Why there is a modulation in case of frequency offset?

There is modulation in a last graph of a signal (figure 1). At programm there is no such kind of modulation (there is only transfer of a signal on the carrier frequency). Why? Where a mistake?

Image: http://picturepush.com/public/12436413

MATLAB code:

% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
% +++++++++++++++++++++ Doppler Shift ++++++++++++++++++++++++++++++++++++
% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

%%Initialization
clear all; close all; clc; format longE;
% ~~~~~~~~~~~~~~~~~~~ Begin of the Programm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

% Initial parameters of the Signal
f=100; %Frequency of the signal (sin), [Hz];
Fc=1e3; %Carrier frequency, [Hz];
Fd=30*f; %Sampling frequencym, [Hz];
A=1; %Amplitude

%%~~~~~~~~~~~~~~~~~~Initializing Doppler Shift~~~~~~~~~~~~~~~~~~~~~~~~~~
c=3e8; %Speed of light, [m/a];

%````````````````````````````````Speeds: ```````````````````````````````
        Fdoppler=10; %Maximal Doppler Shift, [Hz]
        Speed=(Fdoppler*c)/(Fc+f); %Necessary speed in [m/s] for receiving necessary shift.
        speeedfordisp=(Fdoppler*3.6*c)/(Fc+f); %Necessary speed in [m/s] for receiving necessary shift.
        fprintf('\n\n\n\nAt the set parameters for receiving shift on %d Hz\nspeed is necessary: \n \t -- [m/s]:\t %.3f;\n \t -- [km/h]:\t %3f.\n\n', Fdoppler, Speed, speeedfordisp);
%``````````````````````````````````````````````

%New wave frequency (after Doppler Effect):
    betta=Speed/c;
           alpha=0;
           f_dopl=((Speed*(Fc+f))/c)*cosd(alpha);
%%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%~~~~~~~ Passing a signal throught the channel with a Doppler Effect ~~~~~
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%Parametrs of the signal
    t=(0:1/Fd:1);
    N = length(t);
    NFFT = 16000;
    FrequencyAxe=Fd*linspace(0,1,NFFT); %Frequency Axe;
%Initializing the signal
    Signal=A*sin(2*pi*f.*t);

%Modulating the signal:
    SignalModulated=Signal.*exp(1i*2*pi*(Fc).*t);
%Add Doppler Effect:
    Spectr = fft(SignalModulated); %Conver to the Frequency Domain
    SignalModulated_D=ifft(circshift(Spectr, [0 f_dopl])); %Add shift
%Demodulation the signal
    SignalDemodulated=SignalModulated_D.*exp(-1i*2*pi*(Fc).*t);
%`````````````````````````````````````````````````````````````````````````
%```````````````````````` END OF PROGRAMM'S PART `````````````````````````
%`````````````````````````````````````````````````````````````````````````

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%~~~~~~~~~~~~~~~~~ Calculating a spectr of the signals ~~~~~~~~~~~~~~~~~~~
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%Spectr of the initial signal:
    SpectrInitialSignal=(1/NFFT).*fft(Signal, NFFT); Plot_SpectrInitialSignal=abs(SpectrInitialSignal(1:NFFT));
%Spectr of the modulated signal:
    SpectrModulatedSignal=(1/NFFT).*fft(SignalModulated, NFFT); Plot_SpectrModulatedSignal=abs(SpectrModulatedSignal(1:NFFT));
%Spectr of the modulated signal with a Doppler Effect:
    SpectrModulatedSignal_D=(1/NFFT).*fft(SignalModulated_D, NFFT); Plot_SpectrModulatedSignal_D=abs(SpectrModulatedSignal_D(1:NFFT));
%Spectr of the demodulated signal (with a Doppler Effect):
    SpectrDemodulatedSignal=(1/NFFT).*fft(SignalDemodulated, NFFT); Plot_SpectrDemodulatedSignal=abs(SpectrDemodulatedSignal(1:NFFT));
%`````````````````````````````````````````````````````````````````````````
%```````````````````````` END OF PROGRAMM'S PART `````````````````````````
%`````````````````````````````````````````````````````````````````````````

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Plot ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%Intializing of label variables:
    TimeAxe='Time, [s]';
    FreqAxe='Frequency, [Hz]';

%   TESTING PART (All plots are separated from each other)

      figure(1)
      %~~ SIGNALS ~~
          %Plot the initial signal:
              subplot(4,2,1);
                  plot(t, Signal);
              xlabel(TimeAxe);
              title('Initial signal');
          %Plot the modulated signal
              subplot(4,2,3);
                  plot(t,SignalModulated);
              xlabel(TimeAxe);
              title('Modulated signal');
          %Plot the modulated signal with a Doppler Effect
              subplot(4,2,5);
                  plot(t,SignalModulated_D);
              xlabel(TimeAxe);
              title('Modulated signal with a Doppler Effect');
          %Plot the demodulated signal with a Doppler Effect
              subplot(4,2,7);
                  plot(t,SignalDemodulated);
              xlabel(TimeAxe);
              title('Demodulated signal');
      %`````````````
      %~~ SPECTRS ~~       
          %Plot the SPECTR of the initial signal
              subplot(4,2,2);
                  plot(FrequencyAxe,Plot_SpectrInitialSignal);
              xlabel(FreqAxe);
              title('Spectr of the initial signal');
          %Plot the SPECTR of the modulated signal:
              subplot(4,2,4);
                  plot(FrequencyAxe,Plot_SpectrModulatedSignal);
              xlabel(FreqAxe);
              title('Spectr of the modulated signal');
          %Plot SPECTR of the modulated signal with Doppler Effect
              subplot(4,2,6);
                  plot(FrequencyAxe,Plot_SpectrModulatedSignal_D);
              xlabel(FreqAxe);
              title('SPECTR of the modulated signal \bf with Doppler Effect');
          %Plot SPECTR of the demodulated signal:
              subplot(4,2,8);
                  plot(FrequencyAxe,Plot_SpectrDemodulatedSignal);
              xlabel(FreqAxe);
              title('Spectr of the demodulated signal');
      %`````````````

      %GRAPHICS
      %~~ Signals ~~
      figure(2);
      %Plot the initial signal and demodulated signal:
      subplot(2,1,1);
      hold on;
          %Plot the initial signal:
           plot(t,Signal,'m');
          %Plot the recieved signal:
           plot(t,SignalDemodulated, 'r');
      hold off;
      xlabel(TimeAxe);
      legend('Initial signal','Demodulated signal');
      title('Initial signal and demodulated signal');
      %Plot the modulated signal and the modulated signal with the Doppler Effect:
      subplot(2,1,2);
      hold on;
          %Plot the modulated signal:
           plot(t,SignalModulated, 'm');
          %Plot the modulated signal with a Doppler Effect:
           plot(t,SignalModulated_D, 'r');
      xlabel(TimeAxe);
      legend('Modulated signal','Modulated signal \bfwith Doppler Effect');
      title('Modulated signal with/without Doppler Effect');
  %~~ SPECTRS ~~
  figure(3);
      hold on;
          %Plot the SPECTR of initial signal
           plot(FrequencyAxe,Plot_SpectrInitialSignal, 'm');
          %Plot the SPECTR of the recieved signal
           plot(FrequencyAxe,Plot_SpectrDemodulatedSignal, 'r');
      hold off;
      xlabel(TimeAxe);
      legend('Spectr of the initial signal','Spectr of the demodulated signal');
      title('Spectrs of the initial signal and demodulated signal');

Answer 1

[Youssef Khmou]

hi Alex,

Your question is acceptable, you expect to visualize the Demodulated signal as the original, but the Doppler shift acts also as modulation, try this small example :

Fs=20;
f=5;
t=0:1/Fs:5;
y=sin(2*pi*t*f);
FF=fft(y);
FF2=circshift(FF,[0 10]);
y2=ifft(FF2);
subplot(1,2,1), plot(t,y), 
title(' Original Signal');
subplot(122), plot(t,y2);
title(' Recovered Signal after shifting the Spectrum +> with 10 steps');

Then its alright :

Your method for Modulating /Demodulating is correct, but you can do the same with fmmod and fmdemod functions .

Last thing, there is a small error in the last graph , 3rd figure , the xlabel is FreqAxe

I hope this helps

[Alex]

Thank you.

Answer 2

[Masoud Ghanbari]

you know what... Too Long Buddy