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dsp/window: new package for functions to control spectral leakage of FFT
This commit is contained in:
Andrei Blinnikov
325
dsp/window/window_complex.go
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325
dsp/window/window_complex.go
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// Copyright ©2020 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package window
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import "math"
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// Rectangular modifies seq in place by the Rectangular window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Rectangular_window and
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// https://www.recordingblogs.com/wiki/rectangular-window for details.
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//
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// The rectangular window has the lowest width of the main lobe and largest
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// level of the side lobes. The result corresponds to a selection of
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// limited length sequence of values without any modification.
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//
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// The sequence weights are
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// w[k] = 1,
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 2, ΔF_0.5 = 0.89, K = 1, ɣ_max = -13, β = 0.
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func RectangularComplex(seq []complex128) []complex128 {
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return seq
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}
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// SineComplex modifies seq in place by the Sine window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Sine_window and
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// https://www.recordingblogs.com/wiki/sine-window for details.
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//
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// Sine window is a high-resolution window.
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//
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// The sequence weights are
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// w[k] = sin(π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 3, ΔF_0.5 = 1.23, K = 1.5, ɣ_max = -23, β = -3.93.
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func SineComplex(seq []complex128) []complex128 {
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k := math.Pi / float64(len(seq)-1)
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for i := range seq {
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seq[i] *= complex(math.Sin(k*float64(i)), 0)
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}
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return seq
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}
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// LanczosComplex modifies seq in place by the Lanczos window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Lanczos_window and
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// https://www.recordingblogs.com/wiki/lanczos-window for details.
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//
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// The Lanczos window is a high-resolution window.
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//
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// The sequence weights are
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// w[k] = sinc(2*k/(N-1) - 1),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 3.24, ΔF_0.5 = 1.3, K = 1.62, ɣ_max = -26.4, β = -4.6.
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func LanczosComplex(seq []complex128) []complex128 {
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k := 2 / float64(len(seq)-1)
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for i := range seq {
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x := math.Pi * (k*float64(i) - 1)
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if x == 0 {
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// Avoid NaN.
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continue
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}
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seq[i] *= complex(math.Sin(x)/x, 0)
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}
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return seq
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}
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// TriangularComplex modifies seq in place by the Triangular window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Triangular_window and
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// https://www.recordingblogs.com/wiki/triangular-window for details.
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//
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// The Triangular window is a high-resolution window.
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//
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// The sequence weights are
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// w[k] = 1 - |k/A -1|, A=(N-1)/2,
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.33, K = 2, ɣ_max = -26.5, β = -6.
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func TriangularComplex(seq []complex128) []complex128 {
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a := float64(len(seq)-1) / 2
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for i := range seq {
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seq[i] *= complex(1-math.Abs(float64(i)/a-1), 0)
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}
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return seq
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}
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// HannComplex modifies seq in place by the Hann window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Hann_and_Hamming_windows
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// and https://www.recordingblogs.com/wiki/hann-window for details.
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//
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// The Hann window is a high-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.5*(1 - cos(2*π*k/(N-1))),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.5, K = 2, ɣ_max = -31.5, β = -6.
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func HannComplex(seq []complex128) []complex128 {
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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seq[i] *= complex(0.5*(1-math.Cos(k*float64(i))), 0)
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}
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return seq
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}
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// BartlettHannComplex modifies seq in place by the Bartlett-Hann window and returns result.
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// See https://en.wikipedia.org/wiki/Window_function#Bartlett%E2%80%93Hann_window
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// and https://www.recordingblogs.com/wiki/bartlett-hann-window for details.
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//
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// The Bartlett-Hann window is a high-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.62 - 0.48*|k/(N-1)-0.5| - 0.38*cos(2*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.45, K = 2, ɣ_max = -35.9, β = -6.
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func BartlettHannComplex(seq []complex128) []complex128 {
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const (
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a0 = 0.62
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a1 = 0.48
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a2 = 0.38
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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seq[i] *= complex(a0-a1*math.Abs(float64(i)/float64(len(seq)-1)-0.5)-a2*math.Cos(k*float64(i)), 0)
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}
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return seq
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}
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// HammingComplex modifies seq in place by the Hamming window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Hann_and_Hamming_windows
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// and https://www.recordingblogs.com/wiki/hamming-window for details.
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//
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// The Hamming window is a high-resolution window. Among K=2 windows it has
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// the highest ɣ_max.
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//
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// The sequence weights are
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// w[k] = 25/46 - 21/46 * cos(2*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.33, K = 2, ɣ_max = -42, β = -5.37.
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func HammingComplex(seq []complex128) []complex128 {
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const (
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a0 = 25.0 / 46.0
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a1 = 1 - a0
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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seq[i] *= complex(a0-a1*math.Cos(k*float64(i)), 0)
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}
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return seq
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}
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// BlackmanComplex modifies seq in place by the Blackman window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Blackman_window and
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// https://www.recordingblogs.com/wiki/blackman-window for details.
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//
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// The Blackman window is a high-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.42 - 0.5*cos(2*π*k/(N-1)) + 0.08*cos(4*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 6, ΔF_0.5 = 1.7, K = 3, ɣ_max = -58, β = -7.54.
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func BlackmanComplex(seq []complex128) []complex128 {
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const (
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a0 = 0.42
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a1 = 0.5
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a2 = 0.08
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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x := k * float64(i)
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seq[i] *= complex(a0-a1*math.Cos(x)+a2*math.Cos(2*x), 0)
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}
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return seq
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}
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// BlackmanHarrisComplex modifies seq in place by the Blackman-Harris window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Blackman%E2%80%93Harris_window
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// and https://www.recordingblogs.com/wiki/blackman-harris-window for details.
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//
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// The Blackman-Harris window is a low-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.35875 - 0.48829*cos(2*π*k/(N-1)) +
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// 0.14128*cos(4*π*k/(N-1)) - 0.01168*cos(6*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 8, ΔF_0.5 = 1.97, K = 4, ɣ_max = -92, β = -8.91.
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func BlackmanHarrisComplex(seq []complex128) []complex128 {
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const (
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a0 = 0.35875
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a1 = 0.48829
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a2 = 0.14128
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a3 = 0.01168
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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x := k * float64(i)
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seq[i] *= complex(a0-a1*math.Cos(x)+a2*math.Cos(2*x)-a3*math.Cos(3*x), 0)
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}
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return seq
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}
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// NuttallComplex modifies seq in place by the Nuttall window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Nuttall_window,_continuous_first_derivative
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// and https://www.recordingblogs.com/wiki/nuttall-window for details.
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//
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// The Nuttall window is a low-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.355768 - 0.487396*cos(2*π*k/(N-1)) + 0.144232*cos(4*π*k/(N-1)) -
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// 0.012604*cos(6*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 8, ΔF_0.5 = 1.98, K = 4, ɣ_max = -93, β = -9.
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func NuttallComplex(seq []complex128) []complex128 {
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const (
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a0 = 0.355768
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a1 = 0.487396
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a2 = 0.144232
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a3 = 0.012604
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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x := k * float64(i)
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seq[i] *= complex(a0-a1*math.Cos(x)+a2*math.Cos(2*x)-a3*math.Cos(3*x), 0)
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}
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return seq
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}
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// BlackmanNuttallComplex modifies seq in place by the Blackman-Nuttall window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Blackman%E2%80%93Nuttall_window
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// and https://www.recordingblogs.com/wiki/blackman-nuttall-window for details.
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//
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// The Blackman-Nuttall window is a low-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.3635819 - 0.4891775*cos(2*π*k/(N-1)) + 0.1365995*cos(4*π*k/(N-1)) -
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// 0.0106411*cos(6*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 8, ΔF_0.5 = 1.94, K = 4, ɣ_max = -98, β = -8.8.
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func BlackmanNuttallComplex(seq []complex128) []complex128 {
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const (
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a0 = 0.3635819
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a1 = 0.4891775
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a2 = 0.1365995
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a3 = 0.0106411
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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x := k * float64(i)
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seq[i] *= complex(a0-a1*math.Cos(x)+a2*math.Cos(2*x)-a3*math.Cos(3*x), 0)
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}
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return seq
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}
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// FlatTopComplex modifies seq in place by the Flat Top window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Flat_top_window and
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// https://www.recordingblogs.com/wiki/flat-top-window for details.
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//
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// The Flat Top window is a low-resolution window.
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//
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// The sequence weights are
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// w[k] = 0.21557895 - 0.41663158*cos(2*π*k/(N-1)) +
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// 0.277263158*cos(4*π*k/(N-1)) - 0.083578947*cos(6*π*k/(N-1)) +
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// 0.006947368*cos(4*π*k/(N-1)),
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// Spectral leakage parameters: ΔF_0 = 10, ΔF_0.5 = 3.72, K = 5, ɣ_max = -93.0, β = -13.34.
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func FlatTopComplex(seq []complex128) []complex128 {
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const (
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a0 = 0.21557895
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a1 = 0.41663158
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a2 = 0.277263158
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a3 = 0.083578947
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a4 = 0.006947368
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)
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k := 2 * math.Pi / float64(len(seq)-1)
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for i := range seq {
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x := k * float64(i)
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seq[i] *= complex(a0-a1*math.Cos(x)+a2*math.Cos(2*x)-a3*math.Cos(3*x)+a4*math.Cos(4*x), 0)
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}
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return seq
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}
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// GaussianComplex modifies seq in place by the Gaussian window and returns the result.
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// See https://en.wikipedia.org/wiki/Window_function#Gaussian_window
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// and https://www.recordingblogs.com/wiki/gaussian-window for details.
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//
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// The Gaussian window is an adjustable window.
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//
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// The sequence weights are
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// w[k] = exp(-0.5 * ((k-M)/(σ*M))² ), M = (N-1)/2,
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// for k=0,1,...,N-1 where N is the length of the window.
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//
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// The properties of window depends on the σ (sigma) argument.
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// It can be used as high or low resolution window, depending of the σ value.
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//
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// Spectral leakage parameters are summarized in the table:
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// | σ=0.3 | σ=0.5 | σ=1.2 |
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// -------|---------------------------|
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// ΔF_0 | 8 | 3.4 | 2.2 |
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// ΔF_0.5 | 1.82 | 1.2 | 0.94 |
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// K | 4 | 1.7 | 1.1 |
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// ɣ_max | -65 | -31.5 | -15.5 |
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// β | -8.52 | -4.48 | -0.96 |
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func GaussianComplex(seq []complex128, sigma float64) []complex128 {
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a := float64(len(seq)-1) / 2
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for i := range seq {
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x := -0.5 * math.Pow((float64(i)-a)/(sigma*a), 2)
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seq[i] *= complex(math.Exp(x), 0)
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}
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return seq
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}
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