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Example 2: Noisy Transient Signal
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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()

    .. plot:: _gallery/plot_1_3_3_transient.py

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)

    .. plot:: _gallery/plot_1_3_3_transient_spectrogram.py

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