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We already have DTFT so why do we need DFT?

Please don't write DTFT is used for discrete, aperiodic signal which transfers in the frequency domain to periodic and continuous, but in DFT is discrete periodic transfer in discrete periodic, these things I already know.

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    $\begingroup$ Because we have efficient algorithms to compute the DFT, and the DFT can be used to approximate the DTFT and the CTFT. $\endgroup$
    – Matt L.
    Dec 13, 2020 at 16:17
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    $\begingroup$ To help answers keeping focused: why are you suggesting that the DFT could be "not useful enough" ? $\endgroup$
    – Laurent Duval
    Dec 13, 2020 at 16:42

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The answer is the same to the question: "Why do we need computers to process data when we have paper and pencil?"

DTFT as well as the continuous-time Fourier Transform is a theoretical tool for infinitely long hypothetical signals.

the DFT is to observe the spectrum of actual data that is finite in size.

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    $\begingroup$ Don't forget the rulers and compasses! $\endgroup$
    – Laurent Duval
    Dec 13, 2020 at 21:45
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    $\begingroup$ So paper and pencil is a theoretical tool for infinitely long hypothetical signals? $\endgroup$
    – user253751
    Dec 14, 2020 at 15:51
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    $\begingroup$ @user253751 sure, write out the analytical form of the signal and then work out whatever algebra/calculus you need by hand. $\endgroup$
    – hobbs
    Dec 14, 2020 at 17:01
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TL, DR: world pervasive algorithms (FFT-related)!

The continuous Fourier transform, the Discrete-time Fourier transform (DTFT) and the Discrete Fourier transform (DFT) share conceptually similar traits (regarding energy, convolution, shift, scale, etc.)

The DFT unveiled a very scalable and fast algorithm to put those concepts into practice: the FFT. It is considered by many to be one of the most important algorithms, and is in the lists below:

Of course, Google PageRank, the Euclidian division, Quicksort, Kalman filtering are used daily. But if we include digital data compression, for instance: MP3 Compression/The 10 Algorithms That Dominate Our World, or JPEG decoding, as while not directly DFTs, but very close avatars which might not have been discovered without the DFT, I am pretty sure that billions of DFT-inspired algorithms are run every hour. In 2012 (source: A Faster Fast Fourier Transform: New algorithm crunches sparse data with speed), Richard Baraniuk was already credited with "FFTs are run billions of times a day", and Gilbert Strang:

once referred to the fast Fourier transform, or FFT, as “the most important numerical algorithm in our lifetime"

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  • $\begingroup$ How many quicksorts run per hour? $\endgroup$
    – user253751
    Dec 15, 2020 at 19:00
  • $\begingroup$ People used tables by ID so often... I could not find numbers $\endgroup$
    – Laurent Duval
    Dec 15, 2020 at 20:28
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To overcome the limitation of DTFT we use DFT tool. it is almost similar to DTFT but only limitation is the infinite length sequence.

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