Example 2: Noisy Transient Signal

The second introductory example is a transient signal embedded in a -5 dB white gaussian noise. This transient signal is a constant frequency modulated by a one-sided exponential amplitude. The signal and its spectrum are generated as follows:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from tftb.generators import amexpos, fmconst, sigmerge, noisecg
>>>
>>> # Generate a noisy transient signal.
>>> transsig = amexpos(64, kind='unilateral') * fmconst(64)[0]
>>> signal = np.hstack((np.zeros((100,)), transsig, np.zeros((92,))))
>>> signal = sigmerge(signal, noisecg(256), -5)
>>> fig, ax = plt.subplots(2, 1)
>>> ax1, ax2 = ax
>>> ax1.plot(np.real(signal))
>>> ax1.grid()
>>> ax1.set_title('Noisy Transient Signal')
>>> ax1.set_xlabel('Time')
>>> ax1.set_xlim((0, 256))
>>> ax1.set_ylim((np.real(signal).max(), np.real(signal.min())))
>>>
>>> # Energy spectrum of the signal
>>> dsp = np.fft.fftshift(np.abs(np.fft.fft(signal)) ** 2)
>>> ax2.plot(np.arange(-128, 128, dtype=float) / 256, dsp)
>>> ax2.set_title('Energy spectrum of noisy transient signal')
>>> ax2.set_xlabel('Normalized frequency')
>>> ax2.grid()
>>> ax2.set_xlim(-0.5, 0.5)
>>>
>>> plt.subplots_adjust(hspace=0.5)
>>>
>>> plt.show()

(Source code, png, hires.png, pdf)

From these representations, it is difficult to localize precisely the signal in the time-domain as well as in the frequency domain. Now let us have a look at the spectrogram of this signal:

>>> from scipy.signal import hamming
>>> from tftb.processing import Spectrogram
>>> fwindow = hamming(65)
>>> spec = Spectrogram(signal, n_fbins=128, fwindow=fwindow)
>>> spec.run()
>>> spec.plot(kind="contour", threshold=0.1, show_tf=False)

(Source code, png, hires.png, pdf)

the transient signal appears distinctly around the normalized frequency 0.25, and between time points 125 and 160.