I'm learning digital signal processing to implement filters and am using python to easily implement a test ideas. So I just started using the scipy.signal library to find the impulse response and frequency response of different filters.
Currently I am working through the book "Digital Signals, Processors and Noise by Paul A. Lynn (1992)" (and finding it an amazing resource for learning this stuff). In this book they have a filter with the transfer functions shown below:
I divided the numerator and denominator by in order to get the following equation:
I then implemented this with Scipy using:
NumeratorZcoefs = [1, -1, 1, -1]
DenominatorZcoefs = [1, 0.54048, -0.62519, -0.66354, 0.60317, 0.69341]
FreqResponse = scipy.signal.freqz(NumeratorZcoefs, DenominatorZcoefs)
fig = plt.figure(figsize = [8, 6])
ax = fig.add_subplot(111)
ax.plot(FreqResponse[0], abs(np.array(FreqResponse[1])))
ax.set_xlim(0, 2*np.pi)
ax.set_xlabel("$\Omega$")
and produce the plot shown below:
However in the book the frequency response is shown to be the following:
They are the same shape but the ratio of the peaks at ~2.3 and 0.5 are very different for the 2 plots, could someone suggest why this is?
Edit:
To add to this, I've just implemented a function to calculate the frequency response by hand (by calculating the distance from the poles and zeros of the function) and I get a similar ratio to the plot generated by scipy.signal, however the numbers are not the same, does anyone know why this might by?
Implementation is as follows:
def H(omega):
z1 = np.array([0,0]) # zero at 0, 0
z2 = np.array([0,0]) # Another zero at 0, 0
z3 = np.array([0, 1]) # zero at i
z4 = np.array([0, -1]) # zero at -i
z5 = np.array([1, 0]) # zero at 1
z = np.array([z1, z2, z3, z4, z5])
p1 = np.array([-0.8, 0])
p = cmath.rect(0.98, np.pi/4)
p2 = np.array([p.real, p.imag])
p = cmath.rect(0.98, -np.pi/4)
p3 = np.array([p.real, p.imag])
p = cmath.rect(0.95, 5*np.pi/6)
p4 = np.array([p.real, p.imag])
p = cmath.rect(0.95, -5*np.pi/6)
p5 = np.array([p.real, p.imag])
p = np.array([p1, p2, p3, p4, p5])
a = cmath.rect(1,omega)
a_2dvector = np.array([a.real, a.imag])
dz = z-a_2dvector
dp = p-a_2dvector
dzmag = []
for dis in dz:
dzmag.append(np.sqrt(dis.dot(dis)))
dpmag = []
for dis in dp:
dpmag.append(np.sqrt(dis.dot(dis)))
return(np.product(dzmag)/np.product(dpmag))
I then plot the frequency response like so:
omegalist = np.linspace(0,2*np.pi,5000)
Hlist = []
for omega in omegalist:
Hlist.append(H(omega))
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(omegalist, Hlist)
ax.set_xlabel("$\Omega$")
ax.set_ylabel("$|H(\Omega)|$")
and get the following plot: