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Lab 4_ Quadrature Amplitude Modulation

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					KEEE494: 2nd Semester 2009                                                                                          Lab 4

                    Lab 4: Quadrature Amplitude Modulation

1    Modulation
A quadrature amplitude-modulated (QAM) signal employs two quadrature carriers, cos 2πfc t, sin 2πfc t, each of
which is modulated by an independent sequence of information bits. The transmitted signal waveforms have the form

                       um (t) = Amc gT (t) cos 2πfc t + Ams gT (t) sin 2πfc t, 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) = Am gT (t) cos(2πfc t + 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 M1 M2 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 = (    Es Amc       Es Ams ),   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.


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) = Amc gT (t) cos(2πfc t + φ) + Ams gT (t) sin(2πfc t + φ) + n(t)                          (4)

where φ is the carrier-phase offset and

                                       n(t) = nc (t) cos 2πfc t − ns (t) sin 2πfc t


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

                                            ψ1 (t) =    gT (t) cos(2πfc t + φ)
                                            φ2 (t) =    gT (t) sin(2πfc t + 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.


                                                       Transmitting               Balanced
                                                         filter g T (t)           modulator




                       Binary     Serial-to-             Oscillator
                        data
                                   parallel
                                  converter
                                                                                  90 o phase
                                                                                                        +   Transmitted
                                                                                                            QAM signal
                                                                                      shift




                                                       Transmitting               Balanced
                                                         filter g T (t)           modulator




                                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
                                                                                  T
                                                                          1
                                                       nc        =                    nc (t)gT (t) dt
                                                                          2   0
                                                                                  T
                                                                          1
                                                       ns        =                    ns (t)gT (t) dt                     (7)
                                                                          2   0

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
                                                  X                            Sampler




                                                  X


                                                                                Clock

                       Received
                        Signal                               g T (t)                         Computes        Output
                                     PLL
                                                                                         distance metrics   decision
                                                                                               D(s m )
                                              90 o phase
                                                  shift




                                                  X


                                                  X                            Sampler




                                  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 , 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

                                                   1                               3 Eav
                                      P√M = 2 1 − √                        Q                                           (10)
                                                    M                            M − 1 No

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
                                           3



                                           2


                                           1



                    -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’)




                                               5
  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*dˆ2;                  % energy per symbol
snr=10ˆ(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))ˆ2+ (r(i,2)-mapping(j,2))ˆ2;
    end;
    [min_metric decis]=min(metrics);
    if (decis˜=dsource(i)),
         numoferr=numoferr+1;


                                   6
    end;
end; p=numoferr/(N);


                                 0
                                10



                                 −1
                                10



                                 −2
                                10
            Symbol Error Rate




                                 −3
                                10



                                 −4
                                10



                                 −5
                                10



                                 −6
                                10



                                 −7
                                10
                                      0   5                 10              15
                                              Eb/No in dB



         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),

                                                         8
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;


                       9
    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.




                                                          10

				
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Description: QAM (Quadrature Amplitude Modulation) digital modulator for DVB system, front-end device, receiving data from the encoder, multiplexer, DVB gateway, video server and other equipment of the TS stream, the RS coding, convolution coding and QAM modulation, RF signals can be output directly in the cable TV transmission over the Internet, but also can be selected IF output. With its flexible configuration and superior performance, widely used in the field of digital cable TV transmission and digital MMDS system.