Return the Blackman window.
The Blackman window is a taper formed by using the the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only slightly worse than a Kaiser window.
Parameters :  M : int


Returns :  out : ndarray

Notes
The Blackman window is defined as
Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means “removing the foot”, i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a “near optimal” tapering function, almost as good (by some measures) as the kaiser window.
References
Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York.
Oppenheim, A.V., and R.W. Schafer. DiscreteTime Signal Processing. Upper Saddle River, NJ: PrenticeHall, 1999, pp. 468471.
Examples
>>> np.blackman(12)
array([ 1.38777878e17, 3.26064346e02, 1.59903635e01,
4.14397981e01, 7.36045180e01, 9.67046769e01,
9.67046769e01, 7.36045180e01, 4.14397981e01,
1.59903635e01, 3.26064346e02, 1.38777878e17])
Plot the window and the frequency response:
>>> from numpy.fft import fft, fftshift
>>> window = np.blackman(51)
>>> plt.plot(window)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Blackman window")
<matplotlib.text.Text object at 0x...>
>>> plt.ylabel("Amplitude")
<matplotlib.text.Text object at 0x...>
>>> plt.xlabel("Sample")
<matplotlib.text.Text object at 0x...>
>>> plt.show()
>>> plt.figure()
<matplotlib.figure.Figure object at 0x...>
>>> A = fft(window, 2048) / 25.5
>>> mag = np.abs(fftshift(A))
>>> freq = np.linspace(0.5, 0.5, len(A))
>>> response = 20 * np.log10(mag)
>>> response = np.clip(response, 100, 100)
>>> plt.plot(freq, response)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Frequency response of Blackman window")
<matplotlib.text.Text object at 0x...>
>>> plt.ylabel("Magnitude [dB]")
<matplotlib.text.Text object at 0x...>
>>> plt.xlabel("Normalized frequency [cycles per sample]")
<matplotlib.text.Text object at 0x...>
>>> plt.axis('tight')
(0.5, 0.5, 100.0, ...)
>>> plt.show()