# jax.numpy.bartlettÂ¶

jax.numpy.bartlett(*args, **kwargs)Â¶

Return the Bartlett window.

LAX-backend implementation of bartlett(). Original docstring below.

The Bartlett window is very similar to a triangular window, except that the end points are at zero. It is often used in signal processing for tapering a signal, without generating too much ripple in the frequency domain.

Mint

Number of points in the output window. If zero or less, an empty array is returned.

outarray

The triangular window, with the maximum value normalized to one (the value one appears only if the number of samples is odd), with the first and last samples equal to zero.

blackman, hamming, hanning, kaiser

The Bartlett window is defined as

$w(n) = \frac{2}{M-1} \left( \frac{M-1}{2} - \left|n - \frac{M-1}{2}\right| \right)$

Most references to the Bartlett window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. Note that convolution with this window produces linear interpolation. It is also known as an apodization (which meansâ€ťremoving the footâ€ť, i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. The fourier transform of the Bartlett is the product of two sinc functions. Note the excellent discussion in Kanasewich.

1

M.S. Bartlett, â€śPeriodogram Analysis and Continuous Spectraâ€ť, Biometrika 37, 1-16, 1950.

2

E.R. Kanasewich, â€śTime Sequence Analysis in Geophysicsâ€ť, The University of Alberta Press, 1975, pp. 109-110.

3

A.V. Oppenheim and R.W. Schafer, â€śDiscrete-Time Signal Processingâ€ť, Prentice-Hall, 1999, pp. 468-471.

4

Wikipedia, â€śWindow functionâ€ť, https://en.wikipedia.org/wiki/Window_function

5

W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, â€śNumerical Recipesâ€ť, Cambridge University Press, 1986, page 429.

>>> import matplotlib.pyplot as plt
>>> np.bartlett(12)
array([ 0.        ,  0.18181818,  0.36363636,  0.54545455,  0.72727273, # may vary
0.90909091,  0.90909091,  0.72727273,  0.54545455,  0.36363636,
0.18181818,  0.        ])


Plot the window and its frequency response (requires SciPy and matplotlib):

>>> from numpy.fft import fft, fftshift
>>> window = np.bartlett(51)
>>> plt.plot(window)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Bartlett window")
Text(0.5, 1.0, 'Bartlett window')
>>> plt.ylabel("Amplitude")
Text(0, 0.5, 'Amplitude')
>>> plt.xlabel("Sample")
Text(0.5, 0, 'Sample')
>>> plt.show()

>>> plt.figure()
<Figure size 640x480 with 0 Axes>
>>> A = fft(window, 2048) / 25.5
>>> mag = np.abs(fftshift(A))
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> with np.errstate(divide='ignore', invalid='ignore'):
...     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 Bartlett window")
Text(0.5, 1.0, 'Frequency response of Bartlett window')
>>> plt.ylabel("Magnitude [dB]")
Text(0, 0.5, 'Magnitude [dB]')
>>> plt.xlabel("Normalized frequency [cycles per sample]")
Text(0.5, 0, 'Normalized frequency [cycles per sample]')
>>> _ = plt.axis('tight')
>>> plt.show()