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How can I design an all pass filter to have a constant phase shift over a bandwidth centered around a carrier? I don't care about phase shift outside the band. I would like to have the filter in time domain. This is not a straightforward job right? Any keywords, design methods, external links are appreciated.
Edit: i need it to be causal. Data comes in real time in time frames. thanks
It's instructive to see what an ideal filter adding a constant phase shift would look like. If $\theta$ is the desired phase shift, the corresponding ideal frequency response is
$$H(e^{j\omega})=\begin{cases}e^{-j\theta}&,\quad 0<\omega<\pi\\ e^{j\theta}&,\quad-\pi<\omega<0\end{cases}\tag{1}$$
Using the sign function $\text{sign}(\omega)$, the frequency response $(1)$ can be rewritten as
$$H(e^{j\omega})=\cos\theta-j\,\text{sign}(\omega)\sin\theta,\quad -\pi<\omega <\pi\tag{2}$$
With the DTFT correspondence
$$-j\,\text{sign}(\omega)\Longleftrightarrow g[n]=\begin{cases}\frac{2}{\pi n},&\quad n\text{ odd}\\ 0,&\quad n\text{ even}\end{cases}\tag{3}$$
the impulse response corresponding to $H(e^{j\omega})$ is given by
$$h[n]=\cos\theta\cdot\delta[n]+\sin\theta\cdot g[n]\tag{4}$$
where $g[n]$ is the sequence on the right-hand side of Eq. $(3)$, which is the impulse response of an ideal discrete-time Hilbert transformer. Eq. $(4)$ shows that an ideal phase shifter can be implemented as a weighted parallel connection of a wire ($\cos\theta\cdot\delta[n]$) and a Hilbert transformer.
So your problem can be solved by using any of the many available designs of discrete-time Hilbert transformers. Note that you can get much better performance for a given filter order by taking into account that the approximation needs only be accurate in the given frequency band. For a frequency-domain design method this just means that in the formulation of the desired response given in $(1)$ you replace the positive frequencies by the frequency interval of interest and leave the rest as a "don't care" region. The same is done for the negative frequencies.
