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.