Beginning with a causal response, we switched cosines and sines in the frequency domain. Here we do so again, except that we interchange the time and frequency domains, getting a more physical interpretation.
A filter that converts sines into cosines is called a ``phase-shift filter" or a ``quadrature filter."
More specifically, if the input is ,then the output should be .An example is given in Figure 2.
Figure 2 Input (top) filtered
 with quadrature filter yields
 phase-shifted signal (bottom).
Let U(Z) denote the Z-transform of a real signal input and Q(Z) denote a quadrature filter. Then the output signal is
Let us find the numerical values of qt. The time-derivative operation has the phase-shifting property we need. The trouble with a differentiator is that higher frequencies are amplified with respect to lower frequencies. Recall the FT and take its time derivative:
Since qn does not vanish for negative n, the quadrature filter is nonrealizable (that is, it requires future inputs to create its present output). If we were discussing signals in continuous time rather than sampled time, the filter would be of the form 1/t, a function that has a singularity at t=0 and whose integral over positive t is divergent. Convolution with the filter coefficients qn is therefore painful because the infinite sequence drops off slowly. Convolution with the filter qt is called ``Hilbert transformation."