Using a spectral transformation a filter $G(z^{cap})$ is
designed to be a low pass filter from an existing lowpass filter $G_{lp}(z)$ via $G(z^{cap}) = G(z)z=H(z^{cap})$
. Which property should H(z) have in order to preserve
stability of the designed filter?
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The filter is stable if the all the poles of the transformed filter, i.e., filter obtained after applying spectral transformation, lie inside the unit circle. For example, if $H(z)$ is the original filter and $F(z)$ is the filter response of the transformation filter, then the spectrally transformed filter response is given as \begin{align} G(z) = H(F(z)) = H(z)|_{z^{-1}\leftarrow 1/F(z)} \end{align} where \begin{align} \frac{1}{F(z)} = \frac{1-\beta z}{z - \beta}. \end{align}
Check if the poles of $G(z)$ lie inside the unit circle. If yes, then $G(z)$ is stable. Note that $F(z)$ is an all-pass filter. Therefore, the regions of stability and instability of $H(z)$ are preserved in $G(z)$.
