Let's assume we have an ideal discrete time Hilbert Trasnformation system (90-degree phase shifter) with a frequency response over one period:
$$H(e^{jw}) = \left\{ \begin{array}{ll} -j & \mbox{if}\;\;\;\;\;\; 0 < w < \pi \\ \;\;j & \mbox{if}\;\; -\pi < w < 0 \end{array} \right.$$
I want to define an ideal frequency response $H_d(e^{jw})$ of an ideal discrete time Hilbert Trasnformation system that has a non-zero constant group delay.
I consider that $$H_d(e^{jw})= [1-2u(w)]e^{j(\frac{\pi}{2}-τ\cdot w)}$$ since $$\left | H_d(e^{jw}) \right | = 1 \; \; \; \forall w$$ and $$\measuredangle H_d(e^{jw}) = \left\{ \begin{array}{ll} \frac{\pi}{2}-τ\cdot w & \mbox{if} \;-\pi < w < 0 \\ -\frac{\pi}{2}-τ\cdot w & \mbox{if}\;\;\;\;\;\; 0 < w < \pi \end{array} \right.$$
So, it seems an acceptable answer. However, I didn't make any calculations to find it. Just guessed it. So my question is: which is the approach to such a problem is order to define $H_d(e^{jw})$? Thanks in advance!