transforms.py 40.1 KB
 Michel Lespinasse committed Mar 19, 2000 1 2 3 4 5 ``````# Lossy compression algorithms very often make use of DCT or DFT calculations, # or variations of these calculations. This file is intended to be a short # reference about classical DCT and DFT algorithms. `````` Rafaël Carré committed Jul 07, 2010 6 7 8 ``````from random import random from math import pi, sin, cos, sqrt from cmath import exp `````` Michel Lespinasse committed Mar 19, 2000 9 10 `````` def exp_j (alpha): `````` Rafaël Carré committed Jul 07, 2010 11 `````` return exp (alpha * 1j) `````` Michel Lespinasse committed Mar 19, 2000 12 13 `````` def conjugate (c): `````` Michel Lespinasse committed Mar 28, 2000 14 15 `````` c = c + 0j return c.real - 1j * c.imag `````` Michel Lespinasse committed Mar 19, 2000 16 17 `````` def vector (N): `````` Michel Lespinasse committed Mar 28, 2000 18 `````` return [0j] * N `````` Michel Lespinasse committed Mar 19, 2000 19 20 `````` `````` Michel Lespinasse committed Mar 28, 2000 21 22 ``````# Let us start withthe canonical definition of the unscaled DFT algorithm : # (I can not draw sigmas in a text file so I'll use python code instead) :) `````` Michel Lespinasse committed Mar 19, 2000 23 24 25 26 27 28 29 30 `````` def W (k, N): return exp_j ((-2*pi*k)/N) def unscaled_DFT (N, input, output): for o in range(N): # o is output index output[o] = 0 for i in range(N): `````` Michel Lespinasse committed Mar 28, 2000 31 `````` output[o] = output[o] + input[i] * W (i*o, N) `````` Michel Lespinasse committed Mar 19, 2000 32 33 `````` # This algorithm takes complex input and output. There are N*N complex `````` Michel Lespinasse committed Mar 28, 2000 34 ``````# multiplications and N*(N-1) complex additions. `````` Michel Lespinasse committed Mar 19, 2000 35 36 `````` `````` Michel Lespinasse committed Mar 28, 2000 37 ``````# Of course this algorithm is an extremely naive implementation and there are `````` Michel Lespinasse committed Mar 19, 2000 38 ``````# some ways to use the trigonometric properties of the coefficients to find `````` Michel Lespinasse committed Mar 28, 2000 39 40 41 ``````# some decompositions that can accelerate the calculation by several orders # of magnitude... This is a well known and studied problem. One of the # available explanations of this process is at this url : `````` Michel Lespinasse committed Mar 19, 2000 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ``````# www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fft.html # Let's start with the radix-2 decimation-in-time algorithm : def unscaled_DFT_radix2_time (N, input, output): even_input = vector(N/2) odd_input = vector(N/2) even_output = vector(N/2) odd_output = vector(N/2) for i in range(N/2): even_input[i] = input[2*i] odd_input[i] = input[2*i+1] unscaled_DFT (N/2, even_input, even_output) unscaled_DFT (N/2, odd_input, odd_output) for i in range(N/2): `````` Michel Lespinasse committed Mar 28, 2000 61 `````` odd_output[i] = odd_output[i] * W (i, N) `````` Michel Lespinasse committed Mar 19, 2000 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 `````` for i in range(N/2): output[i] = even_output[i] + odd_output[i] output[i+N/2] = even_output[i] - odd_output[i] # This algorithm takes complex input and output. # We divide the DFT calculation into 2 DFT calculations of size N/2 # We then do N/2 complex multiplies followed by N complex additions. # (actually W(0,N) = 1 and W(N/4,N) = -j so we can skip a few of these complex # multiplies... we will skip 1 for i=0 and 1 for i=N/4. Also for i=N/8 and for # i=3*N/8 the W(i,N) values can be special-cased to implement the complex # multiplication using only 2 real additions and 2 real multiplies) # Also note that all the basic stages of this DFT algorithm are easily # reversible, so we can calculate the IDFT with the same complexity. `````` Eric Engestrom committed Apr 02, 2016 80 ``````# A variant of this is the radix-2 decimation-in-frequency algorithm : `````` Michel Lespinasse committed Mar 19, 2000 81 82 83 84 85 86 87 88 89 90 91 92 `````` def unscaled_DFT_radix2_freq (N, input, output): even_input = vector(N/2) odd_input = vector(N/2) even_output = vector(N/2) odd_output = vector(N/2) for i in range(N/2): even_input[i] = input[i] + input[i+N/2] odd_input[i] = input[i] - input[i+N/2] for i in range(N/2): `````` Michel Lespinasse committed Mar 28, 2000 93 `````` odd_input[i] = odd_input[i] * W (i, N) `````` Michel Lespinasse committed Mar 19, 2000 94 95 96 97 98 99 100 101 102 103 104 105 `````` unscaled_DFT (N/2, even_input, even_output) unscaled_DFT (N/2, odd_input, odd_output) for i in range(N/2): output[2*i] = even_output[i] output[2*i+1] = odd_output[i] # Note that the decimation-in-time and the decimation-in-frequency varients # have exactly the same complexity, they only do the operations in a different # order. `````` Eric Engestrom committed Apr 02, 2016 106 ``````# Actually, if you look at the decimation-in-time variant of the DFT, and `````` Michel Lespinasse committed Mar 19, 2000 107 108 109 110 111 112 ``````# reverse it to calculate an IDFT, you get something that is extremely close # to the decimation-in-frequency DFT algorithm... # The radix-4 algorithms are slightly more efficient : they take into account # the fact that with complex numbers, multiplications by j and -j are also `````` Michel Lespinasse committed Mar 28, 2000 113 114 ``````# "free"... i.e. when you code them using real numbers, they translate into # a few data moves but no real operation. `````` Michel Lespinasse committed Mar 19, 2000 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 `````` # Let's start with the radix-4 decimation-in-time algorithm : def unscaled_DFT_radix4_time (N, input, output): input_0 = vector(N/4) input_1 = vector(N/4) input_2 = vector(N/4) input_3 = vector(N/4) output_0 = vector(N/4) output_1 = vector(N/4) output_2 = vector(N/4) output_3 = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) tmp_2 = vector(N/4) tmp_3 = vector(N/4) for i in range(N/4): input_0[i] = input[4*i] input_1[i] = input[4*i+1] input_2[i] = input[4*i+2] input_3[i] = input[4*i+3] unscaled_DFT (N/4, input_0, output_0) unscaled_DFT (N/4, input_1, output_1) unscaled_DFT (N/4, input_2, output_2) unscaled_DFT (N/4, input_3, output_3) for i in range(N/4): `````` Michel Lespinasse committed Mar 28, 2000 144 145 146 `````` output_1[i] = output_1[i] * W (i, N) output_2[i] = output_2[i] * W (2*i, N) output_3[i] = output_3[i] * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 `````` for i in range(N/4): tmp_0[i] = output_0[i] + output_2[i] tmp_1[i] = output_0[i] - output_2[i] tmp_2[i] = output_1[i] + output_3[i] tmp_3[i] = output_1[i] - output_3[i] for i in range(N/4): output[i] = tmp_0[i] + tmp_2[i] output[i+N/4] = tmp_1[i] - 1j * tmp_3[i] output[i+N/2] = tmp_0[i] - tmp_2[i] output[i+3*N/4] = tmp_1[i] + 1j * tmp_3[i] # This algorithm takes complex input and output. # We divide the DFT calculation into 4 DFT calculations of size N/4 # We then do 3*N/4 complex multiplies followed by 2*N complex additions. # (actually W(0,N) = 1 and W(N/4,N) = -j so we can skip a few of these complex # multiplies... we will skip 3 for i=0 and 1 for i=N/8. Also for i=N/8 # the remaining W(i,N) and W(3*i,N) multiplies can be implemented using only # 2 real additions and 2 real multiplies. For i=N/16 and i=3*N/16 we can also # optimise the W(2*i/N) multiply this way. # If we wanted to do the same decomposition with one radix-2 decomposition # of size N and 2 radix-2 decompositions of size N/2, the total cost would be # N complex multiplies and 2*N complex additions. Thus we see that the # decomposition of one DFT calculation of size N into 4 calculations of size # N/4 using the radix-4 algorithm instead of the radix-2 algorithm saved N/4 # complex multiplies... # The radix-4 decimation-in-frequency algorithm is similar : def unscaled_DFT_radix4_freq (N, input, output): input_0 = vector(N/4) input_1 = vector(N/4) input_2 = vector(N/4) input_3 = vector(N/4) output_0 = vector(N/4) output_1 = vector(N/4) output_2 = vector(N/4) output_3 = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) tmp_2 = vector(N/4) tmp_3 = vector(N/4) for i in range(N/4): tmp_0[i] = input[i] + input[i+N/2] tmp_1[i] = input[i+N/4] + input[i+3*N/4] tmp_2[i] = input[i] - input[i+N/2] tmp_3[i] = input[i+N/4] - input[i+3*N/4] for i in range(N/4): input_0[i] = tmp_0[i] + tmp_1[i] input_1[i] = tmp_2[i] - 1j * tmp_3[i] input_2[i] = tmp_0[i] - tmp_1[i] input_3[i] = tmp_2[i] + 1j * tmp_3[i] for i in range(N/4): `````` Michel Lespinasse committed Mar 28, 2000 207 208 209 `````` input_1[i] = input_1[i] * W (i, N) input_2[i] = input_2[i] * W (2*i, N) input_3[i] = input_3[i] * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 `````` unscaled_DFT (N/4, input_0, output_0) unscaled_DFT (N/4, input_1, output_1) unscaled_DFT (N/4, input_2, output_2) unscaled_DFT (N/4, input_3, output_3) for i in range(N/4): output[4*i] = output_0[i] output[4*i+1] = output_1[i] output[4*i+2] = output_2[i] output[4*i+3] = output_3[i] # Once again the complexity is exactly the same as for the radix-4 # decimation-in-time DFT algorithm, only the order of the operations is # different. # Now let us reorder the radix-4 algorithms in a different way : #def unscaled_DFT_radix4_time (N, input, output): # input_0 = vector(N/4) # input_1 = vector(N/4) # input_2 = vector(N/4) # input_3 = vector(N/4) # output_0 = vector(N/4) # output_1 = vector(N/4) # output_2 = vector(N/4) # output_3 = vector(N/4) # tmp_0 = vector(N/4) # tmp_1 = vector(N/4) # tmp_2 = vector(N/4) # tmp_3 = vector(N/4) # # for i in range(N/4): # input_0[i] = input[4*i] # input_2[i] = input[4*i+2] # # unscaled_DFT (N/4, input_0, output_0) # unscaled_DFT (N/4, input_2, output_2) # # for i in range(N/4): `````` Michel Lespinasse committed Mar 28, 2000 251 ``````# output_2[i] = output_2[i] * W (2*i, N) `````` Michel Lespinasse committed Mar 19, 2000 252 253 254 255 256 257 258 259 260 261 262 263 264 ``````# # for i in range(N/4): # tmp_0[i] = output_0[i] + output_2[i] # tmp_1[i] = output_0[i] - output_2[i] # # for i in range(N/4): # input_1[i] = input[4*i+1] # input_3[i] = input[4*i+3] # # unscaled_DFT (N/4, input_1, output_1) # unscaled_DFT (N/4, input_3, output_3) # # for i in range(N/4): `````` Michel Lespinasse committed Mar 28, 2000 265 266 ``````# output_1[i] = output_1[i] * W (i, N) # output_3[i] = output_3[i] * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 267 268 269 270 271 272 273 274 275 276 277 ``````# # for i in range(N/4): # tmp_2[i] = output_1[i] + output_3[i] # tmp_3[i] = output_1[i] - output_3[i] # # for i in range(N/4): # output[i] = tmp_0[i] + tmp_2[i] # output[i+N/4] = tmp_1[i] - 1j * tmp_3[i] # output[i+N/2] = tmp_0[i] - tmp_2[i] # output[i+3*N/4] = tmp_1[i] + 1j * tmp_3[i] `````` Eric Engestrom committed Apr 02, 2016 278 ``````# We didn't do anything here, only reorder the operations. But now, look at the `````` Michel Lespinasse committed Mar 19, 2000 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 ``````# first part of this function, up to the calculations of tmp0 and tmp1 : this # is extremely similar to the radix-2 decimation-in-time algorithm ! or more # precisely, it IS the radix-2 decimation-in-time algorithm, with size N/2, # applied on a vector representing the even input coefficients, and giving # an output vector that is the concatenation of tmp0 and tmp1. # This is important to notice, because this means we can now choose to # calculate tmp0 and tmp1 using any DFT algorithm that we want, and we know # that some of them are more efficient than radix-2... # This leads us directly to the split-radix decimation-in-time algorithm : def unscaled_DFT_split_radix_time (N, input, output): even_input = vector(N/2) input_1 = vector(N/4) input_3 = vector(N/4) even_output = vector(N/2) output_1 = vector(N/4) output_3 = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) for i in range(N/2): even_input[i] = input[2*i] for i in range(N/4): input_1[i] = input[4*i+1] input_3[i] = input[4*i+3] unscaled_DFT (N/2, even_input, even_output) unscaled_DFT (N/4, input_1, output_1) unscaled_DFT (N/4, input_3, output_3) for i in range(N/4): `````` Michel Lespinasse committed Mar 28, 2000 312 313 `````` output_1[i] = output_1[i] * W (i, N) output_3[i] = output_3[i] * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 `````` for i in range(N/4): tmp_0[i] = output_1[i] + output_3[i] tmp_1[i] = output_1[i] - output_3[i] for i in range(N/4): output[i] = even_output[i] + tmp_0[i] output[i+N/4] = even_output[i+N/4] - 1j * tmp_1[i] output[i+N/2] = even_output[i] - tmp_0[i] output[i+3*N/4] = even_output[i+N/4] + 1j * tmp_1[i] # This function performs one DFT of size N/2 and two of size N/4, followed by # N/2 complex multiplies and 3*N/2 complex additions. # (actually W(0,N) = 1 and W(N/4,N) = -j so we can skip a few of these complex # multiplies... we will skip 2 for i=0. Also for i=N/8 the W(i,N) and W(3*i,N) # multiplies can be implemented using only 2 real additions and 2 real # multiplies) # We can similarly define the split-radix decimation-in-frequency DFT : def unscaled_DFT_split_radix_freq (N, input, output): even_input = vector(N/2) input_1 = vector(N/4) input_3 = vector(N/4) even_output = vector(N/2) output_1 = vector(N/4) output_3 = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) for i in range(N/2): even_input[i] = input[i] + input[i+N/2] for i in range(N/4): tmp_0[i] = input[i] - input[i+N/2] tmp_1[i] = input[i+N/4] - input[i+3*N/4] for i in range(N/4): input_1[i] = tmp_0[i] - 1j * tmp_1[i] input_3[i] = tmp_0[i] + 1j * tmp_1[i] for i in range(N/4): `````` Michel Lespinasse committed Mar 28, 2000 357 358 `````` input_1[i] = input_1[i] * W (i, N) input_3[i] = input_3[i] * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 359 360 361 362 363 364 365 366 367 368 369 370 `````` unscaled_DFT (N/2, even_input, even_output) unscaled_DFT (N/4, input_1, output_1) unscaled_DFT (N/4, input_3, output_3) for i in range(N/2): output[2*i] = even_output[i] for i in range(N/4): output[4*i+1] = output_1[i] output[4*i+3] = output_3[i] `````` Eric Engestrom committed Apr 02, 2016 371 ``````# The complexity is again the same as for the decimation-in-time variant. `````` Michel Lespinasse committed Mar 19, 2000 372 373 374 375 376 377 378 379 `````` # Now let us now summarize our various algorithms for DFT decomposition : # radix-2 : DFT(N) -> 2*DFT(N/2) using N/2 multiplies and N additions # radix-4 : DFT(N) -> 4*DFT(N/2) using 3*N/4 multiplies and 2*N additions # split-radix : DFT(N) -> DFT(N/2) + 2*DFT(N/4) using N/2 muls and 3*N/2 adds `````` Michel Lespinasse committed Mar 28, 2000 380 381 382 383 ``````# (we are always speaking of complex multiplies and complex additions... a # complex addition is implemented with 2 real additions, and a complex # multiply is implemented with either 2 adds and 4 muls or 3 adds and 3 muls, # so we will keep a separate count of these) `````` Michel Lespinasse committed Mar 19, 2000 384 385 386 387 388 389 390 `````` # If we want to take into account the special values of W(i,N), we can remove # a few complex multiplies. Supposing N>=16 we can remove : # radix-2 : remove 2 complex multiplies, simplify 2 others # radix-4 : remove 4 complex multiplies, simplify 4 others # split-radix : remove 2 complex multiplies, simplify 2 others `````` Michel Lespinasse committed Mar 28, 2000 391 ``````# This gives the following table for the complexity of a complex DFT : `````` Michel Lespinasse committed Mar 19, 2000 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 ``````# N real additions real multiplies complex multiplies # 1 0 0 0 # 2 4 0 0 # 4 16 0 0 # 8 52 4 0 # 16 136 8 4 # 32 340 20 16 # 64 808 40 52 # 128 1876 84 144 # 256 4264 168 372 # 512 9556 340 912 # 1024 21160 680 2164 # 2048 46420 1364 5008 # 4096 101032 2728 11380 # 8192 218452 5460 25488 # 16384 469672 10920 56436 # 32768 1004884 21844 123792 # 65536 2140840 43688 269428 `````` Michel Lespinasse committed Mar 28, 2000 411 412 413 ``````# If we chose to implement complex multiplies with 3 real muls + 3 real adds, # then these results are consistent with the table at the end of the # www.cmlab.csie.ntu.edu.tw DFT tutorial that I mentionned earlier. `````` Michel Lespinasse committed Mar 19, 2000 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 `````` # Now another important case for the DFT is the one where the inputs are # real numbers instead of complex ones. We have to find ways to optimize for # this important case. # If the DFT inputs are real-valued, then the DFT outputs have nice properties # too : output[0] and output[N/2] will be real numbers, and output[N-i] will # be the conjugate of output[i] for i in 0...N/2-1 # Likewise, if the DFT inputs are purely imaginary numbers, then the DFT # outputs will have special properties too : output[0] and output[N/2] will be # purely imaginary, and output[N-i] will be the opposite of the conjugate of # output[i] for i in 0...N/2-1 # We can use these properties to calculate two real-valued DFT at once : def two_real_unscaled_DFT (N, input1, input2, output1, output2): input = vector(N) output = vector(N) for i in range(N): input[i] = input1[i] + 1j * input2[i] unscaled_DFT (N, input, output) output1[0] = output[0].real + 0j output2[0] = output[0].imag + 0j for i in range(N/2)[1:]: output1[i] = 0.5 * (output[i] + conjugate(output[N-i])) output2[i] = -0.5j * (output[i] - conjugate(output[N-i])) output1[N-i] = conjugate(output1[i]) output2[N-i] = conjugate(output2[i]) output1[N/2] = output[N/2].real + 0j output2[N/2] = output[N/2].imag + 0j # This routine does a total of N-2 complex additions and N-2 complex # multiplies by 0.5 # This routine can also be inverted to calculate the IDFT of two vectors at # once if we know that the outputs will be real-valued. # If we have only one real-valued DFT calculation to do, we can still cut this # calculation in several parts using one of the decimate-in-time methods # (so that the different parts are still real-valued) # As with complex DFT calculations, the best method is to use a split radix. # There are a lot of symetries in the DFT outputs that we can exploit to # reduce the number of operations... `````` Michel Lespinasse committed Mar 28, 2000 468 ``````def real_unscaled_DFT_split_radix_time_1 (N, input, output): `````` Michel Lespinasse committed Mar 19, 2000 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 `````` even_input = vector(N/2) even_output = vector(N/2) input_1 = vector(N/4) output_1 = vector(N/4) input_3 = vector(N/4) output_3 = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) for i in range(N/2): even_input[i] = input[2*i] for i in range(N/4): input_1[i] = input[4*i+1] input_3[i] = input[4*i+3] unscaled_DFT (N/2, even_input, even_output) # this is again a real DFT ! # we will only use even_output[i] for i in 0 ... N/4 included. we know that # even_output[N/2-i] is the conjugate of even_output[i]... also we know # that even_output[0] and even_output[N/4] are purely real. unscaled_DFT (N/4, input_1, output_1) unscaled_DFT (N/4, input_3, output_3) # these are real DFTs too... with symetries in the outputs... once again tmp_0[0] = output_1[0] + output_3[0] # real numbers tmp_1[0] = output_1[0] - output_3[0] # real numbers tmp__0 = (output_1[N/8] + output_3[N/8]) * sqrt(0.5) # real numbers tmp__1 = (output_1[N/8] - output_3[N/8]) * sqrt(0.5) # real numbers tmp_0[N/8] = tmp__1 - 1j * tmp__0 # real + 1j * real tmp_1[N/8] = tmp__0 - 1j * tmp__1 # real + 1j * real for i in range(N/8)[1:]: `````` Michel Lespinasse committed Mar 28, 2000 504 505 `````` output_1[i] = output_1[i] * W (i, N) output_3[i] = output_3[i] * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 `````` tmp_0[i] = output_1[i] + output_3[i] tmp_1[i] = output_1[i] - output_3[i] tmp_0[N/4-i] = -1j * conjugate(tmp_1[i]) tmp_1[N/4-i] = -1j * conjugate(tmp_0[i]) output[0] = even_output[0] + tmp_0[0] # real numbers output[N/4] = even_output[N/4] - 1j * tmp_1[0] # real + 1j * real output[N/2] = even_output[0] - tmp_0[0] # real numbers output[3*N/4] = even_output[N/4] + 1j * tmp_1[0] # real + 1j * real for i in range(N/4)[1:]: output[i] = even_output[i] + tmp_0[i] output[i+N/4] = conjugate(even_output[N/4-i]) - 1j * tmp_1[i] output[N-i] = conjugate(output[i]) output[3*N/4-i] = conjugate(output[i+N/4]) # This function performs one real DFT of size N/2 and two real DFT of size # N/4, followed by 6 real additions, 2 real multiplies, 3*N/4-4 complex # additions and N/4-2 complex multiplies. # We can also try to combine the two real DFT of size N/4 into a single complex # DFT : `````` Michel Lespinasse committed Mar 28, 2000 533 ``````def real_unscaled_DFT_split_radix_time_2 (N, input, output): `````` Michel Lespinasse committed Mar 19, 2000 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 `````` even_input = vector(N/2) even_output = vector(N/2) odd_input = vector(N/4) odd_output = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) for i in range(N/2): even_input[i] = input[2*i] for i in range(N/4): odd_input[i] = input[4*i+1] + 1j * input[4*i+3] unscaled_DFT (N/2, even_input, even_output) # this is again a real DFT ! # we will only use even_output[i] for i in 0 ... N/4 included. we know that # even_output[N/2-i] is the conjugate of even_output[i]... also we know # that even_output[0] and even_output[N/4] are purely real. unscaled_DFT (N/4, odd_input, odd_output) # but this one is a complex DFT so no special properties here output_1 = odd_output[0].real output_3 = odd_output[0].imag tmp_0[0] = output_1 + output_3 # real numbers tmp_1[0] = output_1 - output_3 # real numbers output_1 = odd_output[N/8].real output_3 = odd_output[N/8].imag tmp__0 = (output_1 + output_3) * sqrt(0.5) # real numbers tmp__1 = (output_1 - output_3) * sqrt(0.5) # real numbers tmp_0[N/8] = tmp__1 - 1j * tmp__0 # real + 1j * real tmp_1[N/8] = tmp__0 - 1j * tmp__1 # real + 1j * real for i in range(N/8)[1:]: output_1 = odd_output[i] + conjugate(odd_output[N/4-i]) output_3 = odd_output[i] - conjugate(odd_output[N/4-i]) `````` Michel Lespinasse committed Mar 28, 2000 572 573 `````` output_1 = output_1 * 0.5 * W (i, N) output_3 = output_3 * -0.5j * W (3*i, N) `````` Michel Lespinasse committed Mar 19, 2000 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 `````` tmp_0[i] = output_1 + output_3 tmp_1[i] = output_1 - output_3 tmp_0[N/4-i] = -1j * conjugate(tmp_1[i]) tmp_1[N/4-i] = -1j * conjugate(tmp_0[i]) output[0] = even_output[0] + tmp_0[0] # real numbers output[N/4] = even_output[N/4] - 1j * tmp_1[0] # real + 1j * real output[N/2] = even_output[0] - tmp_0[0] # real numbers output[3*N/4] = even_output[N/4] + 1j * tmp_1[0] # real + 1j * real for i in range(N/4)[1:]: output[i] = even_output[i] + tmp_0[i] output[i+N/4] = conjugate(even_output[N/4-i]) - 1j * tmp_1[i] output[N-i] = conjugate(output[i]) output[3*N/4-i] = conjugate(output[i+N/4]) # This function performs one real DFT of size N/2 and one complex DFT of size # N/4, followed by 6 real additions, 2 real multiplies, N-6 complex additions # and N/4-2 complex multiplies. # After comparing the performance, it turns out that for real-valued DFT, the # version of the algorithm that subdivides the calculation into one real # DFT of size N/2 and two real DFT of size N/4 is the most efficient one. # The other version gives exactly the same number of multiplies and a few more # real additions. `````` Michel Lespinasse committed Mar 28, 2000 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 ``````# Now we can also try the decimate-in-frequency method for a real-valued DFT. # Using the split-radix algorithm, and by taking into account the symetries of # the outputs : def real_unscaled_DFT_split_radix_freq (N, input, output): even_input = vector(N/2) input_1 = vector(N/4) even_output = vector(N/2) output_1 = vector(N/4) tmp_0 = vector(N/4) tmp_1 = vector(N/4) for i in range(N/2): even_input[i] = input[i] + input[i+N/2] for i in range(N/4): tmp_0[i] = input[i] - input[i+N/2] tmp_1[i] = input[i+N/4] - input[i+3*N/4] for i in range(N/4): input_1[i] = tmp_0[i] - 1j * tmp_1[i] for i in range(N/4): input_1[i] = input_1[i] * W (i, N) unscaled_DFT (N/2, even_input, even_output) # This is still a real-valued DFT unscaled_DFT (N/4, input_1, output_1) # But that one is a complex-valued DFT for i in range(N/2): output[2*i] = even_output[i] for i in range(N/4): output[4*i+1] = output_1[i] output[N-1-4*i] = conjugate(output_1[i]) # I think this implementation is much more elegant than the decimate-in-time # version ! It looks very much like the complex-valued version, all we had to # do was remove one of the complex-valued internal DFT calls because we could # deduce the outputs by using the symetries of the problem. # As for performance, we did N real additions, N/4 complex multiplies (a bit # less actually, because W(0,N) = 1 and W(N/8,N) is a "simple" multiply), then # one real DFT of size N/2 and one complex DFT of size N/4. # It turns out that even if the methods are so different, the number of # operations is exactly the same as for the best of the two decimation-in-time # methods that we tried. # This gives us the following performance for real-valued DFT : `````` Michel Lespinasse committed Mar 19, 2000 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 ``````# N real additions real multiplies complex multiplies # 2 2 0 0 # 4 6 0 0 # 8 20 2 0 # 16 54 4 2 # 32 140 10 8 # 64 342 20 26 # 128 812 42 72 # 256 1878 84 186 # 512 4268 170 456 # 1024 9558 340 1082 # 2048 21164 682 2504 # 4096 46422 1364 5690 # 8192 101036 2730 12744 # 16384 218454 5460 28218 # 32768 469676 10922 61896 # 65536 1004886 21844 134714 `````` Michel Lespinasse committed Mar 28, 2000 676 ``````# As an example, this is an implementation of the real-valued DFT8 : `````` Michel Lespinasse committed Mar 19, 2000 677 678 `````` def DFT8 (input, output): `````` Michel Lespinasse committed Mar 28, 2000 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 `````` even_0 = input[0] + input[4] even_1 = input[1] + input[5] even_2 = input[2] + input[6] even_3 = input[3] + input[7] tmp_0 = even_0 + even_2 tmp_1 = even_0 - even_2 tmp_2 = even_1 + even_3 tmp_3 = even_1 - even_3 output[0] = tmp_0 + tmp_2 output[2] = tmp_1 - 1j * tmp_3 output[4] = tmp_0 - tmp_2 odd_0_r = input[0] - input[4] odd_0_i = input[2] - input[6] tmp_0 = input[1] - input[5] tmp_1 = input[3] - input[7] odd_1_r = (tmp_0 - tmp_1) * sqrt(0.5) odd_1_i = (tmp_0 + tmp_1) * sqrt(0.5) output[1] = (odd_0_r + odd_1_r) - 1j * (odd_0_i + odd_1_i) output[5] = (odd_0_r - odd_1_r) - 1j * (odd_0_i - odd_1_i) output[3] = conjugate(output[5]) output[6] = conjugate(output[2]) `````` Michel Lespinasse committed Mar 19, 2000 706 707 708 709 710 711 712 713 714 715 716 `````` output[7] = conjugate(output[1]) # Also a basic implementation of the real-valued DFT4 : def DFT4 (input, output): tmp_0 = input[0] + input[2] tmp_1 = input[0] - input[2] tmp_2 = input[1] + input[3] tmp_3 = input[1] - input[3] `````` Michel Lespinasse committed Mar 28, 2000 717 718 719 720 `````` output[0] = tmp_0 + tmp_2 output[1] = tmp_1 - 1j * tmp_3 output[2] = tmp_0 - tmp_2 output[3] = tmp_1 + 1j * tmp_3 `````` Michel Lespinasse committed Mar 19, 2000 721 722 `````` `````` Michel Lespinasse committed Mar 28, 2000 723 724 725 726 727 728 729 ``````# A similar idea might be used to calculate only the real part of the output # of a complex DFT : we take an DFT algorithm for real inputs and complex # outputs and we simply reverse it. The resulting algorithm will only work # with inputs that satisfy the conjugaison rule (input[i] is the conjugate of # input[N-i]) so we can do a first pass to modify the input so that it follows # this rule. An example implementation is as follows (adapted from the # unscaled_DFT_split_radix_time algorithm) : `````` Michel Lespinasse committed Mar 19, 2000 730 `````` `````` Michel Lespinasse committed Mar 28, 2000 731 ``````def complex2real_unscaled_DFT_split_radix_time (N, input, output): `````` Michel Lespinasse committed Mar 19, 2000 732 `````` even_input = vector(N/2) `````` Michel Lespinasse committed Mar 28, 2000 733 `````` input_1 = vector(N/4) `````` Michel Lespinasse committed Mar 19, 2000 734 `````` even_output = vector(N/2) `````` Michel Lespinasse committed Mar 28, 2000 735 `````` output_1 = vector(N/4) `````` Michel Lespinasse committed Mar 19, 2000 736 737 738 739 `````` for i in range(N/2): even_input[i] = input[2*i] `````` Michel Lespinasse committed Mar 28, 2000 740 741 742 `````` for i in range(N/4): input_1[i] = input[4*i+1] + conjugate(input[N-1-4*i]) `````` Michel Lespinasse committed Mar 19, 2000 743 `````` unscaled_DFT (N/2, even_input, even_output) `````` Michel Lespinasse committed Mar 28, 2000 744 `````` unscaled_DFT (N/4, input_1, output_1) `````` Michel Lespinasse committed Mar 19, 2000 745 `````` `````` Michel Lespinasse committed Mar 28, 2000 746 747 `````` for i in range(N/4): output_1[i] = output_1[i] * W (i, N) `````` Michel Lespinasse committed Mar 19, 2000 748 `````` `````` Michel Lespinasse committed Mar 28, 2000 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 `````` for i in range(N/4): output[i] = even_output[i] + output_1[i].real output[i+N/4] = even_output[i+N/4] + output_1[i].imag output[i+N/2] = even_output[i] - output_1[i].real output[i+3*N/4] = even_output[i+N/4] - output_1[i].imag # This algorithm does N/4 complex additions, N/4-1 complex multiplies # (including one "simple" multiply for i=N/8), N real additions, one # "complex-to-real" DFT of size N/2, and one complex DFT of size N/4. # Also, in the complex DFT of size N/4, we do not care about the imaginary # part of output_1[0], which in practice allows us to save one real addition. # This gives us the following performance for complex DFT with real outputs : # N real additions real multiplies complex multiplies # 1 0 0 0 # 2 2 0 0 # 4 8 0 0 # 8 25 2 0 # 16 66 4 2 # 32 167 10 8 # 64 400 20 26 # 128 933 42 72 # 256 2126 84 186 # 512 4771 170 456 # 1024 10572 340 1082 # 2048 23201 682 2504 # 4096 50506 1364 5690 # 8192 109215 2730 12744 # 16384 234824 5460 28218 # 32768 502429 10922 61896 # 65536 1070406 21844 134714 # Now let's talk about the DCT algorithm. The canonical definition for it is # as follows : def C (k, N): return cos ((k*pi)/(2*N)) `````` Michel Lespinasse committed Mar 19, 2000 787 `````` `````` Michel Lespinasse committed Mar 28, 2000 788 789 790 791 792 ``````def unscaled_DCT (N, input, output): for o in range(N): # o is output index output[o] = 0 for i in range(N): # i is input index output[o] = output[o] + input[i] * C ((2*i+1)*o, N) `````` Michel Lespinasse committed Mar 19, 2000 793 `````` `````` Michel Lespinasse committed Mar 28, 2000 794 ``````# This trivial algorithm uses N*N multiplications and N*(N-1) additions. `````` Michel Lespinasse committed Mar 19, 2000 795 796 `````` `````` Michel Lespinasse committed Mar 28, 2000 797 798 ``````# One possible decomposition on this calculus is to use the fact that C (i, N) # and C (2*N-i, N) are opposed. This can lead to this decomposition : `````` Michel Lespinasse committed Mar 19, 2000 799 `````` `````` Michel Lespinasse committed Mar 28, 2000 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 ``````#def unscaled_DCT (N, input, output): # even_input = vector (N) # odd_input = vector (N) # even_output = vector (N) # odd_output = vector (N) # # for i in range(N/2): # even_input[i] = input[i] + input[N-1-i] # odd_input[i] = input[i] - input[N-1-i] # # unscaled_DCT (N, even_input, even_output) # unscaled_DCT (N, odd_input, odd_output) # # for i in range(N/2): # output[2*i] = even_output[2*i] # output[2*i+1] = odd_output[2*i+1] # Now the even part can easily be calculated : by looking at the C(k,N) # formula, we see that the even part is actually an unscaled DCT of size N/2. # The odd part looks like a DCT of size N/2, but the coefficients are # actually C ((2*i+1)*(2*o+1), 2*N) instead of C ((2*i+1)*o, N). # We use a trigonometric relation here : # 2 * C ((a+b)/2, N) * C ((a-b)/2, N) = C (a, N) + C (b, N) # Thus with a = (2*i+1)*o and b = (2*i+1)*(o+1) : # 2 * C((2*i+1)*(2*o+1),2N) * C(2*i+1,2N) = C((2*i+1)*o,N) + C((2*i+1)*(o+1),N) # This leads us to the Lee DCT algorithm : def unscaled_DCT_Lee (N, input, output): even_input = vector(N/2) odd_input = vector(N/2) even_output = vector(N/2) odd_output = vector(N/2) `````` Michel Lespinasse committed Mar 19, 2000 834 `````` `````` Michel Lespinasse committed Mar 28, 2000 835 836 837 `````` for i in range(N/2): even_input[i] = input[i] + input[N-1-i] odd_input[i] = input[i] - input[N-1-i] `````` Michel Lespinasse committed Mar 19, 2000 838 `````` `````` Michel Lespinasse committed Mar 28, 2000 839 840 `````` for i in range(N/2): odd_input[i] = odd_input[i] * (0.5 / C (2*i+1, N)) `````` Michel Lespinasse committed Mar 19, 2000 841 `````` `````` Michel Lespinasse committed Mar 28, 2000 842 843 `````` unscaled_DCT (N/2, even_input, even_output) unscaled_DCT (N/2, odd_input, odd_output) `````` Michel Lespinasse committed Mar 19, 2000 844 `````` `````` Michel Lespinasse committed Mar 28, 2000 845 846 `````` for i in range(N/2-1): odd_output[i] = odd_output[i] + odd_output[i+1] `````` Michel Lespinasse committed Mar 19, 2000 847 848 `````` for i in range(N/2): `````` Michel Lespinasse committed Mar 28, 2000 849 850 `````` output[2*i] = even_output[i] output[2*i+1] = odd_output[i]; `````` Michel Lespinasse committed Mar 19, 2000 851 `````` `````` Michel Lespinasse committed Mar 28, 2000 852 ``````# Notes about this algorithm : `````` Michel Lespinasse committed Mar 19, 2000 853 `````` `````` Michel Lespinasse committed Mar 28, 2000 854 855 ``````# The algorithm can be easily inverted to calculate the IDCT instead : # each of the basic stages are separately inversible... `````` Michel Lespinasse committed Mar 19, 2000 856 `````` `````` Michel Lespinasse committed Mar 28, 2000 857 858 859 860 861 862 863 864 ``````# This function does N adds, then N/2 muls, then 2 recursive calls with # size N/2, then N/2-1 adds again. If we apply it recursively, the total # number of operations will be N*log2(N)/2 multiplies and N*(3*log2(N)/2-1) + 1 # additions. So this is much faster than the canonical algorithm. # Some of the multiplication coefficients 0.5/cos(...) can get quite large. # This means that a small error in the input will give a large error on the # output... For a DCT of size N the biggest coefficient will be at i=N/2-1 `````` Eric Engestrom committed Apr 02, 2016 865 ``````# and it will be slightly more than N/pi which can be large for large N's. `````` Michel Lespinasse committed Mar 28, 2000 866 867 868 869 870 871 872 873 874 875 876 877 878 879 `````` # In the IDCT however, the multiplication coefficients for the reverse # transformation are of the form 2*cos(...) so they can not get big and there # is no accuracy problem. # You can find another description of this algorithm at # http://www.intel.com/drg/mmx/appnotes/ap533.htm # Another idea is to observe that the DCT calculation can be made to look like # the DFT calculation : C (k, N) is the real part of W (k, 4*N) or W (-k, 4*N). # We can use this idea translate the DCT algorithm into a call to the DFT # algorithm : `````` Michel Lespinasse committed Mar 19, 2000 880 `````` `````` Michel Lespinasse committed Mar 28, 2000 881 882 883 ``````def unscaled_DCT_DFT (N, input, output): DFT_input = vector (4*N) DFT_output = vector (4*N) `````` Michel Lespinasse committed Mar 19, 2000 884 `````` `````` Michel Lespinasse committed Mar 28, 2000 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 `````` for i in range(N): DFT_input[2*i+1] = input[i] #DFT_input[4*N-2*i-1] = input[i] # We could use this instead unscaled_DFT (4*N, DFT_input, DFT_output) for i in range(N): output[i] = DFT_output[i].real # We can then use our knowledge of the DFT calculation to optimize for this # particular case. For example using the radix-2 decimation-in-time method : #def unscaled_DCT_DFT (N, input, output): # DFT_input = vector (2*N) # DFT_output = vector (2*N) # # for i in range(N): # DFT_input[i] = input[i] # #DFT_input[2*N-1-i] = input[i] # We could use this instead # # unscaled_DFT (2*N, DFT_input, DFT_output) # # for i in range(N): # DFT_output[i] = DFT_output[i] * W (i, 4*N) # # for i in range(N): # output[i] = DFT_output[i].real # This leads us to the AAN implementation of the DCT algorithm : if we set # both DFT_input[i] and DFT_input[2*N-1-i] to input[i], then the imaginary # parts of W(2*i+1) and W(-2*i-1) will compensate, and output_DFT[i] will # already be a real after the multiplication by W(i,4*N). Which means that # before the multiplication, it is the product of a real number and W(-i,4*N). # This leads to the following code, called the AAN algorithm : def unscaled_DCT_AAN (N, input, output): DFT_input = vector (2*N) DFT_output = vector (2*N) for i in range(N): DFT_input[i] = input[i] DFT_input[2*N-1-i] = input[i] symetrical_unscaled_DFT (2*N, DFT_input, DFT_output) for i in range(N): output[i] = DFT_output[i].real * (0.5 / C (i, N)) # Notes about the AAN algorithm : # The cost of this function is N real multiplies and a DFT of size 2*N. The # DFT to calculate has special properties : the inputs are real and symmetric. # Also, we only need to calculate the real parts of the N first DFT outputs. # We can try to take advantage of all that. # We can invert this algorithm to calculate the IDCT. The final multiply # stage is trivially invertible. The DFT stage is invertible too, but we have # to take into account the special properties of this particular DFT for that. # Once again we have to take care of numerical precision for the DFT : the # output coefficients can get large, so that a small error in the input will # give a large error on the output... For a DCT of size N the biggest # coefficient will be at i=N/2-1 and it will be slightly more than N/pi `````` Michel Lespinasse committed Mar 19, 2000 949 `````` `````` Michel Lespinasse committed Mar 28, 2000 950 951 952 953 954 955 956 957 958 959 960 ``````# You can find another description of this algorithm at this url : # www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fastdct.html # (It is the same server where we already found a description of the fast DFT) # To optimize the DFT calculation, we can take a lot of specific things into # account : the input is real and symetric, and we only care about the real # part of the output. Also, we only care about the N first output coefficients, # but that one does not save operations actually, because the other # coefficients are the conjugates of the ones we look anyway. `````` Eric Engestrom committed Apr 02, 2016 961 ``````# One useful way to use the symmetry of the input is to use the radix-2 `````` Michel Lespinasse committed Mar 28, 2000 962 ``````# decimation-in-frequency algorithm. We can write a version of `````` Eric Engestrom committed Apr 02, 2016 963 ``````# unscaled_DFT_radix2_freq for the case where the input is symmetrical : `````` Michel Lespinasse committed Mar 28, 2000 964 ``````# we have removed a few additions in the first stages because even_input `````` Eric Engestrom committed Apr 02, 2016 965 ``````# is symmetrical and odd_input is antisymetrical. Also, we have modified the `````` Michel Lespinasse committed Mar 28, 2000 966 967 968 ``````# odd_input vector so that the second half of it is set to zero and the real # part of the DFT output is not modified. After that modification, the second # part of the odd_input was null so we used the radix-2 decimation-in-frequency `````` Eric Engestrom committed Apr 02, 2016 969 ``````# again on the odd DFT. Also odd_output is symmetrical because input is real... `````` Michel Lespinasse committed Mar 28, 2000 970 971 972 973 974 975 976 `````` def symetrical_unscaled_DFT (N, input, output): even_input = vector(N/2) odd_tmp = vector(N/2) odd_input = vector(N/2) even_output = vector(N/2) odd_output = vector(N/2) `````` Michel Lespinasse committed Mar 19, 2000 977 `````` `````` Michel Lespinasse committed Mar 28, 2000 978 979 `````` for i in range(N/4): even_input[N/2-i-1] = even_input[i] = input[i] + input[N/2-1-i] `````` Michel Lespinasse committed Mar 19, 2000 980 `````` `````` Michel Lespinasse committed Mar 28, 2000 981 982 983 984 985 986 987 988 `````` for i in range(N/4): odd_tmp[i] = input[i] - input[N/2-1-i] odd_input[0] = odd_tmp[0] for i in range(N/4)[1:]: odd_input[i] = (odd_tmp[i] + odd_tmp[i-1]) * W (i, N) unscaled_DFT (N/2, even_input, even_output) `````` Eric Engestrom committed Apr 02, 2016 989 `````` # symmetrical real inputs, real outputs `````` Michel Lespinasse committed Mar 28, 2000 990 991 992 993 994 995 996 997 998 999 1000 `````` unscaled_DFT (N/4, odd_input, odd_output) # complex inputs, real outputs for i in range(N/2): output[2*i] = even_output[i] for i in range(N/4): output[N-1-4*i] = output[4*i+1] = odd_output[i] # This procedure takes 3*N/4-1 real additions and N/2-3 real multiplies, `````` Eric Engestrom committed Apr 02, 2016 1001 ``````# followed by another symmetrical real DFT of size N/2 and a "complex to real" `````` Michel Lespinasse committed Mar 28, 2000 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 ``````# DFT of size N/4. # We thus get the following performance results : # N real additions real multiplies complex multiplies # 1 0 0 0 # 2 0 0 0 # 4 2 0 0 # 8 9 1 0 # 16 28 6 0 # 32 76 21 0 # 64 189 54 2 # 128 451 125 10 # 256 1042 270 36 # 512 2358 565 108 # 1024 5251 1158 294 # 2048 11557 2349 750 # 4096 25200 4734 1832 # 8192 54544 9509 4336 # 16384 117337 19062 10026 # 32768 251127 38173 22770 # 65536 535102 76398 50988 # We thus get a better performance with the AAN DCT algorithm than with the # Lee DCT algorithm : we can do a DCT of size 32 with 189 additions, 54+32 real # multiplies, and 2 complex multiplies. The Lee algorithm would have used 209 # additions and 80 multiplies. With the AAN algorithm, we also have the # advantage that a big number of the multiplies are actually grouped at the # output stage of the algorithm, so if we want to do a DCT followed by a # quantization stage, we will be able to group the multiply of the output with # the multiply of the quantization stage, thus saving 32 more operations. In # the mpeg audio layer 1 or 2 processing, we can also group the multiply of the # output with the multiply of the convolution stage... # Another source code for the AAN algorithm (implemented on 8 points, and # without all of the explanations) can be found at this URL : # http://developer.intel.com/drg/pentiumII/appnotes/aan_org.c . This # implementation uses 28 adds and 6+8 muls instead of 29 adds and 5+8 muls - # the difference is that in the symetrical_unscaled_DFT procedure, they noticed # how odd_input[i] and odd_input[N/4-i] will be combined at the start of the # complex-to-real DFT and they took advantage of this to convert 2 real adds # and 4 real muls into one complex multiply. # TODO : write about multi-dimentional DCT # TEST CODE `````` Michel Lespinasse committed Mar 19, 2000 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 `````` def dump (vector): str = "" for i in range(len(vector)): if i: str = str + ", " vector[i] = vector[i] + 0j realstr = "%+.4f" % vector[i].real imagstr = "%+.4fj" % vector[i].imag if (realstr == "-0.0000"): realstr = "+0.0000" if (imagstr == "-0.0000j"): imagstr = "+0.0000j" `````` Michel Lespinasse committed Mar 28, 2000 1063 `````` str = str + realstr #+ imagstr `````` Michel Lespinasse committed Mar 19, 2000 1064 1065 1066 1067 1068 1069 1070 1071 `````` return "[%s]" % str def test(N): input = vector(N) output = vector(N) verify = vector(N) for i in range(N): `````` Rafaël Carré committed Jul 07, 2010 1072 `````` input[i] = random() + 1j * random() `````` Michel Lespinasse committed Mar 19, 2000 1073 `````` `````` Michel Lespinasse committed Mar 28, 2000 1074 1075 `````` unscaled_DFT (N, input, output) unscaled_DFT (N, input, verify) `````` Michel Lespinasse committed Mar 19, 2000 1076 `````` `````` Michel Lespinasse committed Mar 28, 2000 1077 1078 `````` if (dump(output) != dump(verify)): print dump(output) `````` Michel Lespinasse committed Mar 19, 2000 1079 1080 `````` print dump(verify) `````` Michel Lespinasse committed Mar 28, 2000 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 ``````#test (64) # PERFORMANCE ANALYSIS CODE def display (table): N = 1 print "#\tN\treal additions\treal multiplies\tcomplex multiplies" while table.has_key(N): print "#%8d%16d%16d%16d" % (N, table[N][0], table[N][1], table[N][2]) N = 2*N print best_complex_DFT = {} def complex_DFT (max_N): best_complex_DFT[1] = (0,0,0) best_complex_DFT[2] = (4,0,0) best_complex_DFT[4] = (16,0,0) N = 8 while (N<=max_N): # best method = split radix best2 = best_complex_DFT[N/2] best4 = best_complex_DFT[N/4] best_complex_DFT[N] = (best2[0] + 2*best4[0] + 3*N + 4, best2[1] + 2*best4[1] + 4, best2[2] + 2*best4[2] + N/2 - 4) N = 2*N best_real_DFT = {} def real_DFT (max_N): best_real_DFT[1] = (0,0,0) best_real_DFT[2] = (2,0,0) best_real_DFT[4] = (6,0,0) N = 8 while (N<=max_N): # best method = split radix decimate-in-frequency best2 = best_real_DFT[N/2] best4 = best_complex_DFT[N/4] best_real_DFT[N] = (best2[0] + best4[0] + N + 2, best2[1] + best4[1] + 2, best2[2] + best4[2] + N/4 - 2) N = 2*N best_complex2real_DFT = {} def complex2real_DFT (max_N): best_complex2real_DFT[1] = (0,0,0) best_complex2real_DFT[2] = (2,0,0) best_complex2real_DFT[4] = (8,0,0) N = 8 while (N<=max_N): best2 = best_complex2real_DFT[N/2] best4 = best_complex_DFT[N/4] best_complex2real_DFT[N] = (best2[0] + best4[0] + 3*N/2 + 1, best2[1] + best4[1] + 2, best2[2] + best4[2] + N/4 - 2) N = 2*N best_real_symetric_DFT = {} def real_symetric_DFT (max_N): best_real_symetric_DFT[1] = (0,0,0) best_real_symetric_DFT[2] = (0,0,0) best_real_symetric_DFT[4] = (2,0,0) N = 8 while (N<=max_N): best2 = best_real_symetric_DFT[N/2] best4 = best_complex2real_DFT[N/4] best_real_symetric_DFT[N] = (best2[0] + best4[0] + 3*N/4 - 1, best2[1] + best4[1] + N/2 - 3, best2[2] + best4[2]) N = 2*N complex_DFT (65536) real_DFT (65536) complex2real_DFT (65536) real_symetric_DFT (65536) print "complex DFT" display (best_complex_DFT) print "real DFT" display (best_real_DFT) print "complex2real DFT" display (best_complex2real_DFT) print "real symetric DFT" display (best_real_symetric_DFT)``````