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One of the books on "Conceptual Wavelets" by Fugal explains some major differences between the undecimated discrete wavelet transform (UDWT) vs. discrete wavelet transform (DWT). In UDWT the scale of wavelet is increased continuously just like the continuous wavelet transform, but the scale increases in dyads (powers of 2). In the DWT, which is the most commonly used in MATLAB, the filter size remains the same, but the data is reduced dyadically.

His exact wordings are "As discussed briefly in the preview, instead of dyadically stretching the filters, the conventional (decimated) DWT dyadically shrinks the signal instead"

He is using the example of Haar wavelets on a very small set of data, simply exam scores as a signal of eight exam scores [ 80 80 80 80 0 0 0 0].

So the question is when we downsample by 2, which one is more correct to throw away even samples or odd samples?

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Assuming that you have a sufficient number of samples, throwing away odd / even samples does not matter.

The DWT can be thought of as measuring time/frequency content with varying levels of time/frequency resolution. For non frequency varying signals (like chirps), the odd and even samples both contain the same frequency and time content. (Unless we are talking about some explosive transient that occurs in one sample) then you would need some a priori knowledge about that.

If the signal is of odd length, taking the odd samples would give you one more sample than the even samples but wouldn't make a difference in terms of information gained.

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  • $\begingroup$ Is this also correct that in DWT, the filter size remains the same and only the data size is reduced? This is analogous to increasing the scale on a fixed data length. $\endgroup$ – M. Farooq Nov 6 at 1:55
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    $\begingroup$ Both are equivalent implementations. Convolution has the effect of "spreading" in time. Doubling the length of the filter while keeping the signal length the same will create the same amount of scaling as downsampling the signal while keeping the number of taps constant. $\endgroup$ – Samuel Nov 6 at 2:11

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