The way to avoid complex-valued signals is to handle negative frequency the same way we handle .To do this we use a filter with two roots, one at and one at .The filter (1+iZ)(1-iZ)= 1+Z2 has real-valued time-domain coefficients, namely, (1,0,1). The factor (1+iZ) vanishes when Z=i or ,and (1-iZ) vanishes at .Notice what happens when the filter (1,0,1) is convolved with the time series :the output is zero at all times. This is because bt is a sinusoid at the half-Nyquist frequency ,and the filter (1,0,1) has zeros at plus and minus half-Nyquist.
Let us work out the general case for a root anywhere in the complex plane. Let the root Z0 be decomposed into its real and imaginary parts:
Recall that to keep the filter response real, any root on the positive -axis must have a twin on the negative -axis. In the figures I show here, the negative axis is not plotted, so we must remember the twin. Figure 10 shows a discrete approximation to the second derivative.
It is like (1-Z)2, but since both its roots are in the same place at Z=1, I pushed them a little distance apart, one going to positive frequencies and one to negative.