Comments on: Discrete-time Fourier transform (DTFT) https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/?s_tid=feedtopost Retired from MathWorks in 2024 after 30 years of service. Can now be found at MATLAB Central, https://hornjourney.com, and https://matrixvalues.com. MathWorks career included image processing, toolbox development, MATLAB development and design, development team management, MATLAB design standards, Steve on Image Processing blog (https://blogs.mathworks.com/steve). Co-author of Digital Image Processing Using MATLAB (https://www.imageprocessingplace.com/DIPUM-3E/dipum3e_main_page.htm). French horn enthusiast, member of Concord Orchestra and Melrose Symphony, member of the board of Cormont Music and the Kendall Betts Horn Camp. Sun, 21 Mar 2010 22:27:10 +0000 hourly 1 https://wordpress.org/?v=6.2.2 By: Steve https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22891 Sun, 21 Mar 2010 22:27:10 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22891 Ahsan—Are you talking about the formulas? I looked those up in a book. You can also find them in Wikipedia.

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By: Ahsan https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22886 Sat, 20 Mar 2010 06:53:30 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22886 What matlab code computed these DTFTs? Does matlab has functions to compute DTFT?

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By: Matteo https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22589 Thu, 07 Jan 2010 17:15:03 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22589 Thanks for both clarifications Steve, my question was about the latter point, but both help. Cheers!

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By: Steve https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22581 Wed, 06 Jan 2010 16:08:04 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22581 Matteo—Perhaps you were asking why there is a pair of impulses at +/- w0 and not just one at w0? Roughly speaking, the Fourier transform of a signal is an expansion in terms of complex exponential sinusoids, exp(j w n), and cos(w0 n) = (1/2) (exp(j w0 n) + exp(-j w0 n)). That’s why cos(w0 n) shows up as a pair of impulses at +/- w0 in the frequency domain.

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By: Steve https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22580 Wed, 06 Jan 2010 15:59:56 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22580 Matteo—Thanks for reading and for taking the time to comment with your question. I’m relying on feedback from readers to help me adjust both the pacing and the choice of topics in this series.

The pair of impulses near the origin (w=0) correspond to the cosine frequency, +/- w0. (To make these plots I used a frequency of pi/10.) All the other impulses are just from the 2*pi periodicity. Normally a DTFT is plotted showing only one period (-pi to pi), or sometimes a half-period (0 to pi), so you would only see the impulses near the origin.

Your question made me realize I should have included more labels on that particular plot, so I updated the plot to show w0, -w0, and w0+2pi.

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By: Matteo https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22579 Wed, 06 Jan 2010 15:40:25 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22579 Hi Steve,

my math is rusty: I am looking at the plot of DTFT of Cosine and understand the 2Pi peridiocity, but I fail to visualize why I train of 2 pulses around each 2Pi period. Can you elaborate on that a bit more for us snails? Thank you. Matteo

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By: Steve https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22561 Wed, 06 Jan 2010 00:23:38 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22561 Ben—Yes, I’m heading in that direction. I’ll be discussing the relationships between the continuous-time Fourier transform, discrete-time Fourier transform, and discrete Fourier transform.

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By: Ben https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22560 Wed, 06 Jan 2010 00:05:53 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22560 Steve,

I just started reading through your collection of posts on Fourier transforms, thank you for posting these.

Perhaps you will get to this in future posts, but my question for you is the following:

When working with signal processing in MATLAB (a digital environment where all signals are in some way discretized and finite), what is the difference between x(t) = cos(t) and x[n] = cos(w0*n)? In other words, the continuous function x(t) can’t really exist in MATLAB, and neither can the continuous transform X(w). Doesn’t this mean that you are always dealing (in some way) with the DFT when working in MATLAB?

– Ben

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By: Steve https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22556 Mon, 04 Jan 2010 17:42:29 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22556 Thanks, John, I fixed the link.

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By: John A https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22555 Mon, 04 Jan 2010 17:37:58 +0000 https://blogs.mathworks.com/steve/2009/12/31/discrete-time-fourier-transform-dtft/#comment-22555 Steve, the back track link in the first sentence is broken…other than that thanks for keeping the series going in the new year!

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