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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) | |