Functions for frequency, phase, impulse and step response

Let's first define functions to plot filter properties.

from pylab import *
import scipy.signal as signal

#Plot frequency and phase response
def mfreqz(b,a=1):
    w,h = signal.freqz(b,a)
    h_dB = 20 * log10 (abs(h))
    subplot(211)
    plot(w/max(w),h_dB)
    ylim(-150, 5)
    ylabel('Magnitude (db)')
    xlabel(r'Normalized Frequency (x$\pi$rad/sample)')
    title(r'Frequency response')
    subplot(212)
    h_Phase = unwrap(arctan2(imag(h),real(h)))
    plot(w/max(w),h_Phase)
    ylabel('Phase (radians)')
    xlabel(r'Normalized Frequency (x$\pi$rad/sample)')
    title(r'Phase response')
    subplots_adjust(hspace=0.5)

#Plot step and impulse response
def impz(b,a=1):
    l = len(b)
    impulse = repeat(0.,l); impulse[0] =1.
    x = arange(0,l)
    response = signal.lfilter(b,a,impulse)
    subplot(211)
    stem(x, response)
    ylabel('Amplitude')
    xlabel(r'n (samples)')
    title(r'Impulse response')
    subplot(212)
    step = cumsum(response)
    stem(x, step)
    ylabel('Amplitude')
    xlabel(r'n (samples)')
    title(r'Step response')
    subplots_adjust(hspace=0.5)

Lowpass FIR filter

Designing a lowpass FIR filter is very simple to do with SciPy, all you need to do is to define the window length, cut off frequency and the window.

The Hamming window is defined as: w(n) = α − βcos(2πn)/(N − 1), where α = 0.54 and β = 0.46

The next code chunk is executed in term mode, see the source document for syntax. Notice also that Pweave can now catch multiple figures/code chunk.

n = 61
a = signal.firwin(n, cutoff = 0.3, window = "hamming")
#Frequency and phase response
mfreqz(a)

show()
#Impulse and step response
figure(2)

<matplotlib.figure.Figure at 0x7f83f3f52b00>

<matplotlib.figure.Figure at 0x7f83f3f52b00>

impz(a)

show()

Highpass FIR Filter

Let's define a highpass FIR filter:

n = 101
a = signal.firwin(n, cutoff = 0.3, window = "hanning", pass_zero=False)
mfreqz(a)
show()

Bandpass FIR filter

Notice that the plot has a caption defined in code chunk options.

n = 1001
a = signal.firwin(n, cutoff = [0.2, 0.5], window = 'blackmanharris', pass_zero = False)
mfreqz(a)
show()