# Time response of complex frequency response function

In the frequency domain I have the following function:

$$H(f)=A(f)\cdot e^{i\beta(f)}$$

Where both $$A(f)$$ and $$\beta(f)$$ are known functions. The thing I would like to do is to model the response of this filter in the time domain to an incoming discrete signal ($$x[n]$$) according to:

$$y[n]=h[n]\otimes x[n]$$

Where:

$$h[n]=\frac{1}{f_{s}}\int_{-\frac{f_{s}}{2}}^{\frac{f_{s}}{2}}H(f)\cdot e^{2\pi i f n/f_{s}}df$$

When I use a real valued filter, e.g. $$h[n]=0.5$$ for all values of n, the resulting signal is also real valued. However, when I add an imaginary part to the filter, e.g. $$h[n]=0.5+0.5i$$ the resulting real part of the signal is the same as for $$h[n]=0.5$$ and the output signal contains an imaginary part, even though this should only be a phase shift.

How do I convert this to a real signal with a phase shift?

A real signal with a phase shift does not technically exist since phase is $$e^{j\phi}$$, so requires an imaginary component except for phases that are modulo $$\pi$$. (The phase of a real signal is $$0$$ or $$\pi$$ radians only). You may be actually looking for a time delay. A constant time delay has a linear phase with frequency, and for the result to be real in time, the frequency must be complex conjugate symmetric. So to create a real only time delay use a linear phase in frequency that is complex conjugate symmetric such as the plot below. See my related "DSP Puzzle" focusing on this confusion of phase and delay: $9\rm V$ Battery with $45^\circ$ phase