package window

import "gonum.org/v1/gonum/dsp/window"

Package window provides a set of functions to perform the transformation of sequence by different window functions.

Window functions can be used to control spectral leakage parameters when performing a Fourier transform on a signal of limited length. See https://en.wikipedia.org/wiki/Window_function for more details.

Spectral leakage parameters

Application of window functions to an input will result in changes to the frequency content of the signal in an effect called spectral leakage. See https://en.wikipedia.org/wiki/Spectral_leakage.

The characteristic changes associated with each window function may be described using a set of spectral leakage parameters; β, ΔF_0, ΔF_0.5, K and ɣ_max.

The β, attenuation, coefficient of a window is the ratio of the constant component of the spectrum resulting from use of the window compared to that produced using the rectangular window, expressed in a logarithmic scale. 

β_w = 20 log10(A_w / A_rect) dB

The ΔF_0 parameter describes the normalized width of the main lobe of the frequency spectrum at zero amplitude.

The ΔF_0.5 parameter describes the normalized width of the main lobe of the frequency spectrum at -3 dB (half maximum amplitude).

The K parameter describes the relative width of the main lobe of the frequency spectrum produced by the window compared with the rectangular window. The rectangular window has the lowest ΔF_0 at a value of 2. 

K_w = ΔF_0_w/ΔF_0_rect.

The value of K divides windows into high resolution windows (K≤3) and low resolution windows (K>3).

The ɣ_max parameter is the maximum level of the side lobes of the normalized spectrum, in decibels.

Example

Code:play  

package main

import (
	"fmt"
	"math"
	"math/cmplx"

	"gonum.org/v1/gonum/dsp/fourier"
	"gonum.org/v1/gonum/dsp/window"
)

func main() {
	// The input sequence is a 2.5 period of the Sin function.
	src := make([]float64, 20)
	k := 5 * math.Pi / float64(len(src)-1)
	for i := range src {
		src[i] = math.Sin(k * float64(i))
	}

	// Initialize an FFT and perform the analysis.
	fft := fourier.NewFFT(len(src))
	coeff := fft.Coefficients(nil, src)

	// The result shows that width of the main lobe with center
	// between frequencies 0.1 and 0.15 is small, but that the
	// height of the side lobes is large.
	fmt.Println("Rectangular window (or no window):")
	for i, c := range coeff {
		fmt.Printf("freq=%.4f\tcycles/period, magnitude=%.4f,\tphase=%.4f\n",
			fft.Freq(i), cmplx.Abs(c), cmplx.Phase(c))
	}

	// Initialize an FFT and perform the analysis on a sequence
	// transformed by the Hamming window function.
	fft = fourier.NewFFT(len(src))
	coeff = fft.Coefficients(nil, window.Hamming(src))

	// The result shows that width of the main lobe is wider,
	// but height of the side lobes is lower.
	fmt.Println("Hamming window:")
	// The magnitude of all bins has been decreased by β.
	// Generally in an analysis amplification may be omitted, but to
	// make a comparable data, the result should be amplified by -β
	// of the window function — +5.37 dB for the Hamming window.
	//  -β = 20 log_10(amplifier).
	amplifier := math.Pow(10, 5.37/20.0)
	for i, c := range coeff {
		fmt.Printf("freq=%.4f\tcycles/period, magnitude=%.4f,\tphase=%.4f\n",
			fft.Freq(i), amplifier*cmplx.Abs(c), cmplx.Phase(c))
	}
}

Output:

Rectangular window (or no window):
freq=0.0000	cycles/period, magnitude=2.2798,	phase=0.0000
freq=0.0500	cycles/period, magnitude=2.6542,	phase=0.1571
freq=0.1000	cycles/period, magnitude=5.3115,	phase=0.3142
freq=0.1500	cycles/period, magnitude=7.3247,	phase=-2.6704
freq=0.2000	cycles/period, magnitude=1.6163,	phase=-2.5133
freq=0.2500	cycles/period, magnitude=0.7681,	phase=-2.3562
freq=0.3000	cycles/period, magnitude=0.4385,	phase=-2.1991
freq=0.3500	cycles/period, magnitude=0.2640,	phase=-2.0420
freq=0.4000	cycles/period, magnitude=0.1530,	phase=-1.8850
freq=0.4500	cycles/period, magnitude=0.0707,	phase=-1.7279
freq=0.5000	cycles/period, magnitude=0.0000,	phase=0.0000
Hamming window:
freq=0.0000	cycles/period, magnitude=0.0542,	phase=3.1416
freq=0.0500	cycles/period, magnitude=0.8458,	phase=-2.9845
freq=0.1000	cycles/period, magnitude=7.1519,	phase=0.3142
freq=0.1500	cycles/period, magnitude=8.5907,	phase=-2.6704
freq=0.2000	cycles/period, magnitude=2.0804,	phase=0.6283
freq=0.2500	cycles/period, magnitude=0.0816,	phase=0.7854
freq=0.3000	cycles/period, magnitude=0.0156,	phase=-2.1991
freq=0.3500	cycles/period, magnitude=0.0224,	phase=-2.0420
freq=0.4000	cycles/period, magnitude=0.0163,	phase=-1.8850
freq=0.4500	cycles/period, magnitude=0.0083,	phase=-1.7279
freq=0.5000	cycles/period, magnitude=0.0000,	phase=0.0000

Index

Examples

Functions

func BartlettHann 

func BartlettHann(seq []float64) []float64

BartlettHann 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 

func BartlettHannComplex(seq []complex128) []complex128

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 Blackman 

func Blackman(seq []float64) []float64

Blackman 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 

func BlackmanComplex(seq []complex128) []complex128

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 BlackmanHarris 

func BlackmanHarris(seq []float64) []float64

BlackmanHarris 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 

func BlackmanHarrisComplex(seq []complex128) []complex128

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 BlackmanNuttall 

func BlackmanNuttall(seq []float64) []float64

BlackmanNuttall 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 

func BlackmanNuttallComplex(seq []complex128) []complex128

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 FlatTop 

func FlatTop(seq []float64) []float64

FlatTop 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 

func FlatTopComplex(seq []complex128) []complex128

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 Hamming 

func Hamming(seq []float64) []float64

Hamming 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.

Example

Code:play  

package main

import (
	"fmt"

	"gonum.org/v1/gonum/dsp/window"
)

func main() {
	src := []float64{1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
		1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

	// Window functions change data in place. So, if input data
	// needs to stay unchanged, it must be copied.
	srcCpy := append([]float64(nil), src...)
	// Apply window function to srcCpy.
	dst := window.Hamming(srcCpy)

	// src is unchanged.
	fmt.Printf("src:    %f\n", src)
	// srcCpy is altered.
	fmt.Printf("srcCpy: %f\n", srcCpy)
	// dst mirrors the srcCpy slice.
	fmt.Printf("dst:    %f\n", dst)

}

Output:

src:    [1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000]
srcCpy: [0.080000 0.104924 0.176995 0.288404 0.427077 0.577986 0.724780 0.851550 0.944558 0.993726 0.993726 0.944558 0.851550 0.724780 0.577986 0.427077 0.288404 0.176995 0.104924 0.080000]
dst:    [0.080000 0.104924 0.176995 0.288404 0.427077 0.577986 0.724780 0.851550 0.944558 0.993726 0.993726 0.944558 0.851550 0.724780 0.577986 0.427077 0.288404 0.176995 0.104924 0.080000]

func HammingComplex 

func HammingComplex(seq []complex128) []complex128

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 Hann 

func Hann(seq []float64) []float64

Hann 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 

func HannComplex(seq []complex128) []complex128

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 Lanczos 

func Lanczos(seq []float64) []float64

Lanczos 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 

func LanczosComplex(seq []complex128) []complex128

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 Nuttall 

func Nuttall(seq []float64) []float64

Nuttall 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 

func NuttallComplex(seq []complex128) []complex128

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 Rectangular 

func Rectangular(seq []float64) []float64

Rectangular 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 

func RectangularComplex(seq []complex128) []complex128

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 Sine 

func Sine(seq []float64) []float64

Sine 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 

func SineComplex(seq []complex128) []complex128

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 Triangular 

func Triangular(seq []float64) []float64

Triangular 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 

func TriangularComplex(seq []complex128) []complex128

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.

Types

type Gaussian 

type Gaussian struct {
	Sigma float64
}

Gaussian can modify a sequence using the Gaussian window and return the result. See https://en.wikipedia.org/wiki/Window_function#Gaussian_window and https://www.recordingblogs.com/wiki/gaussian-window for details.

The Gaussian window is an adjustable window.

The sequence weights are 

w[k] = exp(-0.5 * ((k-M)/(σ*M))² ), M = (N-1)/2,

for k=0,1,...,N-1 where N is the length of the window.

The properties of the window depend on the value of σ (sigma). It can be used as high or low resolution window, depending of the σ value.

Spectral leakage parameters are summarized in the table: 

       |  σ=0.3  |  σ=0.5 |  σ=1.2 |
-------|---------------------------|
ΔF_0   |   8     |   3.4  |   2.2  |
ΔF_0.5 |   1.82  |   1.2  |   0.94 |
K      |   4     |   1.7  |   1.1  |
ɣ_max  | -65     | -31.5  | -15.5  |
β      |  -8.52  |  -4.48 |  -0.96 |

func (Gaussian) Transform 

func (g Gaussian) Transform(seq []float64) []float64

Transform applies the Gaussian transformation to seq in place, using the value of the receiver as the sigma parameter, and returning the result.

func (Gaussian) TransformComplex 

func (g Gaussian) TransformComplex(seq []complex128) []complex128

TransformComplex applies the Gaussian transformation to seq in place, using the value of the receiver as the sigma parameter, and returning the result.

type Tukey 

type Tukey struct {
	Alpha float64
}

Tukey can modify a sequence using the Tukey window and return the result. See https://en.wikipedia.org/wiki/Window_function#Tukey_window and https://www.recordingblogs.com/wiki/tukey-window for details.

The Tukey window is an adjustable window.

The sequence weights are 

w[k] = 0.5 * (1 + cos(π*(|k - M| - αM)/((1-α) * M))), |k - M| ≥ αM
     = 1, |k - M| < αM

with M = (N - 1)/2 for k=0,1,...,N-1 where N is the length of the window.

Spectral leakage parameters are summarized in the table: 

       |  α=0.3 |  α=0.5 |  α=0.7 |
-------|--------------------------|
ΔF_0   |   1.33 |   1.22 |   1.13 |
ΔF_0.5 |   1.28 |   1.16 |   1.04 |
K      |   0.67 |   0.61 |   0.57 |
ɣ_max  | -18.2  | -15.1  | -13.8  |
β      |  -1.41 |  -2.50 |  -3.74 |

func (Tukey) Transform 

func (t Tukey) Transform(seq []float64) []float64

Transform applies the Tukey transformation to seq in place, using the value of the receiver as the Alpha parameter, and returning the result.

func (Tukey) TransformComplex 

func (t Tukey) TransformComplex(seq []complex128) []complex128

TransformComplex applies the Tukey transformation to seq in place, using the value of the receiver as the Alpha parameter, and returning the result.

type Values 

type Values []float64

Values is an arbitrary real window function.

Example

Code:play  

package main

import (
	"fmt"

	"gonum.org/v1/gonum/dsp/window"
)

func main() {
	src := []float64{1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
		1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

	// Create a Sine Window lookup table.
	sine := window.NewValues(window.Sine, len(src))

	// Apply the transformation to the src.
	fmt.Printf("dst: %f\n", sine.Transform(src))

}

Output:

dst: [0.000000 0.164595 0.324699 0.475947 0.614213 0.735724 0.837166 0.915773 0.969400 0.996584 0.996584 0.969400 0.915773 0.837166 0.735724 0.614213 0.475947 0.324699 0.164595 0.000000]

func NewValues 

func NewValues(window func([]float64) []float64, n int) Values

NewValues returns a Values of length n with weights corresponding to the provided window function.

func (Values) Transform 

func (v Values) Transform(seq []float64) []float64

Transform applies the weights in the receiver to seq in place, returning the result. If v is nil, Transform is a no-op, otherwise the length of v must match the length of seq.

func (Values) TransformComplex 

func (v Values) TransformComplex(seq []complex128) []complex128

TransformComplex applies the weights in the receiver to seq in place, returning the result. If v is nil, TransformComplex is a no-op, otherwise the length of v must match the length of seq.

func (Values) TransformComplexTo 

func (v Values) TransformComplexTo(dst, src []complex128)

TransformComplexTo applies the weights in the receiver to src placing the result in dst. If v is nil, TransformComplexTo is a no-op, otherwise the length of v must match the length of src and dst.

func (Values) TransformTo 

func (v Values) TransformTo(dst, src []float64)

TransformTo applies the weights in the receiver to src placing the result in dst. If v is nil, TransformTo is a no-op, otherwise the length of v must match the length of src and dst.

Example (Gabor)

Code:play  

package main

import (
	"fmt"

	"gonum.org/v1/gonum/dsp/window"
)

func main() {
	src := []float64{1, 2, 1, 0, -1, -1, -2, -2, -1, -1,
		0, 1, 1, 2, 1, 0, -1, -2, -1, 0}

	// Create a Gaussian Window lookup table for 4 samples.
	gaussian := window.NewValues(window.Gaussian{0.5}.Transform, 4)

	// Prepare a destination.
	dst := make([]float64, 8)

	// Apply the transformation to the src, placing it in dst.
	for i := 0; i < len(src)-len(gaussian); i++ {
		gaussian.TransformTo(dst[0:len(gaussian)], src[i:i+len(gaussian)])

		// To perform the Gabor transform, we would calculate
		// the FFT on dst for each iteration.
		fmt.Printf("FFT(%f)\n", dst)
	}

}

Output:

FFT([0.135335 1.601475 0.800737 0.000000 0.000000 0.000000 0.000000 0.000000])
FFT([0.270671 0.800737 0.000000 -0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([0.135335 0.000000 -0.800737 -0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([0.000000 -0.800737 -0.800737 -0.270671 0.000000 0.000000 0.000000 0.000000])
FFT([-0.135335 -0.800737 -1.601475 -0.270671 0.000000 0.000000 0.000000 0.000000])
FFT([-0.135335 -1.601475 -1.601475 -0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([-0.270671 -1.601475 -0.800737 -0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([-0.270671 -0.800737 -0.800737 0.000000 0.000000 0.000000 0.000000 0.000000])
FFT([-0.135335 -0.800737 0.000000 0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([-0.135335 0.000000 0.800737 0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([0.000000 0.800737 0.800737 0.270671 0.000000 0.000000 0.000000 0.000000])
FFT([0.135335 0.800737 1.601475 0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([0.135335 1.601475 0.800737 0.000000 0.000000 0.000000 0.000000 0.000000])
FFT([0.270671 0.800737 0.000000 -0.135335 0.000000 0.000000 0.000000 0.000000])
FFT([0.135335 0.000000 -0.800737 -0.270671 0.000000 0.000000 0.000000 0.000000])
FFT([0.000000 -0.800737 -1.601475 -0.135335 0.000000 0.000000 0.000000 0.000000])

Source Files

doc.go window.go window_complex.go window_parametric.go

Directories

PathSynopsis
dsp/window/cmd
dsp/window/cmd/leakageThe leakage program provides summary characteristics and a plot of spectral response for window functions or csv input.
Version
v0.15.1 (latest)
Published
Aug 16, 2024
Platform
linux/amd64
Imports
1 packages
Last checked
2 days ago –

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