def test_djbfft(self):
from numpy.fft import fft as numpy_fft
for i in range(2,14):
n = 2**i
x = range(n)
y2 = numpy_fft(x)
y1 = zeros((n,),dtype=double)
y1[0] = y2[0].real
y1[-1] = y2[n/2].real
for k in range(1, int(n/2)):
y1[2*k-1] = y2[k].real
y1[2*k] = y2[k].imag
y = fftpack.drfft(x)
assert_array_almost_equal(y,y1)
def test_djbfft(self):
from numpy.fft import ifft as numpy_ifft
for i in range(2,14):
n = 2**i
x = range(n)
x1 = zeros((n,),dtype=cdouble)
x1[0] = x[0]
for k in range(1, int(n/2)):
x1[k] = x[2*k-1]+1j*x[2*k]
x1[n-k] = x[2*k-1]-1j*x[2*k]
x1[n/2] = x[-1]
y1 = numpy_ifft(x1)
y = fftpack.drfft(x,direction=-1)
assert_array_almost_equal(y,y1)
def test_djbfft(self):
from numpy.fft import ifft as numpy_ifft
for i in range(2,14):
n = 2**i
x = list(range(n))
x1 = zeros((n,),dtype=cdouble)
x1[0] = x[0]
for k in range(1, n//2):
x1[k] = x[2*k-1]+1j*x[2*k]
x1[n-k] = x[2*k-1]-1j*x[2*k]
x1[n//2] = x[-1]
y1 = numpy_ifft(x1)
y = fftpack.drfft(x,direction=-1)
assert_array_almost_equal(y,y1)
def test_djbfft(self):
from numpy.fft import fft as numpy_fft
for i in range(2,14):
n = 2**i
x = list(range(n))
y2 = numpy_fft(x)
y1 = zeros((n,),dtype=double)
y1[0] = y2[0].real
y1[-1] = y2[n//2].real
for k in range(1, n//2):
y1[2*k-1] = y2[k].real
y1[2*k] = y2[k].imag
y = fftpack.drfft(x)
assert_array_almost_equal(y,y1)
def fhtq(a, xsave, tdir=1):
"""Kernel routine of FFTLog.
This is the basic FFTLog routine.
fhtq computes a discrete version of the biased Hankel transform
infinity
/ q
ã(k) = | a(r) (k r) J (k r) k dr
/ mu
0
fhti must be called before the first call to fhtq.
A call to fhtq with dir=1 followed by a call to fhtq with dir=-1, or vice
versa, leaves the array a unchanged.
Parameters
----------
a : array
Periodic array a(r) to transform: a(j) is a(r_j) at r_j = r_c
exp[(j-jc) dlnr] where jc = (n+1)/2 = central index of array.
xsave : array
Working array set up by fhti.
tdir : int, optional; {1, -1}
- 1 for forward transform (default),
- -1 for backward transform.
A backward transform (dir = -1) is the same as a forward transform with
q -> -q, for any kr if n is odd, for low-ringing kr if n is even.
Returns
-------
a : array
Transformed periodic array ã(k): a(j) is ã(k_j) at k_j = k_c exp[(j-jc)
dlnr].
"""
fct = a.copy()
q = xsave[0]
n = fct.size
# normal FFT
# fct = rfft(fct)
# _raw_fft(fct, n, -1, 1, 1, _fftpack.drfft)
fct = drfft(fct, n, 1, 0)
m = np.arange(1, n / 2, dtype=int) # index variable
if q == 0: # unbiased (q = 0) transform
# multiply by (kr)^[- i 2 m pi/(n dlnr)] U_mu[i 2 m pi/(n dlnr)]
ar = fct[2 * m - 1]
ai = fct[2 * m]
fct[2 * m - 1] = ar * xsave[2 * m + 1] - ai * xsave[2 * m + 2]
fct[2 * m] = ar * xsave[2 * m + 2] + ai * xsave[2 * m + 1]
# problem(2*m)atical last element, for even n
if np.mod(n, 2) == 0:
ar = xsave[-2]
if (tdir == 1): # forward transform: multiply by real part
# Why? See http://casa.colorado.edu/~ajsh/FFTLog/index.html#ure
fct[-1] *= ar
elif (tdir == -1): # backward transform: divide by real part
# Real part ar can be zero for maximally bad choice of kr.
# This is unlikely to happen by chance, but if it does, policy
# is to let it happen. For low-ringing kr, imaginary part ai
# is zero by construction, and real part ar is guaranteed
# nonzero.
fct[-1] /= ar
else: # biased (q != 0) transform
# multiply by (kr)^[- i 2 m pi/(n dlnr)] U_mu[q + i 2 m pi/(n dlnr)]
# phase
ar = fct[2 * m - 1]
ai = fct[2 * m]
fct[2 * m - 1] = ar * xsave[3 * m + 2] - ai * xsave[3 * m + 3]
fct[2 * m] = ar * xsave[3 * m + 3] + ai * xsave[3 * m + 2]
if tdir == 1: # forward transform: multiply by amplitude
fct[0] *= xsave[3]
fct[2 * m - 1] *= xsave[3 * m + 1]
fct[2 * m] *= xsave[3 * m + 1]
elif tdir == -1: # backward transform: divide by amplitude
# amplitude of m=0 element
ar = xsave[3]
if ar == 0:
# Amplitude of m=0 element can be zero for some mu, q
# combinations (singular inverse); policy is to drop
# potentially infinite constant.
fct[0] = 0
else:
fct[0] /= ar
# remaining amplitudes should never be zero
fct[2 * m - 1] /= xsave[3 * m + 1]
fct[2 * m] /= xsave[3 * m + 1]
# problematical last element, for even n
if np.mod(n, 2) == 0:
m = int(n / 2)
ar = xsave[3 * m + 2] * xsave[3 * m + 1]
if tdir == 1: # forward transform: multiply by real part
fct[-1] *= ar
elif (tdir == -1): # backward transform: divide by real part
# Real part ar can be zero for maximally bad choice of kr.
# This is unlikely to happen by chance, but if it does, policy
# is to let it happen. For low-ringing kr, imaginary part ai
# is zero by construction, and real part ar is guaranteed
# nonzero.
fct[-1] /= ar
# normal FFT back
# fct = irfft(fct)
# _raw_fft(fct, n, -1, -1, 1, _fftpack.drfft)
fct = drfft(fct, n, -1, 1)
# reverse the array and at the same time undo the FFTs' multiplication by n
# => Just reverse the array, the rest is already done in drfft.
fct = fct[::-1]
return fct
def fftpack_drfftb(buf, s, drfft=drfft): return drfft(buf, None, -1, 1, 1)
def fftpack_drfftf(buf, s, drfft=drfft): return drfft(buf, None, 1, 0, 1)
