, cos2fct, sin2fct, each ofwhich is modulated by an independent sequence of information bits. The transmitted signal waveforms have the form

um(t) = AmcgT(t)cos2fct + AmsgT(t)sin2fct, m= 1, 2,...,M (1)

where{Amc}and{Ams}are the sets of amplitude. For example, Fig. ?? illustrates a 16-QAM signal constellationthat is obtained by amplitude modulating each quadrature carrier by M = 4 PAM. In general, rectangular signalconstellations result when two quadrature carriers are each modulated by PAM.

More generally, QAM may be viewed as a form of combined digital amplitude and digital phase modulation. Thus

the transmitted QAM signal waveforms may be expressed as

umn(t) = AmgT(t) cos(2fct + n), m= 1, 2,...,M1, n= 1, 2,...,M2 (2)

IfM1 = 2k1 andM2 = 2

k2 , the combined amplitude- and phase-modulation method results in the simultaneous

transmission ofk1+ k2 = log2 M1M2 binary digits occurring at a symbol rate Rb/(k1+ k2).

It is clear that the geometric signal representation of the signal given by ( ??) and (??) is in the terms of two-

dimensional signal vectors of the form

sm= (

EsAmc

EsAms), m= 1, 2,...,M (3)

Examples of signal space constellation for QAM are shown in Fig. ??. Note thatM= 4QAM is identical to M= 4PSK.

2 Demodulation and Detection of QAM

Let us assume that a carrier-phase offset is introduced in the transmission of the signal through the channel. In addition,

the received signal is corrupted by additive white Gaussian noise. Hence,r(t)may be expressed as

r(t) = AmcgT(t) cos(2fct + ) + AmsgT(t) sin(2fct + ) + n(t) (4)

whereis the carrier-phase offset and

n(t) =nc(t)cos2fct ns(t)sin2fct

The received signal is correlated with the two phase-shifted basis functions

1(t) = gT(t) cos(2fct + )

2(t) = gT(t) sin(2fct + ) (5)

as illustrated in Fig. ??, and the outputs of the correlators are sampled and passed to the detector. The phase-locked

loop (PLL) shown in Fig.?? estimates the carrier-phase offsetof the received signal and compensates for this phaseoffset by phase shifting1(t)and2(t)as indicated in??. The clock shown in Fig. ?? is assumed to be synchronized

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Figure 1:M=16-QAM signal constellation.

Serial-to-

parallel

converter

Transmitting

filter gT(t)

Transmitting

filter gT(t)

Balanced

modulator

Balanced

modulator

90 o phase

shift

Oscillator

+ TransmittedQAM signalBinary

data

Figure 2: Functional block diagram of modulator for QAM

to the received signal so that the correlator outputs are sampled at the proper instant in time. Under these conditions,

the outputs from the two correlators are

r1 = Amc+ nccos nssin r2 = Ams+ nccos + nssin (6)

where

nc = 12 T0

nc(t)gT(t) dt

ns = 1

2

T0

ns(t)gT(t) dt (7)

The noise components are zero-mean, uncorrelated Gaussian random variable with variance No/2.

The optimum detector computes the distance metrics

D(r, sm) =|r sm|2, m= 1, 2,...,M (8)

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Sampler

Computes

distance metrics

D(sm

)

Sampler

X

X

X

X

90 o phase

shift

PLL

Clock

gT(t)

ReceivedSignal Output

decision

Figure 3: Demodulation and detection of QAM signal

where r= (r1, r2)and sm is given by??.

3 Probability of Error for QAM in an AWGN Channel

In this section, we consider the performance of QAM systems that employ rectangular signal constellations. Rectan-

gular QAM signal constellations have the distinct advantage of being easily generated as two PAM signals impressed

on phase quadrature carriers. In addition, they are easily demodulated. For rectangular signal constellations in which

M= 2k, wherek is even, the QAM signal constellation is equivalent to two PAM signals on quadrature carriers, eachhaving

M = 2k/2 signal points. Because the signals in the phase quadrature components are perfectly separated

by coherent detection, the probability of error for QAM is easily determined from the probability of error for PAM.

Specifically, the probability of a correct decision for theM-ary QAM system is

Pc= (1 PM)2 (9)

wherePMis the probability of error of a

M-ary PAM with one-half the average power in each quadrature signalof the equivalent QAM systems. By appropriately modifying the probability of error for M-ary PAM, we obtain

PM = 2

1 1

M

Q

3

M1EavNo

(10)

whereEav/No is the average SNR per symbol. Therefore, the probability of a symbol error for theM-ary QAM is

PM= 1(1 PM)2 (11)

We note that this result is exact for M= 2k

whenk is even.

MATLAB Perform a Monte Carlo simulation of an M=16-QAM communication system using a rectangular signalconstellation. The model of the system to be simulated is hown in Fig.??.

Solution The uniform random number generator (RNG) is used to generate the sequence of information symbols

corresponding to the 16 possible 4-bit combinations ofb1, b2, b3, b4. The information symbols are mapped intothe corresponding signal points, as illustrated in Fig. ??, which have the coordinates[Amc Ams]. Two Gaussian

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= 64= 32

= 16

= 8

= 4

Figure 4: Signal space diagram for QAM signals

RNG are used to generate the noise components [nc ns]. The channel-phase shiftis set to 0 for convenience.Consequently, the received signal-plus-noise vector is

r= [Amc+ nc Ams+ ns]

The detector computes the distance metric given by ?? and decides in favor of the signal point that is closest to

the received vector r. The error counter counts the symbol errors in the detected sequence. Fig. ?? illustrates

the results of the Monte Carlos simulation for the transmission ofN= 10000symbols at different values of theSNR parameterEb/No, whereEb =Es/4is the bit energy. Also, shown in Fig. ??is the theoretical value ofthe symbol-error probability given by (??) and (??).

echo on

SNRindB1=0:2:15;

SNRindB2=0:0.1:15;

M=16;

k=log2(M);for i=1:1:length(SNRindB1),

smld_err_prb(i)=qam_sim(SNRindB1(i)); % simulated error value

echo off;

end;

echo on;

for i=1:length(SNRindB2),

SNR = exp(SNRindB2(i)*log(10)/10); % signal-to-noise ratio

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1

2

3

1 2 3-1-2-3

-1

-2

-3

Figure 5: Block diagram of an M=16-QAM system for the Monte Carlo simulation

% theoretical symbol error rate

theo_err_prb(i)=4*Qfunct(sqrt(3*k*SNR/(M-1)));

echo off;

end;

echo on;% Plotting commands follow.

semilogy(SNRindB1,smld_err_prb, *)

hold

semilogy(SNRindB2,theo_err_prb);

grid on

xlabel(E_b/N_o in dB)

ylabel(Symbol Error Rate)

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function [p]=qam_sim(snr_in_dB)

% [p]=qam_sim(snr_in_dB)

% finds the probability of error for the given value of snr_in_dB,

% SNR in dB.

N=10000;

d=1; % min. distance between symbolsEav=10*d2; % energy per symbol

snr=10(snr_in_dB/10); % SNR per bit (given)

sgma=sqrt(Eav/(8*snr)); % noise variance

M=16;

% Genreation of the data source follows.

for i=1:N,

temp=rand;

dsource(i)=1+floor(M*temp);

end;

% Mapping to the signal constellation follows

mapping=[-3*d 3*d;

-d 3*d;d 3*d;

3*d 3*d;

-3*d d;

-d d;

d d;

3*d d;

-3*d -d;

-d -d;

d -d;

3*d -d;

-3*d -3*d;

-d -3*d;

d -3*d;

3*d -3*d];

for i=1:N,

qam_sig(i,:)=mapping(dsource(i),:);

end;

% received signal

for i=1:N,

[n(1) n(2)]=gngauss(sgma);

r(i,:)=qam_sig(i,:)+n;

end;

% detection and error probability calculation

numoferr=0;

for i=1:N,% Metric computation follows.

for j=1:M,

metrics(j)=(r(i,1)-mapping(j,1))2+ (r(i,2)-mapping(j,2))2;

end;

[min_metric decis]=min(metrics);

if (decis=dsource(i)),

numoferr=numoferr+1;

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end;

end; p=numoferr/(N);

0 5 10 1510

7

106

105

104

103

102

101

100

Eb/N

oin dB

SymbolErrorRate

Figure 6:M = 16-QAM signal constellation for the Monte Carlo simulation

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Lab Homework

In this homework, we want to perform a Monte Carlo simulation of an M=16-QAM communication systems for theperformance of bit error rate (not a symbol error rate) for the SNR range of SNR=015 dB.

Use the following hint: For this simulation, you have to generate not only the symbols but also the bits such as

s0000=[3*d 3*d];

s0001=[d 3*d];

s0011=[-d 3*d];

s0010=[-3*d 3*d];

s1000=[3*d d];

s1001=[d d];

s1011=[-d d];

s1010=[-3*d d];

s1100=[3*d -d];

s1101=[d -d];

s1111=[-d -d];

s1110=[-3*d -d];

s0100=[3*d -3*d];s0101=[d -3*d];

s0111=[-d -3*d];

s0110=[-3*d -3*d];

for i=1:1:N,

temp=rand;

if (temp

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dsource1(i)=0;

dsource2(i)=1;

dsource3(i)=0;

dsource4(i)=1;

elseif (temp

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dsource2(i)=1;

dsource3(i)=1;

dsource4(i)=1;

end;

end;

Then, the received signal at the detector for theith symbol (in Matlab form) is

n(1)=gngauss(sgma);

n(2)=gngauss(sgma);

if ((dsource1(i)==0) & (dsource2(i)==0) & (dsource3(i)==0) & (dsource4(i)==0)),

r=s0000+n;

elseif ((dsource1(i)==0) & (dsource2(i)==0) & (dsource3(i)==0) & (dsource4(i)==1))

r=s0001+n;


r=s0010+n;


r=s0011+n;

elseif ((dsource1(i)==0) & (dsource2(i)==1) & (dsource3(i)==0) & (dsource4(i)==0))r=s0100+n;


r=s0101+n;


r=s0110+n;


r=s0111+n;

elseif ((dsource1(i)==0) & (dsource2(i)==0) & (dsource3(i)==0) & (dsource4(i)==0)),

r=s1000+n;


r=s1001+n;


r=s1010+n;


r=s1011+n;


r=s1100+n;


r=s1101+n;


r=s1110+n;

else

r=s1111+n;

end;

Then, the correlation metrics will be followed.

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