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I determined the PSD of a BPSK modulated signal using the MATLAB pwelch function and default parameters. My code looks like the following:

N = 2000;                   %no of bits
Rb = 1000;                  %bit rate
k = 20;                     %samples per bit
Tb = 1/Rb;
fc1 = 3*Rb;                 %carrier frequency
Fs = k*Rb;                  %sampling frequency
Ts = 1/Fs;
b = round(rand(1,N));       %bits randomly generated 
A0 = 5; 
t = 0:Ts:N*Tb-Ts;           %time frame
m1 = sin(2*pi*fc1*t);       %carrier1
y = [ ];                    %time sampled bit stream
for i=1:N   
      y = [ y b(i)*ones(1,k)];       
end

%% Modulation
modulated = A0y.m1 - A0~y.m1;
%% PSD
[pxx f]=pwelch(modulated,[],[],[],Fs);
pxx=pxx/max(pxx);    %normalization
figure(2);
plot(f, pxx,'LineWidth', 2); grid on;
title('PSD of BPSK SIGNAL');

When I ran the simulation, I noticed that the left side lobe was higher than the right one. Why is that?

Plot of PSD showing asymmetric side lobes.

Peter K.
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Elin Das
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1 Answers1

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You're "mixing up" with a real-valued sine; hence, your PSD estimate only shows half of the whole story. Remember that $\sin (2\pi f_{c,1} t) = \frac{1}{2j}\left(e^{+j2\pi f_c t} - e^{-j2\pi f_c t}\right)$ (with $j=\sqrt{-1}$), i.e. your sine shifts the original sinc-shaped baseband spectrum not only up to $+f_{c,1}$, but also down to $-f_{c,1}$, and adds both.

Now, what that effects is that the sidelobes of the sinc you've created through your rectangular pulse shape overlay constructively (due to your choice of $k$ (which is the inverse of the sinc's side lobe widths) and $f_{c,1}$ as rationally related, and $f_{c,1}\not\gg \frac 1k$) and hence become larger between the two main lobes.

Marcus Müller
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  • Thank you. But I still don't understand why the left lobe is higher than the right lobe. No matter how many times I simulate, each time I get a higher left lobe. – Elin Das Nov 29 '23 at 08:48
  • I addressed that - think about the whole spectrum, not just the positive side. See how the side lobes from the negative side reach into the positive side of the spectrum. – Marcus Müller Nov 29 '23 at 09:32