# Frequency spectrum of signal - Matlab

Here is the code I use to plot a function in frequency domain in Matlab:

    dt = 1/10000; % sampling rate
et = 0.1; % end of the interval
t = 0:dt:et; % sampling range
y = 2+sin(2.*pi.*50.*t)+18.*sin(2.*pi.*90.*t)+6.*sin(2.*pi.*180.*t); %     sample the signal
subplot(2,1,1); % first of two plots
plot(t,y); grid on % plot with grid
xlabel('Time (s)'); % time expressed in seconds
ylabel('Amplitude'); % amplitude as function of time
Y = fft(y); % compute Fourier transform
n = size(y,2)/2; % 2nd half are complex conjugates
amp_spec = abs(Y)/n; % absolute value and normalize
subplot(2,1,2); % second of two plots
freq = (0:100)/(2*n*dt); % abscissa viewing window
stem(freq,amp_spec(1:101)); grid on % plot amplitude spectrum
xlabel('Frequency (Hz)'); % 1 Herz = number of cycles/second
ylabel('Amplitude'); % amplitude as function of frequency


The problem is, when I zoom in my graph I don't see peaks exactly on 50Hz, 90Hz and 180Hz.

What did I do wrong in my code?

## 2 Answers

Discrete Fourier transform doesn't actually decompose a signal into sum of sinusoids it was composed from. DFT projects signal onto a specific set of discrete sinusoids known as Fourier basis. Sinusoids used to compose the signal may or may not coincide with the DFT basis. If they are not, they are being decomposed into a number of DFT basis elements.

There's another way of how to look at it. DFT deals with periodic functions. If you plot 3-4 periods of your signal interval, you'll see it doesn't look like a sum of pure sinusoids and has discontinuities - if sinusoids it was composed from don't coincide with a Fourier basis.

• I think you mean specifically DFT X[k] of x[n]. Because otherwise DTFT $X(e^{j\omega})$ would compute whatever frequency there is in the discrete signal in exactness, due to its continuous frequency variable. – Fat32 Apr 8 '15 at 1:01
• Yes, indeed. Corrected. – Yuri Nenakhov Apr 8 '15 at 1:58

Well it does in Octave, (obtained by pasting your code, unmodified into a script file):