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# scipy.signal.exponential

`scipy.signal.``exponential`(M, center=None, tau=1.0, sym=True)

  Return an exponential (or Poisson) window.

  Parameters: | M : int Number of points in the output window. If zero or less, an empty array is returned. center : float, optional Parameter defining the center location of the window function. The default value if not given is `center = (M-1) / 2`. This parameter must take its default value for symmetric windows. tau : float, optional Parameter defining the decay. For `center = 0` use `tau = -(M-1) / ln(x)` if `x` is the fraction of the window remaining at the end. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.
  Returns: | w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if M is even and sym is True).

  Notes

  The Exponential window is defined as

  \[w(n) = e^{-|n-center| / \tau}\]

  References

  S. Gade and H. Herlufsen, “Windows to FFT analysis (Part I)”, Technical Review 3, Bruel & Kjaer, 1987.

  Examples

  Plot the symmetric window and its frequency response:

      >>> from scipy import signal
      >>> from scipy.fftpack import fft, fftshift
      >>> import matplotlib.pyplot as plt

      >>> M = 51
      >>> tau = 3.0
      >>> window = signal.exponential(M, tau=tau)
      >>> plt.plot(window)
      >>> plt.title("Exponential Window (tau=3.0)")
      >>> plt.ylabel("Amplitude")
      >>> plt.xlabel("Sample")

      >>> plt.figure()
      >>> A = fft(window, 2048) / (len(window)/2.0)
      >>> freq = np.linspace(-0.5, 0.5, len(A))
      >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
      >>> plt.plot(freq, response)
      >>> plt.axis([-0.5, 0.5, -35, 0])
      >>> plt.title("Frequency response of the Exponential window (tau=3.0)")
      >>> plt.ylabel("Normalized magnitude [dB]")
      >>> plt.xlabel("Normalized frequency [cycles per sample]")

  This function can also generate non-symmetric windows:

      >>> tau2 = -(M-1) / np.log(0.01)
      >>> window2 = signal.exponential(M, 0, tau2, False)
      >>> plt.figure()
      >>> plt.plot(window2)
      >>> plt.ylabel("Amplitude")
      >>> plt.xlabel("Sample")

  (Source code)