{"id":315,"date":"2010-02-22T19:58:32","date_gmt":"2010-02-23T00:58:32","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/2010\/02\/22\/aliasing-and-the-discrete-time-fourier-transform\/"},"modified":"2019-10-29T13:21:29","modified_gmt":"2019-10-29T17:21:29","slug":"aliasing-and-the-discrete-time-fourier-transform","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2010\/02\/22\/aliasing-and-the-discrete-time-fourier-transform\/","title":{"rendered":"Aliasing and the discrete-time Fourier transform"},"content":{"rendered":"
\r\n

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

<\/p>\r\n

As it happened, I started with a 1 rad\/second cosine, , :\r\n <\/p>

t = linspace(-2.3*pi, 2.3*pi, 800);\r\nalpha = 1;\r\nx_c = cos(alpha * t);\r\nplot(t, x_c, 'k'<\/span>)<\/pre> 
And I sampled that signal to get   with   so that  .\r\n   <\/p>
T = 1;\r\nn = -7:7;\r\nnt = n * T;\r\nx = cos(alpha*n*T);\r\nhold on<\/span>\r\nplot(nt, x, 'ok'<\/span>)\r\nhold off<\/span><\/pre> 
But I could have started with a completely different frequency and still ended up with exactly the same samples<\/b>.  For example, let's see what happens when we try  :\r\n   <\/p>
alpha_2 = 2*pi - 1;\r\nx2_c = cos(alpha_2 * t);\r\nplot(t, x2_c, 'k'<\/span>)\r\n\r\nx2 = cos(alpha_2 * n * T);\r\nhold on<\/span>\r\nplot(nt, x2, 'ok'<\/span>)\r\nhold off<\/span><\/pre> 
The two sets of samples are the same (within floating-point round-off error):<\/p>
max(abs(x - x2))<\/pre>
\r\nans =\r\n\r\n  1.1657e-015\r\n\r\n<\/pre>
Let's look at this phenomenon in terms of the relationship between the continuous-time Fourier transform and the discrete-time Fourier transform (DTFT)<\/a>. Below is the continuous-time Fourier transform of  .\r\n   <\/p>\r\n   
 <\/p>\r\n   
The DTFT is a collection of copies of the continuous-time Fourier transform, spaced apart by the sampling frequency, and with\r\n      the frequency axis scaled so that the sampling frequency becomes  . Here's the DTFT of  .\r\n   <\/p>\r\n   
 <\/p>\r\n   
Now   has a higher frequency as you can see in the plot of its continuous-time Fourier transform:\r\n   <\/p>\r\n   
 <\/p>\r\n   
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  .\r\n   <\/p>\r\n   
 <\/p>\r\n   
Usually only a single period of the DTFT is plotted:<\/p>\r\n   
 <\/p>\r\n   
In other words, when you use a sampling rate of  , the frequencies 1 and   are indistinguishable.  This is called aliasing<\/i>.  In general, the continuous-time frequency   is indistinguishable from any other frequency of the form  , where   is an integer.\r\n   <\/p>\r\n   
So far<\/a> we've talked about the continuous-time Fourier transform, the discrete-time Fourier transform, their relationship, and a\r\n      little bit about aliasing.  Next time we'll bring the discrete Fourier transform (DFT) into the discussion.  That's what the\r\n      MATLAB function fft<\/tt> actually computes.\r\n   <\/p>