This notebook contains solutions to exercises in Chapter 9: Differentiation and Integration
Copyright 2015 Allen Downey
# Get thinkdsp.py
import os
if not os.path.exists('thinkdsp.py'):
!wget https://github.com/AllenDowney/ThinkDSP/raw/master/code/thinkdsp.py
import numpy as np
import matplotlib.pyplot as plt
from thinkdsp import decorate
The goal of this exercise is to explore the effect of diff and differentiate on a signal. Create a triangle wave and plot it. Apply the diff operator and plot the result. Compute the spectrum of the triangle wave, apply differentiate, and plot the result. Convert the spectrum back to a wave and plot it. Are there differences between the effect of diff and differentiate for this wave?
Solution: Here's the triangle wave.
from thinkdsp import TriangleSignal
in_wave = TriangleSignal(freq=50).make_wave(duration=0.1, framerate=44100)
in_wave.plot()
decorate(xlabel='Time (s)')
The diff of a triangle wave is a square wave, which explains why the harmonics in a square wave drop off like $1/f$, compared to the triangle wave, which drops off like $1/f^2$.
out_wave = in_wave.diff()
out_wave.plot()
decorate(xlabel='Time (s)')
When we take the spectral derivative, we get "ringing" around the discontinuities: https://en.wikipedia.org/wiki/Ringing_(signal)
Mathematically speaking, the problem is that the derivative of the triangle wave is undefined at the points of the triangle.
out_wave2 = in_wave.make_spectrum().differentiate().make_wave()
out_wave2.plot()
decorate(xlabel='Time (s)')
The goal of this exercise is to explore the effect of cumsum and integrate on a signal. Create a square wave and plot it. Apply the cumsum operator and plot the result. Compute the spectrum of the square wave, apply integrate, and plot the result. Convert the spectrum back to a wave and plot it. Are there differences between the effect of cumsum and integrate for this wave?
Solution: Here's the square wave.
from thinkdsp import SquareSignal
in_wave = SquareSignal(freq=50).make_wave(duration=0.1, framerate=44100)
in_wave.plot()
decorate(xlabel='Time (s)')
The cumulative sum of a square wave is a triangle wave. After the previous exercise, that should come as no surprise.
out_wave = in_wave.cumsum()
out_wave.plot()
decorate(xlabel='Time (s)')
The spectral integral is also a triangle wave, although the amplitude is very different.
spectrum = in_wave.make_spectrum().integrate()
spectrum.hs[0] = 0
out_wave2 = spectrum.make_wave()
out_wave2.plot()
decorate(xlabel='Time (s)')
If we unbias and normalize the two waves, they are visually similar.
out_wave.unbias()
out_wave.normalize()
out_wave2.normalize()
out_wave.plot()
out_wave2.plot()
And they are numerically similar, but with only about 3 digits of precision.
out_wave.max_diff(out_wave2)
The goal of this exercise is the explore the effect of integrating twice. Create a sawtooth wave, compute its spectrum, then apply integrate twice. Plot the resulting wave and its spectrum. What is the mathematical form of the wave? Why does it resemble a sinusoid?
Here's the sawtooth.
from thinkdsp import SawtoothSignal
in_wave = SawtoothSignal(freq=50).make_wave(duration=0.1, framerate=44100)
in_wave.plot()
decorate(xlabel='Time (s)')
The first cumulative sum of a sawtooth is a parabola:
out_wave = in_wave.cumsum()
out_wave.unbias()
out_wave.plot()
decorate(xlabel='Time (s)')
The second cumulative sum is a cubic curve:
out_wave = out_wave.cumsum()
out_wave.plot()
decorate(xlabel='Time (s)')
Integrating twice also yields a cubic curve.
spectrum = in_wave.make_spectrum().integrate().integrate()
spectrum.hs[0] = 0
out_wave2 = spectrum.make_wave()
out_wave2.plot()
decorate(xlabel='Time (s)')
At this point, the result looks more and more like a sinusoid. The reason is that integration acts like a low pass filter. At this point we have filtered out almost everything except the fundamental, as shown in the spectrum below:
out_wave2.make_spectrum().plot(high=500)
The goal of this exercise is to explore the effect of the 2nd difference and 2nd derivative. Create a CubicSignal, which is defined in thinkdsp. Compute the second difference by applying diff twice. What does the result look like. Compute the second derivative by applying differentiate twice. Does the result look the same?
Plot the filters that corresponds to the 2nd difference and the 2nd derivative and compare them. Hint: In order to get the filters on the same scale, use a wave with framerate 1.
Solution: Here's the cubic signal
from thinkdsp import CubicSignal
in_wave = CubicSignal(freq=0.0005).make_wave(duration=10000, framerate=1)
in_wave.plot()
The first difference is a parabola and the second difference is a sawtooth wave (no surprises so far):
out_wave = in_wave.diff()
out_wave.plot()
out_wave = out_wave.diff()
out_wave.plot()
When we differentiate twice, we get a sawtooth with some ringing. Again, the problem is that the deriviative of the parabolic signal is undefined at the points.
spectrum = in_wave.make_spectrum().differentiate().differentiate()
out_wave2 = spectrum.make_wave()
out_wave2.plot()
decorate(xlabel='Time (s)')
The window of the second difference is -1, 2, -1. By computing the DFT of the window, we can find the corresponding filter.
from thinkdsp import zero_pad
from thinkdsp import Wave
diff_window = np.array([-1.0, 2.0, -1.0])
padded = zero_pad(diff_window, len(in_wave))
diff_wave = Wave(padded, framerate=in_wave.framerate)
diff_filter = diff_wave.make_spectrum()
diff_filter.plot(label='2nd diff')
decorate(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio')
And for the second derivative, we can find the corresponding filter by computing the filter of the first derivative and squaring it.
PI2 = np.pi * 2
deriv_filter = in_wave.make_spectrum()
deriv_filter.hs = (PI2 * 1j * deriv_filter.fs)**2
deriv_filter.plot(label='2nd deriv')
decorate(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio')
Here's what the two filters look like on the same scale:
diff_filter.plot(label='2nd diff')
deriv_filter.plot(label='2nd deriv')
decorate(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio')
Both are high pass filters that amplify the highest frequency components. The 2nd derivative is parabolic, so it amplifies the highest frequencies the most. The 2nd difference is a good approximation of the 2nd derivative only at the lowest frequencies, then it deviates substantially.