def hilbert(x,
_cache=_cache):
""" hilbert(x) -> y
Return Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = sqrt(-1)*sign(j) * x_j
y_0 = 0
Notes:
If sum(x,axis=0)==0 then
hilbert(ihilbert(x)) == x
For even len(x), the Nyquist mode of x is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return hilbert(tmp.real)+1j*hilbert(tmp.imag)
n = len(x)
omega = _cache.get(n)
if omega is None:
if len(_cache)>20:
while _cache: _cache.popitem()
def kernel(k):
if k>0: return 1.0
elif k<0: return -1.0
return 0.0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[n] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def itilbert(x, h, period=None, _cache=_cache):
"""
Return inverse h-Tilbert transform of a periodic sequence x.
If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j
y_0 = 0
For more details, see `tilbert`.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return itilbert(tmp.real, h, period) + \
1j * itilbert(tmp.imag, h, period)
if period is not None:
h = h * 2 * pi / period
n = len(x)
omega = _cache.get((n, h))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, h=h):
if k:
return -tanh(h * k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n, h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp, omega, swap_real_imag=1, overwrite_x=overwrite_x)
def hilbert(x, _cache=_cache):
"""
Return Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*sign(j) * x_j
y_0 = 0
Parameters
----------
x : array_like
The input array, should be periodic.
_cache : dict, optional
Dictionary that contains the kernel used to do a convolution with.
Returns
-------
y : ndarray
The transformed input.
See Also
--------
scipy.signal.hilbert : Compute the analytic signal, using the Hilbert
transform.
Notes
-----
If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``.
For even len(x), the Nyquist mode of x is taken zero.
The sign of the returned transform does not have a factor -1 that is more
often than not found in the definition of the Hilbert transform. Note also
that `scipy.signal.hilbert` does have an extra -1 factor compared to this
function.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return hilbert(tmp.real) + 1j * hilbert(tmp.imag)
n = len(x)
omega = _cache.get(n)
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k):
if k > 0:
return 1.0
elif k < 0:
return -1.0
return 0.0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[n] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp, omega, swap_real_imag=1, overwrite_x=overwrite_x)
def itilbert(x,h,period=None,
_cache = _cache):
""" itilbert(x, h, period=2*pi) -> y
Return inverse h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j
y_0 = 0
Optional input: see tilbert.__doc__
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return itilbert(tmp.real,h,period)+\
1j*itilbert(tmp.imag,h,period)
if period is not None:
h = h*2*pi/period
n = len(x)
omega = _cache.get((n,h))
if omega is None:
if len(_cache)>20:
while _cache: _cache.popitem()
def kernel(k,h=h):
if k: return -tanh(h*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def hilbert(x, _cache=_cache):
"""
Return Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*sign(j) * x_j
y_0 = 0
Parameters
----------
x : array_like
The input array, should be periodic.
_cache : dict, optional
Dictionary that contains the kernel used to do a convolution with.
Returns
-------
y : ndarray
The transformed input.
See Also
--------
scipy.signal.hilbert : Compute the analytic signal, using the Hilbert
transform.
Notes
-----
If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``.
For even len(x), the Nyquist mode of x is taken zero.
The sign of the returned transform does not have a factor -1 that is more
often than not found in the definition of the Hilbert transform. Note also
that `scipy.signal.hilbert` does have an extra -1 factor compared to this
function.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return hilbert(tmp.real)+1j*hilbert(tmp.imag)
n = len(x)
omega = _cache.get(n)
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k):
if k > 0:
return 1.0
elif k < 0:
return -1.0
return 0.0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[n] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def diff(x, order=1, period=None, _cache=_cache):
"""
Return k-th derivative (or integral) of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j
y_0 = 0 if order is not 0.
Parameters
----------
x : array_like
Input array.
order : int, optional
The order of differentiation. Default order is 1. If order is
negative, then integration is carried out under the assumption
that ``x_0 == 0``.
period : float, optional
The assumed period of the sequence. Default is ``2*pi``.
Notes
-----
If ``sum(x, axis=0) = 0`` then ``diff(diff(x, k), -k) == x`` (within
numerical accuracy).
For odd order and even ``len(x)``, the Nyquist mode is taken zero.
"""
tmp = asarray(x)
if order == 0:
return tmp
if iscomplexobj(tmp):
return diff(tmp.real, order, period) + 1j * diff(tmp.imag, order, period)
if period is not None:
c = 2 * pi / period
else:
c = 1.0
n = len(x)
omega = _cache.get((n, order, c))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, order=order, c=c):
if k:
return pow(c * k, order)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=order,
zero_nyquist=1)
_cache[(n, order, c)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp, omega, swap_real_imag=order % 2,
overwrite_x=overwrite_x)
def cs_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence.
If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a, b : float
Defines the parameters of the cosh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence. Default period is ``2*pi``.
Returns
-------
cs_diff : ndarray
Pseudo-derivative of periodic sequence `x`.
Notes
-----
For even len(`x`), the Nyquist mode of `x` is taken as zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cs_diff(tmp.real,a,b,period) + \
1j*cs_diff(tmp.imag,a,b,period)
if period is not None:
a = a * 2 * pi / period
b = b * 2 * pi / period
n = len(x)
omega = _cache.get((n, a, b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, a=a, b=b):
if k:
return -cosh(a * k) / sinh(b * k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n, a, b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,
omega,
swap_real_imag=1,
overwrite_x=overwrite_x)
def diff(x,order=1,period=None, _cache=_cache):
"""
Return k-th derivative (or integral) of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j
y_0 = 0 if order is not 0.
Parameters
----------
x : array_like
Input array.
order : int, optional
The order of differentiation. Default order is 1. If order is
negative, then integration is carried out under the assumption
that ``x_0 == 0``.
period : float, optional
The assumed period of the sequence. Default is ``2*pi``.
Notes
-----
If ``sum(x, axis=0) = 0`` then ``diff(diff(x, k), -k) == x`` (within
numerical accuracy).
For odd order and even ``len(x)``, the Nyquist mode is taken zero.
"""
tmp = asarray(x)
if order == 0:
return tmp
if iscomplexobj(tmp):
return diff(tmp.real,order,period)+1j*diff(tmp.imag,order,period)
if period is not None:
c = 2*pi/period
else:
c = 1.0
n = len(x)
omega = _cache.get((n,order,c))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,order=order,c=c):
if k:
return pow(c*k,order)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=order,
zero_nyquist=1)
_cache[(n,order,c)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=order % 2,
overwrite_x=overwrite_x)
def __fix_shape(x, n, axis, dct_or_dst):
tmp = _asfarray(x)
copy_made = _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made2 = _fix_shape(tmp, n, axis)
copy_made = copy_made or copy_made2
if n < 1:
raise ValueError("Invalid number of %s data points "
"(%d) specified." % (dct_or_dst, n))
return tmp, n, copy_made
def __fix_shape(x, n, axis, dct_or_dst):
tmp = _asfarray(x)
copy_made = _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made2 = _fix_shape(tmp, n, axis)
copy_made = copy_made or copy_made2
if n < 1:
raise ValueError("Invalid number of %s data points "
"(%d) specified." % (dct_or_dst, n))
return tmp, n, copy_made
def sc_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
Input array.
a,b : float
Defines the parameters of the sinh/cosh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is 2*pi.
Notes
-----
``sc_diff(cs_diff(x,a,b),b,a) == x``
For even ``len(x)``, the Nyquist mode of x is taken as zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return sc_diff(tmp.real,a,b,period) + \
1j*sc_diff(tmp.imag,a,b,period)
if period is not None:
a = a * 2 * pi / period
b = b * 2 * pi / period
n = len(x)
omega = _cache.get((n, a, b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, a=a, b=b):
if k:
return sinh(a * k) / cosh(b * k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n, a, b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,
omega,
swap_real_imag=1,
overwrite_x=overwrite_x)
def shift(x, a, period=None, _cache=_cache):
"""
Shift periodic sequence x by a: y(u) = x(u+a).
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a : float
Defines the parameters of the sinh/sinh pseudo-differential
period : float, optional
The period of the sequences x and y. Default period is ``2*pi``.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return shift(tmp.real, a, period) + 1j * shift(tmp.imag, a, period)
if period is not None:
a = a * 2 * pi / period
n = len(x)
omega = _cache.get((n, a))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel_real(k, a=a):
return cos(a * k)
def kernel_imag(k, a=a):
return sin(a * k)
omega_real = convolve.init_convolution_kernel(n,
kernel_real,
d=0,
zero_nyquist=0)
omega_imag = convolve.init_convolution_kernel(n,
kernel_imag,
d=1,
zero_nyquist=0)
_cache[(n, a)] = omega_real, omega_imag
else:
omega_real, omega_imag = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve_z(tmp,
omega_real,
omega_imag,
overwrite_x=overwrite_x)
def cs_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence.
If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a, b : float
Defines the parameters of the cosh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence. Default period is ``2*pi``.
Returns
-------
cs_diff : ndarray
Pseudo-derivative of periodic sequence `x`.
Notes
-----
For even len(`x`), the Nyquist mode of `x` is taken as zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cs_diff(tmp.real,a,b,period) + \
1j*cs_diff(tmp.imag,a,b,period)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
if k:
return -cosh(a*k)/sinh(b*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def cc_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a,b : float
Defines the parameters of the sinh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is ``2*pi``.
Returns
-------
cc_diff : ndarray
Pseudo-derivative of periodic sequence `x`.
Notes
-----
``cc_diff(cc_diff(x,a,b),b,a) == x``
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cc_diff(tmp.real,a,b,period) + \
1j*cc_diff(tmp.imag,a,b,period)
if period is not None:
a = a * 2 * pi / period
b = b * 2 * pi / period
n = len(x)
omega = _cache.get((n, a, b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, a=a, b=b):
return cosh(a * k) / cosh(b * k)
omega = convolve.init_convolution_kernel(n, kernel)
_cache[(n, a, b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp, omega, overwrite_x=overwrite_x)
def tilbert(x, h, period=None, _cache=_cache):
""" tilbert(x, h, period=2*pi) -> y
Return h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j
y_0 = 0
Input:
h
Defines the parameter of the Tilbert transform.
period
The assumed period of the sequence. Default period is 2*pi.
Notes:
If sum(x,axis=0)==0 and n=len(x) is odd then
tilbert(itilbert(x)) == x
If 2*pi*h/period is approximately 10 or larger then numerically
tilbert == hilbert
(theoretically oo-Tilbert == Hilbert).
For even len(x), the Nyquist mode of x is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return tilbert(tmp.real,h,period)+\
1j*tilbert(tmp.imag,h,period)
if period is not None:
h = h * 2 * pi / period
n = len(x)
omega = _cache.get((n, h))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, h=h):
if k: return 1.0 / tanh(h * k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n, h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,
omega,
swap_real_imag=1,
overwrite_x=overwrite_x)
def cc_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a,b : float
Defines the parameters of the sinh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is ``2*pi``.
Returns
-------
cc_diff : ndarray
Pseudo-derivative of periodic sequence `x`.
Notes
-----
``cc_diff(cc_diff(x,a,b),b,a) == x``
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cc_diff(tmp.real,a,b,period) + \
1j*cc_diff(tmp.imag,a,b,period)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
return cosh(a*k)/cosh(b*k)
omega = convolve.init_convolution_kernel(n,kernel)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,overwrite_x=overwrite_x)
def shift(x, a, period=None, _cache=_cache):
""" shift(x, a, period=2*pi) -> y
Shift periodic sequence x by a: y(u) = x(u+a).
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f
Optional input:
period
The period of the sequences x and y. Default period is 2*pi.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return shift(tmp.real, a, period) + 1j * shift(tmp.imag, a, period)
if period is not None:
a = a * 2 * pi / period
n = len(x)
omega = _cache.get((n, a))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel_real(k, a=a):
return cos(a * k)
def kernel_imag(k, a=a):
return sin(a * k)
omega_real = convolve.init_convolution_kernel(n,
kernel_real,
d=0,
zero_nyquist=0)
omega_imag = convolve.init_convolution_kernel(n,
kernel_imag,
d=1,
zero_nyquist=0)
_cache[(n, a)] = omega_real, omega_imag
else:
omega_real, omega_imag = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve_z(tmp,
omega_real,
omega_imag,
overwrite_x=overwrite_x)
def sc_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
Input array.
a,b : float
Defines the parameters of the sinh/cosh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is 2*pi.
Notes
-----
``sc_diff(cs_diff(x,a,b),b,a) == x``
For even ``len(x)``, the Nyquist mode of x is taken as zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return sc_diff(tmp.real,a,b,period) + \
1j*sc_diff(tmp.imag,a,b,period)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
if k:
return sinh(a*k)/cosh(b*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def cs_diff(x, a, b, period=None, _cache=_cache):
""" cs_diff(x, a, b, period=2*pi) -> y
Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
y_0 = 0
Input:
a,b
Defines the parameters of the cosh/sinh pseudo-differential
operator.
period
The period of the sequence. Default period is 2*pi.
Notes:
For even len(x), the Nyquist mode of x is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cs_diff(tmp.real,a,b,period)+\
1j*cs_diff(tmp.imag,a,b,period)
if period is not None:
a = a * 2 * pi / period
b = b * 2 * pi / period
n = len(x)
omega = _cache.get((n, a, b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, a=a, b=b):
if k: return -cosh(a * k) / sinh(b * k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n, a, b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,
omega,
swap_real_imag=1,
overwrite_x=overwrite_x)
def shift(x, a, period=None, _cache=_cache):
"""
Shift periodic sequence x by a: y(u) = x(u+a).
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a : float
Defines the parameters of the sinh/sinh pseudo-differential
period : float, optional
The period of the sequences x and y. Default period is ``2*pi``.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return shift(tmp.real,a,period)+1j*shift(tmp.imag,a,period)
if period is not None:
a = a*2*pi/period
n = len(x)
omega = _cache.get((n,a))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel_real(k,a=a):
return cos(a*k)
def kernel_imag(k,a=a):
return sin(a*k)
omega_real = convolve.init_convolution_kernel(n,kernel_real,d=0,
zero_nyquist=0)
omega_imag = convolve.init_convolution_kernel(n,kernel_imag,d=1,
zero_nyquist=0)
_cache[(n,a)] = omega_real,omega_imag
else:
omega_real,omega_imag = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve_z(tmp,omega_real,omega_imag,
overwrite_x=overwrite_x)
def tilbert(x,h,period=None,
_cache = _cache):
""" tilbert(x, h, period=2*pi) -> y
Return h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j
y_0 = 0
Input:
h
Defines the parameter of the Tilbert transform.
period
The assumed period of the sequence. Default period is 2*pi.
Notes:
If sum(x,axis=0)==0 and n=len(x) is odd then
tilbert(itilbert(x)) == x
If 2*pi*h/period is approximately 10 or larger then numerically
tilbert == hilbert
(theoretically oo-Tilbert == Hilbert).
For even len(x), the Nyquist mode of x is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return tilbert(tmp.real,h,period)+\
1j*tilbert(tmp.imag,h,period)
if period is not None:
h = h*2*pi/period
n = len(x)
omega = _cache.get((n,h))
if omega is None:
if len(_cache)>20:
while _cache: _cache.popitem()
def kernel(k,h=h):
if k: return 1.0/tanh(h*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def cc_diff(x, a, b, period=None, _cache=_cache):
""" cc_diff(x, a, b, period=2*pi) -> y
Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
Input:
a,b
Defines the parameters of the sinh/sinh pseudo-differential
operator.
Optional input:
period
The period of the sequence x. Default is 2*pi.
Notes:
cc_diff(cc_diff(x,a,b),b,a) == x
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cc_diff(tmp.real,a,b,period)+\
1j*cc_diff(tmp.imag,a,b,period)
if period is not None:
a = a * 2 * pi / period
b = b * 2 * pi / period
n = len(x)
omega = _cache.get((n, a, b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, a=a, b=b):
return cosh(a * k) / cosh(b * k)
omega = convolve.init_convolution_kernel(n, kernel)
_cache[(n, a, b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp, omega, overwrite_x=overwrite_x)
def cs_diff(x, a, b, period=None,
_cache = _cache):
""" cs_diff(x, a, b, period=2*pi) -> y
Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
y_0 = 0
Input:
a,b
Defines the parameters of the cosh/sinh pseudo-differential
operator.
period
The period of the sequence. Default period is 2*pi.
Notes:
For even len(x), the Nyquist mode of x is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cs_diff(tmp.real,a,b,period)+\
1j*cs_diff(tmp.imag,a,b,period)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache)>20:
while _cache: _cache.popitem()
def kernel(k,a=a,b=b):
if k: return -cosh(a*k)/sinh(b*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def cc_diff(x, a, b, period=None,
_cache = _cache):
""" cc_diff(x, a, b, period=2*pi) -> y
Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
Input:
a,b
Defines the parameters of the sinh/sinh pseudo-differential
operator.
Optional input:
period
The period of the sequence x. Default is 2*pi.
Notes:
cc_diff(cc_diff(x,a,b),b,a) == x
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return cc_diff(tmp.real,a,b,period)+\
1j*cc_diff(tmp.imag,a,b,period)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache)>20:
while _cache: _cache.popitem()
def kernel(k,a=a,b=b):
return cosh(a*k)/cosh(b*k)
omega = convolve.init_convolution_kernel(n,kernel)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,overwrite_x=overwrite_x)
def hilbert(x, _cache=_cache):
""" hilbert(x) -> y
Return Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = sqrt(-1)*sign(j) * x_j
y_0 = 0
Notes:
If sum(x,axis=0)==0 then
hilbert(ihilbert(x)) == x
For even len(x), the Nyquist mode of x is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return hilbert(tmp.real) + 1j * hilbert(tmp.imag)
n = len(x)
omega = _cache.get(n)
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k):
if k > 0: return 1.0
elif k < 0: return -1.0
return 0.0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[n] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,
omega,
swap_real_imag=1,
overwrite_x=overwrite_x)
def shift(x, a, period=None,
_cache = _cache):
""" shift(x, a, period=2*pi) -> y
Shift periodic sequence x by a: y(u) = x(u+a).
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then
y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f
Optional input:
period
The period of the sequences x and y. Default period is 2*pi.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return shift(tmp.real,a,period)+1j*shift(tmp.imag,a,period)
if period is not None:
a = a*2*pi/period
n = len(x)
omega = _cache.get((n,a))
if omega is None:
if len(_cache)>20:
while _cache: _cache.popitem()
def kernel_real(k,a=a): return cos(a*k)
def kernel_imag(k,a=a): return sin(a*k)
omega_real = convolve.init_convolution_kernel(n,kernel_real,d=0,
zero_nyquist=0)
omega_imag = convolve.init_convolution_kernel(n,kernel_imag,d=1,
zero_nyquist=0)
_cache[(n,a)] = omega_real,omega_imag
else:
omega_real,omega_imag = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve_z(tmp,omega_real,omega_imag,
overwrite_x=overwrite_x)
def _dst(x, type, n=None, axis=-1, overwrite_x=0, normalize=None):
"""
Return Discrete Sine Transform of arbitrary type sequence x.
Parameters
----------
x : array-like
input array.
n : int, optional
Length of the transform.
axis : int, optional
Axis along which the dst is computed. (default=-1)
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
z : real ndarray
"""
tmp = np.asarray(x)
if not np.isrealobj(tmp):
raise TypeError("1st argument must be real sequence")
if n is None:
n = tmp.shape[axis]
else:
raise NotImplemented("Padding/truncating not yet implemented")
if tmp.dtype == np.double:
if type == 1:
f = _fftpack.ddst1
elif type == 2:
f = _fftpack.ddst2
elif type == 3:
f = _fftpack.ddst3
else:
raise ValueError("Type %d not understood" % type)
elif tmp.dtype == np.float32:
if type == 1:
f = _fftpack.dst1
elif type == 2:
f = _fftpack.dst2
elif type == 3:
f = _fftpack.dst3
else:
raise ValueError("Type %d not understood" % type)
else:
raise ValueError("dtype %s not supported" % tmp.dtype)
if normalize:
if normalize == "ortho":
nm = 1
else:
raise ValueError("Unknown normalize mode %s" % normalize)
else:
nm = 0
if type == 1 and n < 2:
raise ValueError("DST-I is not defined for size < 2")
overwrite_x = overwrite_x or _datacopied(tmp, x)
if axis == -1 or axis == len(tmp.shape) - 1:
return f(tmp, n, nm, overwrite_x)
#else:
# raise NotImplementedError("Axis arg not yet implemented")
tmp = np.swapaxes(tmp, axis, -1)
tmp = f(tmp, n, nm, overwrite_x)
return np.swapaxes(tmp, axis, -1)
def fft(x, n=None, axis=-1, overwrite_x=False):
"""
Return discrete Fourier transform of real or complex sequence.
The returned complex array contains ``y(0), y(1),..., y(n-1)`` where
``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``.
Parameters
----------
x : array_like
Array to Fourier transform.
n : int, optional
Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the fft's are computed; the default is over the
last axis (i.e., ``axis=-1``).
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
z : complex ndarray
with the elements::
[y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even
[y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd
where::
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1
See Also
--------
ifft : Inverse FFT
rfft : FFT of a real sequence
Notes
-----
The packing of the result is "standard": If ``A = fft(a, n)``, then
``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the
positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency
terms, in order of decreasingly negative frequency. So for an 8-point
transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1].
To rearrange the fft output so that the zero-frequency component is
centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use `fftshift`.
Both single and double precision routines are implemented. Half precision
inputs will be converted to single precision. Non floating-point inputs
will be converted to double precision. Long-double precision inputs are
not supported.
This function is most efficient when `n` is a power of two, and least
efficient when `n` is prime.
Note that if ``x`` is real-valued then ``A[j] == A[n-j].conjugate()``.
If ``x`` is real-valued and ``n`` is even then ``A[n/2]`` is real.
If the data type of `x` is real, a "real FFT" algorithm is automatically
used, which roughly halves the computation time. To increase efficiency
a little further, use `rfft`, which does the same calculation, but only
outputs half of the symmetrical spectrum. If the data is both real and
symmetrical, the `dct` can again double the efficiency, by generating
half of the spectrum from half of the signal.
Examples
--------
# >>> from scipy.fftpack import fft, ifft
# >>> x = np.arange(5)
# >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy.
True
"""
tmp = _asfarray(x)
try:
work_function = _DTYPE_TO_FFT[tmp.dtype]
except KeyError:
raise ValueError("type %s is not supported" % tmp.dtype)
if not (istype(tmp, numpy.complex64) or istype(tmp, numpy.complex128)):
overwrite_x = 1
overwrite_x = overwrite_x or _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made = _fix_shape(tmp, n, axis)
overwrite_x = overwrite_x or copy_made
if n < 1:
raise ValueError("Invalid number of FFT data points "
"(%d) specified." % n)
if axis == -1 or axis == len(tmp.shape) - 1:
return work_function(tmp, n, 1, 0, overwrite_x)
tmp = swapaxes(tmp, axis, -1)
tmp = work_function(tmp, n, 1, 0, overwrite_x)
return swapaxes(tmp, axis, -1)
def tilbert(x, h, period=None, _cache=_cache):
"""
Return h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
The input array to transform.
h : float
Defines the parameter of the Tilbert transform.
period : float, optional
The assumed period of the sequence. Default period is ``2*pi``.
Returns
-------
tilbert : ndarray
The result of the transform.
Notes
-----
If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd then
``tilbert(itilbert(x)) == x``.
If ``2 * pi * h / period`` is approximately 10 or larger, then
numerically ``tilbert == hilbert``
(theoretically oo-Tilbert == Hilbert).
For even ``len(x)``, the Nyquist mode of ``x`` is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return tilbert(tmp.real, h, period) + \
1j * tilbert(tmp.imag, h, period)
if period is not None:
h = h * 2 * pi / period
n = len(x)
omega = _cache.get((n, h))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, h=h):
if k:
return 1.0/tanh(h*k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n,h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def tilbert(x, h, period=None, _cache=_cache):
"""
Return h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
The input array to transform.
h : float
Defines the parameter of the Tilbert transform.
period : float, optional
The assumed period of the sequence. Default period is ``2*pi``.
Returns
-------
tilbert : ndarray
The result of the transform.
Notes
-----
If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd then
``tilbert(itilbert(x)) == x``.
If ``2 * pi * h / period`` is approximately 10 or larger, then
numerically ``tilbert == hilbert``
(theoretically oo-Tilbert == Hilbert).
For even ``len(x)``, the Nyquist mode of ``x`` is taken zero.
"""
tmp = asarray(x)
if iscomplexobj(tmp):
return tilbert(tmp.real, h, period) + \
1j * tilbert(tmp.imag, h, period)
if period is not None:
h = h * 2 * pi / period
n = len(x)
omega = _cache.get((n, h))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, h=h):
if k:
return 1.0 / tanh(h * k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n, h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,
omega,
swap_real_imag=1,
overwrite_x=overwrite_x)
def _dst(x, type, n=None, axis=-1, overwrite_x=0, normalize=None):
"""
Return Discrete Sine Transform of arbitrary type sequence x.
Parameters
----------
x : array-like
input array.
n : int, optional
Length of the transform.
axis : int, optional
Axis along which the dst is computed. (default=-1)
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
z : real ndarray
"""
tmp = np.asarray(x)
if not np.isrealobj(tmp):
raise TypeError("1st argument must be real sequence")
if n is None:
n = tmp.shape[axis]
else:
raise NotImplemented("Padding/truncating not yet implemented")
if tmp.dtype == np.double:
if type == 1:
f = _fftpack.ddst1
elif type == 2:
f = _fftpack.ddst2
elif type == 3:
f = _fftpack.ddst3
else:
raise ValueError("Type %d not understood" % type)
elif tmp.dtype == np.float32:
if type == 1:
f = _fftpack.dst1
elif type == 2:
f = _fftpack.dst2
elif type == 3:
f = _fftpack.dst3
else:
raise ValueError("Type %d not understood" % type)
else:
raise ValueError("dtype %s not supported" % tmp.dtype)
if normalize:
if normalize == "ortho":
nm = 1
else:
raise ValueError("Unknown normalize mode %s" % normalize)
else:
nm = 0
if type == 1 and n < 2:
raise ValueError("DST-I is not defined for size < 2")
overwrite_x = overwrite_x or _datacopied(tmp, x)
if axis == -1 or axis == len(tmp.shape) - 1:
return f(tmp, n, nm, overwrite_x)
#else:
# raise NotImplementedError("Axis arg not yet implemented")
tmp = np.swapaxes(tmp, axis, -1)
tmp = f(tmp, n, nm, overwrite_x)
return np.swapaxes(tmp, axis, -1)
