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A $2$-channel stage of a wavelet transform, combines two filters in parallel, followed by a down-sampling by two. The later is the cause for shift-invariance, as the filters are time-invariant. Signals $$x_0[n] = \{\ldots,0,1,0,1,0,1,\ldots\}$$ and $$x_1[n] = \{\ldots,1,0,1,0,1,0,\ldots\}$$ which are shifted by only one sample, yield respectively $$y_0[n] =...

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