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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]) |
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shape = property(lambda self: (self.n, self.n)) |
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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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