FIR filters¶
import matplotlib.pyplot as plt
import numpy as np
import sdr
%config InlineBackend.print_figure_kwargs = {"facecolor" : "w"}
%matplotlib inline
# %matplotlib widget
sdr.plot.use_style()
Create an FIR filter¶
The user creates an FIR filter with the sdr.FIR
class by specifying the feedforward coefficients \(h_i\).
Below is an square-root raised cosine FIR filter.
h = sdr.root_raised_cosine(0.5, 6, 10)
fir = sdr.FIR(h)
Examine the impulse response, \(h[n]\)¶
The impulse response of the FIR filter is computed and returned from the sdr.FIR.impulse_response()
method.
The impulse response \(h[n]\) is the output of the filter when the input is an impulse \(\delta[n]\).
For FIR filters, the impulse response is the same as the feedforward taps.
h = fir.impulse_response()
print(h)
[ 0.00095883 -0.00175012 -0.00423921 -0.0058825 -0.006151 -0.00474595
-0.0017044 0.00254816 0.00721645 0.0112324 0.01342358 0.01273202
0.00845058 0.0004368 -0.01073669 -0.02372977 -0.03650247 -0.04650654
-0.05098525 -0.04734644 -0.03355896 -0.00851486 0.02769991 0.07367348
0.12670446 0.18301321 0.23810898 0.28727058 0.32607991 0.35093841
0.35949665 0.35093841 0.32607991 0.28727058 0.23810898 0.18301321
0.12670446 0.07367348 0.02769991 -0.00851486 -0.03355896 -0.04734644
-0.05098525 -0.04650654 -0.03650247 -0.02372977 -0.01073669 0.0004368
0.00845058 0.01273202 0.01342358 0.0112324 0.00721645 0.00254816
-0.0017044 -0.00474595 -0.006151 -0.0058825 -0.00423921 -0.00175012
0.00095883]
The impulse response is conveniently plotted using the sdr.plot.impulse_response()
function.
plt.figure()
sdr.plot.impulse_response(fir)
plt.show()
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Examine the step response, \(s[n]\)¶
The step response of the FIR filter is computed and returned from the sdr.FIR.step_response()
method.
The step response \(s[n]\) is the output of the filter when the input is a unit step \(u[n]\).
s = fir.step_response()
print(s)
[ 0.00095883 -0.00175012 -0.00423921 -0.0058825 -0.006151 -0.00474595
-0.0017044 0.00254816 0.00721645 0.0112324 0.01342358 0.01273202
0.00845058 0.0004368 -0.01073669 -0.02372977 -0.03650247 -0.04650654
-0.05098525 -0.04734644 -0.03355896 -0.00851486 0.02769991 0.07367348
0.12670446 0.18301321 0.23810898 0.28727058 0.32607991 0.35093841
0.35949665 0.35093841 0.32607991 0.28727058 0.23810898 0.18301321
0.12670446 0.07367348 0.02769991 -0.00851486 -0.03355896 -0.04734644
-0.05098525 -0.04650654 -0.03650247 -0.02372977 -0.01073669 0.0004368
0.00845058 0.01273202 0.01342358 0.0112324 0.00721645 0.00254816
-0.0017044 -0.00474595 -0.006151 -0.0058825 -0.00423921 -0.00175012
0.00095883]
The step response is conveniently plotted using the sdr.plot.step_response()
function.
plt.figure()
sdr.plot.step_response(fir)
plt.show()
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Examine the frequency response, \(H(\omega)\)¶
The frequency response is the transfer function \(H(z)\) evaluated at the complex exponential \(e^{j \omega}\), where \(\omega = 2 \pi f / f_s\).
The two-sided frequency response is conveniently plotted using the sdr.plot.magnitude_response()
function.
plt.figure()
sdr.plot.magnitude_response(fir)
plt.show()
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The one-sided frequency response, with logarithmic scale, can be plotted using the x_axis="log"
keyword argument.
plt.figure()
sdr.plot.magnitude_response(fir, x_axis="log", decades=2)
plt.show()
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Examine the group delay, \(\tau_g(\omega)\)¶
The group delay \(\tau_g(\omega)\) is the time shift of the envelope of a signal passed through the filter as a function of its frequency \(\omega\).
The group delay is conveniently plotted using the sdr.plot.group_delay()
function.
plt.figure()
sdr.plot.group_delay(fir)
plt.ylim(29, 31)
plt.show()
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plt.figure()
sdr.plot.group_delay(fir, x_axis="log")
plt.ylim(29, 31)
plt.show()
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Fully analyze a FIR filter¶
The user can easily analyze the perform of a given IIR filter using the sdr.plot.filter()
function.
Here is an FIR filter with one real zero and 8 complex poles.
h = sdr.root_raised_cosine(0.1, 12, 10)
fir = sdr.FIR(h)
plt.figure(figsize=(8, 6))
sdr.plot.filter(fir)
plt.show()
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