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Diffstat (limited to 'numpy/fft/fftpack.py')
| -rw-r--r-- | numpy/fft/fftpack.py | 126 |
1 files changed, 63 insertions, 63 deletions
diff --git a/numpy/fft/fftpack.py b/numpy/fft/fftpack.py index de675936f..d0df6fb48 100644 --- a/numpy/fft/fftpack.py +++ b/numpy/fft/fftpack.py @@ -177,19 +177,17 @@ def fft(a, n=None, axis=-1, norm=None): Examples -------- >>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8)) - array([ -3.44505240e-16 +1.14383329e-17j, - 8.00000000e+00 -5.71092652e-15j, - 2.33482938e-16 +1.22460635e-16j, - 1.64863782e-15 +1.77635684e-15j, - 9.95839695e-17 +2.33482938e-16j, - 0.00000000e+00 +1.66837030e-15j, - 1.14383329e-17 +1.22460635e-16j, - -1.64863782e-15 +1.77635684e-15j]) + array([-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j, + 2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j, + -1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j, + 1.14423775e-17+1.14423775e-17j, 0.00000000e+00+1.22464680e-16j]) In this example, real input has an FFT which is Hermitian, i.e., symmetric in the real part and anti-symmetric in the imaginary part, as described in the `numpy.fft` documentation: + >>> import matplotlib + >>> matplotlib.use('Agg') >>> import matplotlib.pyplot as plt >>> t = np.arange(256) >>> sp = np.fft.fft(np.sin(t)) @@ -278,19 +276,21 @@ def ifft(a, n=None, axis=-1, norm=None): Examples -------- >>> np.fft.ifft([0, 4, 0, 0]) - array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) + array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) # may vary Create and plot a band-limited signal with random phases: + >>> import matplotlib + >>> matplotlib.use('agg') >>> import matplotlib.pyplot as plt >>> t = np.arange(400) >>> n = np.zeros((400,), dtype=complex) >>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,))) >>> s = np.fft.ifft(n) >>> plt.plot(t, s.real, 'b-', t, s.imag, 'r--') - ... + [<matplotlib.lines.Line2D object at ...>, <matplotlib.lines.Line2D object at ...>] >>> plt.legend(('real', 'imaginary')) - ... + <matplotlib.legend.Legend object at ...> >>> plt.show() """ @@ -374,9 +374,9 @@ def rfft(a, n=None, axis=-1, norm=None): Examples -------- >>> np.fft.fft([0, 1, 0, 0]) - array([ 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j]) + array([ 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j]) # may vary >>> np.fft.rfft([0, 1, 0, 0]) - array([ 1.+0.j, 0.-1.j, -1.+0.j]) + array([ 1.+0.j, 0.-1.j, -1.+0.j]) # may vary Notice how the final element of the `fft` output is the complex conjugate of the second element, for real input. For `rfft`, this symmetry is @@ -465,9 +465,9 @@ def irfft(a, n=None, axis=-1, norm=None): Examples -------- >>> np.fft.ifft([1, -1j, -1, 1j]) - array([ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) + array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary >>> np.fft.irfft([1, -1j, -1]) - array([ 0., 1., 0., 0.]) + array([0., 1., 0., 0.]) Notice how the last term in the input to the ordinary `ifft` is the complex conjugate of the second term, and the output has zero imaginary @@ -543,16 +543,16 @@ def hfft(a, n=None, axis=-1, norm=None): -------- >>> signal = np.array([1, 2, 3, 4, 3, 2]) >>> np.fft.fft(signal) - array([ 15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j]) + array([15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j]) # may vary >>> np.fft.hfft(signal[:4]) # Input first half of signal - array([ 15., -4., 0., -1., 0., -4.]) + array([15., -4., 0., -1., 0., -4.]) >>> np.fft.hfft(signal, 6) # Input entire signal and truncate - array([ 15., -4., 0., -1., 0., -4.]) + array([15., -4., 0., -1., 0., -4.]) >>> signal = np.array([[1, 1.j], [-1.j, 2]]) >>> np.conj(signal.T) - signal # check Hermitian symmetry - array([[ 0.-0.j, 0.+0.j], + array([[ 0.-0.j, -0.+0.j], # may vary [ 0.+0.j, 0.-0.j]]) >>> freq_spectrum = np.fft.hfft(signal) >>> freq_spectrum @@ -616,9 +616,9 @@ def ihfft(a, n=None, axis=-1, norm=None): -------- >>> spectrum = np.array([ 15, -4, 0, -1, 0, -4]) >>> np.fft.ifft(spectrum) - array([ 1.+0.j, 2.-0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.-0.j]) + array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.+0.j]) # may vary >>> np.fft.ihfft(spectrum) - array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) + array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) # may vary """ # The copy may be required for multithreading. @@ -732,17 +732,17 @@ def fftn(a, s=None, axes=None, norm=None): -------- >>> a = np.mgrid[:3, :3, :3][0] >>> np.fft.fftn(a, axes=(1, 2)) - array([[[ 0.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j]], - [[ 9.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j]], - [[ 18.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j]]]) + array([[[ 0.+0.j, 0.+0.j, 0.+0.j], # may vary + [ 0.+0.j, 0.+0.j, 0.+0.j], + [ 0.+0.j, 0.+0.j, 0.+0.j]], + [[ 9.+0.j, 0.+0.j, 0.+0.j], + [ 0.+0.j, 0.+0.j, 0.+0.j], + [ 0.+0.j, 0.+0.j, 0.+0.j]], + [[18.+0.j, 0.+0.j, 0.+0.j], + [ 0.+0.j, 0.+0.j, 0.+0.j], + [ 0.+0.j, 0.+0.j, 0.+0.j]]]) >>> np.fft.fftn(a, (2, 2), axes=(0, 1)) - array([[[ 2.+0.j, 2.+0.j, 2.+0.j], + array([[[ 2.+0.j, 2.+0.j, 2.+0.j], # may vary [ 0.+0.j, 0.+0.j, 0.+0.j]], [[-2.+0.j, -2.+0.j, -2.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]]]) @@ -838,10 +838,10 @@ def ifftn(a, s=None, axes=None, norm=None): -------- >>> a = np.eye(4) >>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,)) - array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]]) + array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], # may vary + [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j], + [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], + [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]]) Create and plot an image with band-limited frequency content: @@ -934,16 +934,16 @@ def fft2(a, s=None, axes=(-2, -1), norm=None): -------- >>> a = np.mgrid[:5, :5][0] >>> np.fft.fft2(a) - array([[ 50.0 +0.j , 0.0 +0.j , 0.0 +0.j , - 0.0 +0.j , 0.0 +0.j ], - [-12.5+17.20477401j, 0.0 +0.j , 0.0 +0.j , - 0.0 +0.j , 0.0 +0.j ], - [-12.5 +4.0614962j , 0.0 +0.j , 0.0 +0.j , - 0.0 +0.j , 0.0 +0.j ], - [-12.5 -4.0614962j , 0.0 +0.j , 0.0 +0.j , - 0.0 +0.j , 0.0 +0.j ], - [-12.5-17.20477401j, 0.0 +0.j , 0.0 +0.j , - 0.0 +0.j , 0.0 +0.j ]]) + array([[ 50. +0.j , 0. +0.j , 0. +0.j , # may vary + 0. +0.j , 0. +0.j ], + [-12.5+17.20477401j, 0. +0.j , 0. +0.j , + 0. +0.j , 0. +0.j ], + [-12.5 +4.0614962j , 0. +0.j , 0. +0.j , + 0. +0.j , 0. +0.j ], + [-12.5 -4.0614962j , 0. +0.j , 0. +0.j , + 0. +0.j , 0. +0.j ], + [-12.5-17.20477401j, 0. +0.j , 0. +0.j , + 0. +0.j , 0. +0.j ]]) """ @@ -1028,10 +1028,10 @@ def ifft2(a, s=None, axes=(-2, -1), norm=None): -------- >>> a = 4 * np.eye(4) >>> np.fft.ifft2(a) - array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j], - [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], - [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]) + array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], # may vary + [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j], + [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], + [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]) """ @@ -1110,16 +1110,16 @@ def rfftn(a, s=None, axes=None, norm=None): -------- >>> a = np.ones((2, 2, 2)) >>> np.fft.rfftn(a) - array([[[ 8.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j]], - [[ 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j]]]) + array([[[8.+0.j, 0.+0.j], # may vary + [0.+0.j, 0.+0.j]], + [[0.+0.j, 0.+0.j], + [0.+0.j, 0.+0.j]]]) >>> np.fft.rfftn(a, axes=(2, 0)) - array([[[ 4.+0.j, 0.+0.j], - [ 4.+0.j, 0.+0.j]], - [[ 0.+0.j, 0.+0.j], - [ 0.+0.j, 0.+0.j]]]) + array([[[4.+0.j, 0.+0.j], # may vary + [4.+0.j, 0.+0.j]], + [[0.+0.j, 0.+0.j], + [0.+0.j, 0.+0.j]]]) """ # The copy may be required for multithreading. @@ -1247,12 +1247,12 @@ def irfftn(a, s=None, axes=None, norm=None): >>> a = np.zeros((3, 2, 2)) >>> a[0, 0, 0] = 3 * 2 * 2 >>> np.fft.irfftn(a) - array([[[ 1., 1.], - [ 1., 1.]], - [[ 1., 1.], - [ 1., 1.]], - [[ 1., 1.], - [ 1., 1.]]]) + array([[[1., 1.], + [1., 1.]], + [[1., 1.], + [1., 1.]], + [[1., 1.], + [1., 1.]]]) """ # The copy may be required for multithreading. |