# Source code for jax._src.scipy.fft

```# Copyright 2021 The JAX Authors.
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and

from __future__ import annotations

from collections.abc import Sequence
from functools import partial
import math

import scipy.fft as osp_fft
from jax import lax
import jax.numpy as jnp
from jax._src.util import canonicalize_axis
from jax._src.numpy.util import implements, promote_dtypes_complex
from jax._src.typing import Array

def _W4(N: int, k: Array) -> Array:
N_arr, k = promote_dtypes_complex(N, k)
return jnp.exp(-.5j * jnp.pi * k / N_arr)

def _dct_interleave(x: Array, axis: int) -> Array:
v0 = lax.slice_in_dim(x, None, None, 2, axis)
v1 = lax.rev(lax.slice_in_dim(x, 1, None, 2, axis), (axis,))
return lax.concatenate([v0, v1], axis)

def _dct_ortho_norm(out: Array, axis: int) -> Array:
factor = lax.concatenate([lax.full((1,), 4, out.dtype), lax.full((out.shape[axis] - 1,), 2, out.dtype)], 0)
factor = lax.expand_dims(factor, [a for a in range(out.ndim) if a != axis])
return out / lax.sqrt(factor * out.shape[axis])

# Implementation based on
# John Makhoul: A Fast Cosine Transform in One and Two Dimensions (1980)

[docs]
@implements(osp_fft.dct)
def dct(x: Array, type: int = 2, n: int | None = None,
axis: int = -1, norm: str | None = None) -> Array:
if type != 2:
raise NotImplementedError('Only DCT type 2 is implemented.')

axis = canonicalize_axis(axis, x.ndim)
if n is not None:
[(0, n - x.shape[axis] if a == axis else 0, 0)
for a in range(x.ndim)])

N = x.shape[axis]
v = _dct_interleave(x, axis)
V = jnp.fft.fft(v, axis=axis)
k = lax.expand_dims(jnp.arange(N, dtype=V.real.dtype), [a for a in range(x.ndim) if a != axis])
out = V * _W4(N, k)
out = 2 * out.real
if norm == 'ortho':
out = _dct_ortho_norm(out, axis)
return out

def _dct2(x: Array, axes: Sequence[int], norm: str | None) -> Array:
axis1, axis2 = map(partial(canonicalize_axis, num_dims=x.ndim), axes)
N1, N2 = x.shape[axis1], x.shape[axis2]
v = _dct_interleave(_dct_interleave(x, axis1), axis2)
V = jnp.fft.fftn(v, axes=axes)
k1 = lax.expand_dims(jnp.arange(N1, dtype=V.dtype),
[a for a in range(x.ndim) if a != axis1])
k2 = lax.expand_dims(jnp.arange(N2, dtype=V.dtype),
[a for a in range(x.ndim) if a != axis2])
out = _W4(N1, k1) * (_W4(N2, k2) * V + _W4(N2, -k2) * jnp.roll(jnp.flip(V, axis=axis2), shift=1, axis=axis2))
out = 2 * out.real
if norm == 'ortho':
return _dct_ortho_norm(_dct_ortho_norm(out, axis1), axis2)
return out

[docs]
@implements(osp_fft.dctn)
def dctn(x: Array, type: int = 2,
s: Sequence[int] | None=None,
axes: Sequence[int] | None = None,
norm: str | None = None) -> Array:
if type != 2:
raise NotImplementedError('Only DCT type 2 is implemented.')

if axes is None:
axes = range(x.ndim)

if len(axes) == 1:
return dct(x, n=s[0] if s is not None else None, axis=axes[0], norm=norm)

if s is not None:
ns = dict(zip(axes, s))
pads = [(0, ns[a] - x.shape[a] if a in ns else 0, 0) for a in range(x.ndim)]

if len(axes) == 2:
return _dct2(x, axes=axes, norm=norm)

# compose high-D DCTs from 2D and 1D DCTs:
for axes_block in [axes[i:i+2] for i in range(0, len(axes), 2)]:
x = dctn(x, axes=axes_block, norm=norm)
return x

[docs]
@implements(osp_fft.idct)
def idct(x: Array, type: int = 2, n: int | None = None,
axis: int = -1, norm: str | None = None) -> Array:
if type != 2:
raise NotImplementedError('Only DCT type 2 is implemented.')

axis = canonicalize_axis(axis, x.ndim)
if n is not None:
[(0, n - x.shape[axis] if a == axis else 0, 0)
for a in range(x.ndim)])
N = x.shape[axis]
x = x.astype(jnp.float32)
if norm is None:
x = _dct_ortho_norm(x, axis)
x = _dct_ortho_norm(x, axis)

k = lax.expand_dims(jnp.arange(N, dtype=jnp.float32), [a for a in range(x.ndim) if a != axis])
# everything is complex from here...
w4 = _W4(N,k)
x = x.astype(w4.dtype)
x = x / (_W4(N, k))
x = x * 2 * N

x = jnp.fft.ifft(x, axis=axis)
# convert back to reals..
out = _dct_deinterleave(x.real, axis)
return out

[docs]
@implements(osp_fft.idctn)
def idctn(x: Array, type: int = 2,
s: Sequence[int] | None=None,
axes: Sequence[int] | None = None,
norm: str | None = None) -> Array:
if type != 2:
raise NotImplementedError('Only DCT type 2 is implemented.')

if axes is None:
axes = range(x.ndim)

if len(axes) == 1:
return idct(x, n=s[0] if s is not None else None, axis=axes[0], norm=norm)

if s is not None:
ns = dict(zip(axes, s))
pads = [(0, ns[a] - x.shape[a] if a in ns else 0, 0) for a in range(x.ndim)]

# compose high-D DCTs from 1D DCTs:
for axis in axes:
x = idct(x, axis=axis, norm=norm)
return x

def _dct_deinterleave(x: Array, axis: int) -> Array:
empty_slice = slice(None, None, None)
ix0 = tuple(
slice(None, math.ceil(x.shape[axis]/2), 1) if i == axis else empty_slice
for i in range(len(x.shape)))
ix1  = tuple(
slice(math.ceil(x.shape[axis]/2), None, 1) if i == axis else empty_slice
for i in range(len(x.shape)))
v0 = x[ix0]
v1 = lax.rev(x[ix1], (axis,))
out = jnp.zeros(x.shape, dtype=x.dtype)
evens = tuple(
slice(None, None, 2) if i == axis else empty_slice for i in range(len(x.shape)))
odds = tuple(
slice(1, None, 2) if i == axis else empty_slice for i in range(len(x.shape)))
out =  out.at[evens].set(v0)
out = out.at[odds].set(v1)
return out
```