a question about z-transform

[liao0009]

Any one can show what is the relationship between H(z) and H(-z) using mathematical expression and graphs of spectra? Similarly H(z) and H(z^2)?
H(z) is the z-transform of h(z), where h(z) could be any function.

 

[deepu_s_s]

r u testing for the odd and even properties of the z-transform in the first case ?

 

[liao0009]

Hi deepu_s_s,
actually i am doing a split band coding in a speech project, the z-transform of the lower band and upper band filter must satisfy the relationship of
upperband{H(z)}=lowerband{H(-z)}
in order to cancel the aliasing at the end,
thus i am trying to see what is the mathematical relationship between them, it will be clear if we can show it with a spectra graph, but currently i am confused about it.
for example, by X_lower(z)=H(z^0.5)*X(z), we are performing decimation, but what is means by U{H(z)}=L{H(-z)} I am not quite sure.

 

[liao0009]

Hi mathuranathan, I am sure in a two channel split band coding it is required that H1(z)=H0(-z) to achieve alias free.
I guess the best way to conclude is the maginitude of H(-z) is similar to that of H(z), but shifted to left by pi. It could be verified by substituting z with j*w.