def _sph_harm(m: jnp.ndarray, n: jnp.ndarray, theta: jnp.ndarray, phi: jnp.ndarray, n_max: int) -> jnp.ndarray: """Computes the spherical harmonics.""" cos_colatitude = jnp.cos(phi) legendre = _gen_associated_legendre(n_max, cos_colatitude, True) legendre_val = legendre[abs(m), n, jnp.arange(len(n))] angle = abs(m) * theta vandermonde = lax.complex(jnp.cos(angle), jnp.sin(angle)) harmonics = lax.complex(legendre_val * jnp.real(vandermonde), legendre_val * jnp.imag(vandermonde)) # Negative order. harmonics = jnp.where(m < 0, (-1.0)**abs(m) * jnp.conjugate(harmonics), harmonics) return harmonics
import numpy as np import textwrap import operator from typing import Tuple, Union, cast from jax import jit, custom_jvp from jax import lax from jax._src.lax import linalg as lax_linalg from jax._src import dtypes from jax._src.numpy.util import _wraps from jax._src.numpy import lax_numpy as jnp from jax._src.util import canonicalize_axis _T = lambda x: jnp.swapaxes(x, -1, -2) _H = lambda x: jnp.conjugate(jnp.swapaxes(x, -1, -2)) def _promote_arg_dtypes(*args): """Promotes `args` to a common inexact type.""" dtype, weak_type = dtypes._lattice_result_type(*args) if not jnp.issubdtype(dtype, jnp.inexact): dtype, weak_type = jnp.float_, False dtype = dtypes.canonicalize_dtype(dtype) args = [lax._convert_element_type(arg, dtype, weak_type) for arg in args] if len(args) == 1: return args[0] else: return args
def _spectral_helper(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, detrend_type='constant', return_onesided=True, scaling='density', axis=-1, mode='psd', boundary=None, padded=False): """LAX-backend implementation of `scipy.signal._spectral_helper`. Unlike the original helper function, `y` can be None for explicitly indicating auto-spectral (non cross-spectral) computation. In addition to this, `detrend` argument is renamed to `detrend_type` for avoiding internal name overlap. """ if mode not in ('psd', 'stft'): raise ValueError(f"Unknown value for mode {mode}, " "must be one of: ('psd', 'stft')") def make_pad(mode, **kwargs): def pad(x, n, axis=-1): pad_width = [(0, 0) for unused_n in range(x.ndim)] pad_width[axis] = (n, n) return jnp.pad(x, pad_width, mode, **kwargs) return pad boundary_funcs = { 'even': make_pad('reflect'), 'odd': odd_ext, 'constant': make_pad('edge'), 'zeros': make_pad('constant', constant_values=0.0), None: lambda x, *args, **kwargs: x } # Check/ normalize inputs if boundary not in boundary_funcs: raise ValueError(f"Unknown boundary option '{boundary}', " f"must be one of: {list(boundary_funcs.keys())}") axis = jax.core.concrete_or_error(operator.index, axis, "axis of windowed-FFT") axis = canonicalize_axis(axis, x.ndim) if nperseg is not None: # if specified by user nperseg = jax.core.concrete_or_error(int, nperseg, "nperseg of windowed-FFT") if nperseg < 1: raise ValueError('nperseg must be a positive integer') # parse window; if array like, then set nperseg = win.shape win, nperseg = signal_helper._triage_segments(window, nperseg, input_length=x.shape[axis]) if noverlap is None: noverlap = nperseg // 2 else: noverlap = jax.core.concrete_or_error(int, noverlap, "noverlap of windowed-FFT") if nfft is None: nfft = nperseg else: nfft = jax.core.concrete_or_error(int, nfft, "nfft of windowed-FFT") _check_arraylike("_spectral_helper", x) x = jnp.asarray(x) if y is None: outdtype = jax.dtypes.canonicalize_dtype( np.result_type(x, np.complex64)) else: _check_arraylike("_spectral_helper", y) y = jnp.asarray(y) outdtype = jax.dtypes.canonicalize_dtype( np.result_type(x, y, np.complex64)) if mode != 'psd': raise ValueError( "two-argument mode is available only when mode=='psd'") if x.ndim != y.ndim: raise ValueError( "two-arguments must have the same rank ({x.ndim} vs {y.ndim})." ) # Check if we can broadcast the outer axes together try: outershape = jnp.broadcast_shapes(tuple_delete(x.shape, axis), tuple_delete(y.shape, axis)) except ValueError as e: raise ValueError('x and y cannot be broadcast together.') from e # Special cases for size == 0 if y is None: if x.size == 0: return jnp.zeros(x.shape), jnp.zeros(x.shape), jnp.zeros(x.shape) else: if x.size == 0 or y.size == 0: outshape = tuple_insert(outershape, min([x.shape[axis], y.shape[axis]]), axis) emptyout = jnp.zeros(outshape) return emptyout, emptyout, emptyout # Move time-axis to the end if x.ndim > 1: if axis != -1: x = jnp.moveaxis(x, axis, -1) if y is not None and y.ndim > 1: y = jnp.moveaxis(y, axis, -1) # Check if x and y are the same length, zero-pad if necessary if y is not None: if x.shape[-1] != y.shape[-1]: if x.shape[-1] < y.shape[-1]: pad_shape = list(x.shape) pad_shape[-1] = y.shape[-1] - x.shape[-1] x = jnp.concatenate((x, jnp.zeros(pad_shape)), -1) else: pad_shape = list(y.shape) pad_shape[-1] = x.shape[-1] - y.shape[-1] y = jnp.concatenate((y, jnp.zeros(pad_shape)), -1) if nfft < nperseg: raise ValueError('nfft must be greater than or equal to nperseg.') if noverlap >= nperseg: raise ValueError('noverlap must be less than nperseg.') nstep = nperseg - noverlap # Apply paddings if boundary is not None: ext_func = boundary_funcs[boundary] x = ext_func(x, nperseg // 2, axis=-1) if y is not None: y = ext_func(y, nperseg // 2, axis=-1) if padded: # Pad to integer number of windowed segments # I.e make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg nadd = (-(x.shape[-1] - nperseg) % nstep) % nperseg zeros_shape = list(x.shape[:-1]) + [nadd] x = jnp.concatenate((x, jnp.zeros(zeros_shape)), axis=-1) if y is not None: zeros_shape = list(y.shape[:-1]) + [nadd] y = jnp.concatenate((y, jnp.zeros(zeros_shape)), axis=-1) # Handle detrending and window functions if not detrend_type: def detrend_func(d): return d elif not hasattr(detrend_type, '__call__'): def detrend_func(d): return detrend(d, type=detrend_type, axis=-1) elif axis != -1: # Wrap this function so that it receives a shape that it could # reasonably expect to receive. def detrend_func(d): d = jnp.moveaxis(d, axis, -1) d = detrend_type(d) return jnp.moveaxis(d, -1, axis) else: detrend_func = detrend_type if np.result_type(win, np.complex64) != outdtype: win = win.astype(outdtype) # Determine scale if scaling == 'density': scale = 1.0 / (fs * (win * win).sum()) elif scaling == 'spectrum': scale = 1.0 / win.sum()**2 else: raise ValueError(f'Unknown scaling: {scaling}') if mode == 'stft': scale = jnp.sqrt(scale) # Determine onesided/ two-sided if return_onesided: sides = 'onesided' if jnp.iscomplexobj(x) or jnp.iscomplexobj(y): sides = 'twosided' warnings.warn('Input data is complex, switching to ' 'return_onesided=False') else: sides = 'twosided' if sides == 'twosided': freqs = jax.numpy.fft.fftfreq(nfft, 1 / fs) elif sides == 'onesided': freqs = jax.numpy.fft.rfftfreq(nfft, 1 / fs) # Perform the windowed FFTs result = _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides) if y is not None: # All the same operations on the y data result_y = _fft_helper(y, win, detrend_func, nperseg, noverlap, nfft, sides) result = jnp.conjugate(result) * result_y elif mode == 'psd': result = jnp.conjugate(result) * result result *= scale if sides == 'onesided' and mode == 'psd': end = None if nfft % 2 else -1 result = result.at[..., 1:end].mul(2) time = jnp.arange(nperseg / 2, x.shape[-1] - nperseg / 2 + 1, nperseg - noverlap) / fs if boundary is not None: time -= (nperseg / 2) / fs result = result.astype(outdtype) # All imaginary parts are zero anyways if y is None and mode != 'stft': result = result.real # Move frequency axis back to axis where the data came from result = jnp.moveaxis(result, -1, axis) return freqs, time, result
def _H(x): return jnp.conjugate(jnp.swapaxes(x, -1, -2))