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: $&lt;g_n,g_{n-2k}&gt;_n = \delta_k$
(Filters with impulse responses orthogonal to their even shifts)

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

Thank in advance.
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.