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// Copyright ©2018 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 fourier_test
import (
"fmt"
"math"
"math/cmplx"
"gonum.org/v1/gonum/dsp/fourier"
"gonum.org/v1/gonum/floats/scalar"
"gonum.org/v1/gonum/mat"
)
func ExampleFFT_Coefficients() {
// Samples is a set of samples over a given period.
samples := []float64{1, 0, 2, 0, 4, 0, 2, 0}
period := float64(len(samples))
// Initialize an FFT and perform the analysis.
fft := fourier.NewFFT(len(samples))
coeff := fft.Coefficients(nil, samples)
for i, c := range coeff {
fmt.Printf("freq=%v cycles/period, magnitude=%v, phase=%.4g\n",
fft.Freq(i)*period, cmplx.Abs(c), cmplx.Phase(c))
}
// Output:
//
// freq=0 cycles/period, magnitude=9, phase=0
// freq=1 cycles/period, magnitude=3, phase=3.142
// freq=2 cycles/period, magnitude=1, phase=-0
// freq=3 cycles/period, magnitude=3, phase=3.142
// freq=4 cycles/period, magnitude=9, phase=0
}
func ExampleFFT_Coefficients_tone() {
// Tone is a set of samples over a second of a pure Middle C.
const (
mC = 261.625565 // Hz
samples = 44100
)
tone := make([]float64, samples)
for i := range tone {
tone[i] = math.Sin(mC * 2 * math.Pi * float64(i) / samples)
}
// Initialize an FFT and perform the analysis.
fft := fourier.NewFFT(samples)
coeff := fft.Coefficients(nil, tone)
var maxFreq, magnitude, mean float64
for i, c := range coeff {
m := cmplx.Abs(c)
mean += m
if m > magnitude {
magnitude = m
maxFreq = fft.Freq(i) * samples
}
}
fmt.Printf("freq=%v Hz, magnitude=%.0f, mean=%.4f\n", maxFreq, magnitude, mean/samples)
// Output:
//
// freq=262 Hz, magnitude=17296, mean=2.7835
}
func ExampleCmplxFFT_Coefficients() {
// Samples is a set of samples over a given period.
samples := []complex128{1, 0, 2, 0, 4, 0, 2, 0}
period := float64(len(samples))
// Initialize a complex FFT and perform the analysis.
fft := fourier.NewCmplxFFT(len(samples))
coeff := fft.Coefficients(nil, samples)
for i := range coeff {
// Center the spectrum.
i = fft.ShiftIdx(i)
fmt.Printf("freq=%v cycles/period, magnitude=%v, phase=%.4g\n",
fft.Freq(i)*period, cmplx.Abs(coeff[i]), cmplx.Phase(coeff[i]))
}
// Output:
//
// freq=-4 cycles/period, magnitude=9, phase=0
// freq=-3 cycles/period, magnitude=3, phase=3.142
// freq=-2 cycles/period, magnitude=1, phase=0
// freq=-1 cycles/period, magnitude=3, phase=3.142
// freq=0 cycles/period, magnitude=9, phase=0
// freq=1 cycles/period, magnitude=3, phase=3.142
// freq=2 cycles/period, magnitude=1, phase=0
// freq=3 cycles/period, magnitude=3, phase=3.142
}
func Example_fFT2() {
// This example shows how to perform a 2D fourier transform
// on an image. The transform identifies the lines present
// in the image.
// Image is a set of diagonal lines.
image := mat.NewDense(11, 11, []float64{
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
})
// Make appropriately sized real and complex FFT types.
r, c := image.Dims()
fft := fourier.NewFFT(c)
cfft := fourier.NewCmplxFFT(r)
// Only c/2+1 coefficients will be returned for
// the real FFT.
c = c/2 + 1
// Perform the first axis transform.
rows := make([]complex128, r*c)
for i := 0; i < r; i++ {
fft.Coefficients(rows[c*i:c*(i+1)], image.RawRowView(i))
}
// Perform the second axis transform, storing
// the result in freqs.
freqs := mat.NewDense(c, c, nil)
column := make([]complex128, r)
for j := 0; j < c; j++ {
for i := 0; i < r; i++ {
column[i] = rows[i*c+j]
}
cfft.Coefficients(column, column)
for i, v := range column[:c] {
freqs.Set(i, j, scalar.Round(cmplx.Abs(v), 1))
}
}
fmt.Printf("%v\n", mat.Formatted(freqs))
// Output:
//
// ⎡ 40 0.4 0.5 1.4 3.2 1.1⎤
// ⎢ 0.4 0.5 0.7 1.8 4 1.2⎥
// ⎢ 0.5 0.7 1.1 2.8 5.9 1.7⎥
// ⎢ 1.4 1.8 2.8 6.8 14.1 3.8⎥
// ⎢ 3.2 4 5.9 14.1 27.5 6.8⎥
// ⎣ 1.1 1.2 1.7 3.8 6.8 1.6⎦
}
func Example_cmplxFFT2() {
// Image is a set of diagonal lines.
image := mat.NewDense(11, 11, []float64{
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0,
0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0,
0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
})
// Make appropriately sized complex FFT.
// Rows and columns are the same, so the same
// CmplxFFT can be used for both axes.
r, c := image.Dims()
cfft := fourier.NewCmplxFFT(r)
// Perform the first axis transform.
rows := make([]complex128, r*c)
for i := 0; i < r; i++ {
row := rows[c*i : c*(i+1)]
for j, v := range image.RawRowView(i) {
row[j] = complex(v, 0)
}
cfft.Coefficients(row, row)
}
// Perform the second axis transform, storing
// the result in freqs.
freqs := mat.NewDense(c, c, nil)
column := make([]complex128, r)
for j := 0; j < c; j++ {
for i := 0; i < r; i++ {
column[i] = rows[i*c+j]
}
cfft.Coefficients(column, column)
for i, v := range column {
// Center the frequencies.
freqs.Set(cfft.UnshiftIdx(i), cfft.UnshiftIdx(j), scalar.Round(cmplx.Abs(v), 1))
}
}
fmt.Printf("%v\n", mat.Formatted(freqs))
// Output:
//
// ⎡ 1.6 6.8 3.8 1.7 1.2 1.1 1.1 1.4 2.6 3.9 1.1⎤
// ⎢ 6.8 27.5 14.1 5.9 4 3.2 3 3 3.9 3.2 3.9⎥
// ⎢ 3.8 14.1 6.8 2.8 1.8 1.4 1.2 1.1 1.4 3.9 2.6⎥
// ⎢ 1.7 5.9 2.8 1.1 0.7 0.5 0.5 0.5 1.1 3 1.4⎥
// ⎢ 1.2 4 1.8 0.7 0.5 0.4 0.4 0.5 1.2 3 1.1⎥
// ⎢ 1.1 3.2 1.4 0.5 0.4 40 0.4 0.5 1.4 3.2 1.1⎥
// ⎢ 1.1 3 1.2 0.5 0.4 0.4 0.5 0.7 1.8 4 1.2⎥
// ⎢ 1.4 3 1.1 0.5 0.5 0.5 0.7 1.1 2.8 5.9 1.7⎥
// ⎢ 2.6 3.9 1.4 1.1 1.2 1.4 1.8 2.8 6.8 14.1 3.8⎥
// ⎢ 3.9 3.2 3.9 3 3 3.2 4 5.9 14.1 27.5 6.8⎥
// ⎣ 1.1 3.9 2.6 1.4 1.1 1.1 1.2 1.7 3.8 6.8 1.6⎦
}
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