Why is the Fourier (or cosine) transform decorrelating?

Question

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?

Answer 1

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.