# Fourier Transforms **URL:** \ **Category:** Factual Questions\ **Created:** [March 9, 2004, 11:37am UTC](https://boards.straightdope.com/t/fourier-transforms/233288 "2004-03-09T11:37:01Z")\ **Posts on this page:** 3\ **Page:** 1
**Author:** ![Baa-Baas](https://avatars.discourse-cdn.com/v4/letter/b/4da419/32.png) [@Baa-Baas](https://boards.straightdope.com/u/Baa-Baas)\ **Post date:** [March 9, 2004, 11:37am UTC](https://boards.straightdope.com/t/fourier-transforms/233288/1 "2004-03-09T11:37:01Z")
Can anybody clearly explain the differnce between the Discrete Time Fourier Transform and the Discrete Fourier Transform? Also for what type of input signal should you use each? Cheers ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [March 9, 2004, 12:27pm UTC](https://boards.straightdope.com/t/fourier-transforms/233288/2 "2004-03-09T12:27:48Z")
The Fourier Transform is the way to get from the “time domain” to the “frequency domain” and back again (_Inverse_ Fourier Transform). Suppose your input signal is a simple tone of, say, 440Hz. In the time domain (_eg._ on an oscilliscope), this looks like a single sine wave. If you Fourier transform this signal into the _frequency_ domain and you will see a single vertical spike at f=440. Inverse Fourier transform that back and you recover your original sinewave. Now, The Fourier transform is an _integral_ over all time. Computers can only _approximate_ integrals by merely **summing** the contribution from all available **discrete** data points, hence the [Discrete Fourier Transform](http://ccrma-www.stanford.edu/~jos/mdft/DFT_Definition.html). Now these data points might be limited to a brief period of time. The [Discrete **Time** Fourier Transform](http://ccrma-www.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html) sums over _all_ time (_ie._ N approaches infinity), and so is a function of _continuous_ frequencies w rather than sampled frequencies w[sub]k[/sub]. ---
**Author:** ![crozell](https://avatars.discourse-cdn.com/v4/letter/c/3da27b/32.png) [@crozell](https://boards.straightdope.com/u/crozell)\ **Post date:** [March 9, 2004, 2:54pm UTC](https://boards.straightdope.com/t/fourier-transforms/233288/3 "2004-03-09T14:54:53Z")
Both the Discrete Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) are to be used on signals that are discrete in time (e.g., samples of an analog signal, or naturally discrete signals). This is in opposition to the regular Fourier Transform and the Fourier Series, which are to be used on continuous time signals. If you look at the formula for the DTFT, you will notice that the summation is over all time (an infinite number of points). Also, in the inverse DTFT formula, it is an integral over all frequency. Both of these operations are impossible on a digital computer, and the DFT was developed as something that we actually can calculate on data. The DFT assumes that the signal is some finite length (N), and the summation is only carried out over these points. Since only N points are valuable, we can assume that the signal is periodic with period N and no harm is done. Essentially, the math works out that by doing this, the DFT spectrum is actually a sampled version of the DTFT spectrum. This also makes the inverse DFT a summation instead of an integral. The FFT (which is what you see most often in practice) is equivalent to the DFT. The FFT is simply a clever algorithm for calculating the DFT faster than just computing the standard formula. So, to sum up: DTFT is a theoretical construct for discrete-time signals. It is useful for analytic exercises, but we can’t actually compute it for data. The DFT amounts to sampling the DTFT in the frequency domain, and is calculable because it assumes a finite length signal. The FFT is the commonly used implementation of the DFT. Hope that helps!