# Spectrum of fm modulaton

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AbdAlla Hefny on 8 Apr 2017
Answered: santy vega on 28 Aug 2019
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]?

David Goodmanson on 9 Apr 2017
Edited: David Goodmanson on 9 Apr 2017
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.
venu dunde on 26 Mar 2018
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,

### More Answers (1)

santy vega on 28 Aug 2019
hello,
excuse me and if I want to get the fm bandwitch of this signal,How could I do it? thanks