Solution 1.
Symbol Mapper for 16QAM.
a)Sketch the constellation for 16QAM.
Or
b)
Now for dmin
For square signal constellations, where si1 and si2 take values on
,
The minimum distance between signal points reduces to,
c)
The constellation mapping is usually done by Gray encoding, where the messages associated with signal amplitudes that are adjacent to each other differ by one bit value.
With this encoding method, if noise causes the demodulation process to mistake one symbol for an adjacent one (the most likely type of error), this results in only a single bit error in the sequence of Kbits. Gray codes can be designed for square MQAM constellation but not rectangular MQAM.
Hence cannot be applied for the 16 QAM constellations.
Solution 2
a)
Rate of code is
Intuitively, the greater the distance between code words in a given code, the less chance that errors introduced by the channel will cause a transmitted codeword to be decoded as a different codeword. We define the Hamming distance between two code words Ci and Cj, denoted as d(Ci,Cj) or dij, as the number of elements in which they differ:
b)
The rate of the code is information bits per codeword symbol. If we assume that codeword symbols are transmitted across the channel at a rate of Rs symbols/second , then the information rate associated with an(n, k) block code is
Thus we see that block coding reduces the data rate compared to what we obtain with uncoded modulation by the code rate Rc.
Solution 3
QPSK modulation consists of BPSK modulation on both the in-phase and quadrature components of the signal. With perfect phase and carrier recovery, the received signal components corresponding to each of these branches are orthogonal. Therefore, the bit error probability on each branch is the same as for BPSK
MATLAB CODE
clear all
N=10^3;
tx_sym=1+j;
M=4;
n=[0:M-1];
p=[0:5:35];
count1=zeros(1,36);
count2=zeros(1,36);
for EbNo_dB=0:5:25
Gno=1/sqrt(2)*10^(-EbNo_dB/20);
for i=1:N
y=zeros(1,M);
z1=zeros(1,M);
noise=(randn(1,M)+j*randn(1,M))*Gno;
for k=0:M-1
y=y+(1/sqrt(M))*tx_sym*exp((j*2*pi*k.*n)/M);
end
h = 1/sqrt(2)*[randn(1,1) + j*randn(1,1)];
sig_no=h.*y+noise;
sig_no1=sig_no.*conj(h)
for k=0:M-1
z1=z1+(1/sqrt(M))*sig_no1(k+1)*exp((-j*2*pi*k.*n)/M);
end
for t=1:M
if (real(z1(t))<0)
count1(EbNo_dB+1)=count1(EbNo_dB+1)+1;
end
if (imag(z1(t))<0);
count2(EbNo_dB+1)=count2(EbNo_dB+1)+1;
end
end
end
end
tmp_ber=(count1+count2)/2/(4*N);
ber=tmp_ber(1,1:5:end);
semilogy(p,ber,’mx-‘,’linewidth’,2);
xlabel(‘E/b in dB———–>’);
ylabel(‘BER——–>’)
axis([0 25 10^-5 0.5])
grid on
OUTPUT
MATLAB CODE
KC=1;
gamma=0.01; % Stepsize of adjusting the phase estimate
eth=0.5; % Error threshold
b=2; M=2^b; % Number of bits and symbols
Tb=1e-5; Ts=b*Tb; % Bit/Symbol time
Nb=40; Ns=b*Nb; % Number of sample times in Tb and Ts
T=Ts/Ns; % Sampling interval
LB=4*Ns; Ns_2=Ns/2; % Buffer size
g=[0.1 -0.2 0.6 0.9 1 0.9 0.6 -0.2 0.1]; g=g/norm(g)^2;
Ng=length(g); zi_XMTR=zeros(1,Ng-1);
ndd=1; % Detection delay due to XMTR and channel
% QPSK signal waveforms
ss=[0 0; 0 1; 1 1; 1 0];
wc=2*pi/Ts; wcT=wc*T; t=[0:Ns-1]*T; wct=wc*t;
pi2=pi*2; phases=[0:M-1]*(pi2/M);
su=sqrt(2/Ts)*[cos(wct);-sin(wct)]; suT=su*T; % Basis signal Eq.(7.3.2)
thd=-pi/10; % Phase offset between XMTR and RCVR
Es=2; % Energy of signal waveform representing the symbols
for m=1:M % PSK signal waveforms
sw(m,:)=sqrt(2*Es/Ts)*cos(wct+phases(m)+thd); % PSK signal (7.3.1)
end
interval=2*Ns+[1:Ns-1]; % Regular integration interval [k-2,k-1]*Ts
interval1=interval+Ns_2; % Later integration interval [k-3/2,k-1/2]*Ts
SNRbdB=9; SNRb=10^(SNRbdB/10); N0=2*(Es/b)/SNRb; sgmsT=sqrt(N0/2/T);
pobet= prob_error(SNRbdB,’PSK’,b,’bit’) % Theoretical BER
yr= zeros(2,LB); % Multiplier output buffer
s0s=zeros(ndd,size(ss,2)); % Transmitted symbol buffer
MaxIter=4; % # of iterations for getting the probability of error
nobe= 0; % Number of bit errors to be accumulated
thh(1)=0; nh(1)=-1; % Initial estimate of phase offset/sampling instant
ycsk0=[0 0]; % Previous correlator output samples initialized
rand(‘twister’,5489); randn(‘state’,0);
for k=1:MaxIter
i= ceil(rand*M); s=ss(i,:); % Signal index, Data bits to transmit
[ch_input,zi_XMTR]= filter(g,1,sw(i,:),zi_XMTR);
wct=wc*T*[0:Ns-1];
noise= sgmsT*(randn(1,Ns).*cos(wct)-randn(1,Ns).*sin(wct));
r = ch_input + noise; % Received signal
yr=[yr(:,Ns+1:LB) suT.*[r;r]]; % Multiplier output (buffer)
ycsk=sum(yr(:,interval+nh(k))’); % Correlator output at (k-1)Ts
ycsk1=sum(yr(:,interval1+nh(k))’); % Correlator output at (k-1/2)Ts
e(k)=ycsk*(ycsk1-ycsk0).’; % Eq.(8.5.3a)
dnh=0; % Stepsize of adjusting the sampling instant
if KC>1 % For timing recovery
if e(k)>eth, dnh=1; elseif e(k)<-eth, dnh=-1; end
end
nh(k+1)=nh(k)+dnh; % Eq.(8.5.2) for timing recovery
ycsk0=ycsk1; % Correlator output sample buffer updated
ynejth= ycsk(1)+j*ycsk(2);
if KC>0, ynejth=ynejth*exp(-j*thh(k)); end % Phase compensation
th=angle(ynejth); % Compensated phase of received signal waveform
if th<-pi/M, th=th+2*pi; end
[themin,imin]=min(abs(th-phases)); D=ss(imin,:); % Detected data
thh(k+1)=thh(k)+gamma*imag(exp(-j*phases(imin))*ynejth); %(8.4.10)
if k>ndd, nobe=nobe+sum(s0s(1,:)~=D); end
if nobe>100, break; end
s0s=[s0s(2:end,:); s]; % Transmitted symbol buffer updated
end
pobe= nobe/((k-ndd)*b) % The BER
OUTPUT
pobet = 0.0024
pobe = 0.1667
Section 2
PART A, B, C, D
SEQUENCE GENERATION
clc
clear
% Generating the bit pattern with each bit 6 samples long
b=round(rand(1,20));
pattern=[];
for k=1:20
if b(1,k)==0
sig=zeros(1,6);
else
sig=ones(1,6);
end
pattern=[pattern sig];
end
plot(pattern);
axis([-1 130 -.5 1.5]);
title(‘\bf\it Original Bit Sequence’);
% Generating the pseudo random bit pattern for spreading
spread_sig=round(rand(1,120));
figure,plot(spread_sig);
axis([-1 130 -.5 1.5]);
title(‘\bf\it Pseudorandom Bit Sequence’);
% XORing the pattern with the spread signal
hopped_sig=xor(pattern,spread_sig);
% Modulating the hopped signal
dsss_sig=[];
t=[0:100];
fc=.1
c1=cos(2*pi*fc*t);
c2=cos(2*pi*fc*t+pi);
for k=1:120
if hopped_sig(1,k)==0
dsss_sig=[dsss_sig c1];
else
dsss_sig=[dsss_sig c2];
end
end
figure,plot([1:12120],dsss_sig);
axis([-1 12220 -1.5 1.5]);
title(‘\bf\it DSSS Signal’);
% Plotting the FFT of DSSS signal
figure,plot([1:12120],abs(fft(dsss_sig)))
GRAPH
QPSK BER WITH TRANSMITTER AND RECEIVER
%TRANSMITTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear
nr_data_bits=1000000;% 0’s and 1’s, keep even number-Takes ~1 minute for a run of 1 million
nr_symbols=nr_data_bits/2;
b_data = (randn(1, nr_data_bits) > .5);%random 0’s and 1’s
b = (b_data);
% |
d=zeros(1,length(b)/2);
%definition of the QPSK symbols using Gray coding.
for n=1:length(b)/2
p=b(2*n);
imp=b(2*n-1);
if (imp==0)&&(p==0)
d(n)=exp(j*pi/4);%45 degrees
end
if (imp==1)&&(p==0)
d(n)=exp(j*3*pi/4);%135 degrees
end
if (imp==1)&&(p==1)
d(n)=exp(j*5*pi/4);%225 degrees
end
if (imp==0)&&(p==1)
d(n)=exp(j*7*pi/4);%315 degrees
end
end
qpsk=d;
SNR=0:30;%change SNR values
BER1=[];
SNR1=[];
SER=[];
SER1=[];
sigma1=[];
%Rayleigh multipath/AWGN(Additive White Gaussian Noise)
for SNR=0:length(SNR);%loop over SNR-change SNR values (0,5,10 etc dB)
sigma = sqrt(10.0^(-SNR/10.0));
sigma=sigma/2;%Required a division by 2 to get close to exact solutions(Notes)-WHY?
%Is dividing by two(2) legitimate?
%sigma1=[sigma1 sigma];
%add Rayleigh multipath(no LOS) to signal(qpsk)
x=randn(1,nr_symbols);
y=randn(1,nr_symbols);
ray=sqrt(0.5*(x.^2+y.^2));%variance=0.5-Tracks theoritical PDF closely
mpqpsk=qpsk.*ray;
mpsnqpsk=(real(mpqpsk)+sigma.*randn(size(mpqpsk))) +i.*(imag(mpqpsk)+sigma.*randn(size(mpqpsk)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Receiver
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
r=mpsnqpsk;%received signal plus noise and multipath
bhat=[real(r)<0;imag(r)<0];%detector
bhat=bhat(:)’;
bhat1=bhat;%0’s and 1’s
ne=sum(b~=bhat1);%number of errors
BER=ne/nr_data_bits;
SER=ne/nr_symbols;%consider this to be Ps=log2(4)*Pb=2*Pb
SER1=[SER1 SER];
BER1=[BER1 BER];
SNR1=[SNR1 SNR];
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Plots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure(1);
plot(d,’o’);%plot constellation without noise
axis([-2 2 -2 2]);
grid on;
xlabel(‘real’); ylabel(‘imag’);
title(‘QPSK constellation’);
figure(2);
semilogy(SNR1,BER1,’*’,SNR1,SER1,’o’);
grid on;
xlabel(‘SNR=Eb/No(dB)’); ylabel(‘BER/SER’);
title(‘Simulation of BER/SER for QPSK with Gray coding( Rayleigh multipath and AWGN)’);
legend(‘BER-simulated’,’SER-simulated’);
figure(3)
plot(real(qpsk));
grid on;
axis([1 200 -2 2]);
title(‘QPSK symbols’);
xlabel(‘symbols’);ylabel(‘Amplitude’);
figure(4)
plot(20*log10(abs(ray)));
grid on;
axis([1 200 -30 10]);
title(‘Rayleigh Fading Envelope(variance=0.5)’);
xlabel(‘symbols’);ylabel(‘Amplitude/RMS(dB)’);
GRAPH
