Lab 4: Quadrature Amplitude Modulation
1 Modulation
A quadrature amplitude-modulated (QAM) signal employs two quadrature carriers, cos 2πfct, sin 2πfct, each of
which is modulated by an independent sequence of information bits. The transmitted signal waveforms have the form
um(t) = AmcgT (t) cos 2πfct + AmsgT (t) sin 2πfct, m = 1, 2, ..., M (1)
where {Amc} and {Ams} are the sets of amplitude. For example, Fig. illustrates a 16-QAM signal constellation
that is obtained by amplitude modulating each quadrature carrier by M = 4 PAM. In general, rectangular signal
constellations 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(2πfct + θn), m = 1, 2, ..., M1, n = 1, 2, ..., M2 (2)
If M1 = 2k1 and M2 = 2k2
, the combined amplitude- and phase-modulation method results in the simultaneous
transmission of k1 + 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 twodimensional
signal vectors of the form
sm = (p
EsAmc p
EsAms), m = 1, 2, ..., M (3)
Examples of signal space constellation for QAM are shown in Fig. . Note that M = 4 QAM is identical to M = 4
PSK.Lab 4留学生作业代写、代做Matlab程序语言作业、代做QAM作业、代写MATLAB实验作业
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(2πfct + φ) + AmsgT (t) sin(2πfct + φ) + n(t) (4)
where φ is the carrier-phase offset and
n(t) = nc(t) cos 2πfct ns(t) sin 2πfct
The received signal is correlated with the two phase-shifted basis functions
ψ1(t) = gT (t) cos(2πfct + φ)
φ2(t) = gT (t) sin(2πfct + φ) (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 offset φ of the received signal and compensates for this phase
offset by phase shifting ψ1(t) and ψ2(t) as indicated in. The clock shown in Fig. is assumed to be synchronized
1
Figure 1: M=16-QAM signal constellation.
Serial-to- parallel converter Transmitting filter g
T
(t) Transmitting filter g
T
(t) Balanced modulator Balanced modulator 90
o
phase shift Oscillator +
Transmitted QAM signal Binary 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 + nc cos φ ? ns sin φ
r2 = Ams + nc cos φ + ns sin φ (6)
where
nc(t)gT (t) dt
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)
2
Sampler Computes distance metrics D(s
m ) Sampler 90
o
phase shift PLL Clock g
T
(t) Received Signal
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. Rectangular
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
, where k is even, the QAM signal constellation is equivalent to two PAM signals on quadrature carriers, each
having √
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 the M-ary QAM system is
Pc = (1 P√
M)
2
(9)
where P√
M is the probability of error of a √
M-ary PAM with one-half the average power in each quadrature signal
of the equivalent QAM systems. By appropriately modifying the probability of error for M-ary PAM, we obtain
(10)
where Eav/No is the average SNR per symbol. Therefore, the probability of a symbol error for the M-ary QAM is
PM = 1 (1 P√
M)
2
(11)
We note that this result is exact for M = 2k when k is even.
MATLAB Perform a Monte Carlo simulation of an M=16-QAM communication system using a rectangular signal
constellation. 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 of b1, b2, b3, b4. The information symbols are mapped into
the corresponding signal points, as illustrated in Fig. ??, which have the coordinates [Amc Ams]. Two Gaussian
3
= 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 shift φ is 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 of N = 10000 symbols at different values of the
SNR parameter Eb/No, where Eb = Es/4 is the bit energy. Also, shown in Fig. is the theoretical value of
the 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
4
1 2 3 -3
-2
-1
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 symbols
Eav=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;
6
end;
end; p=numoferr/(N);
0 5 10 15
Eb
/No
in dB
Symbol Error Rate
Figure 6: M = 16-QAM signal constellation for the Monte Carlo simulation
7
Lab Homework
In this homework, we want to perform a Monte Carlo simulation of an M=16-QAM communication systems for the
performance of bit error rate (not a symbol error rate) for the SNR range of SNR=0～ 15 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<1/16),
dsource1(i)=0;
dsource2(i)=0;
dsource3(i)=0;
dsource4(i)=0;
elseif (temp<2/16),
dsource1(i)=0;
dsource2(i)=0;
dsource3(i)=0;
dsource4(i)=1;
elseif (temp<3/16),
dsource1(i)=0;
dsource2(i)=0;
dsource3(i)=1;
dsource4(i)=0;
elseif (temp<4/16),
dsource1(i)=0;
dsource2(i)=0;
dsource3(i)=1;
dsource4(i)=1;
elseif (temp<5/16),
dsource1(i)=0;
dsource2(i)=1;
dsource3(i)=0;
dsource4(i)=0;
elseif (temp<6/16),
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dsource1(i)=0;
dsource2(i)=1;
dsource3(i)=0;
dsource4(i)=1;
elseif (temp<7/16),
dsource1(i)=0;
dsource2(i)=1;
dsource3(i)=1;
dsource4(i)=0;
elseif (temp<8/16),
dsource1(i)=0;
dsource2(i)=1;
dsource3(i)=1;
dsource4(i)=1;
elseif (temp<9/16),
dsource1(i)=1;
dsource2(i)=0;
dsource3(i)=0;
dsource4(i)=0;
elseif (temp<10/16),
dsource1(i)=1;
dsource2(i)=0;
dsource3(i)=0;
dsource4(i)=1;
elseif (temp<11/16),
dsource1(i)=1;
dsource2(i)=0;
dsource3(i)=1;
dsource4(i)=0;
elseif (temp<12/16),
dsource1(i)=1;
dsource2(i)=0;
dsource3(i)=1;
dsource4(i)=1;
elseif (temp<13/16),
dsource1(i)=1;
dsource2(i)=1;
dsource3(i)=0;
dsource4(i)=0;
elseif (temp<14/16),
dsource1(i)=1;
dsource2(i)=1;
dsource3(i)=0;
dsource4(i)=1;
elseif (temp<15/16),
dsource1(i)=1;
dsource2(i)=1;
dsource3(i)=1;
dsource4(i)=0;
else (temp<15/16),
dsource1(i)=1;
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dsource2(i)=1;
dsource3(i)=1;
dsource4(i)=1;
end;
end;
Then, the received signal at the detector for the ith 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;
elseif ((dsource1(i)==0) & (dsource2(i)==0) & (dsource3(i)==1) & (dsource4(i)==0))
r=s0010+n;
elseif ((dsource1(i)==0) & (dsource2(i)==0) & (dsource3(i)==1) & (dsource4(i)==1))
r=s0011+n;
elseif ((dsource1(i)==0) & (dsource2(i)==1) & (dsource3(i)==0) & (dsource4(i)==0))
r=s0100+n;
elseif ((dsource1(i)==0) & (dsource2(i)==1) & (dsource3(i)==0) & (dsource4(i)==1))
r=s0101+n;
elseif ((dsource1(i)==0) & (dsource2(i)==1) & (dsource3(i)==1) & (dsource4(i)==0))
r=s0110+n;
elseif ((dsource1(i)==0) & (dsource2(i)==1) & (dsource3(i)==1) & (dsource4(i)==1))
r=s0111+n;
elseif ((dsource1(i)==0) & (dsource2(i)==0) & (dsource3(i)==0) & (dsource4(i)==0)),
r=s1000+n;
elseif ((dsource1(i)==1) & (dsource2(i)==0) & (dsource3(i)==0) & (dsource4(i)==1))
r=s1001+n;
elseif ((dsource1(i)==1) & (dsource2(i)==0) & (dsource3(i)==1) & (dsource4(i)==0))
r=s1010+n;
elseif ((dsource1(i)==1) & (dsource2(i)==0) & (dsource3(i)==1) & (dsource4(i)==1))
r=s1011+n;
elseif ((dsource1(i)==1) & (dsource2(i)==1) & (dsource3(i)==0) & (dsource4(i)==0))
r=s1100+n;
elseif ((dsource1(i)==1) & (dsource2(i)==1) & (dsource3(i)==0) & (dsource4(i)==1))
r=s1101+n;
elseif ((dsource1(i)==1) & (dsource2(i)==1) & (dsource3(i)==1) & (dsource4(i)==0))
r=s1110+n;
else
r=s1111+n;
end;
Then, the correlation metrics will be followed.
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