Fast Algorithm and Common Structure Design of Recursive Analysis and Synthesis Quadrature Mirror Filterbanks for Digital Radio Mondiale

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(1)Li An Kai. Fast Algorithm and Common Structure Design of Recursive Analysis and Synthesis QMF for DRM Presenter : An-Kai Li. 01/16/2022. 應用系統晶片設計實驗室. 1.

(2) Outline. Li An Kai. Introduction Proposed QMF Algorithm Comparison & Analyses Conclusion Reference 01/16/2022. 應用系統晶片設計實驗室. 2.


(4) Li An Kai. Introduction. 01/16/2022. Specification. Long window. Short window. MPEG-4 AAC < MDCT >. 1920. 240. MPEG-4 HVXC < DFT >. 320. 160. SBR 、 PS 、 MPS < QMF >. 64. 32. 應用系統晶片設計實驗室. 4.

(5) Outline. Li An Kai. Introduction Proposed QMF Algorithm AQMF SQMF Complex Multiplication. Comparison & Analyses Conclusion Reference 01/16/2022. 應用系統晶片設計實驗室. 5.

(6) Li An Kai. Proposed AQMF(1/6) X [k ] . N 1.  x[n]  e. j ( k  0.5)( 2 n  0.25) N. ,0  k . n 0. e.  j ( k  0.5) 4N. N 1.   x[n]  e. N 1 2. j ( 2 k 1) n N. n 0.  A1[ k ]  X 1[k ]. x[n]. 0. x1[n], 0  n  N/ 4  1. 01/16/2022. N/4. +. x2 [n], N/ 4  n  N/ 2  1. N/2. +. 3N/4. x3 [n], N/ 2  n  3 N/ 4  1. +. N-1. x4 [n], 3 N/ 4  n  N  1. 應用系統晶片設計實驗室. 6.

(7) Proposed AQMF(2/6) X 1[k]  . N / 4 1. . N 1.  x[n] e. j ( 2 k 1) n N. n 0. x[n] e. j ( 2 k 1) n N. . n 0. . x[n] e. j ( 2 k 1) n N. . n 0. . x[n] e. j ( 2 k 1) n N. . n 0. . N / 4 1.  n 0. . n 0. j ( 2 k 1)(n  N/ 4) j ( 2 k 1)(n  N/ 2) j ( 2 k 1)(n  3 N/ 4) N / 4 1 N / 4 1 N N 3N N N N ]e   x[n  ]e   x[n  ]e 4 2 4 n0 n0. N / 4 1. x[n . j ( 2 k 1) N j ( 2 k 1) n j ( 2 k 1) N j ( 2 k 1) n j ( 2 k 1) 3 N j ( 2 k 1) n N / 4 1 N / 4 1    N N 3N 4 N 2 N 4 N ]e N e   x[n  ]e N e   x[n  ]e N e 4 2 4 n 0 n 0. . N / 4 1. x[n . N  k j4  j ( 2Nk 1) n ]  j e  e  4  .  n0.  N  k  ]  j  e  x[n]  x[n  4   . N / 4 1. . . x re [n] e. 01/16/2022. N 1 2. x[n . n 0. N / 4 1. . N / 4 1 n0. N / 4 1. . ,0  k . Li An Kai. j 4. N / 4 1.  n 0. x[n . j ( 2 k 1) n j (2 k 1) n N / 41 N N N N ]  ( 1) k j  e   x[n  ]  ( 1) k j  e 2 2 n 0. j ( 2 k 1) n j   N 3N   k k N 4  x [n  ] (  1)  j  x [n  ] (  1)  j  e e      2 4    . j ( 2 k 1) n N. 應用系統晶片設計實驗室. 7.

(8) Li An Kai. Proposed AQMF(3/6) X1[k]  . N / 4 1.  n0. N 1.  x[n]e. j (2 k 1) n N. ,0  k . n 0. N 1 2. j j   j (2Nk 1) n   N  k N 3N   k k 4 4 ]   j  e   x[n  ]  ( 1)  j   x[n  ]  ( j )  j  e  e  x[n]  x[n  4  2 4        InputSignal _Regulation. 01/16/2022. k. X[n]. X[n+N/4]. 4p. 1. e. 4p+1. 1. je. 4p+2. 1. e. 4p+3. 1.  je. j 4 j 4. j 4 j 4. X[n+N/2]. X[n+3N/4]. J. je. -j. e. J.  je. -j. e. 應用系統晶片設計實驗室. j 4. j 4 j 4. j 4. 8.

(9) Proposed AQMF(4/6) N 1. X1[k]   x[n]e. j 2 (k  0.5) n N. ,0  k . n 0. . N / 4 1.  n 0. Li An Kai. N 1 2. j 2 (k  0.5) n j j    N  k N 3N    k k N 4 4 ]   j  e   x[n  ]  (1)  j   x[n  ]  ( 1)  j  e   e  x[n]  x[n  4 2 4         InputSignal _Regulation. . N / 4 1.  n 0. . N / 4 1.  n 0. . N / 4 1.  n 0. . N / 4 1.  n 0. x regu [n]  e x regu [n]  e. j 2 (k  0.5) n N. j 2 (k  0.5) n N. N x regu [  1  n]  e 4 j 2 (k  0.5)(. ,0  k . N 1 2. ,0  k . N 1 2. j 2 (k  0.5)(. N 1 n) 4. N. N 1 n) N 4 u[ 1 n] 4. ,0  k . N x re [n]  e  , 0  k .  xre [p]  h[p] 01/16/2022. h[. N 1 n] 4. N 1 2. N 1 2. 應用系統晶片設計實驗室. 9.

(10) Li An Kai. Proposed AQMF(5/6) X1[k]  Yre [m]  xre [p]  h[p], 0  k . N 2.  Z transform Yre (z)  X re (z)  H (z) 1. Yre (z)  X re (z)  1 e. j 2 (k  0.5) N.  z 1.  j 2 (k  0.5) N. 1 e  z 1 Yre (z)  X re (z)  1  2 cos[2 (k  0.5)]  z 1  z 2 Yre (z)  2 cos[2 (k  0.5)]  z. 1.  Yre (z)  z. 2.  Yre (z)  X re (z)  e.  j 2 (k  0.5) N.  z 1  X re (z).  Inverse  Z  transform Yre [m]  2 cos[2 (k  0.5)]  Yre [ m  1]  Yre [ m  2]  xre [ m]  e. 01/16/2022.  j 2 (k  0.5) N.  xre [ m  1]. 應用系統晶片設計實驗室. 10.

(11) Li An Kai. Proposed AQMF(6/6) N/4 0. N 2 1 0. 0. E. C1[k]. N 4. C2 [k]. N 2. C3 [k]. 3N 4. C4 [k]. E. E. N/4 2 1 0. N/2 2 1 0.  A1[k]. N/4. 01/16/2022. 應用系統晶片設計實驗室. 11.

(12) Outline. Li An Kai. Introduction Proposed QMF Algorithm AQMF SQMF Complex Multiplication. Comparison & Analyses Conclusion Reference 01/16/2022. 應用系統晶片設計實驗室. 12.

(13) Li An Kai. Proposed SQMF(1/7) j (k  0.5)(2 n  2 N 1)  N / 21  N x[n]  Re   X [k]  e  , 0  n  N 1  k 0 . • Definition :  x [n]  Re   X [k]  e •  N / 2 1. j (k  0.5)( 2 n  2 N 1) N. k 0.  , 0  n  N 1 . j 2 (k  0.5)(n  0.5)  N / 2 1  N  ( 1)  Re   X [k]  e  , 0  n  N 1  k 0 . x[n]. 0. x1[n], 0  n  N/ 4  1. 01/16/2022. N/4. x2 [n], N/ 4  n  N/ 2  1. N/2. 3N/4. x3 [n], N/ 2  n  3 N/ 4  1. N-1. x4 [n], 3 N/ 4  n  N  1. 應用系統晶片設計實驗室. 13.

(14) Proposed SQMF(2/7) j 2 (k  0.5)(n  0.5)  N / 2 1  N x1[n]  ( 1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  0.5)  N / 21  N xS 1[n]  ( 1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  0.5)  N / 2 1  N x2 [n]  (1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  N/ 4  0.5)  N / 2 1  N xS 2 [n]  (1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  0.5)  N / 21  N x3 [n]  ( 1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  N/ 2  0.5)  N / 2 1  N xS 3 [n]  ( 1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  0.5)  N / 2 1  N x4 [n]  (1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  3 N/ 4  0.5)  N / 2 1  N xS 4 [n]  ( 1)  Re   X [k]  e   k 0 . 01/16/2022. 應用系統晶片設計實驗室. Li An Kai. 14.

(15) Proposed SQMF(3/7). Li An Kai. j 2 (k  0.5)(n  0.5)  N / 21  N xS 1[n]  (1)  Re   X [k]  e   k 0 . j 2 (k  0.5)(n  N/ 4  0.5) j 2  (k  0.5)(n  0.5)  N / 2 1 j  N / 2 1   k  N N xS 2 [n]  ( 1)  Re   X [k]  e   ( 1)  Re  j  e 4   X [k]  e  k 0  k 0   . j 2 (k  0.5)(n  N/ 2  0.5) j 2  (k  0.5)(n  0.5) N / 2 1  N / 2 1    k N N xS 3 [n]  ( 1)  Re   X [k]  e  (  1)  Re j  (  1)  X [k]  e     k 0  k 0   . j 2 (k  0.5)(n  3 N/ 4  0.5) j 2 (k  0.5)(n  0.5)  N / 2 1 j  N / 2 1    k N N xS 4 [n]  ( 1)  Re   X [k]  e   ( 1)  Re ( 1)  (  j)  e 4   X [k]  e  k 0  k 0   . 01/16/2022. 應用系統晶片設計實驗室. 15.

(16) Li An Kai. Proposed SQMF(4/7) � � �1 [ ¿ ] ¿. Coefficient. k 4p 4p+1 4p+2 4p+3. 1. � � �1 [ ¿ ] ¿. 1 1 1 1. � � �2 [ ¿ ] ¿ �. �. � ×�. �. � � �2 [ ¿ ] ¿ j 4. e. je e. j 4 j 4.  je 01/16/2022. 4. j 4. � � � 3 [¿ ] ¿. � × ( −1 ). �. � � �4 [ ¿ ] ¿. �. − ( − � ) ×�. � � � 3 [¿ ] ¿. J -j J -j. � 4. � � �4 [ ¿ ] ¿. je. e. j 4. j 4.  je. e 應用系統晶片設計實驗室. �. j 4. j 4. 16.

(17) Proposed SQMF(5/7). Li An Kai. j 2 (4 k  0.5)(n  0.5)  N /81  N x[n]kfirst  (  1)  Re X [4 k]  e   0  k 0  j 2 ( 4 k)(n  0.5)  j 2 (0.5Nn  0.25) N /81  first N  ( 1)  Re e   X [4 k]  e   ( 1)  Re  x p [n]k 0  k 0  . j 2 ( 4 k  0.5)(n  0.5)  N /81  N x[n]kfirst  (  1)  Re X [4 k  1]  e   1  k 0  j 2 (1.5n  0.75) j 2 ( 4 k)(n  0.5) N /8 1   first N N  (1)  Re e   X [4 k  1]  e   ( 1)  Re  x p [n]k 1  k 0   j 2 ( 4 k  0.5)(n  0.5)  N /81  N x[n]kfirst  (  1)  Re X [4 k  2]  e    2  k 0  j 2 ( 2.5 n 1.25) j 2 ( 4 k)(n  0.5) N /8 1   first N N  ( 1)  Re e   X [4 k  2]  e   (1)  Re  x p [n]k 2  k 0   j 2 ( 4 k  0.5)(n  0.5)  N /81  N x[n]kfirst  (  1)  Re X [4 k  3]  e   3  k 0  j 2 (3.5 n 1.75) j 2 ( 4 k)(n  0.5) N /8 1   first N N  ( 1)  Re e   X [4 k  3]  e   ( 1)  Re  x p [n]k 3  k 0  . 01/16/2022. using lifting scheme. 應用系統晶片設計實驗室. 17.

(18) Proposed SQMF(6/7) First Section. Li An Kai. Second Section. x[n]kfirst  ( 1)  Re  x p [n]kfirst 0 0 . ond x[n]sec  ( 1)  Re  j  x p [n]kfirst k0 0 . x[n]kfirst  ( 1)  Re  x p [n]kfirst 1 1 . ond x[n]sec  ( 1)  Re   j  x p [n]kfirst k1 1  ond x[n]sec  ( 1)  Re  j  x p [n]kfirst k2 2 . x[n]kfirst  ( 1)  Re  x p [n]kfirst 2 2 . ond x[n]sec  ( 1)  Re   j  x p [n]kfirst k3 3 . x[n]kfirst  ( 1)  Re  x p [n]kfirst 3 3  Third Section. Forth Section. x[n]third  ( 1)  Re  e j / 4  x p [n]kfirst k0 0 . x[n]kforth  (1)  Re  je j / 4  x p [n]kfirst 0 0 . x[n]third  ( 1)  Re  je j / 4  x p [n]kfirst k1 1 . x[n]kforth  (1)  Re  e j / 4  x p [n]kfirst 1 1 . x[n]third  ( 1)  Re  e j / 4  x p [n]kfirst k2 2  x[n]third  ( 1)  Re   je j / 4  x p [n]kfirst k3 3 . x[n]kforth  (1)  Re   je j / 4  x p [n]kfirst 2 2  x[n]kforth  (1)  Re  e j / 4  x p [n]kfirst 3 3 . using CSD 01/16/2022. 應用系統晶片設計實驗室. 18.

(19) Proposed SQMF(7/7). Li An Kai. Coef n , k 0.  Coef n , k 1.  Coef n ,k 2.  Coef n ,k 3. 01/16/2022. 應用系統晶片設計實驗室. 19.

(20) Outline. Li An Kai. Introduction Proposed QMF Algorithm AQMF SQMF Complex Multiplication. Comparison Results Conclusion Reference 01/16/2022. 應用系統晶片設計實驗室. 20.

(21) Complex Multiplication(1/3). Li An Kai. • General complex multiplication  4 multiplications, 2 additions. • Lifting Scheme  3 multiplications, 3 additions. • CSD(Canonic-Singed-Digit )  fixed-coefficient only  in hardware, uses adders and shifters only  transfers a multiplication into a series of additions 01/16/2022. 應用系統晶片設計實驗室. 21.

(22) Li An Kai. Complex Multiplication(2/3) • Multiplication of fixed-coefficient e. j. . 4.  cos(. . )  jsin(. . ). 2 2  j 2 2. 4 4  (a  ja)  (b  jc)  (ab  ac)  j(ab  ac)  2  (0.1011, 0101, 0000, 0100,1111, 001) 2  2 multiplications, 2 additions try to implement by hardware 2.  24-bit word-length 11 shifters 10 adders 01/16/2022. 應用系統晶片設計實驗室. 22.

(23) Complex Multiplication(3/3). Li An Kai. • Using CSD   . 2  (0.1011, 0101, 0000, 0100,1111, 001) 2 2     (1.0 10 1, 0101, 0000, 0101, 000 1, 001) CSD.  1=(-1) , totally, the number of -1 and 1 is 9. 8 shifters, 8 adders.  2 multiplications, 2 additions implemented by 8 adders, 8 shifters 01/16/2022. 應用系統晶片設計實驗室. 23.

(24) Outline. Li An Kai. Introduction Proposed QMF Algorithm Comparison & Analyses Computation Complexity Number of Recursive Cycles PSNR value. Conclusion Reference 01/16/2022. 應用系統晶片設計實驗室. 24.

(25) Computation Complexity(1/3) Proposed AQMF algorithm computation complexity Pre. Recursive. Post. Total. Improvement(%). N=64. 0 / 32. 1120. 96. 1216 / 1248. 46.4 / 43.6. N=32. 0 / 16. 304. 48. 352 / 368. 43.2 / 38.1. N=64. 0. 2112. 96. 2208. 47.2 / 45.7. N=32. 0. 544. 48. 592. 44.4 / 41.4. N=64. 0/1. 32. 32. 64 / 65. 66.3 / 65.8. N=32. 0/1. 16. 16. 32 / 65. 66 / 64.9. Filter Bank Mpy. A Q M F. Add.. Coef.. Li An Kai. with CSD w/o CSD. AQMF algorithm computation complexity proposed by Lai Shin-Chi Pre. Recursive. Post. Total. N=64. 192. 1980. 96. 2268. N=32. 96. 476. 48. 620. N=64. 192. 3896. 96. 4184. N=32. 96. 920. 48. 1064. N=64. 128. 30. 32. 190. N=32. 64. 14. 16. 94. Filter Bank Mpy. A Q M F. Add.. Coef.. 01/16/2022. 應用系統晶片設計實驗室. 25.

(26) Computation Complexity(1/3). Li An Kai. AQMF 4500 4000 3500 3000. Original [1] Proposed. 2500 2000 1500 1000 500 0. 01/16/2022. Mpy.[N=64]. Add.[N=64]. Coeff.[N=64]. Mpy.[N=32]. Add.[N=32]. Coeff.[N=32]. 應用系統晶片設計實驗室. 26.

(27) Computation Complexity(2/3) Proposed SQMF algorithm computation complexity Pre. Recursive. Post. Total. Improvement( %). N=64. 0. 288. 192 / 320. 480 / 608. 28.1 / 9. N=32. 0. 80. 96 / 160. 176 / 240. 13.7 / -17.6. N=64. 0. 576. 192 / 320. 768 / 896. 48.8 / 40.3. N=32. 0. 160. 96 / 160. 256 / 320. 48 / 35. N=64. 0. 15. 32 / 33. 47 / 48. --2.2 / 4.3. N=32. 0. 7. 16 / 17. 23 / 24. -4.6 / -9.1. Filter Bank Mpy. SQ M F. Add. Coef.. Li An Kai. with CSD w/o CSD. SQMF algorithm computation complexity proposed by Lai Shin-Chi Pre. Recursive. Post. Total. N=64. 96. 476. 96. 668. N=32. 48. 108. 48. 204. N=64. 160. 1244. 96. 1500. N=32. 80. 364. 48. 492. N=64. 32. 14. 0. 46. N=32. 16. 6. 0. 22. Filter Bank Mpy. S Q M F. Add.. Coef.. 01/16/2022. 應用系統晶片設計實驗室. 27.

(28) Computation Complexity(2/3). Li An Kai. SQMF 4500 4000 3500 3000. Original [1] Proposed. 2500 2000 1500 1000 500 0. 01/16/2022. Mpy.[N=64]. Add.[N=64]. Coeff.[N=64]. Mpy.[N=32]. Add.[N=32]. Coeff.[N=32]. 應用系統晶片設計實驗室. 28.

(29) Computation Complexity(3/3) [1] Filterbank. Mpy. A Q M F. Add. Coeff. Mpy.. S Q M F. Add. Coeff. Mpy.. T O T A L. Add. Coeff.. Original. Li An Kai. Proposed. Pre.. RDFT. Post.. Total. Pre. Post.. Recursive. Improvement[1](%). w/o CSD. with CSD. w/o CSD. with CSD. w/o CSD. with CSD. w/o CSD. with CSD. 4096. 192. 1980. 96. 2268. 64. 0. 1120. 96. 1280. 1216. 68.75. 70.31. 43.56. 46.38. N=32. 1024. 96. 476. 48. 620. 32. 0. 304. 48. 384. 352. 62.50. 65.63. 38.06. 43.23. N=64. 4032. 192. 3896. 96. 4184. 64. 0. 2112. 96. 2272. 2208. 43.65. 45.24. 45.70. 47.23. N=32. 992. 96. 920. 48. 1064. 32. 0. 544. 48. 624. 592. 37.10. 40.32. 41.35. 44.36. N=64. 4096. 128. 30. 32. 190. 1. 0. 32. 32. 65. 64. 98.41. 98.44. 65.79. 66.32. N=32. 1024. 64. 14. 16. 94. 1. 0. 16. 16. 33. 32. 96.78. 96.88. 64.89. 65.96. N=64. 4096. 96. 476. 96. 668. 0. 288. 320. 192. 608. 480. 85.16. 88.28. 8.98. 28.14. N=32. 1024. 48. 108. 48. 204. 0. 80. 160. 96. 240. 176. 76.56. 82.81. -17.65. 13.73. N=64. 4032. 160. 1244. 96. 1500. 0. 576. 320. 192. 896. 768. 77.78. 80.95. 40.27. 48.80. N=32. 992. 80. 364. 48. 492. 0. 160. 160. 96. 320. 256. 67.74. 74.19. 34.96. 47.97. N=64. 4096. 32. 14. 0. 46. 0. 15. 33. 32. 48. 47. 98.83. 98.85. -4.35. -2.17. N=32. 1024. 16. 6. 0. 22. N=64. 8192. 288. 2456. 192. 2936. 64. 0. N=32. 2048. 144. 584. 96. 824. 32. N=64. 8064. 352. 5140. 192. 5684. 64. N=32. 1984. 176. 1284. 96. 1556. N=64. 8192. 160. 44. 32. N=32. 2048. 80. 20. 16. 01/16/2022. with CSD. Improvement[ori](%). N=64. 0. w/o CSD. Total. 7. 17. 16. 24. 23. 97.66. 97.75. -9.09. -4.55. 1408. 416. 288. 1888. 1696. 76.95. 79.30. 35.69. 42.23. 0. 384. 208. 144. 624. 528. 69.53. 74.22. 24.27. 35.92. 0. 2688. 416. 288. 3168. 2976. 60.71. 63.10. 44.26. 47.64. 32. 0. 704. 208. 144. 944. 848. 52.42. 57.26. 39.33. 45.50. 236. 1. 0. 47. 65. 64. 113. 111. 98.62. 98.65. 52.12. 52.97. 116. 1. 0. 23. 33. 32. 57. 55. 97.22. 97.31. 50.86. 52.59. 應用系統晶片設計實驗室. 29.

(30) Computation Complexity(3/3). Li An Kai. Total 9000 8000 7000 6000. Original [1] Proposed. 5000 4000 3000 2000 1000 0. 01/16/2022. Mpy.[N=64]. Add.[N=64]. Coeff.[N=64]. Mpy.[N=32]. Add.[N=32]. Coeff.[N=32]. 應用系統晶片設計實驗室. 30.

(31) Li An Kai. Number of Recursive Cycles Number of Cycles of AQMF Number of Recursive Cycles-AQMF N=32. N=64. [1]. 272. 1056. Proposed. 144. 544. Number of Recursive Cycles-SQMF N=32. N=64. [1]. 144. 544. Proposed. 80. 288. 01/16/2022. 1200 1000 800 600 400 200 0. [1] Proposed. N=64. N=32. Number of Cycles of SQMF 600 500 400 300 200 100 0. [1] Proposed. N=64. N=32. 應用系統晶片設計實驗室. 31.

(32) Li An Kai. PSNR value. AQMF. 01/16/2022. SQMF. 應用系統晶片設計實驗室. 32.


(34) Conclusion. Li An Kai. • Recursive Structure • Multiplication ： 1. Lifting Scheming 2. CSD • Future work ： coefficient, PSNR 01/16/2022. 應用系統晶片設計實驗室. 34.


(36) Reference •. • • •. Li An Kai. [1] S.-C. Lai, M.-K. Lee, A.-K, Li, C.-H Luo, and S.-F. Lei, “An Innovative Fast Algorithm and Structure Design for Analysis and Synthesis Quadrature Mirror Filterbanks on the SBR in DRM”, IEEE Trans. Circuit Syst. II, [2] S. M. Kim, J. G. Chung, and K. K. Parhi, “Low error fixed-width CSD multiplier with e fficient sign extension,” IEEE Trans. Circuits Syst. II, vol. 50, pp. 984–993, Dec. 2003. [3] Kim, S.M., Chung, J.G. and Parhi, K.K., “Design of Low Error CSD Fixed-Width Mul tiplier”, IEEE International Symposium on Circuits and Systems, 2002, pp. 69-72. [4] K. K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementation. New York: Wiley, 1999.. 01/16/2022. 應用系統晶片設計實驗室. 36.

(37) Li An Kai. END Thank you!. 01/16/2022. 應用系統晶片設計實驗室. 37.

(38) Li An Kai. 01/16/2022. 應用系統晶片設計實驗室. 38.

(39) Li An Kai. •. QMF 原式. 原架構：. QMF recursive QMF 係數共用 QMF 定係數 QMF 調整 pole 位置 QMF CSD 01/16/2022. 目標：降低運算量、乘法係數共用結果：. 應用系統晶片設計實驗室. 39.

(40) Li An Kai - j ( n -1/8)    - jN2 nk N /2 X [2k ]  e    xe [n]  e e n 0   jk - j 3 ( n -1/8)  N / 2-1   - jN2 nk * 2N N /2 X [2k  1]  e    xo [n]  e e n0   *. •. jk  2N. N / 2-1. By taking out xe[0] and xo[0] from the above respectively, using Eular formula, and adjusting th e order of the input sequence, the above can be expressed as the following, where xef[n] and xo N N  f[n] are defined in the behine. j ( k 1/4) x [n]  jx [  n] ,1  n   1 X * [2k ]  e. 2N. X [2k  1]  e *. 01/16/2022. N / 2-1  (4k  1)n    xe [0]   xef [n]  cos( ) N n0  . j ( k  3/ 4) 2N. N / 2-1  (4k  3)n    xo [0]   xof [ n]  cos( ) N n0  .  xef [n]   . e. e. 2. 0. 2 ,n  0. N N   x [n]  jxo [  n] ,1  n   1 xof [n]   o 2 2  0 ,n  0 . 應用系統晶片設計實驗室. 40.


(42) Li An Kai. QMF 原式. 原架構：. QMF recursive QMF 係數共用 QMF 定係數 QMF 調整 pole 位置 QMF CSD 01/16/2022. 目標：降低乘法器數量，把架構中的乘法器換成加法器結果：. cos(2 )  (0.09801714033)10. 應用系統晶片設計實驗室. 42.

(43) Li An Kai •. Define the absolute module function amod(s,t) as shown in the following. By using it, the pha se of cosine function will fall in the range [0,π/2] and the even part of index k of equation ca n be expressed as the following, where S1 is a function of variable k and n defined as the follo wing. j ( k 1/ 4) N / 2-1 amod (amod (4nk  n, 2 N ), N )     X * [2k ]  e 2 N   xe [0]   (1) S  xef [ n]  cos( ) N n0   1.  mod ( s, t ) , if mod ( s, t )  t / 2 amod ( s, t )    p  mod ( s, t ) , if mod ( s, t )  t / 2. •. Let(the 1) result x [n ] of be a variable m which is from 0 to N/2-1 and use a matrix to express the ite m , equation can be rewritten as the following, where is and xev(k,m) is the input fu j  ( k 1/ 4) nction. N / 2-1   * X [2k ]  e 2 N  xe [0]   xev (k , m)  cos (m )  S1. ef. . 01/16/2022. 0, if amod (4nk  n, 2 N )  N / 2 S1   1, if amod (4nk  n, 2 N )  N / 2. m0. . 應用系統晶片設計實驗室. 43.

(44) Li An Kai. N /2 1   X *[2k  1]  C[k]   x1[0]   x fo 2 [k, m]  cos(m 1 )  m 0   N /2 1   X *[2k ]  C[k]   x2 [0]   x fo1[k, m]  cos(m 1 )  m0  . 01/16/2022. 應用系統晶片設計實驗室. 44.

(45) Li An Kai. • •. IP ： 16 bits , OP ： 16 bits , 係數 & 內部結點：如上圖所示輸入訊號筆數： 1000000. 01/16/2022. 應用系統晶片設計實驗室. 45.

(46) Li An Kai. •. QMF 原式. 原精準度：. QMF recursive QMF 係數共用 • •. 01/16/2022. QMF 定係數 QMF 調整 pole 位置 QMF CSD. 目標：藉由改變系統的頻率響應來提高精準度. X R ( z) 1  cos ( )  z 1 H A ( z)   X ev ( z ) 1  2cos( )  z 1  z 2 .. 應用系統晶片設計實驗室. 46.

(47) Li An Kai. N / 41    xˆ[2n  1]  Re c1[n]   X 4 [0]   X ffe1[n, m]  cos(m  q )   m0   N / 4 1    xˆ[2n]  Re c2 [ n]   X 3[0]   X ffe 2 [n, m]  cos(m  q )  m 0   . 01/16/2022. 應用系統晶片設計實驗室. 47.


(49) Li An Kai. • • • • • • •. QMF 原式 QMF recursive. 原架構： 11 shifter + 10 adder (AQMF & SQMF) 目標：降低 shifter 以及 adder 數量將原 digit 的範圍 (1,0) 擴充為 (1,0,-1). (0,1,1,1)  (1,0,0,-1) QMF 係數共用結果： 8 shifters + 7 adders(AQMF). QMF 9 shifters + 8 adders(SQMF) 定係數 QMF 調整 pole 位置 QMF CSD. 01/16/2022. 應用系統晶片設計實驗室. 49.

(50) Canonic Signed Digit. Li An Kai. cos(2 )  (0.09801714033)10. 01/16/2022. cos(2 )  (0.000110010001011110100110) 2. ˆˆˆˆ cos(2 )  (0.001010010010100010101010) CSD. 11 shifters 10 adders. 9 shifters 8 adders. 應用系統晶片設計實驗室. 50.

(51) Li An Kai. Computaion Complex [18] Filterbank. Mpy A Q M F. Add Coeff Mpy. S Q M F. Add Coef Mpy. T O T A L. Add Coeff. 01/16/2022. Original. Proposed. Pre.. Recursive DFT. Post.. Total. Pre.. Recursive Kernel. Post.. Total. 192. 1980. 96. 2268. 0. 0. 96. 96. N=64. 4096. N=32. 1024. 96. 476. 48. 620. 0. 0. 48. 48. N=64. 4032. 192. 3896. 96. 4184. 64. 19260. 160. 19484. N=32. 992. 96. 920. 48. 1064. 32. 4508. 80. 4618. N=64. 4096. 128. 30. 32. 190. 0. 0. 32. 32. N=32. 1024. 64. 14. 16. 94. 0. 0. 16. 16. N=64. 4096. 96. 476. 96. 668. 0. 0. 96. 96. N=32. 1024. 48. 108. 48. 204. 0. 0. 48. 48. N=64. 4032. 160. 1244. 96. 1500. 64. 10974. 96. 11134. N=32. 992. 80. 364. 48. 492. 32. 2670. 48. 2750. N=64. 4096. 32. 14. 0. 46. 0. 0. 32. 32. N=32. 1024. 16. 6. 0. 22. 0. 0. 16. 16. N=64. 8192. 288. 2456. 192. 2936. 0. 0. 192. 192. N=32. 2048. 144. 584. 96. 824. 0. 96. 8064. 352. 5140. 192. 5684. 128. 0 30234. 96. N=64. 256. 30618. N=32. 1984. 176. 1284. 96. 1556. 64. 7178. 125. 7367. N=64. 8192. 160. 44. 32. 236. 0. 0. 64. 64. N=32. 2048. 80. 20. 16. 116. 0. 0. 32. 32. 應用系統晶片設計實驗室. 51.



(54) Li An Kai. N 1. • AQMF ：. X [k ]   x[m]  e. j (2 k 1)( n 1/8) N. m0. fl[n] , 0  n  N  1 j (2 k 1)( n 1/8) N f k [n]  fl[n] e. • Low-pass filter ： • The k-th band filter ： • The k-th band signal ：. sk [n]  x[n]  f k [n] N 1.   x[n  m]  fl[m]  e. j (2 k 1)( n 1/8) N. m 0. N 1.   un [ m ]  e. j (2 k 1)( n 1/8) N. m 0. 應用系統晶片設計實驗室.

(55) Li An Kai. N 1. sk [n]  un [m]  e. j (2 k 1)( n 1/8) N. m0. • The k-th band signal ： u n [ m]  x[n  m]  fl[m]. 0 0 • Re-ordered  x[0]  fl[0] input matrix ： . 0 x[0]  fl[1] 0 0  x[1]  fl[0]   x[2]  fl[0] x[1]  fl[1] x[0]  fl[2] 0        x[M  1]  fl[0] x[M  2]  fl[1]   x[0] . 0   0   0  0  fl[N  1] M  N. 應用系統晶片設計實驗室.

(56) Li An Kai N 1. sk [n]   un [m]  e. 1 j (2 k 1)( m  ) 8 N. m0. • The k-th band signal =： u [ N -1-m]  e n. , un [m]  x[n  m]  lp[m]. 1 j (2 k 1)( m  ) 8 N.  令 wn [m]  un [ N -1-m]  wn [m]  e  s0 [0]  s [0]  1   s2 [0]     sk [0]. 1 j (2 k 1)( m  ) 8 N. s0 [1] s0 [2]  s0 [M  1] s1[1] s1[2]  s1[M  1]   s2 [1] s2 [2]  s2 [M  1]       sk [1] sk [2]  sk [M  1] k M 應用系統晶片設計實驗室.

(57) Li An Kai. • The k-th band filter ：. f k [n]  fl[n] e. j (2 k 1)( n 1/8) N j 2 n ( k  0.5) N.  j 2 ( k /8 1/16) N.  fl[n] e   e phase  shift. gain.  DFT Fk ( z )  LP( z  (k  0.5))  e. • Frequency deviation ：.  j 2 ( k /81/16) N. f  (k  0.5)  ( N / 2)  ( fs / 2) fs  (k  0.5)  N 應用系統晶片設計實驗室.

(58) Li An Kai. 應用系統晶片設計實驗室.

(59)