Orthogonality of filter impulse response to its even shift

Question

I meet this problem but still do not know how to solve it. Could you guy give me some guides?

Upsampling by 2 ($U_2$) followed by filtering by $g$, with operator $G$

And given: $<g_n,g_{n-2k}>_n = \delta_k$ (Filters with impulse responses orthogonal to their even shifts)

Prove that: $$ I = U_2^*G^*GU_2 $$

Thank in advance.

Marcus Müller · Accepted Answer · 2020-03-01 22:29:50Z

0

Since this is clearly homework, let's help you with hints:

The set of tricks for such problems is typically pretty limited; try (and that's really what solves the issue)

Multiplying the equation with elements from your right-hand term from left and/or right, so that identity matrices form.
Writing your matrix/operator property down as structure of your matrix
applying transposes, and hermitians until things happen.

answered Mar 1 at 22:29

Marcus Müller

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$\begingroup$ I just don't know what to do with: <𝑔𝑛,𝑔𝑛−2𝑘>𝑛=𝛿𝑘 $\endgroup$ – hminle Mar 2 at 0:26
$\begingroup$ The notation you're using is unfamiliar. What is $\left < g_n,\,g_{n-2k} \right >_n = \delta_k$ supposed to mean? If you don't know, there's your problem, and it should be in your book or the lecture notes. $\endgroup$ – TimWescott Mar 2 at 1:09
$\begingroup$ @TimWescott I'd have said that these brackets signify the signal vector space inner product between an infinite sequence $g[n]$ and it's $2k$-shifted variant $g[n-2k]$. And that's always 0, unless $k=0$. $\endgroup$ – Marcus Müller Mar 2 at 9:01

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