Add more plots for Laplace transform

A plot demonstrating attenuating responses in the Laplace domain was
added. Also, a plot of the Laplace transform of the Fmajor4 chord was
added to demonstrate nonorthogonality.

classical-control-theory/laplace-domain-analysis/the-fourier-transform.tex

@@ -6,7 +6,7 @@ \section{The Fourier transform}
 produce the original signal. That is, we create a sum of waveforms that are
 multiplied by their respective contribution amount.
 
-Think of an F major 4 chord which has the notes $F_4$ ($349.23\,Hz$), $A_4$
+Think of an Fmajor4 chord which has the notes $F_4$ ($349.23\,Hz$), $A_4$
 ($440\,Hz$), and $C_4$ ($261.63\,Hz$). The waveform over time looks like figure
 \ref{fig:fourier_chord}.
 

classical-control-theory/laplace-domain-analysis/the-laplace-transform.tex

@@ -27,14 +27,37 @@ \section{The Laplace transform}
 converges to zero. Figure \ref{fig:impulse_response_poles} shows this for
 various points.
 
+If we move the component frequencies in the Fmajor4 chord example parallel to
+the real axis to $\sigma = -25$, the resulting time domain response attenuates
+according to the decaying exponential $e^{-25t}$ (see figure
+\ref{fig:laplace_chord_attenuating}).
+
+\begin{svg}{build/code/laplace_chord_attenuating}
+  \caption{Fmajor4 chord at $\sigma = 0$ and $\sigma = -25$}
+  \label{fig:laplace_chord_attenuating}
+\end{svg}
+
 Note that this explanation as a basis isn't exact because the Laplace basis
 isn't orthogonal (that is, the x and y coordinates affect each other and have
 cross-talk). In the frequency domain, we had a basis of sine waves that we
 represented as delta functions in the frequency domain. Each frequency
 contribution was independent of the others. In the Laplace domain, this is not
-the case; a pure exponential is $\frac{1}{s -a}$ (a rational function) instead
-of a delta function. This function is nonzero at points that aren't actually
-frequencies present in the time domain.
+the case; a pure exponential is $\frac{1}{s - a}$ (a rational function where $a$
+is a real number) instead of a delta function. This function is nonzero at
+points that aren't actually frequencies present in the time domain. Figure
+\ref{fig:laplace_chord_3d} demonstrates this, which shows the Laplace transform
+of the Fmajor4 chord plotted in 3D.
+
+\begin{svg}{build/code/laplace_chord}
+  \caption{Laplace transform of Fmajor4 chord plotted in 3D}
+  \label{fig:laplace_chord_3d}
+\end{svg}
+
+Notice how the values of the function around each component frequency decrease
+according to $\frac{1}{\sqrt{x^2 + y^2}}$ in the $x$ and $y$ directions (in just
+the $x$ direction, it would be $\frac{1}{x}$).
+
+\subsection{The definition}
 
 The Laplace transform of a function $f(t)$ is defined as
 
@@ -50,7 +73,7 @@ \section{The Laplace transform}
 defined as $0$ for $t < 0$ and $1$ for $t \ge 0$.}, and exponential decay. We
 can see that a derivative is equivalent to multiplying by $s$, and an integral
 is equivalent to multiplying by $\frac{1}{s}$. We'll discuss the decaying
-exponential shortly.
+exponential in subsection \ref{subsec:poles_and_zeroes}.
 
 \begin{booktable}
   \begin{tabular}{|ccc|}

classical-control-theory/laplace-domain-analysis/transfer-functions.tex

@@ -12,13 +12,14 @@ \section{Transfer functions}
 \end{equation}
 
 \subsection{Poles and zeroes}
+\label{subsec:poles_and_zeroes}
 
 The roots of factors in the numerator of a transfer function are called
 \textit{zeroes} because they make the transfer function approach zero. Likewise,
 the roots of factors in the denominator of a transfer function are called
 \textit{poles} because they make the transfer function approach infinity; on a
-3D graph, these look like the poles of a circus tent. See figure
-\ref{fig:tf_3d}.
+3D graph, these look like the poles of a circus tent (see figure
+\ref{fig:tf_3d}).
 
 When the factors of the denominator are broken apart using partial fraction
 expansion into something like $\frac{A}{s + a} + \frac{B}{s + b}$, the constants

code/laplace_chord.py

@@ -0,0 +1,93 @@
+#!/usr/bin/env python3
+
+# Avoid needing display if plots aren't being shown
+import sys
+
+if "--noninteractive" in sys.argv:
+    import matplotlib as mpl
+
+    mpl.use("svg")
+    import latexutils
+
+import matplotlib.pyplot as plt
+from matplotlib import cm
+from mpl_toolkits.mplot3d import Axes3D
+import math
+import numpy as np
+
+plt.rc("text", usetex=True)
+
+
+def sin_tf(freq, s):
+    return freq / ((s - freq * 1j) * (s + freq * 1j))
+
+
+def clamp(val, low, high):
+    return max(low, min(val, high))
+
+
+def main():
+    f_f = 349.23
+    f_a = 440
+    f_c = 261.63
+
+    T = 0.000001
+    xlim = [0, 0.05]
+    ylim = [-3, 3]
+    x = np.arange(xlim[0], xlim[1], T)
+    plt.xlim(xlim)
+    plt.ylim(ylim)
+
+    yf = np.sin(f_f * 2 * math.pi * x)
+    ya = np.sin(f_a * 2 * math.pi * x)
+    yc = np.sin(f_c * 2 * math.pi * x)
+    ysum = yf + ya + yc
+    ysum_attenuating = ysum * np.exp(-25 * x)
+
+    num_plots = 2
+
+    plt.subplot(num_plots, 1, 1)
+    plt.ylim(ylim)
+    plt.ylabel("Fmaj4 ($\sigma = 0$)")
+    plt.plot(x, ysum)
+    plt.gca().axes.get_xaxis().set_ticks([])
+
+    plt.subplot(num_plots, 1, 2)
+    plt.ylim(ylim)
+    plt.ylabel("Attenuating Fmaj4 ($\sigma = -25$)")
+    plt.plot(x, ysum_attenuating)
+    plt.gca().axes.get_xaxis().set_ticks([])
+
+    plt.xlabel("$t$")
+
+    if "--noninteractive" in sys.argv:
+        latexutils.savefig("laplace_chord_attenuating")
+
+    x, y = np.mgrid[-150.0:150.0:500j, 200.0:500.0:500j]
+
+    # Need an (N, 2) array of (x, y) pairs.
+    xy = np.column_stack([x.flat, y.flat])
+
+    z = np.zeros(xy.shape[0])
+    for i, pair in enumerate(xy):
+        s = pair[0] + pair[1] * 1j
+        h = sin_tf(f_f, s) * sin_tf(f_a, s) * sin_tf(f_c, s)
+        z[i] = clamp(math.sqrt(h.real ** 2 + h.imag ** 2), -30, 30)
+    z = z.reshape(x.shape)
+
+    fig = plt.figure(2)
+    ax = fig.add_subplot(111, projection="3d")
+    ax.plot_surface(x, y, z, cmap=cm.coolwarm)
+    ax.set_xlabel("$Re(\sigma)$")
+    ax.set_ylabel("$Im(j\omega)$")
+    ax.set_zlabel("$H(s)$")
+    ax.set_zticks([])
+
+    if "--noninteractive" in sys.argv:
+        latexutils.savefig("laplace_chord_3d")
+    else:
+        plt.show()
+
+
+if __name__ == "__main__":
+    main()