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The discrete Fourier transform (DFT) and the discrete cosine transform (DCT) both decompose a signal into its frequency-domain spectrum. One property that I have seen praised across various domains such as image processing, audio/speech processing and more, is that it tends to produce decorrelated coefficients. For example, subsequent coefficients of a signal might tend to be statistically correlated, but its Fourier- or cosine-transformed counterpart does not exhibit such behaviour. As a result, the statistical properties of the latter (e.g. covariance matrix) can be modeled much more efficiently using diagonal matrices instead of dense matrices, among other benefits.

However, I do not immediately see why this decorrelation should hold true in general. What are the reasons behind this fact? Where is the connection between the DFT or DCT's frequency-decomposing property and the described decorrelating property?

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    $\begingroup$ Can you please provide a few examples or references to support your claim? I've never heard of DFT/DCT being describe as "decorrelators" and I've done a ton of audio processing. $\endgroup$ – Hilmar Jun 11 at 12:56
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Whether that decorrelation happens depends on the signals you put in – it's not a general property of the DFT!

Especially, when you model your signal as sum of narrowband signals, then you'll find that due to the DFT of a narrowband signal being concentrated on one or very few bins, and practically 0 elsewhere, the decorrelation simply stems from the fact that it's easy to represent the same signal in frequency domain with fewer non-zero coefficients than in time domain.

The fact that these non-zero coefficients are then uncorrelated amongst each other depends again on the signal model, and isn't widely true – for example, if you have a line spectrum DFT, for example from observing a rectangular wave digital signal, the lines aren't uncorrelated at all.

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the DFT bins for stationary noise processes tend toward independence as the separation increases. A white noise process is often simulated by using independent bin values.

The Toeplitz structure of covariance matrices comes from assuming stationary.

One might use the assumption of short term stationarity for noise like signals or low SNR signals in nominally stationary noise but the DFT is just a transformation. Whitening decorrelates.

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