Why discrete fourier transform is used

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Using the DFT, we can compose the above signal to a series of sinusoids and each of them will have a different frequency. The following 3D figure shows the idea behind the DFT, that the above signal is actually the results of the sum of 3 different sine waves.

The time domain signal, which is the above signal we saw can be transformed into a figure in the frequency domain called DFT amplitude spectrum, where the signal frequencies are showing as vertical bars. The height of the bar after normalization is the amplitude of the signal in the time domain. You can see that the 3 vertical bars are corresponding the 3 frequencies of the sine wave, which are also plotted in the figure. The DFT can transform a sequence of evenly spaced signal to the information about the frequency of all the sine waves that needed to sum to the time domain signal.

It is defined as:. Therefore, usually we only plot the DFT corresponding to the positive frequencies. The amplitude and phase of the signal can be calculated as:. The amplitudes returned by DFT equal to the amplitudes of the signals fed into the DFT if we normalize it by the number of sample points. TRY IT! This is a direct consequence of the DFT output being discrete. Show 4 more comments. That is a point I always forget.

Has anyone ever performed an infinite-length DFT? From what I remember say, from reading george silov's dover book, if you make the number of fourier coefficients large enough by using a fine enough grid of frequencies, then the fourier series can reproduce a period continuous function arbitrarily closely.

Bob K Bob K 8 8 bronze badges. Add a comment. Is that output a sequence of numbers or a continuous function an equation? Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog. Podcast Explaining the semiconductor shortage, and how it might end.

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