應用於正交多頻通訊系統之具成本效益傅立葉轉換處理器設計

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(5) . . . . . . . Cost-effective Fast Fourier Transform Processor Design for Orthogonal Multi-carrier Communication Systems $. %. &. ! '. ". Student:Zhao-Hong Lai #. (. ). *. +. ,Advisor: Dr. Hsiang-Feng Chi - ./. 0123/ 4,5 4,67 A Thesis Submitted to Department of Communication Engineering College of Electrical Engineering and Computer Science National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Master of Science in Communication Engineering July 2004. . Hsinchu, Taiwan . . . . ii. .

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(11) Cost-effective Fast Fourier Transform Processor Design for Orthogonal Multi-carrier Communication Systems. Student:Zhao-Hong Lai . . . . . . Advisor: Dr. Hsiang-Feng Chi. Department of Communication Engineering National Chiao Tung University Hsinchu, Taiwan. Abstract In this thesis, we implement a multi-mode variable length RFFT(Real-value FFT)/HS-IFFT(Hermitian Symmetric IFFT) processor which based on memory-based architecture. Because there is only one butterfly process element in our hardware design, the architecture is area-efficient. By designing the address generators, the processor has the variable length character. This architecture has efficient computation for real-value FFT and Hermitian Symmetric IFFT. In order to improve the fixed point precision without increasing the word length, block scaling method is used.. v.

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(24) efg.h B)C .............................................................. 42 4.7 Memory-based OP- radix-2 FFT L!M+NXY ............................................. 42 4.8 16 7 radix-2 DIT IFFT e f!B)C ..................................................................... 44 4.9 16 7 radix-2 DIT IFFT e fg.h BDC ............................................................. 44 4.10 Memory-based OP radix-2 IFFT L!M+NXY .............................................. 45 4.11 `ZD4 memory-based OP raidx-2 FFT/IFFT LM+NDXY .......................... 46 4.12 (a) iDjke f (b) #)=>l@mjkef ......................................................... 48 4.13 i : [ a, b] → j : [ b, a ]

(25) .......................................................................... 49 4.14 onpqrstNXY ...................................................................................... 50 4.15 16 7 raidx-2 DIT FFT/IFFT CAG

(26) .................................................... 50 4.16 vu!]!qrstNXY ...................................................................................... 51 4.17 32 7 #2)E)F)89 :;!<wyx3 twiddle factors

(27) ................................... 52 4.18 Radix-r FFT butterfly XY .............................................................................. 53 4.19 16 7 radix-2 FFT .Yz {D

(28) ................................................................ 54 4.20 BFP memory-based OPZD4 radix-2 FFT/IFFT L!M+NDXY ................... 55 4.21 64 7 radix-4 DIT FFT efg.h BDC ............................................................ 57 4.22 64 7 radix-4 DIT IFFT e fgh B)C ........................................................... 58 4.23 BFP memory-based OPZD4 radix-4 FFT/IFFT L!M+NDXY ................... 59 4.24 RFFT/HS-IFFT XY ........................................................................................ 60 4.25 RFFT XY ....................................................................................................... 60 4.27 HS-IFFT XY .................................................................................................. 61 4.26 |bc LM+NBDC .............................................................................................. 62 4.28 \dDc LM+NBDC .............................................................................................. 63 4.29 BFP memory-based OP }4 RFFT/HS-IFFT L!M-NDXY ..................... 64 5.1 Radix-2 }4~ )) RFFT/HS-IFFT L!M-NDXY ....................................... 66 5.2 n3pq r!stNXY ........................................................................................ 67 5.3 k : [a, b, c] → address[ a, c, b] Dq

(29) ....................................... 68 5.4 k : [a, b, c] → address[ a, c, b] )

(30) ................................................... 68 5.5 Radix-2 DAG OP .................................................................................... 68 5.6 Radix-2 R_DAG OP ................................................................................ 69 5.7 Radix-2 UVWYz {XY .......................................................................... 70 5.8 Radix-2 UVWYz {XY .......................................................................... 71 5.9 Radix-2 u!]!q r!stNDXY .......................................................................... 71 5.10 Radix-2 R_CAG OP .............................................................................. 72 5.11 Reduce coefficient memory size OP .............................................................. 73 5.12 Decoder 3)OP ....................................................................................... 73 5.13 o}4 raidx-2 butterfly e fN))R ef4 .................................................. 74 4.6. 16 7. radix-2 DIT FFT. S3 , 5. shuffle shuffle. x.

(31) o}4 radix-2 butterfly e fNOP ...................................................................... 74 5.15 Single port UV!W timing diagram.................................................................. 75 5.16 ?.@ [)]N.[D?.@ clock cycles .e f ]e fOP ........ 75 5.17 Radix-2 butterfly 3!)3y ............................................. 76 5.18 Radix-2 }4~ ) RFFT/HS-IFFT W OP ......................................... 77 5.19 Radix-4 }4~ ) RFFT/HS-IFFT LM+NDXY ..................................... 78 5.20 Radix-4 DAG OP .................................................................................. 79 5.21 Radix-4 R_DAG OP .............................................................................. 80 5.22 Radix-4 UVW.Yz {)XY ........................................................................ 81 5.23 Radix-4 CAG OP ................................................................................... 81 5.24 Radix-4 R_CAG OP .............................................................................. 82 5.25 Basic radix-4 butterfly e fN ............................................................................. 83 5.26 o}4 radix-4 butterfly e fN ef4 .................................................. 84 5.27 Radix-4 butterfly 3!)3y ............................................. 85 5.28 Radix-4 }4~ ) RFFT/HS-IFFT }W OP ..................................... 86 5.29 v f SQNR D

(32) ......................................................................................... 87 5.30 FFT/IFFT !¡¢£ ¤¥ £ ¤+ ..................................................... 89 5.31 FFT/IFFT !¡¢£ ¤¥ SQNR y ........................................................ 89 5.32 RFFT ¡¢£ ¤D¥ D£ ¤.y ........................................................... 90 5.33 RFFT ¡¢£ ¤D¥ SQNR y ............................................................... 90 5.34 HS-IFFT ¡¢£ ¤D¥ D£ ¤y ...................................................... 91 5.35 HS-IFFT ¡¢£ ¤D¥ SQNR + .......................................................... 91 5.36 `¦§¨!!)©ª% FFT/IFFT ¡¢£ ¤D¥ £ ¤+ ..................... 92 5.37 FFT/IFFT ¦§¨!) SQNR ................................................................ 92 5.38 RFFT ¦§¨!) SQNR ....................................................................... 93 5.39 HS-IFFT ¦§¨!!) SQNR .................................................................. 93 5.40 FFT/IFFT !¡¢£ ¤¥ £ ¤+ ..................................................... 95 5.41 FFT/IFFT !¡¢£ ¤¥ SQNR y ........................................................ 95 5.42 `¦§¨!!)©ª% FFT/IFFT ¡¢£ ¤D¥ £ ¤+ ..................... 96 5.43 FFT/IFFT ¦§¨!) SQNR ................................................................ 96 5.44 RFFT ¦§¨!) SQNR ....................................................................... 97 5.45 HS-IFFT ¦§¨!!) SQNR .................................................................. 97 5.46 RTL code «¬ ............................................................................................... 100 5.47 DSP development board, Stratix edition ............................................................ 102 5.48 ®¯X ............................................................................................................ 102 5.14. xi.

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(41) DFT/IDFT . 2.1 8. _. CD. !. ". #. Transform, DFT):u &¸ 8. à _. T. 3. S. £. FFT S. T. `. &æ. ç+ã. Ã. &ù. ". 8. _. #. É. C6. ÷. ³. b. b. &U. ¢. E6. ¦g. . Z. ´{. &·. ”6 s. . ¤/ N Å C<. i. T. +¸. í. Æ+Ê. §P. Ú. ð. Ì. T. ¢. ,§u. ¦S. T. &. Æ&æ. Ã. op. ². T. . Ç. µ. . . û. . «. r. f. ø. s. Ë. &ï. ð. Jg. &. ´<. i. ¤. j. &S. &g. ï. + FFT S. T. DFT û. q. q. b. C§u. ¤9:h. ·. Ã. &+r. :S. C. (Discrete Fourier s. ECæ. FFT(IFFT)S î. `. T. radix-22 J split-radix 2/4 S[4][5][6]9 . op. q. r. 9FFT(IFFT)S. &. (Decimation-in-frequency, DIF)&S Æ9m. Å. r. Ò. b. T. ï. /ü. U. ÷. à(Decimation-in-time, DIT)&S ]. q . d. cà+ú. T +r. §”£ U. r. op. c9. #. q. Ê T. tb. Ã. +:h. h. Å. `. 9<. (FFT)S s. 9. C¦. T. T. +æ. Ê. &·. _. &. O ( N log N ) + . m. /. Ú. ¨. ¶. `. 8. r ¶. op. :ñ. çl. q. &°. S§. +'(Q. VLSI Z. 4. v. ¾§ Eú. O( N 2 ) þ ä. 3. . :. /'(op. . . S. 4. T. % +¼½. ¤ý. _. #. . DFT &S. j +. (¤<. ?. ". '(g. Ji. ð. !. ö. Å æ. Ã. µ. Ê. §³. Æ&r. s. +. . V. <. +¸. q . W. r. ]. 0. ñ. . B. 8. &]. Æ x[n] Å. :ñ. X [k ] s. çl. . / ]. ]. çl. ]. 8. ^ _. B +> raidx-2Oraidx-4O î. +. &g. T. T. op. &:;. s. . . S. T. &g. ï. ð. ¤9. 2.2 Radix-2 FFT/IFFT 2.2.1 FFT . Decimation-in-time . ½. C+X T. DFT à+K¤&i. ¾g. &]. h. 2 ç. i +ï U. °. ³. ¢. ý. ]. çg. l. C W Nkn = e − j ( 2π / N ) kn &@. C Z. j. æ. Ã. Å. Æ x[n] ] l. 8 6. _. `. A. 8. Å. «. _ . . u. Æ+¼A. DFT g. «. &`. . /]. Ø. . EC+S àS. T. 9.

(42) :h. ¤/ N &æ £. X [k ] = Á. N −1. X [k ] =. Æ x[n] +¦æ Å. x[n ]W Nnk , k = 0,1,2,. n =0. § x[n] ] {. Ã. l. N 2 −1 n =0. C. J. C. Ã. oØ. q. , N −1 0. h. ». r. s. ¢. W N = e − j ( 2π. ¦. ]. +½. ,¹/¥. ¾¢. x[2n + 1]W N(2 n +1)k , k = 0,1,2,. n =0. (2.1). ,C¥. N 2 −1. x[2n]W N2 nk +. N). , N −1. (2.2).

(43) g[n] = x[2n] ´ h[n] = x[2n + 1] +È¥ N 2 −1. X [k ] =. g[n] W N2 nk +. n =0. =. N 2 −1. N 2 −1. h[n] W N(2 n +1)k. n=0. N 2 −1. g[n] W Nnk2 + W Nk. n=0. n =0. = G[k ] + W Nk H [k ]. G[k ] J H [k ] ]. ¦. ò. / x[n] . C. N 2 −1. X [k + N 2] =. n=0. J. & N 2. C. g[n] W Nnk2 − W Nk. = G[k ] − W Nk H [k ] ¸(2.1)/ . i. ¸9. #. ]. àæ. +½. ¾¢. Ã. op ,. q . r ³. s. ]. çl. N 2. &æ. h[n] W Nnk2. 0. Ã. op. q. r. s. 9. N 2 −1. h[n] W Nnk2. n =0. h. Ã. æ. (2.3). ó. op. . &]. q. r. àæ. (2.4). s. ¶. ]. Ã. op. çl. 0. q. h. r. . s. &. C/. N 4 &æ Ã op q r s + þ ] çÑ ï +È¢ ] à'(op q r s 9Ñ ¸(2.2) /. . ]. à'(op. q. r. s. . i. ¸9. :h. £. ¤/ N radix-2 DIT FFT &g. ï. ð. ¤>Ñ. d¹¥. M ( N ) = ( N / 2) log 2 N. (2.5). A( N ) = N log 2 N ¦. + M (N ) J A(N ) ] ò. . ï. Cú. T. Jï. 7. (2.6) CÌ. T. &h. C9.

(44) W80 W81 W82 W83. ¸ 2.1¥ . ]. àæ. Ã. op. q. r. s. ]. çl. 0. h. ó. . &]. àæ. Ã. op. q. r. s. . i. ¸. W80 W80 W82. W80. W80 W81. W80. W80. ¸ 2.2¥ . W80. W82. W82. W83. ]. à'(op. q. r. s. . i. ¸. . Decimation-in-frequency à& FFT S. T. U. ¾. Å. Æ X [k ] ]. ] ½ A :h. /] £. ¢. ³ ^S. 2. T. N −1 n =0. 2. Ì. Å çl. Æ x[n] ] 8. _. ·. çl Å. 8. Å. Æ& DFT 6. '(g. DFT+X. _. Æ6. `. ñ. . 9 . FFT S. 9. ¤/ N &æ. X [k ] =. ³. Ã. Å. Æ x[n] +¦æ. x[n ]W Nnk , k = 0,1,2,. Ã. oØ. , N −1 8. q. r ¦. s. ¢. ,¹/¥. W N = e − j ( 2π. N). (2.7). Q + T.

(45) §¦ x[n] ] l. 0. . +È¥ N 2 −1. X [k ] =. x[n]W Nnk +. n =0 N 2 −1. =. n=0. x[n]W. N −1. x[n]W Nnk. n= N 2. + (− 1). nk N. k. N 2 −1. x[n + N 2]W. n =0. (2.8) nk N. X [k ] & C J C ] ò /¥. ¦. X [2r ] = =. N 2 −1 n=0 N 2 −1 n=0. =. n =0. x[n + N 2]W N2 nr. N 2 −1 n=0. x[n + N 2]W Nnr2. (2.9). (x[n] + x[n + N 2])WNnr2. N 2 −1 n =0. x[n]W NnW N2 nr + (− 1). N 2 −1. =. x[n]W Nnr2 +. N 2 −1. 2r. N 2 −1 n=0. X [2r + 1] =. x[n]W N2 nr + (− 1). n=0. N 2 −1. 2 r +1. n=0. x[n + N 2]W Nn W N2 nr. N 2 −1. x[n]W NnW Nnr2 −. x[n + N 2]W NnW Nnr2. n=0. (2.10). N 2 −1. =. (x[n] − x[n + N 2])WNnWNnr2. n=0.

(46) g [n ] = x[n] + x[n + N 2] O h[n ] = ( x[n ] − x[n + N 2])W Nn +È¥. X [2r ] =. . ¸(2.3)/ i. ¸9. . N 2 −1. n=0. n =0. ^æ. g [n]W Nnr2 ´ X [2r + 1] = Ã. # +½. ¾. s. ¸(2.4)/. 9Ñ. ¢. op ,. q ]. r. s. ]. àæ. (op. q. :h. ¤/ N radix-2 DIF FFT ¿ £. r. ]. N 2 −1. . çl. ]. Ã. op. 0. h. q. r. ó s. ^'(op. <. DIT . h[n]W Nnr2. q. . &]. :. . . r. s. &g. (2.11) ^æ. þ . ] i. ï. ò. . ï. Cú. T. Jï. 9. q. r ]. ¸9. ð. ¤¥. (2.12). A( N ) = N log 2 N + M (N ) J A(N ) ]. op. ç+È¢. M ( N ) = ( N / 2) log 2 N. ¦. Ã. (2.13) CÌ. T. &h. C9. s. & ^'.

(47) W80. W81 W82 W83. ¸ 2.3¥ . ]. ^æ. Ã. op. q. r. s. ]. çl. 0. h. ó. . &]. ^æ. Ã. op. q. r. s. . i. ¸. W80 W80 W82. W80. W80 W81. ¸ 2.4¥. W80. W82. W80. W83. W82. . ]. ^'(op. W80. q. r. s. . i. ¸. 2.2.2 IFFT . Decimation-in-time :h. £. ¤/ N &æ. x[n ] = ö]. N −1 k =0. Ã. Å. Æ X [k ] +¦né. X [k ]W N− nk , n = 0,1,2,. à FFT :. +Á. {. § X [k ] ] l. æ. Ã. oØ. , N −1 C. 10. q. ¦. J. C. r. s. ¢. ,¹/¥. W N = e − j ( 2π 0. h. ». ]. N). +½. (2.14) ¾¢. ,C¥.

(48) x[n] =. N 2 −1 k =0. N 2 −1. X [2k ]WN− 2 nk +. k =0. X [2k + 1]W N−( 2 k +1)n , n = 0,1,2,. , N −1. (2.15).

(49) G[k ] = X [2k ] ´ H [k ] = X [2k + 1] +È¥. x[n] =. N 2 −1 k =0. G[k ] W N−2 nk +. N 2 −1. =. k =0. N 2 −1 k =0. H [k ] W N−( 2 k +1)n. G[k ] W N− nk2 + W N− n. N 2 −1 k =0. = g [n] + W N− n h[n]. H [k ] W N− nk2. (2.16). g [n ] J h[n] ] ò / X [k ] C J C & N 2 né æ Ã op q r s 9. ¦. N 2 −1. x[n + N 2] =. G[k ] W. k =0. − nk N 2. −W. N 2 −1. −n N. k =0. H [k ] W N− nk2. = g [n] − W h[n]. (2.17). −n N. ½. é. ¾¢. ,. Ã. op. æ. /. . né. § N 2 . q ]. r. s +. &né. þ. ]. æ. çÑ. à'(op. q. Ã. op. q. r. ï +È¢ r. s. . i. s. ¶. né. ]. ]. çl. 0. h. C/ N 4 . à'(op. q. r. s 9Ñ. &n ¸(2.5). ¸9. W80 W80 W8−2. W80. W80 W8−1. W80. W80. ¸ 2.5¥. :h. ¦. . W80. W8−2. W8−2. W8−3. né. ]. £. ¤/ N radix-2 DIT IFFT &g. à'(op. + M (N ) J A(N ) ] ò. ï. ð. q. r. s. . i. ¸. ¤/¥. M ( N ) = ( N / 2) log 2 N. (2.18). A( N ) = N log 2 N. (2.19). . ï. Cú. T. Jï 11. CÌ. T. &h. C9.

(50) . Decimation-in-frequency :h. ¤/ N &æ £. x[n ] =. N −1 k =0. Æ X [k ] +¦æ Å. Ã. X [k ]W N− nk , n = 0,1,2, +§¦ X [k ] ]. ^ FFT :. ö]. Ã. N 2 −1. x[n] =. k =0. 0. . r. s. ¢. ,¹/¥. W N = e − j ( 2π. ¦. N). (2.20). +È¥ N −1. X [k ]W N− nk. k =N 2. X [k ]WN− nk + (− 1). k =0. q. , N −1. X [k ]W N− nk +. N 2 −1. =. l. oØ. (2.21). N 2 −1. n. X [k + N 2]W N− nk. k =0. x[n] & C J C ] ò /¥. ¦. N 2 −1. x[2r ] =. k =0. =. X [k ]W N− 2 kr + (− 1). N 2 −1 k =0. =. N 2 −1. N 2 −1 k =0. X [k + N 2]W N− kr2. (2.22). ( X [k ] + X [k + N 2])WN− kr2. X [k ]W N− k W N− 2 kr + (− 1). k =0. N 2 −1. =. k =0. X [k + N 2]W N− 2 kr. N 2 −1 k =0. x[2r + 1] =. X [k ]WN− kr2 +. N 2 −1. 2r. k =0. 2 r +1. N 2 −1 k =0. X [k + N 2]W N− k W N− 2 kr. X [k ]WN− k W N− kr2 −. N 2 −1 k =0. X [k + N 2]W N− k W N− knr 2. (2.23). =. N 2 −1 k =0. ( X [k ] − X [k + N 2])WN− kWN−kr2.

(51) G[k ] = X [k ] + X [k + N 2] O H [k ] = ( X [k ] − X [k + N 2])W N− k +È¥. x[2r ] = . § N 2. # +¢. Ã. op . :h. ¦. q. né. r. ^æ. N 2 −1. k =0. k =0. &né. s + ]. N 2 −1. Ã. þ. ] op. G[k ]WN− kr2 ´ x[2r + 1] = æ. Ã. çÑ q. op. q. r. ï +È¢ r. s. . i. s. + M (N ) J A(N ) ]. . ]. né. ]. çl. 0. h. ^'(op. (2.24). C/ N 4 . q. r. s 9Ñ. &né. ï. ð. ¤¥. M ( N ) = ( N / 2) log 2 N. (2.25). A( N ) = N log 2 N. (2.26). ò. . ï. Cú. T. Jï 12. CÌ. T. &h. C9. æ. ¸(2.6)/. ¸9. ¤/ N radix-2 DIF IFFT &g £. . H [k ]W N− kr2.

(52) W80. W80 W8−2. W80. W80 W8−1. ¸ 2.6¥. W80. W8−2. W80. W8−3. W8−2. . né. ]. W80. ^'(op. q. r. s. . i. ¸. 2.3 Radix-4 FFT/IFFT 2.3.1 FFT Decimation-in-time DFT &2. radix-2 S :h. Ì. T. Å 6. g. ¤/ N &æ £. X [k ] =. N −1 n =0. C N / 4 &. Æ. 9² Ã. &° /¦O. m. ¾. x[n ]W Nnk , k = 0,1,2,. butterfly . 9Ñ. T. ¢. ,Ñ. à ( N = 4 v ) + É. ¢. Æ x[n] +¦æ Å. Raidx-2 DIT &ü. >. ñ+½. þ. , Ã. op. . ©< q. r. , N −1. ¦. ó. EÑ. B. +Ø. s. i. ½. ¾. & radix-4 S j. ¢. . ,. T. 6. g. E 9. B. +½. N). ¾¢. (2.27) ,C radix-4. ¸ 2.7(a)/ radix-4 DIT FFT butterfly ¸¹+Ñ. ¸ 2.7(b). ¸9. X [k ]. = F1 [k ] + W Nk F2 [k ] + W N2 k F3 [k ] + W N3k F4 [k ]. X [k + N / 4] = F1 [k ] − jW F2 [k ] − W F3 [k ] + jW F4 [k ]. (2.28). k N. 2k N. 3k N. (2.29). k N. 2k N. 3k N. (2.30). X [k + 2 N / 4] = F1 [k ] − W F2 [k ] + W F3 [k ] − W F4 [k ]. X [k + 3 N / 4] = F1 [k ] + jW Nk F2 [k ] − W N2 k F3 [k ] − jW N3k F4 [k ]. ¦. ¢. ,¹/¥. W N = e − j ( 2π Æó. U. (2.31). f 1[m] = x[4m] O f 2 [m] = x[4m + 1] O f 3 [m] = x[4m + 2] O f 4 [m] = x[4m + 3] ´. F1 [k ] O F2 [k ] O F3 [k ] O F4 [k ] ] ò /¦op q r s 9 13.

(53) . § F1 [k ] O F2 [k ] O F3 [k ] ö F4 [k ] S N / 4 . # +¢. ]. . þ. ]. çÑ. à'(op. radix-4 ]. ï +È¢ q. r. s. . i. à'(op. q. &æ. Ã. op. q. r. s 9Ñ ¸(2.8)/ r. s ö. . . ]. radix-4. ¸9. F1 [k ]. WN0. X [k ]. F2 [k ]. WNk. X [k + N / 4]. F3 [k ]. WN2k. F4 [k ]. WN3k. F1 [k ]. X [k ]. 0. F2 [k ]. X [k + N / 4]. k 2k 3k. F3 [k ]. X [k + 2N / 4]. X [k + 2N / 4]. F4 [k ]. X [k + 3N / 4]. X [k + 3N / 4]. (a). (b). 2.7(a) Radix-4 FFT . butterfly(b) Radix-4 butterfly . . x[0]. X [0]. x[1]. X [ 4]. x[2]. X [8]. x[3]. X [12]. x[4]. X [1]. x[5]. X [5]. x[6]. X [9]. x[7]. X [13]. x[8]. X [ 2]. x[9]. X [6]. x[10]. X [10]. x[11]. X [14]. x[12]. X [3]. x[13]. X [7 ]. x[14]. X [11]. x[15]. X [15]. 2.816 . ç+. radix-4 . 14. .

(54). . . .

(55) . . N radix-4 DIT FFT . . . M ( N ) A( N ) . . . . . . . . . M ( N ) = (3N / 8) log 2 N. (2.32). A( N ) = (3N / 2) log 2 N. (2.33). . . !. ". #. . !. $. #. . . !. %. 2.3.2 Radix-4 IFFT Decimation-in-time . . . N . x[n ] =. N −1 k =0. &. '. (. X [k ]W N− nk , n = 0,1,2,. / radix-4 DIT FFT <. . x[n] * + & '

(56) , ). 0. 1. 2. 3. , N −1 ). 4. 5. . 6. W N = e − j ( 2π . 7. ,. -. x[n]. N). radix-4 . 89. 2.9(a) radix-4 DIT IFFT butterfly .. %3. .. x[n + N / 4] = f 1 [n] + jW. f 2 [n ] − W. x[n + 2 N / 4] = f1 [n] − W. f 2 [n ] + W. −n N. f 3 [n ] − jW. −2 n N. f 3 [n ] − W. −2n N. :. 2.9(b). 3. = f 1 [n ] + W N− n f 2 [n ] + W N−2 n f 3 [n ] + W N−3n f 4 [n] −n N. (2.34). −3 n N −3 n N. f 4 [n]. f 4 [n]. x[n + 3 N / 4] = f 1 [n ] − jW N− n f 2 [n ] − W N− 2 n f 3 [n] + jW N−3n f 4 [n ]. butterfly ; =. %. (2.35) (2.36) (2.37) (2.38). F1[k ] = X [4k ] > F2 [k ] = X [4k + 1] > F3 [k ] = X [4k + 2] > F4 [k ] = X [4k + 3] ?. . . f 1[n] > f 2 [n] > f 3 [n] > f 4 [n] * +

(57) % @. A , FG. H. f 1 [ n ] > f 2 [ n] > f 3 [ n] / f 4 [ n] C N / 4 * + & '

(58) D E F3. radix-4 * . . . . B. . . . +. I J. 8 radix-4 * ,. . .

(59). +. N radix-4 DIT IFFT . M ( N ) A( N ) . . . . .

(60). %3. (2.10)K. % . . . . M ( N ) = (3N / 8) log 2 N. (2.39). A( N ) = (3N / 2) log 2 N. (2.40). . . !. ". #. 15. !. $. #. . . !. %. L.

(61) f1 [n]. WN−0. x[n]. f 2 [n]. WN− n. x[n + N / 4]. f 3 [n]. WN−2 n. f 4 [n]. WN−3n. f1 [n] f 2 [n ]. x[n + N / 4]. −n − 2n − 3n. f 3 [n]. x[n + 2N / 4]. x[n ]. 0. x[n + 2N / 4]. x[n + 3N / 4]. f 4 [n]. x[n + 3N / 4]. (a) 2.9(a) Radix-4 IFFT . (b) . butterfly(b) Radix-4 butterfly . X [0]. x[0]. X [1]. x[4]. X [ 2]. x[8]. X [3]. x[12]. X [ 4]. x[1]. X [5]. x[5]. X [6]. x[9]. X [7 ]. x[13]. X [8]. x[2]. X [9]. x[6]. X [10]. x[10]. X [11]. x[14]. X [12]. x[3]. X [13]. x[7]. X [14]. x[11]. X [15]. x[15]. 2.1016 radix-4 .

(62). 16. . . . . . . . . .

(63) 2.4 Radix-22 FFT . . . N . N −1. X [k ] = '. (. )*. n =0. . x[n] ! " # $% . x[n ]W Nnk , k = 0,1,2,. DIF , +. . -.. N). (2.41). N N n1 + n 2 + n3 1 k = k1 + 2k 2 + 4k 3 23 2 4. 0 n =. /. & W N = e − j ( 2π. , N −1. (. )(2.41)4. X [k1 + 2k 2 + 4k ] =. N −1 1 4. N. 1. n3 = 0 n2 = 0 n1 = 0. =. N −1 1 4. 6. ". twiddle factor W N 7 1. &;. N n2 + n3 k1 4. WN. N n2 + n3 4. (2.42). (2 k 2 + 4 k3 ). N N N N n 2 + n3 = x n 2 + n3 + (−1) k1 x n 2 + n3 + 4 4 4 2. & B Nk1 / 2. 5. N n 2 + n3 W N 4. B Nk1 / 2. n3 = 0 n2 = 0. N. ( n1 + n2 + n3 ) (k1 + 2 k 2 + 4 k) N N n1 + n 2 + n3 W N 2 4 2 4. x. <. =. >. ". ?. N n2 + n3 4. ( k1 + 2 k 2 + 4 k3 ). 7. (2.43). 8)(2.44)9. /. )(2.42). 2:. )(2.45)@. n2 = 0 N. n2 + n3. WN 4. ( k1 + 2 k 2 + 4 k3 ). N. = W NN n2 k3 W N4 = (− j ). X [k1 + 2k 2 + 4k ] =. N −1 4 n3 = 0. n2 ( k1 + 2 k 2 ). W Nn3 ( k1 + 2 k 2 )W N4 n3k3. n2 ( k1 + 2 k 2 ). W. n3 ( k1 + 2 k 2 ) N. W. [H (k , k , n )W 1. 2. 3. (2.44). 4 n3 k3 N. n3 ( k1 + 2 k 2 ) N. ]W. n3 k 3 N/4. (2.45). & BF I. {. }. H (k1 , k 2 , n3 ) = x[n3 ] + (−1) x[n3 + N / 2] + (− j ) k1. {x[n. ( k1 + 2 k 2 ). BF I 3. }. + N / 4] + (−1) k1 x[n3 + 3N / 4]. BF II. (2.46) A. )(2.46)" (. -(2.11)I . . . . . . J . . H (k1 , k 2 , n3 ) " B. 8 BFC BFD. radix-22 K " 7. (2.11)M N. . . . /. (2.12)@ 17. OP. ". ?. . . I. / J. E. F. butterflies G. 8L. L K. radix-22 K. H. . .

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(65) −j 0 16. W. W162. W164 W166. −j. W160 W161 W162 W163. −j. 0 16. −j. W. −j. W163. −j. W166. −j. W169. −j. 2.1216 radix-22 K. . . . . . . . . 2.5 Split-Radix 2/4 FFT 5. 6. ". i. #. l. . f. A. m G. /. t. X ( 2r ) =. radix-2 DIF / 7. n. ?. Ou. o %. p. H. -Q. R. &g @q. T r. K. h. i. j. radix-4 DIF /. &k. split-radix 2/4 DIF ,. )s. X (4 s + 1) = X (4 s + 3) =. h. @. ( x[n] + x[n + 2 N / 4]) W Nnr/ 2. (2.49). N / 4 −1 n=0. K. -. . N / 4 −1 n =0. T. [ x[n] − x[n + 2 N / 4] − j ( x[n + N / 4] − x[n + 3 N / 4])] W NnW N4 ns. (2.50). N / 4 −1 n =0. [ x[n] − x[n + 2 N / 4] + j ( x[n + N / 4] − x[n + 3 N / 4])] W N3nW N4 ns (2.51). & 0 ≤ r ≤ N / 2 − 1 1 0 ≤ s ≤ N / 4 − 1 @(2.13) split-radix 2/4 DIF butterfly %. (2.14) 16 Split-Radix 2/4 K. 19. . . . . . . . @.

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(67) . . . N radix-22 DIF FFT H . -Q. R. . . M ( N ) = [(3log 2 N − 2) × N + 2 × (−1)log2 N ] / 9. (2.52). A( N ) = N log 2 N. (2.53). & M (N ) C A(N ) S. 2$Q. T. U. .. CQ. T. V. .. . T. @. 2.6 X. . v. ~. &5. Q. . H. R. x. . . . [. . @ . . w . . x. . . y . - FFT . ". " G. -Q. radix FFT C IFFT , {. . 6 . H. z . $(2.1)&5. fixed radix p o. . . 6. #. # R. . B. . H. . . . . z. . o. Y. |. y. ,. G @$(2.1). -. high-radix [ . -.. . o. p. @q. U. .. Q. R. , . -. -. '. split-radix , . fixed radix G . @. N2. N2 −N. ( N / 2) log 2 N. N log 2 N. (3N / 8) log 2 N. (3 N / 2) log 2 N. N log 2 N. [(3log 2 N − 2) × N + 2 × (−1)log 2 N ] / 9. N log 2 N. 2.1. . 21. .

(68). u Y. . . -.. split-radix o. (3 N / 8) log 2 N. }. . . . [. `.

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