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On Wikipedia I read this : "The Discrete Wavelet Transform, often used in modern signal processing, is time variant because it makes use of the decimation operation."

Why does decimation makes system time variant?

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    $\begingroup$ On the Wikipedia talk page someone complains about calling the discrete wavelet transform (DWT) a system. Its output is in a different domain than its input, so is it strictly a system? At least it's not a good example of one. $\endgroup$ – Olli Niemitalo Dec 30 '18 at 10:58
  • $\begingroup$ That link doesnt work. $\endgroup$ – Sweeper Dec 30 '18 at 10:59
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    $\begingroup$ try again please. $\endgroup$ – Olli Niemitalo Dec 30 '18 at 11:00
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HINT: Think about the following situation: you have an input signal with alternating ones and zeros, and you decimate by a factor of $2$. Now shift the input signal by one sample and compute a new output signal by decimating by a factor of $2$. Is the output signal a shifted version of the previous output signal? If the system is time-invariant, shifting the input signal causes the output signal to shift by the same amount.

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  • $\begingroup$ I dont understand. $\endgroup$ – Sweeper Dec 30 '18 at 10:58
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    $\begingroup$ @Sweeper: Imagine the signal I gave as an example. If you take, say, all even samples you end up with only ones at the output. If you take all odd samples you end up with all zeros at the output. So shifting the input by 1 sample totally changes the output, hence the system cannot be time-invariant. $\endgroup$ – Matt L. Dec 30 '18 at 10:59
  • $\begingroup$ Ah yes, I get it now. My problem was that I thought decimation of 1 - 0 - 1 - 0 signal would give 0.5 - 0.5 signal. $\endgroup$ – Sweeper Dec 30 '18 at 11:01
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    $\begingroup$ Depending on the source, decimation or downsampling may include a prior filtering step, see for instance dsp.stackexchange.com/questions/45276/… $\endgroup$ – Laurent Duval Dec 31 '18 at 13:09
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The continuous wavelet transform is aimed at being shift-invariant (or equivariant, if you do care about terms, see Difference between “equivariant to translation” and “invariant to translation”). While some redundant versions of their discrete implementations can be made invariant (stationary, undecimated or shift/time-invariant discrete wavelets, see Shift invariant in wavelet), the classical critically sampled discrete wavelet is not.

For an $M$-band (classically $M=2$) discrete wavelet transform, over $L$ levels, the transform is equivariant to multiples of $M^L$ shifts only, and generally not in-between, unless the deterministic signal, or the stochastic noise properties, allows for more.

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Decimation as part of the calculations does not necessarily lead to a time-invariant (shift-invariant) system. Consider the system: DWT → IDWT, where IDWT stands for inverse discrete wavelet transform. Typically, a DWT together with its inverse transform incorporates aliasing cancellation (usually with quadrature mirror filters) that enables to reconstruct the original signal even when decimation is used in DWT, making the full system shift-invariant.

However, typically the transformed signal would be somehow processed before IDWT, which might break the aliasing cancellation and lead to shift-invariance as in user Matt L's example.

Instead of aliasing cancellation, it would also work to have perfect anti-alias filters, but this is typically not practical, because of the infinite length required of the impulse responses of such filters, or because they would introduce Gibbs phenomenon ringing.

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