Spectrum of fm modulaton

Is there a way to plot the spectrum of an fm modulated signal. I know that Fourier transform is not directly applicable for fm signals as they are not linear. So is there a direct method to do that [even for a tone modulation at least]?

Answer 1

[David Goodmanson]

Hello AbdAlla, The generation of a real fm signal is not a linear process, but there is no problem with finding the resulting spectrum by fft since the fft can find the frequency components of any signal. The fft works out best with modulation by a tone or the sum of a small number of tones, AND when the carrier and all the tones have an exact number of cycles in the time array.

(revised) Here is a small example:

F = 1e6;                 % sampling frequency
f0 = 20000;              % carrier
f1 = 200;                % modulating tone
beta = 1;                % modulation index  

% signal, spectrum
N = 1e6;
t = (0:N-1)*(1/F);       % delt = 1us
y = cos(2*pi*f0*t+beta*cos(2*pi*f1*t));
z = fftshift(fft(y))/N;
f = (-N/2:N/2-1)*(F/N);

% predicted sidebands for the positive frequencies only
n = 4;
sidamp = (1/2)*besselj(0:n,beta);
sidamp = [fliplr(sidamp(2:end)), sidamp];
sidf = (f0-n*f1):f1:(f0+n*f1);

figure(1)
plot (f,abs(z))
xlim([-2*f0 2*f0])
figure(2)
plot(f,abs(z),sidf,sidamp,'o')
xlim([f0-f1*10 f0+f1*10])            % sidebands at the pos frequencies

Fig 1 shows positive and negative frequencies. Most of the time people double the abs(amplitudes) and show positive frequencies only, in which case the sideband calculation does not have the factor of 1/2.

[AbdAlla Hefny]

Hi David. Thanks for your reply. I actually tried the fft but it did not work fine. I have only one tone. According to what I studied, the spectrum of such an fm modulated signal should have a component at the carrier and only (beta+1) significant side bands WITH DECAYING POWER. But what I can see as the output of the fft is: equal components at each single frequency (1 Hz, 2 Hz, ...........).

[David Goodmanson]

Hi AbdAlla, see the revised answer above

[venu dunde]

clear all;

clc;

close all;

Fs_fm = 200e3; %%%%%% sampling rate of FM signal bw = 100e3; %%% signal bandwidth

F_carrier = 98e6; %%%%% FM carrier

fc = 98e6; % Operating frequency fs = Fs_fm; % Waveform

[y_song,Fs_audio] = audioread('song.mp3');

%Fs_audio = 44100;

%y_fm = resample(y_song,Fs_fm,Fs_audio);

%%%%%%%%%%%%%%%%% %%%% time of simulations , sec %%%%%%%%%%%%%%%%%%

deltaT =40;

%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% creating FM stereo signal %%%%%%%%%%%%%%%%%%%%%%%%%%%

audio_signal = y_song(1:deltaT*Fs_audio,:);

%rc = rcosine(Fs_audio,16*Fs_audio);

%audio_signal = filter(rc,1,randn(deltaT*Fs_audio,2));

audio_signal = resample(audio_signal, Fs_fm, Fs_audio);

t_vect = [0:length(audio_signal)-1]'/Fs_fm;

sum_ch= 0.5 *(audio_signal(:,1)+audio_signal(:,2));

diff_ch = 0.5 *(audio_signal(:,1)-audio_signal(:,2));

stereophonic_signal = sum_ch + 0.1*cos(2*pi*19e3*t_vect) + diff_ch.*cos(2*pi*38e3*t_vect);

CE_FM_radio_signal = exp(1j*2*pi*75e3*cumsum(stereophonic_signal/Fs_fm));

s_wav = CE_FM_radio_signal( 8.45e4:end);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% Source characterization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

figure(1)

plot(fft(s_wav),Fs_fm,'b');

title('FM signal spectral density');

xlabel('Frequency KHz');

ylabel('Magnitude dB'); grid

figure(2)

fm_acor = abs(xcorr(s_wav));

plot(10*log10(fm_acor));

title('Autocorrelation Function');

xlabel('Autocorrelation Lags');

ylabel('magnitude');

grid

[venu dunde]

plot the frequency spectrum of FM signal ( which is almost real-time signal), i am facing problem at figure(1) plot(s_wav,Fs_fm,'b');title('FM signal spectral density');xlabel('Frequency KHz');ylabel('Magnitude dB');grid, figure-1

update the file (use any short .mp3 file, here i am unable to attach the .mp3 file) thank you,

Answer 2

[santy vega]

hello,

excuse me and if I want to get the fm bandwitch of this signal,How could I do it? thanks