# scipy.signal.windows.exponential

scipy.signal.windows.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, an empty array is returned. An exception is thrown when it is negative.

    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

  [1]

  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:

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

      >>> M = 51
      >>> tau = 3.0
      >>> window = signal.windows.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.windows.exponential(M, 0, tau2, False)
      >>> plt.figure()
      >>> plt.plot(window2)
      >>> plt.ylabel("Amplitude")
      >>> plt.xlabel("Sample")