dsp/window/window_complex.go

// Copyright ©2020 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package window

import "math"

// RectangularComplex modifies seq in place by the Rectangular window and
// returns the result.
// See https://en.wikipedia.org/wiki/Window_function#Rectangular_window and
// https://www.recordingblogs.com/wiki/rectangular-window for details.
//
// The rectangular window has the lowest width of the main lobe and largest
// level of the side lobes. The result corresponds to a selection of
// limited length sequence of values without any modification.
//
// The sequence weights are
//
//	w[k] = 1,
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 2, ΔF_0.5 = 0.89, K = 1, ɣ_max = -13, β = 0.
func RectangularComplex(seq []complex128) []complex128 {
	return seq
}

// SineComplex modifies seq in place by the Sine window and returns the
// result.
// See https://en.wikipedia.org/wiki/Window_function#Sine_window and
// https://www.recordingblogs.com/wiki/sine-window for details.
//
// Sine window is a high-resolution window.
//
// The sequence weights are
//
//	w[k] = sin(π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 3, ΔF_0.5 = 1.23, K = 1.5, ɣ_max = -23, β = -3.93.
func SineComplex(seq []complex128) []complex128 {
	k := math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		w := math.Sin(k * float64(i))
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// LanczosComplex modifies seq in place by the Lanczos window and returns
// the result.
// See https://en.wikipedia.org/wiki/Window_function#Lanczos_window and
// https://www.recordingblogs.com/wiki/lanczos-window for details.
//
// The Lanczos window is a high-resolution window.
//
// The sequence weights are
//
//	w[k] = sinc(2*k/(N-1) - 1),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 3.24, ΔF_0.5 = 1.3, K = 1.62, ɣ_max = -26.4, β = -4.6.
func LanczosComplex(seq []complex128) []complex128 {
	k := 2 / float64(len(seq)-1)
	for i, v := range seq {
		x := math.Pi * (k*float64(i) - 1)
		if x == 0 {
			// Avoid NaN.
			continue
		}
		w := math.Sin(x) / x
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// TriangularComplex modifies seq in place by the Triangular window and
// returns the result.
// See https://en.wikipedia.org/wiki/Window_function#Triangular_window and
// https://www.recordingblogs.com/wiki/triangular-window for details.
//
// The Triangular window is a high-resolution window.
//
// The sequence weights are
//
//	w[k] = 1 - |k/A -1|, A=(N-1)/2,
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.33, K = 2, ɣ_max = -26.5, β = -6.
func TriangularComplex(seq []complex128) []complex128 {
	a := float64(len(seq)-1) / 2
	for i, v := range seq {
		w := 1 - math.Abs(float64(i)/a-1)
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// HannComplex modifies seq in place by the Hann window and returns the result.
// See https://en.wikipedia.org/wiki/Window_function#Hann_and_Hamming_windows
// and https://www.recordingblogs.com/wiki/hann-window for details.
//
// The Hann window is a high-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.5*(1 - cos(2*π*k/(N-1))),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.5, K = 2, ɣ_max = -31.5, β = -6.
func HannComplex(seq []complex128) []complex128 {
	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		w := 0.5 * (1 - math.Cos(k*float64(i)))
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// BartlettHannComplex modifies seq in place by the Bartlett-Hann window
// and returns result.
// See https://en.wikipedia.org/wiki/Window_function#Bartlett%E2%80%93Hann_window
// and https://www.recordingblogs.com/wiki/bartlett-hann-window for details.
//
// The Bartlett-Hann window is a high-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.62 - 0.48*|k/(N-1)-0.5| - 0.38*cos(2*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.45, K = 2, ɣ_max = -35.9, β = -6.
func BartlettHannComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.62
		a1 = 0.48
		a2 = 0.38
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		w := a0 - a1*math.Abs(float64(i)/float64(len(seq)-1)-0.5) - a2*math.Cos(k*float64(i))
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// HammingComplex modifies seq in place by the Hamming window and returns
// the result.
// See https://en.wikipedia.org/wiki/Window_function#Hann_and_Hamming_windows
// and https://www.recordingblogs.com/wiki/hamming-window for details.
//
// The Hamming window is a high-resolution window. Among K=2 windows it has
// the highest ɣ_max.
//
// The sequence weights are
//
//	w[k] = 25/46 - 21/46 * cos(2*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 4, ΔF_0.5 = 1.33, K = 2, ɣ_max = -42, β = -5.37.
func HammingComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.54
		a1 = 0.46
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		w := a0 - a1*math.Cos(k*float64(i))
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// BlackmanComplex modifies seq in place by the Blackman window and returns
// the result.
// See https://en.wikipedia.org/wiki/Window_function#Blackman_window and
// https://www.recordingblogs.com/wiki/blackman-window for details.
//
// The Blackman window is a high-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.42 - 0.5*cos(2*π*k/(N-1)) + 0.08*cos(4*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 6, ΔF_0.5 = 1.7, K = 3, ɣ_max = -58, β = -7.54.
func BlackmanComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.42
		a1 = 0.5
		a2 = 0.08
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		x := k * float64(i)
		w := a0 - a1*math.Cos(x) + a2*math.Cos(2*x)
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// BlackmanHarrisComplex modifies seq in place by the Blackman-Harris window
// and returns the result.
// See https://en.wikipedia.org/wiki/Window_function#Blackman%E2%80%93Harris_window
// and https://www.recordingblogs.com/wiki/blackman-harris-window for details.
//
// The Blackman-Harris window is a low-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.35875 - 0.48829*cos(2*π*k/(N-1)) +
//	       0.14128*cos(4*π*k/(N-1)) - 0.01168*cos(6*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters:  ΔF_0 = 8, ΔF_0.5 = 1.97, K = 4, ɣ_max = -92, β = -8.91.
func BlackmanHarrisComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.35875
		a1 = 0.48829
		a2 = 0.14128
		a3 = 0.01168
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		x := k * float64(i)
		w := a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x)
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// NuttallComplex modifies seq in place by the Nuttall window and returns
// the result.
// See https://en.wikipedia.org/wiki/Window_function#Nuttall_window,_continuous_first_derivative
// and https://www.recordingblogs.com/wiki/nuttall-window for details.
//
// The Nuttall window is a low-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.355768 - 0.487396*cos(2*π*k/(N-1)) + 0.144232*cos(4*π*k/(N-1)) -
//	       0.012604*cos(6*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 8, ΔF_0.5 = 1.98, K = 4, ɣ_max = -93, β = -9.
func NuttallComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.355768
		a1 = 0.487396
		a2 = 0.144232
		a3 = 0.012604
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		x := k * float64(i)
		w := a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x)
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// BlackmanNuttallComplex modifies seq in place by the Blackman-Nuttall
// window and returns the result.
// See https://en.wikipedia.org/wiki/Window_function#Blackman%E2%80%93Nuttall_window
// and https://www.recordingblogs.com/wiki/blackman-nuttall-window for details.
//
// The Blackman-Nuttall window is a low-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.3635819 - 0.4891775*cos(2*π*k/(N-1)) + 0.1365995*cos(4*π*k/(N-1)) -
//	       0.0106411*cos(6*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 8, ΔF_0.5 = 1.94, K = 4, ɣ_max = -98, β = -8.8.
func BlackmanNuttallComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.3635819
		a1 = 0.4891775
		a2 = 0.1365995
		a3 = 0.0106411
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		x := k * float64(i)
		w := a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x)
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}

// FlatTopComplex modifies seq in place by the Flat Top window and returns
// the result.
// See https://en.wikipedia.org/wiki/Window_function#Flat_top_window and
// https://www.recordingblogs.com/wiki/flat-top-window for details.
//
// The Flat Top window is a low-resolution window.
//
// The sequence weights are
//
//	w[k] = 0.21557895 - 0.41663158*cos(2*π*k/(N-1)) +
//	       0.277263158*cos(4*π*k/(N-1)) - 0.083578947*cos(6*π*k/(N-1)) +
//	       0.006947368*cos(4*π*k/(N-1)),
//
// for k=0,1,...,N-1 where N is the length of the window.
//
// Spectral leakage parameters: ΔF_0 = 10, ΔF_0.5 = 3.72, K = 5, ɣ_max = -93.0, β = -13.34.
func FlatTopComplex(seq []complex128) []complex128 {
	const (
		a0 = 0.21557895
		a1 = 0.41663158
		a2 = 0.277263158
		a3 = 0.083578947
		a4 = 0.006947368
	)

	k := 2 * math.Pi / float64(len(seq)-1)
	for i, v := range seq {
		x := k * float64(i)
		w := a0 - a1*math.Cos(x) + a2*math.Cos(2*x) - a3*math.Cos(3*x) + a4*math.Cos(4*x)
		seq[i] = complex(w*real(v), w*imag(v))
	}
	return seq
}