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