# Frequency response of a complex IIR filter

 Hi all! I am trying to make a digital implementation of the One Zero Gammatone Filterbank (OZGF), however, I need the output to be complex instead of real. I have converted the filter equations by R. Lyon with the partial fraction expansion obtaining two conjugate poles, then treated it with the bilinear transform. The final stage is the coupled form of a complex IIR. In order to check the implementation, I would like to plot the frequency response of a single filter channel (= a cascade of three low-pass biquads + one bandpass biquad). But here is where I got a bit puzzled - for a real domain filter,  the way to do it is to plot the magnitude and the phase of the FFT of the impulse response. How to do it however, with a complex output u+jv (<=> r exp (jw0)) ? My intuition says it might be done by computing y= r exp (w0) and then computing abs(fft(y)) and angle(fft(y)). Is this the correct way to do it or shall I treat the real and the imaginary output separately - in that case how to interpret the outcome? I enclose the python prototype code, it requires only the numpy/scipy and the matplotlib(pylab) to run. I'll be very thankful for help. Best Piotr Holonowicz PhD candidate at Music Technology Group Universitat Pompeu Fabra Barcelona,Spain
```#!/usr/bin/env python
#******************************************************************************
#
# Purpose: Bandpass biquad test.
# Date: %date% at %time%
#
#
# by Piotr Holonowicz <pholonow@xxxxxxxxxxx>
# Music Technology Group, University Pompeu Fabra, Barcelona, Spain
# www.mtg.upf.es
# *****************************************************************************

#imports
import sys,os,glob
from optparse import OptionParser
import numpy
import scipy
from scipy import signal
import pylab

def bpbiquad(X, w0, Q, bp=False):
"""
Complex bandpass biquad IIR - plots the impulse response and the magnitude and phase
Coupled form, obtained by the bilinear transform from its Laplace transfer function.
w0 - omega0 - center frequency
Q - quality factor
"""
d=1.0/Q #easier to compute
Y = numpy.zeros(X.shape)
mag = numpy.zeros(X.shape[0])
phs = numpy.zeros(X.shape[0])
om0 = numpy.tan(w0*0.5)
s0 = numpy.sqrt(abs(d*d-4.0))
#bandpass or lowpass?
if(bp):
reZ0 = 0.5*om0
imZ0 = (om0 / d) * s0 #one zero
else:
reZ0 = 1.0
imZ0 = 1.0  #all pole
reP0 = 0.5 * (om0 - d)
imP0 = 0.5 * (om0 * s0)
for i in range(X.shape[0]-1):
Y[i, 0] = reZ0 * (reP0 * Y[i-1, 0] - imP0*Y[i-1, 1] + X[i-1, 0]) #last value is 0 , wrapping
Y[i, 1] = imZ0 * (reP0 * Y[i-1, 1] - imP0*Y[i-1, 0] + X[i-1, 1])
mag[i] = numpy.sqrt(Y[i, 0] * Y[i, 0] + Y[i, 1] * Y[i, 1])
if(Y[i, 0]!=0.0):
phs[i] = numpy.arctan(Y[i, 1] / Y[i, 0])
else:
phs[i] = 0.0
return Y, mag, phs

def magn(x):
n = len(x)/2
return abs(scipy.fft(x))[:n]
def phse(x):
n = len(x)/2
return scipy.angle(scipy.fft(x))[:n]
#main
if __name__ == "__main__":
x = numpy.zeros(128);
x[0]=1.0; #impulse

#x = scipy.signal.chirp(scipy.arange(100), 20.0,  1000.0, 20000.0)
#make a complex version of the input signal, X[n] = [Re{x(n)}, Im{x(n)}]
X = numpy.array([x, numpy.zeros(len(x))]).T
w0 = scipy.pi/4.0;
Q =2.5;
Y, mag, phs = bpbiquad(X, w0, Q, bp=False);
Y, mag, phs = bpbiquad(Y, w0, Q, bp=False);
Y, mag, phs = bpbiquad(Y, w0, Q, bp=False);
Y, mag, phs = bpbiquad(Y, w0, Q, bp=True);
cplx = mag*scipy.exp(phs);
pylab.close('all')
xt = numpy.linspace(0.0, numpy.pi, len(x))
pylab.subplot(121)
pylab.plot(mag)
pylab.title("mag");
pylab.subplot(122)
pylab.plot(phs)
pylab.title("phase")

#pylab.subplot(323)
#pylab.plot(magn(cplx))
#pylab.title("Re_fftMag")
#pylab.subplot(324)
#pylab.plot(phse(cplx))
#pylab.title("Re_fftPhase")
#pylab.subplot(325)
#pylab.plot(magn(Y[:, 1]))
#pylab.title("Im_fftMag")
#pylab.subplot(326)
#pylab.plot(phse(Y[:, 1]))
#pylab.title("Im_fftPhase")
pylab.show()

```