Example 1: Linear Frequency Modulation

Let us consider first a signal with constant amplitude, and with a linear frequency modulation - i.e. a signal such that its amplitude remains constant, but frequency increases linearly with time - varying from 0 to 0.5 in normalized frequency (ratio of the frequency in Hertz to the sampling frequency, with respect to the Shannon sampling theorem). This signal is called a chirp, and as its frequency content is varying with time, it is a non-stationary signal. To obtain such a signal, we can use the function tftb.generators.fmlin, which generates a linear frequency modulation.

>>> from tftb.generators import amgauss, fmlin
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> z = amgauss(128, 50, 40) * fmlin(128, 0.05, 0.3, 50)[0]
>>> plt.plot(np.real(z))
>>> plt.title("Linear Frequency Modulation")
>>> plt.show()

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From this time-domain representation, it is difficult to say what kind of modulation is contained in this signal: what are the initial and final frequencies, is it a linear, parabolic, hyperbolic… frequency modulation?

If we now consider the energy spectrum of the signal z by squaring the modulus of its Fourier transform (using the numpy.fft.fft function):

>>> import numpy as np
>>> dsp1 = np.fft.fftshift(np.abs(np.fft.fft(z)) ** 2)
>>> plt.plot(np.arange(-64, 64, dtype=float) / 128.0, dsp1)
>>> plt.xlim(-0.5, 0.5)
>>> plt.title('Spectrum')
>>> plt.ylabel('Squared modulus')
>>> plt.xlabel('Normalized Frequency')
>>> plt.grid()
>>> plt.show()

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we still can not say, from this plot, anything about the evolution in time of the frequency content. This is due to the fact that the Fourier transform is a decomposition on complex exponentials, which are of infinite duration and completely unlocalized in time. Time information is in fact encoded in the phase of the Fourier transform (which is simply ignored by the energy spectrum), but their interpretation is not straightforward and their direct extraction is faced with a number of difficulties such as phase unwrapping. In order to have a more informative description of such signals, it would be better to directly represent their frequency content while still keeping the time description parameter. This is precisely the aim of time-frequency analysis. To illustrate this, let us try the Wigner-Ville distribution on this signal.

>>> from tftb.processing import WignerVilleDistribution
>>> wvd = WignerVilleDistribution(z)
>>> wvd.run()
>>> wvd.plot(kind='contour', extent=[0, n_points, fmin, fmax])

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we can see that the linear progression of the frequency with time, from 0 to 0.5, is clearly shown.

If we now add some complex white gaussian noise on this signal,

>>> from tftb.generators import sigmerge, noisecg
>>> noisy_signal = sigmerge(z, noisecg(128), 0)
>>> plt.plot(np.real(noisy_signal))
>>> plt.xlim(0, 128)
>>> plt.title('Noisy chirp')
>>> plt.ylabel('Real Part')
>>> plt.xlabel('Time')
>>> plt.grid()
>>> plt.show()

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and consider the spectrum of it,

>>> dsp1 = np.fft.fftshift(np.abs(np.fft.fft(noisy_signal)) ** 2)
>>> plt.plot(np.arange(-64, 64, dtype=float) / 128.0, dsp1)
>>> plt.xlim(-0.5, 0.5)
>>> plt.title('Spectrum of Noisy Chirp')
>>> plt.ylabel('Squared modulus')
>>> plt.xlabel('Normalized Frequency')
>>> plt.grid()
>>> plt.show()

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it is worse than before to interpret these plots. On the other hand, the Wigner-Ville distribution still show quite clearly the linear progression of the frequency with time.

>>> wvd = WignerVilleDistribution(noisy_signal)
>>> wvd.run()
>>> wvd.plot(kind='contour')

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