FFT of a simple sinusoid

The FFT, or ``fast fourier transform'' is an efficient implementation of the discrete fourer transform if the signal length is a power of two. In this example, we will use Matlab to take the FFT. Recall our simple discrete sinusoid is

$$x(t) = \cos(\omega_0 nT + \phi).$$ (1)

which we may implement in Matlab as

N = 64;				% signal length (power of 2)
T = 1;  			% sampling period (and rate) is set to 1
A = 1;				% sinusoid amplitude
phi = 0;			% phase of zero
f = 0.25;			% frequency (under Nyquist limit)
nT = [0:N-1]*T;			% discrete time axis
x = cos(2*pi*f*nT + phi);	% sinusoid
X = fft(x);

In the last line, we use Matlab's fft function to obtain the spectrum of the sinusoid. Since X is complex, we do no usually plot it as is. Rather, to obtain a more meaningful graph, we first obtain the magnitude before plotting. Recall that the magnitude of a complex number $z = x + jy$ is given by

$$z = \vert z\vert = \sqrt{x^2 + y^2}.$$ (2)

In matlab we can use the abs function to obtain the abolute value of the spectrum. Therefore if we add the line

magX = abs(X);

to our code above we will have a sequence of real numbers representing the magnitude of the frequency components.

Likewise, we may obtain the phase using Matlab's angle function:

argX = angle(X);

Alternatively, we could use the following:

angleX = atan2(imag(X), real(X));

which implements the fact that the angle is given by

$$\angle z = \tan^{-1}\left(\frac{\mbox{Im}\left\{X\right\}}{\mbox{Re}\left\{X\right\}}\right).$$

Let's now make 3 plots: one to plot the time-domain evolution, one to plot the magnitude of the fft on a linear scale, and finally one to plot the magnitude on a dB scale.

% Plot time data:
figure(1);
subplot(3,1,1);
plot(n,x,'*k');        
ni = [0:.1:N-1];     % Interpolated time axis
hold on; 
plot(ni,A*cos(2*pi*ni*f*T+phi),'-k'); grid off;
title('Sinusoid at 1/4 the Sampling Rate');
xlabel('Time (samples)'); 
ylabel('Amplitude');
text(-8,1,'a)');
hold off; 

% Plot spectral magnitude:
magX = abs(X);
fn = [0:1/N:1-1/N];  % Normalized frequency axis
subplot(3,1,2);
stem(fn,magX,'ok'); grid on;
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (Linear)');
text(-.11,40,'b)');

% Same thing on a dB scale:
spec = 20*log10(magX);        % Spectral magnitude in dB
spec = spec - max(spec);      % Normalize to 0 db max
subplot(3,1,3);
plot(fn,spec,'--ok'); grid on;
axis([0 1 -350 50]);
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (dB)');
text(-.11,50,'c)');