# Fourier Reconstruction of RR-Lyrae TemplatesΒΆ

Figure 10.1

An example of a truncated Fourier representation of an RR Lyrae light curve. The thick dashed line shows the true curve; the gray lines show the approximation based on 1, 3, and 8 Fourier modes (sinusoids).

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```
```# Author: Jake VanderPlas
#   The figure produced by this code is published in the textbook
#   "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)
#   To report a bug or issue, use the following forum:
import numpy as np
from matplotlib import pyplot as plt

from astroML.datasets import fetch_rrlyrae_templates

#----------------------------------------------------------------------
# This function adjusts matplotlib settings for a uniform feel in the textbook.
# Note that with usetex=True, fonts are rendered with LaTeX.  This may
# result in an error if LaTeX is not installed on your system.  In that case,
# you can set usetex to False.
if "setup_text_plots" not in globals():
from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)

#------------------------------------------------------------
# Load the RR Lyrae template
templates = fetch_rrlyrae_templates()
x, y = templates['115r'].T

#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(5, 5))

kvals = [1, 3, 8]
subplots = [311, 312, 313]

for (k, subplot) in zip(kvals, subplots):

# Use FFT to fit a truncated Fourier series
y_fft = np.fft.fft(y)
y_fft[k + 1:-k] = 0
y_fit = np.fft.ifft(y_fft).real

# plot the true value and the k-term reconstruction
ax.plot(np.concatenate([x, 1 + x]),
np.concatenate([y, y]), '--k', lw=2)
ax.plot(np.concatenate([x, 1 + x]),
np.concatenate([y_fit, y_fit]), color='gray')

label = "%i mode" % k
if k > 1:
label += 's'

ax.text(0.02, 0.1, label, ha='left', va='bottom',
transform=ax.transAxes)

if subplot == subplots[-1]:
ax.set_xlabel('phase')
else:
ax.xaxis.set_major_formatter(plt.NullFormatter())

if subplot == subplots[1]:
ax.set_ylabel('amplitude')
ax.yaxis.set_major_formatter(plt.NullFormatter())

ax.set_xlim(0, 2)
ax.set_ylim(1.1, -0.1)

plt.show()
```