Is The Frequency Domain a Real Place?

When analyzing data, one can use a variety of transformations on the data to massage it into a form that works better to tease out the information one is interested in. One such example is the application of the Fourier transform, which transforms a data set from the time domain into the frequency domain. Yet what is this frequency domain really? After enticing us to follow the white rabbit down a sudden plummet into the intangible question of what is and what is not, [lcamtuf] shows us around aspects of the frequency domain and kin.

One thing about the (discrete) Fourier transform is that it is excellent at analyzing data that consists out of sinewaves, such as audio signals. Yet when using the Fourier transform for square waves, the resulting output is less than useful, almost as if square waves are not real. Similarly, other transforms exist which work great for square waves, but turn everything else into meaningless harmonics. Starting with the discrete cosine transform (DCT), this gets us into Walsh and Hadamard matrices and the Walsh-Hadamard Transform (WHT), and their usage with transforming data from the time into the frequency domain.

Ultimately it would seem that the frequency domain is as real as it needs to be, albeit that its appearance is wholly dependent on the algorithm used to create it, whether this the DFT, DCT, WHT or something else entirely.







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