def convolution_fwht(a, b):
"""
Performs dyadic (*bitwise-XOR*) convolution using Fast Walsh Hadamard
Transform.
The convolution is automatically padded to the right with zeros, as the
*radix-2 FWHT* requires the number of sample points to be a power of 2.
Parameters
==========
a, b : iterables
The sequences for which convolution is performed.
Examples
========
>>> from sympy import symbols, S, I
>>> from sympy.discrete.convolutions import convolution_fwht
>>> u, v, x, y = symbols('u v x y')
>>> convolution_fwht([u, v], [x, y])
[u*x + v*y, u*y + v*x]
>>> convolution_fwht([2, 3], [4, 5])
[23, 22]
>>> convolution_fwht([2, 5 + 4*I, 7], [6*I, 7, 3 + 4*I])
[56 + 68*I, -10 + 30*I, 6 + 50*I, 48 + 32*I]
>>> convolution_fwht([S(33)/7, S(55)/6, S(7)/4], [S(2)/3, 5])
[2057/42, 1870/63, 7/6, 35/4]
References
==========
.. [1] https://www.radioeng.cz/fulltexts/2002/02_03_40_42.pdf
.. [2] https://en.wikipedia.org/wiki/Hadamard_transform
"""
if not a or not b:
return []
a, b = a[:], b[:]
n = max(len(a), len(b))
if n & (n - 1): # not a power of 2
n = 2**n.bit_length()
# padding with zeros
a += [S.Zero] * (n - len(a))
b += [S.Zero] * (n - len(b))
a, b = fwht(a), fwht(b)
a = [expand_mul(x * y) for x, y in zip(a, b)]
a = ifwht(a)
return a
def convolution_fwht(a, b):
"""
Performs dyadic (*bitwise-XOR*) convolution using Fast Walsh Hadamard
Transform.
The convolution is automatically padded to the right with zeros, as the
*radix-2 FWHT* requires the number of sample points to be a power of 2.
Parameters
==========
a, b : iterables
The sequences for which convolution is performed.
Examples
========
>>> from sympy import symbols, S, I
>>> from sympy.discrete.convolutions import convolution_fwht
>>> u, v, x, y = symbols('u v x y')
>>> convolution_fwht([u, v], [x, y])
[u*x + v*y, u*y + v*x]
>>> convolution_fwht([2, 3], [4, 5])
[23, 22]
>>> convolution_fwht([2, 5 + 4*I, 7], [6*I, 7, 3 + 4*I])
[56 + 68*I, -10 + 30*I, 6 + 50*I, 48 + 32*I]
>>> convolution_fwht([S(33)/7, S(55)/6, S(7)/4], [S(2)/3, 5])
[2057/42, 1870/63, 7/6, 35/4]
References
==========
.. [1] https://www.radioeng.cz/fulltexts/2002/02_03_40_42.pdf
.. [2] https://en.wikipedia.org/wiki/Hadamard_transform
"""
if not a or not b:
return []
a, b = a[:], b[:]
n = max(len(a), len(b))
if n&(n - 1): # not a power of 2
n = 2**n.bit_length()
# padding with zeros
a += [S.Zero]*(n - len(a))
b += [S.Zero]*(n - len(b))
a, b = fwht(a), fwht(b)
a = [expand_mul(x*y) for x, y in zip(a, b)]
a = ifwht(a)
return a