Why do signal have mirror signal after pure frequency shift?

Question

In my Matlab code I have a simple simulation of FPGA with the ability to simulate doppler shift, and after I and Q components plugged into IQmod with some carrier freq.

In my Matlab code when I multiply primordial signal $s_{prim}(t)e^{(jwt)}$ and have mirror signal after? And what's strange when I carrying that signal onto carrier frequency this carried signal doesn't have mirror signal.

totally working Matlab code included, will be glad to any ideas.

t = 0:0.000001:0.1;
fs = 1/(t(2) - t(1));
%ss = exp(j*2*pi*100*t) + exp(j*2*pi*200*t) + exp(j*2*pi*300*t) + exp(j*2*pi*900*t) + exp(j*2*pi*600*t); %Generating I-Q signal *.bin file for AWG

N=8;
bit_stream = round(rand(1,N));
f = 1000;
P1= 0;
P2= pi;
PSK_signal = [];

for ii = 1:1:N
PSK_signal = [PSK_signal (bit_stream(ii)==0)*sin(2*pi*f*t + P1)+...
        (bit_stream(ii)==1)*sin(2*pi*f*t + P2)];
end

ss = PSK_signal;
figure(1); plot(ss);



sp = fft(ss);
N = length(ss);
fp = ((0:N-1)/(N-1))*fs;
fp = fp(1:N/2);
sp = abs(sp(1:N/2))/(N/2);
figure(2); hold on; plot(fp,sp); %Spectre of primordial signal

w = 3000*pi*2;
t = (1:length(ss))/fs;

%Real outputs from AWG
I = real(ss); % I Input on FPGA
Q = imag(ss); % Q Input on FPGA

%Adding frequency SHIFT in Simuation FPGA plate 
ss_i_real = I.*cos(w*t) - Q.*sin(w*t); % I component from FPGA
ss_i_imag = I.*sin(w*t) + Q.*cos(w*t); % Q component from FPGA


sp = fft(ss_i_real+1j*ss_i_imag);
N = length(ss_i_real);
fp = ((0:N-1)/(N-1))*fs;
fp = fp(1:N/2);
sp = abs(sp(1:N/2))/(N/2);
figure(2);  plot(fp,sp);


w = 40000*pi*2;
t = (1:length(ss_i_real))/fs;

%ss_c = ss_i.*exp(1j*w*t); %Modulation that carrying signal from FPGA on Carrier Frequency (40KHz)
ss_c_real = ss_i_real.*cos(w*t) - ss_i_imag.*sin(w*t); 
ss_c_imag = ss_i_real.*sin(w*t) + ss_i_imag.*cos(w*t);

ss_c = ss_c_real+ss_c_imag*1j; % Real part of Signal onto carrier <- this is the signal that will be transmitting from modulator

sp = fft(ss_c);
N = length(ss_c);
fp = ((0:N-1)/(N-1))*fs;
fp = fp(1:N/2);
sp = abs(sp(1:N/2))/(N/2);
figure(2); plot(fp,sp); %Spectre of signal onto carrier freq

% figure(4); plot(ss_c_real); %I - component of signal onto carrier freq
% 
% figure(5); plot(ss_c_imag); %Q - component of signal onto carrier freq

edit1: What causes my confusion, when I'm using the signal:

 %ss = exp(j*2*pi*100*t) + exp(j*2*pi*200*t) + exp(j*2*pi*300*t) + exp(j*2*pi*900*t) + exp(j*2*pi*600*t); %Generating I-Q signal *.bin file for AWG

I have this nice picture:

1 primordial signal, 1 shifted signal and 1 carried onto 40KHz shifted signal. When I'm using the BPSK signal I'm shifting this signal I have mirrored signal and shifted (i mean two signals with $F_{shift} \pm F_{prim}$)

To be simple I just want to add shift on intermediate frequency and after carry that shifted signal onto carrying frequency without clone signal at $f_{car}-f_{shifted}$ as I have now.

Answer 1

Your code is (probably) ok. But your way of plotting is slightly wrong.

The complete spectrum of the baseband PSK signal would include two peaks one at positive frequency $f_0=1000$ and the other at negative frequency $f=-1000$. When you IQ modulate this signal with $f_m=3000$ you will create two twins one at $f_{11}=2000$ and $f_{12} = 4000$ and the other at the negative frequencies at $f_{21}=-2000$ and $f_{22} = -4000$.

Since you are taking only the first half of the FFT you do not observe the peaks at negative ones, which causes the confusion. If you plot the full spectrums, you can see the missing components at the second half that you omit...

t = 0:0.000001:0.1;
fs = 1/(t(2) - t(1));
%ss = exp(j*2*pi*100*t) + exp(j*2*pi*200*t) + exp(j*2*pi*300*t) + exp(j*2*pi*900*t) + exp(j*2*pi*600*t); %Generating I-Q signal *.bin file for AWG

N=8;
bit_stream = round(rand(1,N));
f = 1000;
P1= 0;
P2= pi;
PSK_signal = [];

for ii = 1:1:N
PSK_signal = [PSK_signal (bit_stream(ii)==0)*sin(2*pi*f*t + P1)+...
        (bit_stream(ii)==1)*sin(2*pi*f*t + P2)];
end

ss = PSK_signal;
%figure(1); plot(ss); title('The PSK baseband signal');

ff = linspace(-fs/2,fs/2,length(ss));
figure,plot(ff,abs(fftshift(fft(ss))));
title('The PSK baseband SPECTRUM ');
% sp = fft(ss);
% N = length(ss);
% fp = ((0:N-1)/(N-1))*fs;
% fp = fp(1:N/2);
% sp = abs(sp(1:N/2))/(N/2);
% figure(2); plot(fp,sp); %Spectre of primordial signal
                     %title('The PSK baseband SPECTRUM ');

w = 3000*pi*2;
t = (1:length(ss))/fs;

%Real outputs from AWG
I = real(ss); % I Input on FPGA
Q = imag(ss); % Q Input on FPGA

%Adding frequency SHIFT in Simuation FPGA plate 
ss_i_real = I.*cos(w*t) - Q.*sin(w*t); % I component from FPGA
ss_i_imag = I.*sin(w*t) + Q.*cos(w*t); % Q component from FPGA

sp = fft(ss_i_real+1j*ss_i_imag);  % computing the Spectrum of the ANALYTIC signal here! Hence the NEGATIVE frequencies are deleted.
figure,plot(ff,abs(fftshift(sp))); 
title('IQ modulated signal FSK baseband');

% sp = fft(ss_i_real+1j*ss_i_imag);
% N = length(ss_i_real);
% fp = ((0:N-1)/(N-1))*fs;
% fp = fp(1:N/2);
% sp = abs(sp(1:N/2))/(N/2);
%figure(2);  plot(fp,sp);
%figure(3),plot(fp,sp); title('IQ modulated signal FSK baseband');


w = 40000*pi*2;
t = (1:length(ss_i_real))/fs;

%ss_c = ss_i.*exp(1j*w*t); %Modulation that carrying signal from FPGA on Carrier Frequency (40KHz)
ss_c_real = ss_i_real.*cos(w*t) - ss_i_imag.*sin(w*t); 
ss_c_imag = ss_i_real.*sin(w*t) + ss_i_imag.*cos(w*t);

ss_c = ss_c_real+ss_c_imag*1j; % Real part of Signal onto carrier <- this is the signal that will be transmitting from modulator

sp = fft(ss_c);
N = length(ss_c);
fp = ((0:N-1)/(N-1))*fs;
fp = fp(1:N/2);
sp = abs(sp(1:N/2))/(N/2);
figure(4); plot(fp,sp); %Spectre of signal onto carrier freq
title('IQ modulated and up-mixed FSK signal');

% figure(4); plot(ss_c_real); %I - component of signal onto carrier freq
% 
% figure(5); plot(ss_c_imag); %Q - component of signal onto carrier freq