signal.slepian should use definition of digital window

It’s now off by a factor of 2: http://dsp.stackexchange.com/questions/2577/a-few-questions-about-slepian-and-generalized-gaussian-windows

The relation between width and the beta parameter to kaiser should be documented. Here’s a comparison for Matlab, there it looks like width = pi * beta: https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html

There’s a factor 2 between the analog and digital versions, maybe the original implementation worked from a description of the analog window?

See discussion on gh-3679.

[rgommers]

dpss sounds fine to me. maybe better indeed to deprecate slepian.

[endolith]

@Eric89GXL I’m not sure. It doesn’t produce an amplitude-normalized output, so we’d still need to figure that out?

By trial and error, these values all produce similar plots:

• signal.slepian: width=0.3
• signal.kaiser: beta=11.2
• nitime.algorithms.spectral.dpss_windows: NW=3.82

I’m not sure what the relationship is.

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

dpss = nitime.algorithms.spectral.dpss_windows(51, 3.82, 1)[0][0]
dpss /= np.amax(dpss)

for window, label in ((signal.slepian(51, width=0.3), 'signal.slepian'),
                      (dpss, 'nitime...dpss_windows'),
                      (signal.kaiser(51, beta=11.2), 'signal.kaiser')):

    plt.figure('window')
    plt.plot(window, label=label)
    plt.title("Window function")
    plt.ylabel("Amplitude")
    plt.xlabel("Sample")
    plt.legend()

    plt.figure('spectrum')
    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, label=label)
    plt.axis([-0.5, 0.5, -130, 0])
    plt.title("Frequency response")
    plt.ylabel("Normalized magnitude [dB]")
    plt.xlabel("Normalized frequency [cycles per sample]")
    plt.legend()