# Perfect reconstruction ( Single Channel )

I was trying to solve the following problem .

Given an input x[n], consider upsampling by 2, followed by interpolation with a filter having z-transform H(z) for magnification of the signal. Then, to recover the original signal size, apply filtering by a decimation filter G(z) followed by downsampling by 2, in order to obtain a reconstruction ˆx[n].

using the results for upsampling as $Y(z) = X(z^2)$

and downsampling as $Y(z) = X(z^{1/2})+X(-z^{1/2})$

I obtained

$\hat{X}(z) = 1/2\bigg(G(z^{1/2})H(z^{1/2})X(z) + G(-z^{1/2})H(-z^{1/2})X(-z)\bigg)$

however, the solution is listed as ,

$\hat{X}(z) = 1/2X(z)\bigg(G(z^{1/2})H(z^{1/2}) + G(-z^{1/2})H(-z^{1/2})\bigg)$

Where have i gone wrong ?

I solved in the following process ..

$A(z) = X(z^2)$

$B(z) = H(z)A(z)$

$C(z) = G(z)B(z))$

$\hat{X}(z) = 1/2[C(z^{1/2})+C(-z^{1/2})]$

and substituted them in a chain.

HINT: Note that if

$$Y(z)=X(z^2)$$

then obviously

$$Y(z^{1/2})=X(z)$$

but also

$$Y(-z^{1/2})=X((-z^{1/2})^2)=X(z)$$

from which the desired result follows in a straightforward way.