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from sympy.core.sympify import _sympify
from sympy.matrices.expressions import MatrixExpr
from sympy.core.numbers import I
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
class DFT(MatrixExpr):
r"""
Returns a discrete Fourier transform matrix. The matrix is scaled
with :math:`\frac{1}{\sqrt{n}}` so that it is unitary.
Parameters
==========
n : integer or Symbol
Size of the transform.
Examples
========
>>> from sympy.abc import n
>>> from sympy.matrices.expressions.fourier import DFT
>>> DFT(3)
DFT(3)
>>> DFT(3).as_explicit()
Matrix([
[sqrt(3)/3, sqrt(3)/3, sqrt(3)/3],
[sqrt(3)/3, sqrt(3)*exp(-2*I*pi/3)/3, sqrt(3)*exp(2*I*pi/3)/3],
[sqrt(3)/3, sqrt(3)*exp(2*I*pi/3)/3, sqrt(3)*exp(-2*I*pi/3)/3]])
>>> DFT(n).shape
(n, n)
References
==========
.. [1] https://en.wikipedia.org/wiki/DFT_matrix
"""
def __new__(cls, n):
n = _sympify(n)
cls._check_dim(n)
obj = super().__new__(cls, n)
return obj
n = property(lambda self: self.args[0]) # type: ignore
shape = property(lambda self: (self.n, self.n)) # type: ignore
def _entry(self, i, j, **kwargs):
w = exp(-2*S.Pi*I/self.n)
return w**(i*j) / sqrt(self.n)
def _eval_inverse(self):
return IDFT(self.n)
class IDFT(DFT):
r"""
Returns an inverse discrete Fourier transform matrix. The matrix is scaled
with :math:`\frac{1}{\sqrt{n}}` so that it is unitary.
Parameters
==========
n : integer or Symbol
Size of the transform
Examples
========
>>> from sympy.matrices.expressions.fourier import DFT, IDFT
>>> IDFT(3)
IDFT(3)
>>> IDFT(4)*DFT(4)
I
See Also
========
DFT
"""
def _entry(self, i, j, **kwargs):
w = exp(-2*S.Pi*I/self.n)
return w**(-i*j) / sqrt(self.n)
def _eval_inverse(self):
return DFT(self.n)
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