Let f is the corresponding function value for independent data values in another vector x, which is gridded. Sometimes data can be contaminated with noise, meaning its value is off by a small amount from what it would be if it were computed from a pure mathematical function. This can often occur in engineering due to inaccuracies in measurement devices or the data itself can be slightly modified by perturbations outside the system of interest.
#!/usr/bin/env python
""" Numerical differentiation with noise """
import numpy as np
import matplotlib.pyplot as plt
plt.style.use("classic")
plt.rc("text", usetex="True")
plt.rc("pgf", texsystem="pdflatex")
plt.rc("font", family="serif", weight="normal", size=10)
plt.rc("axes", labelsize=12, titlesize=12)
plt.rc("figure", titlesize=12)
OMEGA = 100
EPSILON = 0.01
x = np.linspace(0, 2 * np.pi, 1000)
f1 = np.cos(x)
f2 = np.cos(x) + EPSILON * np.sin(OMEGA * x)
df1 = -np.sin(x)
df2 = -np.sin(x) + EPSILON * OMEGA * np.cos(OMEGA * x)
fig, (ax1, ax2) = plt.subplots(2, 1)
ax1.plot(x, f2, "r", label=r"$cos(x) + \epsilon * sin(\omega * x)$")
ax1.plot(x, f1, "b", label=r"$cos(x)$")
ax1.set(xlabel="$x$", ylabel="$f(x)$")
ax1.set(xlim=(0, 2 * np.pi), ylim=(-2.0, 2.0))
ax1.legend(loc="best")
ax1.grid(True, which="both")
ax1.set_title(r"cosine wave corrupted by a small sine wave")
ax2.plot(x, df2, "r", label=r"$sin(x) + \epsilon * \omega * cos(\omega * x)$")
ax2.plot(x, df1, "b", label=r"$sin(x)$")
ax2.set(xlabel="$x$", ylabel="$df(x)$")
ax2.set(xlim=(0, 2 * np.pi), ylim=(-2.0, 2.0))
ax2.legend(loc="best")
ax2.grid(True, which="both")
ax2.set_title(r"Numerical differentiation with noise ($\omega = 100$, $\epsilon = 0.01$)")
fig.tight_layout()
plt.savefig("diffnoise.svg", dpi=100, bbox_inches="tight", pad_inches=0.1)
#!/usr/bin/env julia
""" Numerical differentiation with noise """
import PyPlot as plt
plt.matplotlib.style.use("classic")
plt.rc("text", usetex="True")
plt.rc("pgf", texsystem="pdflatex")
plt.rc("font", family="serif", weight="normal", size=10)
plt.rc("axes", labelsize=12, titlesize=12)
plt.rc("figure", titlesize=12)
OMEGA = 100
EPSILON = 0.01
x = LinRange(0, 2 * pi, 1000)
f1 = cos.(x)
f2 = cos.(x) + EPSILON * sin.(OMEGA .* x)
df1 = -sin.(x)
df2 = -sin.(x) + EPSILON * OMEGA * cos.(OMEGA .* x)
fig, ax = plt.subplots(2, 1)
ax[1].plot(x, f2, "r", label = raw"$cos(x) + \epsilon * sin(\omega * x)$")
ax[1].plot(x, f1, "b", label = raw"$cos(x)$")
ax[1].set(xlabel = raw"$x$", ylabel = raw"$f(x)$")
ax[1].set(xlim = (0, 2 * pi), ylim = (-2.0, 2.0), yticks = -2.0:0.5:2.0)
ax[1].tick_params(direction = "in")
ax[1].legend(loc="best")
ax[1].grid(true, which = "both")
ax[1].set_title(raw"cosine wave corrupted by a small sine wave")
ax[2].plot(x, df2, "r", label = raw"$sin(x) + \epsilon * \omega * cos(\omega * x)$")
ax[2].plot(x, df1, "b", label = raw"$sin(x)$")
ax[2].set(xlabel = raw"$x$", ylabel = raw"$df(x)$")
ax[2].set(xlim = (0, 2 * pi), ylim = (-2.0, 2.0), yticks = -2.0:0.5:2.0)
ax[2].tick_params(direction = "in")
ax[2].legend(loc="best")
ax[2].grid(true, which = "both")
ax[2].set_title(raw"Numerical differentiation with noise ($\omega = 100$, $\epsilon = 0.01$)")
fig.tight_layout()
plt.savefig("diffnoise.svg", dpi = 100, bbox_inches = "tight", pad_inches = 0.1)
