0
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.