About FFT of cosine function

Hi, I have to calculate DFT of g(t)=cos(0.5*t). So I write the code below

>> N=128; %number of sampled time points

>> t=linspace(0,128,N); %time domain

>> Fs=1/(t(2)-t(1)); %sampling frequency

>> Fn=Fs/2; %Nyquist Frequency

>> g=cos(0.5*t); %input signal

>> G=fftshift(fft(g)/N); %Fourier transform of g

>> w=linspace(-Fn,Fn,N)*2*pi; %angular frequency domain

>> subplot(2,1,1)

>> plot(w,abs(G)); % to see amplitude

>> subplot(2,1,2)

>> plot(w,angle(G)); % to see phase

<The result of above code>

In this situation, my question is

1) In amplitude graph, I expected it has to have formation of two Dirac delta function Aδ(w-0.5)+Aδ(w+0.5), i.e., amplitude shoudn't have values at w=/0.5 and w=/-0.5. But it have values. How can I change the code to make this two delta functions?

2) In amplitude graph, I also expected the coordinates of peaks are (-0.5,0.5) and (0.5,0.5), but it is not. How can I change the code to make peaks at this coordinates??

Thank you..!!

Answer 1

[David Goodmanson]

Hi mk,

In order to get just the two sharp peaks you need to have exactly n oscillations in the time domain. Otherwise the wave is cut off partway through an oscillation and does not represent a continuous oscillatory wave.

The code below uses a total time of 4*pi so there are 8 oscillations in the time record. (Also, cos = 1 at t = 0, and exactly n oscillations means that there can't be a repeated point cos = 1 at the end of the time record).

w0 = .5;
T = 4*pi;                % total time
N = 1000;
t = linspace(0,T,N+1);   % N+1 points, N intervals
t(end) = [];             % eliminate repeated point at the end
y = cos(w0*t);
z = fftshift(fft(y)/N);
% fft golden rule: delta_t*delta_w = 2*pi/N
% delta_w = 2*pi/(N*delta_t) = 2*pi/T
delw = 2*pi/T;
w = (-N/2:N/2-1)*delw;
r = [real(z);imag(z)];
stem(w,[real(z);imag(z)]')
xlim([-10 10])

[MK kim]

Thank you very much. I understand your answer..!!

[MK kim]

I have question..

I want to plot phase graph, so add

stem(w,angle(z))

but the result missed my expection. The peaks are w=0, 1.5,... but I think the graph has to have peaks at only w=0.5,-0.5. What should I do?

[David Goodmanson]

Hello mk,

There is nothing wrong here. The only two nonzero values in the spectrum occur at w = +-(1/2), with amplitude 1//2 and phase 0 (the amplitudes are real). If you look at the stem plot, the phase is zero at w = +-(1/2) as it should be.

For all other values of w, the amplitude is supposed to be zero, but because of usual numercal precision issues the values are actually down around 1e-16 for both the real and imaginary parts. The phase of such a calculaton varies randomly and is meaningless, so the phase plot is actually ok.

If you use sin instead of cos, then the phase at w = +-(1/2) is -+(pi/2) as you can check..

[MK kim]

Thank you for your help..!!