scipy.fft.fht¶
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scipy.fft.fht(a, dln, mu, offset=0.0, bias=0.0)[source]¶ Compute the fast Hankel transform.
Computes the discrete Hankel transform of a logarithmically spaced periodic sequence using the FFTLog algorithm [1], [2].
- Parameters
- aarray_like (…, n)
Real periodic input array, uniformly logarithmically spaced. For multidimensional input, the transform is performed over the last axis.
- dlnfloat
Uniform logarithmic spacing of the input array.
- mufloat
Order of the Hankel transform, any positive or negative real number.
- offsetfloat, optional
Offset of the uniform logarithmic spacing of the output array.
- biasfloat, optional
Exponent of power law bias, any positive or negative real number.
- Returns
- Aarray_like (…, n)
The transformed output array, which is real, periodic, uniformly logarithmically spaced, and of the same shape as the input array.
Notes
This function computes a discrete version of the Hankel transform
\[A(k) = \int_{0}^{\infty} \! a(r) \, J_\mu(kr) \, k \, dr \;,\]where \(J_\mu\) is the Bessel function of order \(\mu\). The index \(\mu\) may be any real number, positive or negative.
The input array a is a periodic sequence of length \(n\), uniformly logarithmically spaced with spacing dln,
\[a_j = a(r_j) \;, \quad r_j = r_c \exp[(j-j_c) \, \mathtt{dln}]\]centred about the point \(r_c\). Note that the central index \(j_c = (n+1)/2\) is half-integral if \(n\) is even, so that \(r_c\) falls between two input elements. Similarly, the output array A is a periodic sequence of length \(n\), also uniformly logarithmically spaced with spacing dln
\[A_j = A(k_j) \;, \quad k_j = k_c \exp[(j-j_c) \, \mathtt{dln}]\]centred about the point \(k_c\).
The centre points \(r_c\) and \(k_c\) of the periodic intervals may be chosen arbitrarily, but it would be usual to choose the product \(k_c r_c = k_j r_{n-1-j} = k_{n-1-j} r_j\) to be unity. This can be changed using the offset parameter, which controls the logarithmic offset \(\log(k_c) = \mathtt{offset} - \log(r_c)\) of the output array. Choosing an optimal value for offset may reduce ringing of the discrete Hankel transform.
If the bias parameter is nonzero, this function computes a discrete version of the biased Hankel transform
\[A(k) = \int_{0}^{\infty} \! a_q(r) \, (kr)^q \, J_\mu(kr) \, k \, dr\]where \(q\) is the value of bias, and a power law bias \(a_q(r) = a(r) \, (kr)^{-q}\) is applied to the input sequence. Biasing the transform can help approximate the continuous transform of \(a(r)\) if there is a value \(q\) such that \(a_q(r)\) is close to a periodic sequence, in which case the resulting \(A(k)\) will be close to the continuous transform.
References