core/bsinc_tables.cpp

   1
   2 #include "bsinc_tables.h"
   3
   4 #include <algorithm>
   5 #include <array>
   6 #include <cassert>
   7 #include <cmath>
   8 #include <cstddef>
   9 #include <limits>
  10 #include <stdexcept>
  11 #include <vector>
  12
  13 #include "alnumbers.h"
  14 #include "alnumeric.h"
  15 #include "alspan.h"
  16 #include "bsinc_defs.h"
  17 #include "opthelpers.h"
  18 #include "resampler_limits.h"
  19
  20
  21 namespace {
  22
  23 using uint = unsigned int;
  24
  25
  26 /* The zero-order modified Bessel function of the first kind, used for the
  27  * Kaiser window.
  28  *
  29  *   I_0(x) = sum_{k=0}^inf (1 / k!)^2 (x / 2)^(2 k)
  30  *          = sum_{k=0}^inf ((x / 2)^k / k!)^2
  31  *
  32  * This implementation only handles nu = 0, and isn't the most precise (it
  33  * starts with the largest value and accumulates successively smaller values,
  34  * compounding the rounding and precision error), but it's good enough.
  35  */
  36 template<typename T, typename U>
  37 constexpr auto cyl_bessel_i(T nu, U x) -> U
  38 {
  39     if(nu != T{0})
  40         throw std::runtime_error{"cyl_bessel_i: nu != 0"};
  41
  42     /* Start at k=1 since k=0 is trivial. */
  43     const double x2{x/2.0};
  44     double term{1.0};
  45     double sum{1.0};
  46     int k{1};
  47
  48     /* Let the integration converge until the term of the sum is no longer
  49      * significant.
  50      */
  51     double last_sum{};
  52     do {
  53         const double y{x2 / k};
  54         ++k;
  55         last_sum = sum;
  56         term *= y * y;
  57         sum += term;
  58     } while(sum != last_sum);
  59     return static_cast<U>(sum);
  60 }
  61
  62 /* This is the normalized cardinal sine (sinc) function.
  63  *
  64  *   sinc(x) = { 1,                   x = 0
  65  *             { sin(pi x) / (pi x),  otherwise.
  66  */
  67 constexpr double Sinc(const double x)
  68 {
  69     constexpr double epsilon{std::numeric_limits<double>::epsilon()};
  70     if(!(x > epsilon || x < -epsilon))
  71         return 1.0;
  72     return std::sin(al::numbers::pi*x) / (al::numbers::pi*x);
  73 }
  74
  75 /* Calculate a Kaiser window from the given beta value and a normalized k
  76  * [-1, 1].
  77  *
  78  *   w(k) = { I_0(B sqrt(1 - k^2)) / I_0(B),  -1 <= k <= 1
  79  *          { 0,                              elsewhere.
  80  *
  81  * Where k can be calculated as:
  82  *
  83  *   k = i / l,         where -l <= i <= l.
  84  *
  85  * or:
  86  *
  87  *   k = 2 i / M - 1,   where 0 <= i <= M.
  88  */
  89 constexpr double Kaiser(const double beta, const double k, const double besseli_0_beta)
  90 {
  91     if(!(k >= -1.0 && k <= 1.0))
  92         return 0.0;
  93     return ::cyl_bessel_i(0, beta * std::sqrt(1.0 - k*k)) / besseli_0_beta;
  94 }
  95
  96 /* Calculates the (normalized frequency) transition width of the Kaiser window.
  97  * Rejection is in dB.
  98  */
  99 constexpr double CalcKaiserWidth(const double rejection, const uint order) noexcept
 100 {
 101     if(rejection > 21.19)
 102         return (rejection - 7.95) / (2.285 * al::numbers::pi*2.0 * order);
 103     /* This enforces a minimum rejection of just above 21.18dB */
 104     return 5.79 / (al::numbers::pi*2.0 * order);
 105 }
 106
 107 /* Calculates the beta value of the Kaiser window. Rejection is in dB. */
 108 constexpr double CalcKaiserBeta(const double rejection)
 109 {
 110     if(rejection > 50.0)
 111         return 0.1102 * (rejection-8.7);
 112     if(rejection >= 21.0)
 113         return (0.5842 * std::pow(rejection-21.0, 0.4)) + (0.07886 * (rejection-21.0));
 114     return 0.0;
 115 }
 116
 117
 118 struct BSincHeader {
 119     double width{};
 120     double beta{};
 121     double scaleBase{};
 122
 123     std::array<uint,BSincScaleCount> a{};
 124     uint total_size{};
 125
 126     constexpr BSincHeader(uint Rejection, uint Order) noexcept
 127         : width{CalcKaiserWidth(Rejection, Order)}, beta{CalcKaiserBeta(Rejection)}
 128         , scaleBase{width / 2.0}
 129     {
 130         uint num_points{Order+1};
 131         for(uint si{0};si < BSincScaleCount;++si)
 132         {
 133             const double scale{lerpd(scaleBase, 1.0, (si+1) / double{BSincScaleCount})};
 134             const uint a_{std::min(static_cast<uint>(num_points / 2.0 / scale), num_points)};
 135             const uint m{2 * a_};
 136
 137             a[si] = a_;
 138             total_size += 4 * BSincPhaseCount * ((m+3) & ~3u);
 139         }
 140     }
 141 };
 142
 143 /* 11th and 23rd order filters (12 and 24-point respectively) with a 60dB drop
 144  * at nyquist. Each filter will scale up the order when downsampling, to 23rd
 145  * and 47th order respectively.
 146  */
 147 constexpr BSincHeader bsinc12_hdr{60, 11};
 148 constexpr BSincHeader bsinc24_hdr{60, 23};
 149
 150
 151 template<const BSincHeader &hdr>
 152 struct SIMDALIGN BSincFilterArray {
 153     alignas(16) std::array<float, hdr.total_size> mTable{};
 154
 155     BSincFilterArray()
 156     {
 157         static constexpr uint BSincPointsMax{(hdr.a[0]*2u + 3u) & ~3u};
 158         static_assert(BSincPointsMax <= MaxResamplerPadding, "MaxResamplerPadding is too small");
 159
 160         using filter_type = std::array<std::array<double,BSincPointsMax>,BSincPhaseCount>;
 161         auto filter = std::vector<filter_type>(BSincScaleCount);
 162
 163         const double besseli_0_beta{::cyl_bessel_i(0, hdr.beta)};
 164
 165         /* Calculate the Kaiser-windowed Sinc filter coefficients for each
 166          * scale and phase index.
 167          */
 168         for(uint si{0};si < BSincScaleCount;++si)
 169         {
 170             const uint m{hdr.a[si] * 2};
 171             const size_t o{(BSincPointsMax-m) / 2};
 172             const double scale{lerpd(hdr.scaleBase, 1.0, (si+1) / double{BSincScaleCount})};
 173             const double cutoff{scale - (hdr.scaleBase * std::max(1.0, scale*2.0))};
 174             const auto a = static_cast<double>(hdr.a[si]);
 175             const double l{a - 1.0/BSincPhaseCount};
 176
 177             for(uint pi{0};pi < BSincPhaseCount;++pi)
 178             {
 179                 const double phase{std::floor(l) + (pi/double{BSincPhaseCount})};
 180
 181                 for(uint i{0};i < m;++i)
 182                 {
 183                     const double x{i - phase};
 184                     filter[si][pi][o+i] = Kaiser(hdr.beta, x/l, besseli_0_beta) * cutoff *
 185                         Sinc(cutoff*x);
 186                 }
 187             }
 188         }
 189
 190         size_t idx{0};
 191         for(size_t si{0};si < BSincScaleCount;++si)
 192         {
 193             const size_t m{((hdr.a[si]*2) + 3) & ~3u};
 194             const size_t o{(BSincPointsMax-m) / 2};
 195
 196             /* Write out each phase index's filter and phase delta for this
 197              * quality scale.
 198              */
 199             for(size_t pi{0};pi < BSincPhaseCount;++pi)
 200             {
 201                 for(size_t i{0};i < m;++i)
 202                     mTable[idx++] = static_cast<float>(filter[si][pi][o+i]);
 203
 204                 /* Linear interpolation between phases is simplified by pre-
 205                  * calculating the delta (b - a) in: x = a + f (b - a)
 206                  */
 207                 if(pi < BSincPhaseCount-1)
 208                 {
 209                     for(size_t i{0};i < m;++i)
 210                     {
 211                         const double phDelta{filter[si][pi+1][o+i] - filter[si][pi][o+i]};
 212                         mTable[idx++] = static_cast<float>(phDelta);
 213                     }
 214                 }
 215                 else
 216                 {
 217                     /* The delta target for the last phase index is the first
 218                      * phase index with the coefficients offset by one. The
 219                      * first delta targets 0, as it represents a coefficient
 220                      * for a sample that won't be part of the filter.
 221                      */
 222                     mTable[idx++] = static_cast<float>(0.0 - filter[si][pi][o]);
 223                     for(size_t i{1};i < m;++i)
 224                     {
 225                         const double phDelta{filter[si][0][o+i-1] - filter[si][pi][o+i]};
 226                         mTable[idx++] = static_cast<float>(phDelta);
 227                     }
 228                 }
 229             }
 230
 231             /* Now write out each phase index's scale and phase+scale deltas,
 232              * to complete the bilinear equation for the combination of phase
 233              * and scale.
 234              */
 235             if(si < BSincScaleCount-1)
 236             {
 237                 for(size_t pi{0};pi < BSincPhaseCount;++pi)
 238                 {
 239                     for(size_t i{0};i < m;++i)
 240                     {
 241                         const double scDelta{filter[si+1][pi][o+i] - filter[si][pi][o+i]};
 242                         mTable[idx++] = static_cast<float>(scDelta);
 243                     }
 244
 245                     if(pi < BSincPhaseCount-1)
 246                     {
 247                         for(size_t i{0};i < m;++i)
 248                         {
 249                             const double spDelta{(filter[si+1][pi+1][o+i]-filter[si+1][pi][o+i]) -
 250                                 (filter[si][pi+1][o+i]-filter[si][pi][o+i])};
 251                             mTable[idx++] = static_cast<float>(spDelta);
 252                         }
 253                     }
 254                     else
 255                     {
 256                         mTable[idx++] = static_cast<float>((0.0 - filter[si+1][pi][o]) -
 257                             (0.0 - filter[si][pi][o]));
 258                         for(size_t i{1};i < m;++i)
 259                         {
 260                             const double spDelta{(filter[si+1][0][o+i-1] - filter[si+1][pi][o+i]) -
 261                                 (filter[si][0][o+i-1] - filter[si][pi][o+i])};
 262                             mTable[idx++] = static_cast<float>(spDelta);
 263                         }
 264                     }
 265                 }
 266             }
 267             else
 268             {
 269                 /* The last scale index doesn't have scale-related deltas. */
 270                 for(size_t i{0};i < BSincPhaseCount*m*2;++i)
 271                     mTable[idx++] = 0.0f;
 272             }
 273         }
 274         assert(idx == hdr.total_size);
 275     }
 276
 277     [[nodiscard]] constexpr auto getHeader() const noexcept -> const BSincHeader& { return hdr; }
 278     [[nodiscard]] constexpr auto getTable() const noexcept { return al::span{mTable}; }
 279 };
 280
 281 const BSincFilterArray<bsinc12_hdr> bsinc12_filter{};
 282 const BSincFilterArray<bsinc24_hdr> bsinc24_filter{};
 283
 284 template<typename T>
 285 constexpr BSincTable GenerateBSincTable(const T &filter)
 286 {
 287     BSincTable ret{};
 288     const BSincHeader &hdr = filter.getHeader();
 289     ret.scaleBase = static_cast<float>(hdr.scaleBase);
 290     ret.scaleRange = static_cast<float>(1.0 / (1.0 - hdr.scaleBase));
 291     for(size_t i{0};i < BSincScaleCount;++i)
 292         ret.m[i] = ((hdr.a[i]*2) + 3) & ~3u;
 293     ret.filterOffset[0] = 0;
 294     for(size_t i{1};i < BSincScaleCount;++i)
 295         ret.filterOffset[i] = ret.filterOffset[i-1] + ret.m[i-1]*4*BSincPhaseCount;
 296     ret.Tab = filter.getTable();
 297     return ret;
 298 }
 299
 300 } // namespace
 301
 302 const BSincTable gBSinc12{GenerateBSincTable(bsinc12_filter)};
 303 const BSincTable gBSinc24{GenerateBSincTable(bsinc24_filter)};