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| 1 | +#ifdef XTENSOR_USE_TBB |
| 2 | +#include <oneapi/tbb.h> |
| 3 | +#endif |
| 4 | +#include <stdexcept> |
| 5 | + |
| 6 | +#include <xtl/xcomplex.hpp> |
| 7 | + |
| 8 | +#include <xtensor/xarray.hpp> |
| 9 | +#include <xtensor/xaxis_slice_iterator.hpp> |
| 10 | +#include <xtensor/xbuilder.hpp> |
| 11 | +#include <xtensor/xcomplex.hpp> |
| 12 | +#include <xtensor/xmath.hpp> |
| 13 | +#include <xtensor/xnoalias.hpp> |
| 14 | +#include <xtensor/xview.hpp> |
| 15 | + |
| 16 | +namespace xt |
| 17 | +{ |
| 18 | + namespace fft |
| 19 | + { |
| 20 | + namespace detail |
| 21 | + { |
| 22 | + template < |
| 23 | + class E, |
| 24 | + typename std::enable_if<xtl::is_complex<typename std::decay<E>::type::value_type>::value, bool>::type = true> |
| 25 | + inline auto radix2(E&& e) |
| 26 | + { |
| 27 | + using namespace xt::placeholders; |
| 28 | + using namespace std::complex_literals; |
| 29 | + using value_type = typename std::decay_t<E>::value_type; |
| 30 | + using precision = typename value_type::value_type; |
| 31 | + auto N = e.size(); |
| 32 | + const bool powerOfTwo = !(N == 0) && !(N & (N - 1)); |
| 33 | + // check for power of 2 |
| 34 | + if (!powerOfTwo || N == 0) |
| 35 | + { |
| 36 | + // TODO: Replace implementation with dft |
| 37 | + XTENSOR_THROW(std::runtime_error, "FFT Implementation requires power of 2"); |
| 38 | + } |
| 39 | + auto pi = xt::numeric_constants<precision>::PI; |
| 40 | + xt::xtensor<value_type, 1> ev = e; |
| 41 | + if (N <= 1) |
| 42 | + { |
| 43 | + return ev; |
| 44 | + } |
| 45 | + else |
| 46 | + { |
| 47 | +#ifdef XTENSOR_USE_TBB |
| 48 | + xt::xtensor<value_type, 1> even; |
| 49 | + xt::xtensor<value_type, 1> odd; |
| 50 | + oneapi::tbb::parallel_invoke( |
| 51 | + [&] |
| 52 | + { |
| 53 | + even = radix2(xt::view(ev, xt::range(0, _, 2))); |
| 54 | + }, |
| 55 | + [&] |
| 56 | + { |
| 57 | + odd = radix2(xt::view(ev, xt::range(1, _, 2))); |
| 58 | + } |
| 59 | + ); |
| 60 | +#else |
| 61 | + auto even = radix2(xt::view(ev, xt::range(0, _, 2))); |
| 62 | + auto odd = radix2(xt::view(ev, xt::range(1, _, 2))); |
| 63 | +#endif |
| 64 | + |
| 65 | + auto range = xt::arange<double>(N / 2); |
| 66 | + auto exp = xt::exp(static_cast<value_type>(-2i) * pi * range / N); |
| 67 | + auto t = exp * odd; |
| 68 | + auto first_half = even + t; |
| 69 | + auto second_half = even - t; |
| 70 | + // TODO: should be a call to stack if performance was improved |
| 71 | + auto spectrum = xt::xtensor<value_type, 1>::from_shape({N}); |
| 72 | + xt::view(spectrum, xt::range(0, N / 2)) = first_half; |
| 73 | + xt::view(spectrum, xt::range(N / 2, N)) = second_half; |
| 74 | + return spectrum; |
| 75 | + } |
| 76 | + } |
| 77 | + |
| 78 | + template <typename E> |
| 79 | + auto transform_bluestein(E&& data) |
| 80 | + { |
| 81 | + using value_type = typename std::decay_t<E>::value_type; |
| 82 | + using precision = typename value_type::value_type; |
| 83 | + |
| 84 | + // Find a power-of-2 convolution length m such that m >= n * 2 + 1 |
| 85 | + const std::size_t n = data.size(); |
| 86 | + size_t m = std::ceil(std::log2(n * 2 + 1)); |
| 87 | + m = std::pow(2, m); |
| 88 | + |
| 89 | + // Trignometric table |
| 90 | + auto exp_table = xt::xtensor<std:: complex<precision>, 1>:: from_shape({n}); |
| 91 | + xt::xtensor<std::size_t, 1> i = xt::pow(xt::linspace<std::size_t>(0, n - 1, n), 2); |
| 92 | + i %= (n * 2); |
| 93 | + |
| 94 | + auto angles = xt::eval(precision{3.141592653589793238463} * i / n); |
| 95 | + auto j = std:: complex<precision>( 0, 1); |
| 96 | + exp_table = xt::exp(-angles * j); |
| 97 | + |
| 98 | + // Temporary vectors and preprocessing |
| 99 | + auto av = xt::empty<std:: complex<precision>>({m}); |
| 100 | + xt::view(av, xt::range(0, n)) = data * exp_table; |
| 101 | + |
| 102 | + |
| 103 | + auto bv = xt::empty<std:: complex<precision>>({m}); |
| 104 | + xt::view(bv, xt::range(0, n)) = ::xt::conj(exp_table); |
| 105 | + xt::view(bv, xt::range(-n + 1, xt::placeholders::_)) = xt::view( |
| 106 | + ::xt::conj(xt::flip(exp_table)), |
| 107 | + xt::range(xt::placeholders::_, -1) |
| 108 | + ); |
| 109 | + |
| 110 | + // Convolution |
| 111 | + auto xv = radix2(av); |
| 112 | + auto yv = radix2(bv); |
| 113 | + auto spectrum_k = xv * yv; |
| 114 | + auto complex_args = xt::conj(spectrum_k); |
| 115 | + auto fft_res = radix2(complex_args); |
| 116 | + auto cv = xt::conj(fft_res) / m; |
| 117 | + |
| 118 | + return xt::eval(xt::view(cv, xt::range(0, n)) * exp_table); |
| 119 | + } |
| 120 | + } // namespace detail |
| 121 | + |
| 122 | + /** |
| 123 | + * @brief 1D FFT of an Nd array along a specified axis |
| 124 | + * @param e an Nd expression to be transformed to the fourier domain |
| 125 | + * @param axis the axis along which to perform the 1D FFT |
| 126 | + * @return a transformed xarray of the specified precision |
| 127 | + */ |
| 128 | + template < |
| 129 | + class E, |
| 130 | + typename std::enable_if<xtl::is_complex<typename std::decay<E>::type::value_type>::value, bool>::type = true> |
| 131 | + inline auto fft(E&& e, std::ptrdiff_t axis = -1) |
| 132 | + { |
| 133 | + using value_type = typename std::decay_t<E>::value_type; |
| 134 | + using precision = typename value_type::value_type; |
| 135 | + const auto saxis = xt::normalize_axis(e.dimension(), axis); |
| 136 | + const size_t N = e.shape(saxis); |
| 137 | + const bool powerOfTwo = !(N == 0) && !(N & (N - 1)); |
| 138 | + xt::xarray<std:: complex<precision>> out = xt::eval(e); |
| 139 | + auto begin = xt::axis_slice_begin(out, saxis); |
| 140 | + auto end = xt::axis_slice_end(out, saxis); |
| 141 | + for (auto iter = begin; iter != end; iter++) |
| 142 | + { |
| 143 | + if (powerOfTwo) |
| 144 | + { |
| 145 | + xt::noalias(*iter) = detail::radix2(*iter); |
| 146 | + } |
| 147 | + else |
| 148 | + { |
| 149 | + xt::noalias(*iter) = detail::transform_bluestein(*iter); |
| 150 | + } |
| 151 | + } |
| 152 | + return out; |
| 153 | + } |
| 154 | + |
| 155 | + /** |
| 156 | + * @breif 1D FFT of an Nd array along a specified axis |
| 157 | + * @param e an Nd expression to be transformed to the fourier domain |
| 158 | + * @param axis the axis along which to perform the 1D FFT |
| 159 | + * @return a transformed xarray of the specified precision |
| 160 | + */ |
| 161 | + template < |
| 162 | + class E, |
| 163 | + typename std::enable_if<!xtl::is_complex<typename std::decay<E>::type::value_type>::value, bool>::type = true> |
| 164 | + inline auto fft(E&& e, std::ptrdiff_t axis = -1) |
| 165 | + { |
| 166 | + using value_type = typename std::decay<E>::type::value_type; |
| 167 | + return fft(xt::cast<std::complex<value_type>>(e), axis); |
| 168 | + } |
| 169 | + |
| 170 | + template < |
| 171 | + class E, |
| 172 | + typename std::enable_if<xtl::is_complex<typename std::decay<E>::type::value_type>::value, bool>::type = true> |
| 173 | + auto ifft(E&& e, std::ptrdiff_t axis = -1) |
| 174 | + { |
| 175 | + // check the length of the data on that axis |
| 176 | + const std::size_t n = e.shape(axis); |
| 177 | + if (n == 0) |
| 178 | + { |
| 179 | + XTENSOR_THROW(std::runtime_error, "Cannot take the iFFT along an empty dimention"); |
| 180 | + } |
| 181 | + auto complex_args = xt::conj(e); |
| 182 | + auto fft_res = xt::fft::fft(complex_args, axis); |
| 183 | + fft_res = xt::conj(fft_res); |
| 184 | + return fft_res; |
| 185 | + } |
| 186 | + |
| 187 | + template < |
| 188 | + class E, |
| 189 | + typename std::enable_if<!xtl::is_complex<typename std::decay<E>::type::value_type>::value, bool>::type = true> |
| 190 | + inline auto ifft(E&& e, std::ptrdiff_t axis = -1) |
| 191 | + { |
| 192 | + using value_type = typename std::decay<E>::type::value_type; |
| 193 | + return ifft(xt::cast<std::complex<value_type>>(e), axis); |
| 194 | + } |
| 195 | + |
| 196 | + /* |
| 197 | + * @brief performs a circular fft convolution xvec and yvec must |
| 198 | + * be the same shape. |
| 199 | + * @param xvec first array of the convolution |
| 200 | + * @param yvec second array of the convolution |
| 201 | + * @param axis axis along which to perform the convolution |
| 202 | + */ |
| 203 | + template <typename E1, typename E2> |
| 204 | + auto convolve(E1&& xvec, E2&& yvec, std::ptrdiff_t axis = -1) |
| 205 | + { |
| 206 | + // we could broadcast but that could get complicated??? |
| 207 | + if (xvec.dimension() != yvec.dimension()) |
| 208 | + { |
| 209 | + XTENSOR_THROW(std::runtime_error, "Mismatched dimentions"); |
| 210 | + } |
| 211 | + |
| 212 | + auto saxis = xt::normalize_axis(xvec.dimension(), axis); |
| 213 | + if (xvec.shape(saxis) != yvec.shape(saxis)) |
| 214 | + { |
| 215 | + XTENSOR_THROW(std::runtime_error, "Mismatched lengths along slice axis"); |
| 216 | + } |
| 217 | + |
| 218 | + const std::size_t n = xvec.shape(saxis); |
| 219 | + |
| 220 | + auto xv = fft(xvec, axis); |
| 221 | + auto yv = fft(yvec, axis); |
| 222 | + |
| 223 | + auto begin_x = xt::axis_slice_begin(xv, saxis); |
| 224 | + auto end_x = xt::axis_slice_end(xv, saxis); |
| 225 | + auto iter_y = xt::axis_slice_begin(yv, saxis); |
| 226 | + |
| 227 | + for (auto iter = begin_x; iter != end_x; iter++) |
| 228 | + { |
| 229 | + (*iter) = (*iter_y++) * (*iter); |
| 230 | + } |
| 231 | + |
| 232 | + auto outvec = ifft(xv, axis); |
| 233 | + |
| 234 | + // Scaling (because this FFT implementation omits it) |
| 235 | + outvec = outvec / n; |
| 236 | + |
| 237 | + return outvec; |
| 238 | + } |
| 239 | + |
| 240 | + } |
| 241 | +} // namespace xt::fft |
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