In [1]:
import numpy as np
from stingray import Lightcurve, Powerspectrum, AveragedPowerspectrum
import matplotlib.pyplot as plt
import matplotlib.font_manager as font_manager
%matplotlib inline
font_prop = font_manager.FontProperties(size=16)
There are two ways to make Lightcurve objects. We'll show one way here. Check out "Lightcurve/Lightcurve\ tutorial.ipynb" for more examples.
Generate an array of relative timestamps that's 8 seconds long, with dt = 0.03125 s, and make two signals in units of counts. The signal is a sine wave with amplitude = 300 cts/s, frequency = 2 Hz, phase offset = 0 radians, and mean = 1000 cts/s. We then add Poisson noise to the light curve.
In [2]:
dt = 0.03125 # seconds
exposure = 8. # seconds
times = np.arange(0, exposure, dt) # seconds
signal = 300 * np.sin(2.*np.pi*times/0.5) + 1000 # counts/s
noisy = np.random.poisson(signal*dt) # counts
Now let's turn noisy into a Lightcurve object.
In [3]:
lc = Lightcurve(times, noisy)
Here we plot it to see what it looks like.
In [4]:
fig, ax = plt.subplots(1,1,figsize=(10,6))
ax.plot(lc.time, lc.counts, lw=2, color='blue')
ax.set_xlabel("Time (s)", fontproperties=font_prop)
ax.set_ylabel("Counts (cts)", fontproperties=font_prop)
ax.tick_params(axis='x', labelsize=16)
ax.tick_params(axis='y', labelsize=16)
ax.tick_params(which='major', width=1.5, length=7)
ax.tick_params(which='minor', width=1.5, length=4)
plt.show()
In [5]:
ps = Powerspectrum(lc)
print(ps)
Since the negative Fourier frequencies (and their associated powers) are discarded, the number of time bins per segment n is twice the length of freq and power.
In [6]:
print("\nSize of positive Fourier frequencies:", len(ps.freq))
print("Number of data points per segment:", ps.n)
A Powerspectrum object has the following properties :
freq : Numpy array of mid-bin frequencies that the Fourier transform samples.power : Numpy array of the power spectrum.df : The frequency resolution.m : The number of power spectra averaged together. For a Powerspectrum of a single segment, m=1.n : The number of data points (time bins) in one segment of the light curve.nphots1 : The total number of photons in the light curve.
In [7]:
print(ps.freq)
print(ps.power)
print(ps.df)
print(ps.m)
print(ps.n)
print(ps.nphots1)
We can plot the power as a function of Fourier frequency. Notice how there's a spike at our signal frequency of 2 Hz!
In [8]:
fig, ax1 = plt.subplots(1,1,figsize=(9,6), sharex=True)
ax1.plot(ps.freq, ps.power, lw=2, color='blue')
ax1.set_ylabel("Frequency (Hz)", fontproperties=font_prop)
ax1.set_ylabel("Power (raw)", fontproperties=font_prop)
ax1.set_yscale('log')
ax1.tick_params(axis='x', labelsize=16)
ax1.tick_params(axis='y', labelsize=16)
ax1.tick_params(which='major', width=1.5, length=7)
ax1.tick_params(which='minor', width=1.5, length=4)
for axis in ['top', 'bottom', 'left', 'right']:
ax1.spines[axis].set_linewidth(1.5)
plt.show()
You'll notice that the power spectrum is a bit noisy. This is because we're only using one segment of data. Let's try averaging together power spectra from multiple segments of data.
You could use a long Lightcurve and have AveragedPowerspectrum chop it into specified segments, or give a list of Lightcurves where each segment of Lightcurve is the same length. We'll show the first way here.
Generate an array of relative timestamps that's 1600 seconds long, and a signal in count units, with the same properties as the previous example. We then add Poisson noise and turn it into a Lightcurve object.
In [9]:
long_dt = 0.03125 # seconds
long_exposure = 1600. # seconds
long_times = np.arange(0, long_exposure, long_dt) # seconds
# In count rate units here
long_signal = 300 * np.sin(2.*np.pi*long_times/0.5) + 1000
# Multiply by dt to get count units, then add Poisson noise
long_noisy = np.random.poisson(long_signal*dt)
long_lc = Lightcurve(long_times, long_noisy)
fig, ax = plt.subplots(1,1,figsize=(10,6))
ax.plot(long_lc.time, long_lc.counts, lw=2, color='blue')
ax.set_xlim(0,20)
ax.set_xlabel("Time (s)", fontproperties=font_prop)
ax.set_ylabel("Counts (cts)", fontproperties=font_prop)
ax.tick_params(axis='x', labelsize=16)
ax.tick_params(axis='y', labelsize=16)
ax.tick_params(which='major', width=1.5, length=7)
ax.tick_params(which='minor', width=1.5, length=4)
plt.show()
In [10]:
avg_ps = AveragedPowerspectrum(long_lc, 8.)
We can check how many segments were averaged together by printing the m attribute.
In [11]:
print("Number of segments: %d" % avg_ps.m)
AveragedPowerspectrum has the same properties as Powerspectrum, but with m $>$1.
Let's plot the averaged power spectrum!
In [12]:
fig, ax1 = plt.subplots(1,1,figsize=(9,6))
ax1.plot(avg_ps.freq, avg_ps.power, lw=2, color='blue')
ax1.set_xlabel("Frequency (Hz)", fontproperties=font_prop)
ax1.set_ylabel("Power (raw)", fontproperties=font_prop)
ax1.set_yscale('log')
ax1.tick_params(axis='x', labelsize=16)
ax1.tick_params(axis='y', labelsize=16)
ax1.tick_params(which='major', width=1.5, length=7)
ax1.tick_params(which='minor', width=1.5, length=4)
for axis in ['top', 'bottom', 'left', 'right']:
ax1.spines[axis].set_linewidth(1.5)
plt.show()
Now we'll show examples of all the things you can do with a Powerspectrum or AveragedPowerspectrum object using built-in stingray methods.
The three kinds of normalization are:
leahy: Leahy normalization. Makes the Poisson noise level $= 2$. See Leahy et al. 1983, ApJ, 266, 160L. frac: Fractional rms-squared normalization, also known as rms normalization. Makes the Poisson noise level $= 2 / meanrate$. See Belloni & Hasinger 1990, A&A, 227, L33, and Miyamoto et al. 1992, ApJ, 391, L21.abs: Absolute rms-squared normalization, also known as absolute normalization. Makes the Poisson noise level $= 2 \times meanrate$. See insert citation.none: No normalization applied. This is the default.
In [13]:
avg_ps_leahy = AveragedPowerspectrum(long_lc, 8, norm='leahy')
avg_ps_frac = AveragedPowerspectrum(long_lc, 8., norm='frac')
avg_ps_abs = AveragedPowerspectrum(long_lc, 8., norm='abs')
In [14]:
fig, [ax1, ax2, ax3] = plt.subplots(3,1,figsize=(6,12))
ax1.plot(avg_ps_leahy.freq, avg_ps_leahy.power, lw=2, color='black')
ax1.set_xlabel("Frequency (Hz)", fontproperties=font_prop)
ax1.set_ylabel("Power (Leahy)", fontproperties=font_prop)
ax1.set_yscale('log')
ax1.tick_params(axis='x', labelsize=14)
ax1.tick_params(axis='y', labelsize=14)
ax1.tick_params(which='major', width=1.5, length=7)
ax1.tick_params(which='minor', width=1.5, length=4)
ax1.set_title("Leahy norm.", fontproperties=font_prop)
ax2.plot(avg_ps_frac.freq, avg_ps_frac.power, lw=2, color='black')
ax2.set_xlabel("Frequency (Hz)", fontproperties=font_prop)
ax2.set_ylabel("Power (rms)", fontproperties=font_prop)
ax2.tick_params(axis='x', labelsize=14)
ax2.tick_params(axis='y', labelsize=14)
ax2.set_yscale('log')
ax2.tick_params(which='major', width=1.5, length=7)
ax2.tick_params(which='minor', width=1.5, length=4)
ax2.set_title("Fractional rms-squared norm.", fontproperties=font_prop)
ax3.plot(avg_ps_abs.freq, avg_ps_abs.power, lw=2, color='black')
ax3.set_xlabel("Frequency (Hz)", fontproperties=font_prop)
ax3.set_ylabel("Power (abs)", fontproperties=font_prop)
ax3.tick_params(axis='x', labelsize=14)
ax3.tick_params(axis='y', labelsize=14)
ax3.set_yscale('log')
ax3.tick_params(which='major', width=1.5, length=7)
ax3.tick_params(which='minor', width=1.5, length=4)
ax3.set_title("Absolute rms-squared norm.", fontproperties=font_prop)
for axis in ['top', 'bottom', 'left', 'right']:
ax1.spines[axis].set_linewidth(1.5)
ax2.spines[axis].set_linewidth(1.5)
ax3.spines[axis].set_linewidth(1.5)
plt.tight_layout()
plt.show()
In [15]:
print("DF before:", avg_ps.df)
# Both of the following ways are allowed syntax:
# lin_rb_ps = Powerspectrum.rebin(avg_ps, 0.25, method='mean')
lin_rb_ps = avg_ps.rebin(0.25, method='mean')
print("DF after:", lin_rb_ps.df)
In [16]:
# Both of the following ways are allowed syntax:
# log_rb_ps, log_rb_freq, binning = Powerspectrum.rebin_log(avg_ps, f=0.02)
log_rb_ps = ps.rebin_log(f=0.02)
Like rebin, rebin_log returns a Powerspectrum or AveragedPowerspectrum object (depending on the input object):
In [17]:
print(type(lin_rb_ps))