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.

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 means"removing the foot", 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.

Return a Bartlett window.

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

window = signal.windows.bartlett(51)
plt.plot(window)
plt.title("Bartlett window")
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, -120, 0])
plt.title("Frequency response of the Bartlett window")
plt.ylabel("Normalized magnitude [dB]")
plt.xlabel("Normalized frequency [cycles per sample]")