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404 lines
13 KiB
Go
404 lines
13 KiB
Go
// 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 Rectangular(seq []float64) []float64 {
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return seq
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}
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// Sine 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 Sine(seq []float64) []float64 {
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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] *= math.Sin(k * float64(i))
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}
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return seq
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}
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// Lanczos 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 Lanczos(seq []float64) []float64 {
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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] *= math.Sin(x) / x
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}
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return seq
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}
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// Triangular 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 Triangular(seq []float64) []float64 {
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a := float64(len(seq)-1) / 2
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for i := range seq {
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seq[i] *= 1 - math.Abs(float64(i)/a-1)
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}
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return seq
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}
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// Hann 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 Hann(seq []float64) []float64 {
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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] *= 0.5 * (1 - math.Cos(k*float64(i)))
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}
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return seq
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}
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// BartlettHann 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 BartlettHann(seq []float64) []float64 {
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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] *= a0 - a1*math.Abs(float64(i)/float64(len(seq)-1)-0.5) - a2*math.Cos(k*float64(i))
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}
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return seq
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}
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// Hamming 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 Hamming(seq []float64) []float64 {
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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] *= a0 - a1*math.Cos(k*float64(i))
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}
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return seq
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}
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// Blackman 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 Blackman(seq []float64) []float64 {
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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] *= a0 - a1*math.Cos(x) + a2*math.Cos(2*x)
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}
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return seq
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}
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// BlackmanHarris 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 BlackmanHarris(seq []float64) []float64 {
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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] *= a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x)
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}
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return seq
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}
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// Nuttall 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 Nuttall(seq []float64) []float64 {
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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] *= a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x)
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}
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return seq
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}
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// BlackmanNuttall 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 BlackmanNuttall(seq []float64) []float64 {
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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] *= a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x)
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}
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return seq
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}
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// FlatTop 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 FlatTop(seq []float64) []float64 {
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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] *= a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x) + a4*math.Cos(4*x)
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}
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return seq
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}
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// Gaussian can modify a sequence by the Gaussian window and return 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 the window depend on the value of σ (sigma).
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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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type Gaussian struct {
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Sigma float64
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}
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// Transform applies the Gaussian transformation to seq in place, using the value
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// of the receiver as the sigma parameter, and returning the result.
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func (g Gaussian) Transform(seq []float64) []float64 {
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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)/(g.Sigma*a), 2)
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seq[i] *= math.Exp(x)
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}
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return seq
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}
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// Tukey can modify a sequence using the Tukey window and return the result.
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// See https://en.wikipedia.org/wiki/Window_function#Tukey_window
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// and https://www.recordingblogs.com/wiki/tukey-window for details.
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//
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// The Tukey window is an adjustable window.
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//
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// The sequence weights are
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// w[k] = 0.5 * (1 + cos(π*(|k - M| - αM)/((1-α) * M))), |k - M| ≥ αM
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// = 1, |k - M| < αM
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// with M = (N - 1)/2 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 are summarized in the table:
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// | α=0.3 | α=0.5 | α=0.7 |
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// -------|--------------------------|
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// ΔF_0 | 1.33 | 1.22 | 1.13 |
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// ΔF_0.5 | 1.28 | 1.16 | 1.04 |
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// K | 0.67 | 0.61 | 0.57 |
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// ɣ_max | -18.2 | -15.1 | -13.8 |
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// β | -1.41 | -2.50 | -3.74 |
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type Tukey struct {
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Alpha float64
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}
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// Transform applies the Tukey transformation to seq in place, using the value
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// of the receiver as the Alpha parameter, and returning the result
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func (t Tukey) Transform(seq []float64) []float64 {
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switch {
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case t.Alpha <= 0:
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return Rectangular(seq)
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case t.Alpha >= 1:
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return Hann(seq)
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default:
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alphaL := t.Alpha * float64(len(seq)-1)
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width := int(0.5*alphaL) + 1
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for i := range seq[:width] {
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w := 0.5 * (1 - math.Cos(2*math.Pi*float64(i)/alphaL))
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seq[i] *= w
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seq[len(seq)-1-i] *= w
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}
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return seq
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}
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}
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// Values is an arbitrary real window function.
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type Values []float64
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// NewValues returns a Values of length n with weights corresponding to the
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// provided window function.
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func NewValues(window func([]float64) []float64, n int) Values {
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v := make(Values, n)
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for i := range v {
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v[i] = 1
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}
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return window(v)
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}
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// Transform applies the weights in the receiver to seq in place, returning the
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// result. If v is nil, Transform is a no-op, otherwise the length of v must
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// match the length of seq.
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func (v Values) Transform(seq []float64) []float64 {
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if v == nil {
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return seq
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}
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if len(v) != len(seq) {
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panic("window: length mismatch")
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}
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for i, w := range v {
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seq[i] *= w
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}
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return seq
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}
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