I have seen this thread: Difference between Fourier transform and Wavelets

AFAIK there is no common usage of wavelets in the real-time DSP world (excluding image and video processing).

I am curious why is it so?

Where on the other hand Fourier analysis is widely common and all the transmitter/receiver block diagrams that I have seen contains an FFT module in it:

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Is it related to runtime complexity issues? I guess not, since Wavelets transform can be formalized as convolution, right?

e.g aren't wavelets a good candidate to replace the Short-time Fourier transform method ?

  • $\begingroup$ Fairly commonly used in image processing - compression for example. $\endgroup$ – Mike Miller Aug 24 '15 at 20:22
  • $\begingroup$ I mean the traditional DSP, like all the IEEE specifications. $\endgroup$ – 0x90 Aug 24 '15 at 20:34
  • $\begingroup$ Wavelets are commonly used in noise removal and data compression for subsurface imaging. But you're right that Fourier analysis is more common, due to the additional property that $\partial_{t}^{2} \mapsto -\omega^{2}$ under the Fourier transform. So it both makes PDEs simpler, and it can remove noise via filtering. $\endgroup$ – user14717 Aug 24 '15 at 20:48
  • $\begingroup$ I'd disagree with your premise. $\endgroup$ – AnonSubmitter85 Aug 24 '15 at 21:01
  • $\begingroup$ I want to expose my ignorance about this topic, so please allow me to make some suggestions (they are probably way off). 1) The little what I've seen about wavelets (in say image compression) suggests to me that they cover a frequency band from zero to some maximum. Can we move them to a fixed band of known location in the RF spectrum? The RF-regulatory authorities only allocate you fixed chunks of BW. 2) It is "easy" to coerce an antenna to transmit a linear combination of cosine waves (ignoring the envelope limiting the signal to a time window), can we do that with wavelet waveforms? $\endgroup$ – Jyrki Lahtonen Aug 24 '15 at 21:20

One key difference between the DFT and discrete wavelet transform (DWT) is that the DFT deals with frequency whereas the wavelet transform deals with "scales". (e.g. you estimate the width of a pulse or a series of pulses in a time series).

A radio signal is more or less a continuous signal when observed locally. A constant waveform signal is very sparse when decoded by FFT. So it is very easy and natural to represent an RF signal using the DFT/frequency domain.

One of the common issues one have to deal with in regards to the DFT is spectral leakage. Unfortunately, the wavelet kernels generally have a good amount of spectral leakage also so this issue cannot be improved by using the DWT.

The WT does have some useful properties. Aside from the examples you listed, there are some applications in time series analysis where the WT trumps the DFT. For example, detecting drift of atomic clocks or measuring changes in tides (see Wavelet Methods, by Percival and Walden). Also, a more limited application: the WT can be used for denoising a signal when you want to preserve the spikes as well as the baseline drift (e.g. to visually make a spectrum look cleaner).

In terms of time-frequency analysis, the Gabor wavelet is probably the closest you get to STFT.

Like the DFT/FFT the DWT is an orthogonal transform. However, the continuous wavelet transform (CWT), which is computed using convolution, is not. It's useful as an analysis tool only.

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