Show that the lowpass-to-bandpass and lowpass-to-bandstop transformations proposed in Section 6.4 are valid. Section 6.4 Usually, in

Show that the lowpass-to-bandpass and lowpass-to-bandstop transformations proposed in Section 6.4 are valid.

Section 6.4

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Usually, in the approximation of a continuous-time filter, we begin by designing a nor- malized lowpass filter and then, through a frequency transformation, the filter with the specified magnitude response is obtained. In the design of digital filters, we can also start by designing a digital lowpass filter and then apply a frequency transformation in the discrete-time domain. The procedure consists of replacing the variable z by an appropriate function g(z) to generate the desired magnitude response. The function g(2) needs to meet some constraints to be a valid transformation (Constantinides, 1970), namely: The function g(z) must be a ratio of polynomials, since the digital filter transfer function must remain a ratio of polynomials after the transformation. The mapping g(2) must be such that the filter stability is maintained; that is, stable filters generate stable transformed filters and unstable filters generate unstable