"""The suite of window functions.""" from __future__ import division, print_function, absolute_import import numpy as np from scipy import special, linalg from scipy.fftpack import fft __all__ = ['boxcar', 'triang', 'parzen', 'bohman', 'blackman', 'nuttall', 'blackmanharris', 'flattop', 'bartlett', 'hanning', 'barthann', 'hamming', 'kaiser', 'gaussian', 'general_gaussian', 'chebwin', 'slepian', 'cosine', 'hann', 'get_window'] def boxcar(M, sym=True): """Return a boxcar or rectangular window. Included for completeness, this is equivalent to no window at all. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional Whether the window is symmetric. (Has no effect for boxcar.) Returns ------- w : ndarray The window, with the maximum value normalized to 1. Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.boxcar(51) >>> plt.plot(window) >>> plt.title("Boxcar 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 boxcar window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ return np.ones(M, float) def triang(M, sym=True): """Return a triangular window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.triang(51) >>> plt.plot(window) >>> plt.title("Triangular 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 triangular window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(1, (M + 1) // 2 + 1) if M % 2 == 0: w = (2 * n - 1.0) / M w = np.r_[w, w[::-1]] else: w = 2 * n / (M + 1.0) w = np.r_[w, w[-2::-1]] if not sym and not odd: w = w[:-1] return w def parzen(M, sym=True): """Return a Parzen window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.parzen(51) >>> plt.plot(window) >>> plt.title("Parzen 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 Parzen window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(-(M - 1) / 2.0, (M - 1) / 2.0 + 0.5, 1.0) na = np.extract(n < -(M - 1) / 4.0, n) nb = np.extract(abs(n) <= (M - 1) / 4.0, n) wa = 2 * (1 - np.abs(na) / (M / 2.0)) ** 3.0 wb = (1 - 6 * (np.abs(nb) / (M / 2.0)) ** 2.0 + 6 * (np.abs(nb) / (M / 2.0)) ** 3.0) w = np.r_[wa, wb, wa[::-1]] if not sym and not odd: w = w[:-1] return w def bohman(M, sym=True): """Return a Bohman window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.bohman(51) >>> plt.plot(window) >>> plt.title("Bohman 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 Bohman window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 fac = np.abs(np.linspace(-1, 1, M)[1:-1]) w = (1 - fac) * np.cos(np.pi * fac) + 1.0 / np.pi * np.sin(np.pi * fac) w = np.r_[0, w, 0] if not sym and not odd: w = w[:-1] return w def blackman(M, sym=True): r""" Return a Blackman window. The Blackman window is a taper formed by using 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 Number of points in the output window. If zero or less, an empty array is returned. 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 Blackman window is defined as .. math:: w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M) 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 ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471. Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.blackman(51) >>> plt.plot(window) >>> plt.title("Blackman 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 Blackman window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ # Docstring adapted from NumPy's blackman function if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) w = (0.42 - 0.5 * np.cos(2.0 * np.pi * n / (M - 1)) + 0.08 * np.cos(4.0 * np.pi * n / (M - 1))) if not sym and not odd: w = w[:-1] return w def nuttall(M, sym=True): """Return a minimum 4-term Blackman-Harris window according to Nuttall. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.nuttall(51) >>> plt.plot(window) >>> plt.title("Nuttall 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 Nuttall window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 a = [0.3635819, 0.4891775, 0.1365995, 0.0106411] n = np.arange(0, M) fac = n * 2 * np.pi / (M - 1.0) w = (a[0] - a[1] * np.cos(fac) + a[2] * np.cos(2 * fac) - a[3] * np.cos(3 * fac)) if not sym and not odd: w = w[:-1] return w def blackmanharris(M, sym=True): """Return a minimum 4-term Blackman-Harris window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.blackmanharris(51) >>> plt.plot(window) >>> plt.title("Blackman-Harris 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 Blackman-Harris window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 a = [0.35875, 0.48829, 0.14128, 0.01168] n = np.arange(0, M) fac = n * 2 * np.pi / (M - 1.0) w = (a[0] - a[1] * np.cos(fac) + a[2] * np.cos(2 * fac) - a[3] * np.cos(3 * fac)) if not sym and not odd: w = w[:-1] return w def flattop(M, sym=True): """Return a flat top window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.flattop(51) >>> plt.plot(window) >>> plt.title("Flat top 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 flat top window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 a = [0.2156, 0.4160, 0.2781, 0.0836, 0.0069] n = np.arange(0, M) fac = n * 2 * np.pi / (M - 1.0) w = (a[0] - a[1] * np.cos(fac) + a[2] * np.cos(2 * fac) - a[3] * np.cos(3 * fac) + a[4] * np.cos(4 * fac)) if not sym and not odd: w = w[:-1] return w def bartlett(M, sym=True): r""" Return a Bartlett window. 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. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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 triangular window, with the first and last samples equal to zero and the maximum value normalized to 1 (though the value 1 does not appear if M is even and sym is True). Notes ----- The Bartlett window is defined as .. math:: 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. References ---------- .. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", Biometrika 37, 1-16, 1950. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal Processing", Prentice-Hall, 1999, pp. 468-471. .. [4] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 429. Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.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]") """ # Docstring adapted from NumPy's bartlett function if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) w = np.where(np.less_equal(n, (M - 1) / 2.0), 2.0 * n / (M - 1), 2.0 - 2.0 * n / (M - 1)) if not sym and not odd: w = w[:-1] return w def hann(M, sym=True): r""" Return a Hann window. The Hann window is a taper formed by using a raised cosine or sine-squared with ends that touch zero. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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 Hann window is defined as .. math:: w(n) = 0.5 - 0.5 \cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1 The window was named for Julius van Hann, an Austrian meteorologist. It is also known as the Cosine Bell. It is sometimes erroneously referred to as the "Hanning" window, from the use of "hann" as a verb in the original paper and confusion with the very similar Hamming window. Most references to the Hann 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. References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 106-108. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425. Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.hann(51) >>> plt.plot(window) >>> plt.title("Hann 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 Hann window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ # Docstring adapted from NumPy's hanning function if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) w = 0.5 - 0.5 * np.cos(2.0 * np.pi * n / (M - 1)) if not sym and not odd: w = w[:-1] return w hanning = hann def barthann(M, sym=True): """Return a modified Bartlett-Hann window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.barthann(51) >>> plt.plot(window) >>> plt.title("Bartlett-Hann 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-Hann window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) fac = np.abs(n / (M - 1.0) - 0.5) w = 0.62 - 0.48 * fac + 0.38 * np.cos(2 * np.pi * fac) if not sym and not odd: w = w[:-1] return w def hamming(M, sym=True): r"""Return a Hamming window. The Hamming window is a taper formed by using a raised cosine with non-zero endpoints, optimized to minimize the nearest side lobe. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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 Hamming window is defined as .. math:: w(n) = 0.54 - 0.46 \cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1 The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and is described in Blackman and Tukey. It was recommended for smoothing the truncated autocovariance function in the time domain. Most references to the Hamming 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. References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425. Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.hamming(51) >>> plt.plot(window) >>> plt.title("Hamming 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 Hamming window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ # Docstring adapted from NumPy's hamming function if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) w = 0.54 - 0.46 * np.cos(2.0 * np.pi * n / (M - 1)) if not sym and not odd: w = w[:-1] return w def kaiser(M, beta, sym=True): r"""Return a Kaiser window. The Kaiser window is a taper formed by using a Bessel function. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. beta : float Shape parameter, determines trade-off between main-lobe width and side lobe level. As beta gets large, the window narrows. 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 Kaiser window is defined as .. math:: w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}} \right)/I_0(\beta) with .. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2}, where :math:I_0 is the modified zeroth-order Bessel function. The Kaiser was named for Jim Kaiser, who discovered a simple approximation to the DPSS window based on Bessel functions. The Kaiser window is a very good approximation to the Digital Prolate Spheroidal Sequence, or Slepian window, which is the transform which maximizes the energy in the main lobe of the window relative to total energy. The Kaiser can approximate many other windows by varying the beta parameter. ==== ======================= beta Window shape ==== ======================= 0 Rectangular 5 Similar to a Hamming 6 Similar to a Hann 8.6 Similar to a Blackman ==== ======================= A beta value of 14 is probably a good starting point. Note that as beta gets large, the window narrows, and so the number of samples needs to be large enough to sample the increasingly narrow spike, otherwise NaNs will get returned. Most references to the Kaiser 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. References ---------- .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285. John Wiley and Sons, New York, (1966). .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 177-178. .. [3] Wikipedia, "Window function", http://en.wikipedia.org/wiki/Window_function Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.kaiser(51, beta=14) >>> plt.plot(window) >>> plt.title(r"Kaiser window ($\beta$=14)") >>> 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(r"Frequency response of the Kaiser window ($\beta$=14)") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ # Docstring adapted from NumPy's kaiser function if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) alpha = (M - 1) / 2.0 w = (special.i0(beta * np.sqrt(1 - ((n - alpha) / alpha) ** 2.0)) / special.i0(beta)) if not sym and not odd: w = w[:-1] return w def gaussian(M, std, sym=True): r"""Return a Gaussian window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. std : float The standard deviation, sigma. 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 Gaussian window is defined as .. math:: w(n) = e^{ -\frac{1}{2}\left(\frac{n}{\sigma}\right)^2 } Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.gaussian(51, std=7) >>> plt.plot(window) >>> plt.title(r"Gaussian window ($\sigma$=7)") >>> 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(r"Frequency response of the Gaussian window ($\sigma$=7)") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) - (M - 1.0) / 2.0 sig2 = 2 * std * std w = np.exp(-n ** 2 / sig2) if not sym and not odd: w = w[:-1] return w def general_gaussian(M, p, sig, sym=True): r"""Return a window with a generalized Gaussian shape. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. p : float Shape parameter. p = 1 is identical to gaussian, p = 0.5 is the same shape as the Laplace distribution. sig : float The standard deviation, sigma. 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 generalized Gaussian window is defined as .. math:: w(n) = e^{ -\frac{1}{2}\left|\frac{n}{\sigma}\right|^{2p} } the half-power point is at .. math:: (2 \log(2))^{1/(2 p)} \sigma Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.general_gaussian(51, p=1.5, sig=7) >>> plt.plot(window) >>> plt.title(r"Generalized Gaussian window (p=1.5, $\sigma$=7)") >>> 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(r"Freq. resp. of the gen. Gaussian window (p=1.5, $\sigma$=7)") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 n = np.arange(0, M) - (M - 1.0) / 2.0 w = np.exp(-0.5 * np.abs(n / sig) ** (2 * p)) if not sym and not odd: w = w[:-1] return w # chebwin contributed by Kumar Appaiah. def chebwin(M, at, sym=True): r"""Return a Dolph-Chebyshev window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. at : float Attenuation (in dB). 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 always normalized to 1 Notes ----- This window optimizes for the narrowest main lobe width for a given order M and sidelobe equiripple attenuation at, using Chebyshev polynomials. It was originally developed by Dolph to optimize the directionality of radio antenna arrays. Unlike most windows, the Dolph-Chebyshev is defined in terms of its frequency response: .. math:: W(k) = \frac {\cos\{M \cos^{-1}[\beta \cos(\frac{\pi k}{M})]\}} {\cosh[M \cosh^{-1}(\beta)]} where .. math:: \beta = \cosh \left [\frac{1}{M} \cosh^{-1}(10^\frac{A}{20}) \right ] and 0 <= abs(k) <= M-1. A is the attenuation in decibels (at). The time domain window is then generated using the IFFT, so power-of-two M are the fastest to generate, and prime number M are the slowest. The equiripple condition in the frequency domain creates impulses in the time domain, which appear at the ends of the window. References ---------- .. [1] C. Dolph, "A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level", Proceedings of the IEEE, Vol. 34, Issue 6 .. [2] Peter Lynch, "The Dolph-Chebyshev Window: A Simple Optimal Filter", American Meteorological Society (April 1997) http://mathsci.ucd.ie/~plynch/Publications/Dolph.pdf .. [3] F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transforms", Proceedings of the IEEE, Vol. 66, No. 1, January 1978 Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.chebwin(51, at=100) >>> plt.plot(window) >>> plt.title("Dolph-Chebyshev window (100 dB)") >>> 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 Dolph-Chebyshev window (100 dB)") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 # compute the parameter beta order = M - 1.0 beta = np.cosh(1.0 / order * np.arccosh(10 ** (np.abs(at) / 20.))) k = np.r_[0:M] * 1.0 x = beta * np.cos(np.pi * k / M) # Find the window's DFT coefficients # Use analytic definition of Chebyshev polynomial instead of expansion # from scipy.special. Using the expansion in scipy.special leads to errors. p = np.zeros(x.shape) p[x > 1] = np.cosh(order * np.arccosh(x[x > 1])) p[x < -1] = (1 - 2 * (order % 2)) * np.cosh(order * np.arccosh(-x[x < -1])) p[np.abs(x) <= 1] = np.cos(order * np.arccos(x[np.abs(x) <= 1])) # Appropriate IDFT and filling up # depending on even/odd M if M % 2: w = np.real(fft(p)) n = (M + 1) // 2 w = w[:n] / w[0] w = np.concatenate((w[n - 1:0:-1], w)) else: p = p * np.exp(1.j * np.pi / M * np.r_[0:M]) w = np.real(fft(p)) n = M // 2 + 1 w = w / w[1] w = np.concatenate((w[n - 1:0:-1], w[1:n])) if not sym and not odd: w = w[:-1] return w def slepian(M, width, sym=True): """Return a digital Slepian (DPSS) window. Used to maximize the energy concentration in the main lobe. Also called the digital prolate spheroidal sequence (DPSS). Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. width : float Bandwidth 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 always normalized to 1 Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.slepian(51, width=0.3) >>> plt.plot(window) >>> plt.title("Slepian (DPSS) window (BW=0.3)") >>> 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 Slepian window (BW=0.3)") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") """ if (M * width > 27.38): raise ValueError("Cannot reliably obtain Slepian sequences for" " M*width > 27.38.") if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 twoF = width / 2.0 alpha = (M - 1) / 2.0 m = np.arange(0, M) - alpha n = m[:, np.newaxis] k = m[np.newaxis, :] AF = twoF * special.sinc(twoF * (n - k)) [lam, vec] = linalg.eig(AF) ind = np.argmax(abs(lam), axis=-1) w = np.abs(vec[:, ind]) w = w / max(w) if not sym and not odd: w = w[:-1] return w def cosine(M, sym=True): """Return a window with a simple cosine shape. .. versionadded:: 0.13.0 Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. 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). Examples -------- Plot the window and its frequency response: >>> from scipy import signal >>> from scipy.fftpack import fft, fftshift >>> import matplotlib.pyplot as plt >>> window = signal.cosine(51) >>> plt.plot(window) >>> plt.title("Cosine 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 cosine window") >>> plt.ylabel("Normalized magnitude [dB]") >>> plt.xlabel("Normalized frequency [cycles per sample]") >>> plt.show() """ if M < 1: return np.array([]) if M == 1: return np.ones(1, 'd') odd = M % 2 if not sym and not odd: M = M + 1 w = np.sin(np.pi / M * (np.arange(0, M) + .5)) if not sym and not odd: w = w[:-1] return w def get_window(window, Nx, fftbins=True): """ Return a window. Parameters ---------- window : string, float, or tuple The type of window to create. See below for more details. Nx : int The number of samples in the window. fftbins : bool, optional If True, create a "periodic" window ready to use with ifftshift and be multiplied by the result of an fft (SEE ALSO fftfreq). Returns ------- get_window : ndarray Returns a window of length Nx and type window Notes ----- Window types: boxcar, triang, blackman, hamming, hann, bartlett, flattop, parzen, bohman, blackmanharris, nuttall, barthann, kaiser (needs beta), gaussian (needs std), general_gaussian (needs power, width), slepian (needs width), chebwin (needs attenuation) If the window requires no parameters, then window can be a string. If the window requires parameters, then window must be a tuple with the first argument the string name of the window, and the next arguments the needed parameters. If window is a floating point number, it is interpreted as the beta parameter of the kaiser window. Each of the window types listed above is also the name of a function that can be called directly to create a window of that type. Examples -------- >>> from scipy import signal >>> signal.get_window('triang', 7) array([ 0.25, 0.5 , 0.75, 1. , 0.75, 0.5 , 0.25]) >>> signal.get_window(('kaiser', 4.0), 9) array([ 0.08848053, 0.32578323, 0.63343178, 0.89640418, 1. , 0.89640418, 0.63343178, 0.32578323, 0.08848053]) >>> signal.get_window(4.0, 9) array([ 0.08848053, 0.32578323, 0.63343178, 0.89640418, 1. , 0.89640418, 0.63343178, 0.32578323, 0.08848053]) """ sym = not fftbins try: beta = float(window) except (TypeError, ValueError): args = () if isinstance(window, tuple): winstr = window[0] if len(window) > 1: args = window[1:] elif isinstance(window, str): if window in ['kaiser', 'ksr', 'gaussian', 'gauss', 'gss', 'general gaussian', 'general_gaussian', 'general gauss', 'general_gauss', 'ggs', 'slepian', 'optimal', 'slep', 'dss', 'chebwin', 'cheb']: raise ValueError("The '" + window + "' window needs one or " "more parameters -- pass a tuple.") else: winstr = window if winstr in ['blackman', 'black', 'blk']: winfunc = blackman elif winstr in ['triangle', 'triang', 'tri']: winfunc = triang elif winstr in ['hamming', 'hamm', 'ham']: winfunc = hamming elif winstr in ['bartlett', 'bart', 'brt']: winfunc = bartlett elif winstr in ['hanning', 'hann', 'han']: winfunc = hann elif winstr in ['blackmanharris', 'blackharr', 'bkh']: winfunc = blackmanharris elif winstr in ['parzen', 'parz', 'par']: winfunc = parzen elif winstr in ['bohman', 'bman', 'bmn']: winfunc = bohman elif winstr in ['nuttall', 'nutl', 'nut']: winfunc = nuttall elif winstr in ['barthann', 'brthan', 'bth']: winfunc = barthann elif winstr in ['flattop', 'flat', 'flt']: winfunc = flattop elif winstr in ['kaiser', 'ksr']: winfunc = kaiser elif winstr in ['gaussian', 'gauss', 'gss']: winfunc = gaussian elif winstr in ['general gaussian', 'general_gaussian', 'general gauss', 'general_gauss', 'ggs']: winfunc = general_gaussian elif winstr in ['boxcar', 'box', 'ones', 'rect', 'rectangular']: winfunc = boxcar elif winstr in ['slepian', 'slep', 'optimal', 'dpss', 'dss']: winfunc = slepian elif winstr in ['cosine', 'halfcosine']: winfunc = cosine elif winstr in ['chebwin', 'cheb']: winfunc = chebwin else: raise ValueError("Unknown window type.") params = (Nx,) + args + (sym,) else: winfunc = kaiser params = (Nx, beta, sym) return winfunc(*params)