Key focus: Compare Performance and spectral efficiency of bandwidth-efficient digital modulation techniques (BPSK,QPSK and QAM) on their theoretical BER over AWGN.
More detailed analysis of Shannon’s theorem and Channel capacity is available in the following book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.
Simulation of various digital modulation techniques are available in these books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
Let’s take up some bandwidth-efficient linear digital modulation techniques (BPSK,QPSK and QAM) and compare its performance based on their theoretical BER over AWGN. (Readers are encouraged to read previous article on Shannon’s theorem and channel capacity).
Table 1 summarizes the theoretical BER (given SNR per bit ration – Eb/N0) for various linear modulations. Note that the Eb/N0 values used in that table are in linear scale [to convert Eb/N0 in dB to linear scale – use Eb/N0(linear) = 10^(Eb/N0(dB)/10) ]. A small script written in Matlab (given below) gives the following output.
The following table is obtained by extracting the values of Eb/N0 to achieve BER=10-6 from Figure-1. (Table data sorted with increasing values of Eb/N0).
where,
is the bandwidth efficiency for linear modulation with M point constellation, meaning that ηB bits can be stuffed in one symbol with Rb bits/sec data rate for a given minimum bandwidth.
is the minimum bandwidth needed for information rate of Rb bits/second. If a pulse shaping technique like raised cosine pulse [with roll off factor (a)] is used then Bmin becomes
Next the data in table 2 is plotted with Eb/N0 on the x-axis and η on the y-axis (see figure 2) along with the well known Shannon’s Capacity equation over AWGN given by,
which can be represented as (refer [1])
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Matlab Code
EbN0dB=-4:1:24;
EbN0lin=10.^(EbN0dB/10);
colors={'b-*','g-o','r-h','c-s','m-d','y-*','k-p','b-->','g:<','r-.d'};
index=1;
%BPSK
BPSK = 0.5*erfc(sqrt(EbN0lin));
plotHandle=plot(EbN0dB,log10(BPSK),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
hold on;
index=index+1;
%M-PSK
m=2:1:5;
M=2.^m;
for i=M,
k=log2(i);
berErr = 1/k*erfc(sqrt(EbN0lin*k)*sin(pi/i));
plotHandle=plot(EbN0dB,log10(berErr),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
index=index+1;
end
%Binary DPSK
Pb = 0.5*exp(-EbN0lin);
plotHandle = plot(EbN0dB,log10(Pb),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
index=index+1;
%Differential QPSK
a=sqrt(2*EbN0lin*(1-sqrt(1/2)));
b=sqrt(2*EbN0lin*(1+sqrt(1/2)));
Pb = marcumq(a,b,1)-1/2.*besseli(0,a.*b).*exp(-1/2*(a.^2+b.^2));
plotHandle = plot(EbN0dB,log10(Pb),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
index=index+1;
%M-QAM
m=2:2:6;
M=2.^m;
for i=M,
k=log2(i);
berErr = 2/k*(1-1/sqrt(i))*erfc(sqrt(3*EbN0lin*k/(2*(i-1))));
plotHandle=plot(EbN0dB,log10(berErr),char(colors(index)));
set(plotHandle,'LineWidth',1.5);
index=index+1;
end
legend('BPSK','QPSK','8-PSK','16-PSK','32-PSK','D-BPSK','D-QPSK','4-QAM','16-QAM','64-QAM');
axis([-4 24 -8 0]);
set(gca,'XTick',-4:2:24); %re-name axis accordingly
ylabel('Probability of BER Error - log10(Pb)');
xlabel('Eb/N0 (dB)');
title('Probability of BER Error log10(Pb) Vs Eb/N0');
grid on;Reference
[1] “Digital Communications” by John G.Proakis , Chapter 7: Channel Capacity and Coding.↗
Related topics
| Digital Modulators and Demodulators - Complex Baseband Equivalent Models ● Introduction ● Complex baseband representation of modulated signal ● Complex baseband representation of channel response ● Modulators for amplitude and phase modulations □ Pulse Amplitude Modulation (M-PAM) □ Phase Shift Keying Modulation (M-PSK) □ Quadrature Amplitude Modulation (M-QAM) ● Demodulators for amplitude and phase modulations □ M-PAM detection □ M-PSK detection □ M-QAM detection □ Optimum detector on IQ plane using minimum Euclidean distance ● M-ary FSK modulation and detection □ Modulator for M orthogonal signals □ M-FSK detection |
Books by the author
 Wireless Communication Systems in Matlab Second Edition(PDF)  (169 votes, average: 3.70 out of 5) |  Digital Modulations using Python (PDF ebook) (125 votes, average: 3.61 out of 5) |  Digital Modulations using Matlab (PDF ebook) (130 votes, average: 3.68 out of 5) |
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hi sir, my project title is modelling and simulation of prime physical layer(ofdm) using matlab.
so kindly pls give me the brief note of it
and which modulation sheme and channels can be used when compared to existing?
can you pls give me the coding for this too…….?