Show that a halfband filter can be designed from a Hilbert transformer as follows:

\[h(n)=0.5\left[\delta(n)+(-1)^{(n-1) / 2} h_{\mathrm{h}}(n)\right]\]

where \(h(n)\) is the halfband impulse response and \(h_{\mathrm{h}}(n)\) is the impulse response of the Hilbert transformer. Note that \(h_{\mathrm{h}}(n)=0\) for even \(n\) (see Equation (5.18)). Show also that its \(z\) transform is equal to

\[H(z)=0.5\left[1-\mathrm{j} H_{\mathrm{h}}(-\mathrm{j} z)\right] .\]