FFT of a Simple Sinusoid

Our first example is an FFT of the simple sinusoid

$\displaystyle x(n) = \cos(\omega_x n T)$

where we choose $\omega_x=2\pi(f_s/4)$ (frequency

Hz) and

(sampling rate

set to 1). Since we're using a Cooley-Tukey FFT, the signal length

should be a power of

for fastest results. Here is the Matlab code:

% Example 1: FFT of a DFT-sinusoid

% Parameters:
N = 64;              % Must be a power of two
T = 1;               % Set sampling rate to 1
A = 1;               % Sinusoidal amplitude
phi = 0;             % Sinusoidal phase
f = 0.25;            % Frequency (cycles/sample)
n = [0:N-1];         % Discrete time axis
x = A*cos(2*pi*n*f*T+phi); % Sampled sinusoid
X = fft(x);          % Spectrum

% 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
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)');
cmd = ['print -deps ', '../eps/example1.eps'];
disp(cmd); eval(cmd);

**Figure 8.1:** Sampled sinusoid at frequency . a) Time waveform. b) Magnitude spectrum. c) DB magnitude spectrum.
$\includegraphics[width=\textwidth]{eps/example1}$

The results are shown in Fig.8.1. The time-domain signal is shown in the upper plot (Fig.8.1a), both in pseudo-continuous and sampled form. In the middle plot (Fig.8.1b), we see two peaks in the magnitude spectrum, each at magnitude on a linear scale, located at normalized frequencies and . A spectral peak amplitude of is what we expect, since

$\displaystyle \hbox{\sc DFT}_k(\cos(\omega_x n)) \isdef \sum_{n=0}^{N-1} \frac{e^{j\omega_x n} + e^{-j\omega_x n}}{2} e^{-j\omega_k n},$

and when $\omega_k=\pm\omega_x$ , this reduces to

$\displaystyle \sum_{n=0}^{N-1}\frac{e^{j 0 n}}{2} = \frac{N}{2}.$

For

and $\omega_x=2\pi f_s/4$ , this happens at bin numbers

and

. However, recall that array indexes in matlab start at

, so that these peaks will really show up at indexes

and

in the magX array.

The spectrum should be exactly zero at the other bin numbers. How accurately this happens can be seen by looking on a dB scale, as shown in Fig.8.1c. We see that the spectral magnitude in the other bins is on the order of dB lower, which is close enough to zero for audio work $(\stackrel{\mbox{.\,.}}{\smile})$ .

Order a Hardcopy of Mathematics of the DFT

Previous: Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and FFT
Next: FFT of a Not-So-Simple Sinusoid

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Mathematics of the DFT
    Example Applications of the DFT
       Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and FFT
          FFT of a Simple Sinusoid