# Orthogonality of filter impulse response to its even shift

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: $$_n = \delta_k$$ (Filters with impulse responses orthogonal to their even shifts)

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

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.
• applying transposes, and hermitians until things happen.
• I just don't know what to do with: <𝑔𝑛,𝑔𝑛−2𝑘>𝑛=𝛿𝑘 – hminle Mar 2 at 0:26
• 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. – TimWescott Mar 2 at 1:09
• @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 $k=0$. – Marcus Müller Mar 2 at 9:01