Steve on Image Processing and MATLAB

Concepts, algorithms & MATLAB

Aliasing and the discrete-time Fourier transform 9

Posted by Steve Eddins,

Many of you were onto me immediately last week. I asked you to estimate the frequency of the sampled cosine signal below, and readers quickly chimed in to guess that this question was really a teaser about aliasing.

As it happened, I started with a 1 rad/second cosine, , :

t = linspace(-2.3*pi, 2.3*pi, 800);
alpha = 1;
x_c = cos(alpha * t);
plot(t, x_c, 'k')

And I sampled that signal to get with so that .

T = 1;
n = -7:7;
nt = n * T;
x = cos(alpha*n*T);
hold on
plot(nt, x, 'ok')
hold off

But I could have started with a completely different frequency and still ended up with exactly the same samples. For example, let's see what happens when we try :

alpha_2 = 2*pi - 1;
x2_c = cos(alpha_2 * t);
plot(t, x2_c, 'k')

x2 = cos(alpha_2 * n * T);
hold on
plot(nt, x2, 'ok')
hold off

The two sets of samples are the same (within floating-point round-off error):

max(abs(x - x2))
ans =


Let's look at this phenomenon in terms of the relationship between the continuous-time Fourier transform and the discrete-time Fourier transform (DTFT). Below is the continuous-time Fourier transform of .

The DTFT is a collection of copies of the continuous-time Fourier transform, spaced apart by the sampling frequency, and with the frequency axis scaled so that the sampling frequency becomes . Here's the DTFT of .

Now has a higher frequency as you can see in the plot of its continuous-time Fourier transform:

But when you make a bunch of copies of spaced apart by the sampling frequency, you find that the DTFT of is exactly the same as the DTFT of .

Usually only a single period of the DTFT is plotted:

In other words, when you use a sampling rate of , the frequencies 1 and are indistinguishable. This is called aliasing. In general, the continuous-time frequency is indistinguishable from any other frequency of the form , where is an integer.

So far we've talked about the continuous-time Fourier transform, the discrete-time Fourier transform, their relationship, and a little bit about aliasing. Next time we'll bring the discrete Fourier transform (DFT) into the discussion. That's what the MATLAB function fft actually computes.

Get the MATLAB code

Published with MATLAB® 7.9

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