Ideal Discrete Time Hilbert Transformer with constant group delay Let's assume we have an ideal discrete time Hilbert Trasnformation system (90-degree phase shifter) with a frequency response over one period:

$$H(e^{jw}) =
\left\{
 \begin{array}{ll}
  -j  &amp; \mbox{if}\;\;\;\;\;\; 0 &lt; w &lt; \pi \\
  \;\;j &amp; \mbox{if}\;\; -\pi &lt; w &lt; 0
 \end{array}
\right.$$

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 
$$\measuredangle H_d(e^{jw})  =
\left\{
 \begin{array}{ll}
  \frac{\pi}{2}-τ\cdot w  &amp; \mbox{if} \;-\pi &lt; w &lt; 0 \\
  -\frac{\pi}{2}-τ\cdot w &amp; \mbox{if}\;\;\;\;\;\;  0 &lt; w &lt; \pi
 \end{array}
\right.$$

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!
There's not much need for complicated calculations. If you have the zero-delay frequency response $H(e^{j\omega})$ as defined in the first equation of your question, the frequency response with a linear-phase (constant delay) is given by

$$H_d(e^{j\omega})=H(e^{j\omega})e^{-j\omega\tau}=e^{-j\left(\frac{\pi}{2}\textrm{sgn}(\omega)+\omega\tau\right)},\qquad \omega\in(-\pi,\pi)\tag{1}$$

where $\tau$ is the group delay in samples.