TOPICS MULTIPLIERLESS FILTER DESIGN ELEMENTARY SCHOOL ALGORITHM MULTIPLICATION

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1 1 2 MULTIPLIERLESS FILTER DESIGN Realzaton of flters wthout full-fledged multplers Some sldes based on support materal by W. Wolf for hs book Modern VLSI Desgn, 3 rd edton. Partly based on followng papers: Hewltt, R.M. and E.S. Swartzlander, "Canoncal Sgned Dgt Representaton for FIR Dgtal Flters", IEEE Workshop on Sgnal Processng Systems, SPS 2, Lafayette, LA, pp , (2). Voronenko, Y. and M. Pueschel, Multplerless Multple Constant Multplcaton, ACM Transactons on Algorthms, Vol. 3(2), (May 27). Aksoy, L., P. Flores and J. Montero, A Tutoral on Multplerless Desgn of FIR Flters: Algorthms and Archtectures, Crcuts, Systems and Sgnal Processng, Vol.33(6), pp , (214). TOPICS Multpler wrap-up: Array multpler Booth multpler Flter structures: drect, transposed and hybrd forms Canoncal sgned dgt Optmal sngle and multple-constant multplcaton Choosng coeffcents 3 4 MULTIPLICATION ELEMENTARY SCHOOL ALGORITHM Dstngush between: Multplcaton of two varables Multplcaton of one varable by a constant (scalng) opportuntes of optmzaton Constants: Can be consdered as gven Can be specally chosen Implementaton: One-to-one Resource sharng In software, on processor wthout hardware multpler Unsgned numbers! 1 1 multplcand x 1 1 multpler partal product

2 5 ARRAY MULTIPLIER 6 ARRAY MULTIPLIER ORGANIZATION Array multpler s an effcent layout of a combnatonal (parallel-parallel) multpler. Array multplers may be ppelned to decrease clock perod at the expense of latency. multplcand multpler product 1 1 x skew array for rectangular layout y y1 y2 y3 7 UNSIGNED 4X4 ARRAY MULTIPLIER x3y3 x3y2 x2y3 x3y1 x2y2 x1y3 x3 x3y x2 x2y x2y1 x1y2 xy3 x1y1 xy2 x1 x1y x xy xy1 8 ARRAY MULTIPLIER COMPONENTS AND gates FULL ADDERs HALF ADDERs A B Carry n A B Full adder Half adder Sum Carry out Sum Carry out Fast multplcaton amounts to reducng the crtcal path. P7 P6 P5 P4 P3 P2 P1 P

3 9 2 S COMPLEMENT MULTIPLICATION (1) An n-bt number X, and an m-bt number Y: X n 2 n x n x m 2 m 1 m 12 2 Y y y 1 2 S COMPLEMENT MULTIPLICATION (2) Product: P XY x y m n 2 n 1 m 12 n 2 m 2 j xy m 2 n 2 n 1 m 1 yx n 1 xy m j 2 j 11 2 S COMPLEMENT MULTIPLICATION (3) x 2 n 2 n x 2 n k k Note that: and: Therefore: m 2 m 2 m 2 n 1 n 1 n 1 yx n 1 yx n 1 m 2 n m 2 n 1 n yx n S COMPLEMENT MULTIPLICATION (4) The product becomes: n 2 m 2 j P XY x y n m 2 n 1 m 12 xy j j n m 1 n 2 m 2 m 2 n 2 n 1 m 1 yx n 1 xy m

4 13 14 BAUGH-WOOLEY MULTIPLIER Algorthm for two s-complement multplcaton. Careful processng of partal products leads to: Array wth only addtons, no subtractons No hardware for sgn extensons n upper left corner Acheved by: Negaton of some partal products Injecton of ones n some array postons 1 y y1 y2 y3 BAUGH-WOOLEY SIGNED 4X4 ARRAY MULTIPLIER x3y3 x3y2 x2y3 x3y1 x2y2 x1y3 x3 x3y 1 x2y1 x1y2 xy3 x2 x2y x1y1 xy2 x1 x1y x xy xy1 P7 P6 P5 P4 P3 P2 P1 P 15 BOOTH MULTIPLIER Encodng scheme to reduce number of stages n multplcaton. Performs two bts of multplcaton at once; requres half the stages. Each stage s slghtly more complex than an adder. 16 The wanted product: x*y. BOOTH ENCODING Two s-complement form of multpler: y = -2 n y n 2 n-1 y n-1 2 n-2 y n-2... Rewrte usng 2 a = 2 a1-2 a : y = 2 n (y n-1 -y n ) 2 n-1 (y n-2 -y n-1 ) 2 n-2 (y n-3 -y n-2 ) 2 n-3 (y n-4 -y n-3 )... y = 2 n-1 (2(y n-1 -y n ) (y n-2 -y n-1 )) 2 n-3 (2(y n-3 -y n-2 ) (y n-4 -y n-3 ))... Consder frst two terms: by lookng at three bts of y, we can determne whether to add x, 2x,-x, -2x,or to partal product.

5 17 BOOTH ACTIONS y y -1 y -2 ncrement (2(y -1 y ) y -2 y -1 ) 1 1x 1 1x 1 1 2x 1-2x 1 1-1x 1 1-1x BOOTH EXAMPLE x = 111 (25 1 ), y = 1111 (-18 1 ). y 1 y y -1 = 1, P 1 = P - (1 111) = y 3 y 2 y 1 = 111, P 2 = P 1 = y 5 y 4 y 3 = 11, P 3 = P = (-45 1 ) x 4 x x 19 BOOTH STRUCTURE multplcand Left shft 2 multplcand 2 FIR-FILTER DIRECT FORM (1) FIR = fnte mpulse response Dfference equaton: Where s the crtcal path? How long s t as functon of N? Left shft 2 multplcand

6 21 FIR-FILTER DIRECT FORM (2) Use a bnary tree structure for the addtons: Crtcal path as functon of N? 22 CLASSICAL RETIMING It s allowed to push delay elements through a computaton: From nputs to outputs or From outputs to nputs Compute-and-then-delay s the same as delay-and-thencompute. Allowed n cyclc DFGs. 23 Generalzaton of classcal retmng. Cut-set = set of edges that cuts a graph n two when removed. Gven a cut-set of any DFG, the DFG s behavor remans unchanged f the same number of delays are added (removed) on ncomng edges as are removed (added) on outgong edges. CUT-SET RETIMING cut lne 24 FIR-FILTER DIRECT FORM (3) Reverse order of addtons:

7 25 CUT-SET RETIMED FIR-FILTER 26 FIR-FILTER TRANSPOSED FORM Computatonally equvalent to drect form Can be obtaned by systematcally applyng cut-set retmng. Now, all multplcatons share one nput 27 FIR FILTER HYBRID FORM The drect-form-mplementaton has all ts delays n the nput lne. The transposed-form mplementaton has all delays on the output lne. Hybrd-form mplementaton has part of the delays n the nput lne and part on the output lne. See paper by Aksoy et al. for more detals. 28 IIR = nfnte mpulse response Dfference equaton: IIR FILTER

8 29 IIR-FILTER DIRECT FORM 1 3 IIR-FILTER TRANSPOSED FORM 31 SCALING: BOUNDS ON ADDITIONS (1) Consder multplcaton of x by 71 = Addtons-only soluton: 71x = (x << 6) (x << 2) (x << 1) x (realzed by means of 3 shfts and 3 addtons; shfts by a constant costs only wres n hardware) Subtractons-only soluton: 71x = ((x << 7) x) (x << 5) (x << 4) (x << 3) (realzed by means of 4 shfts and 4 subtractons) 32 SCALING: BOUNDS ON ADDITIONS (2) In general, f b s the number of bts, z the number of zeros and o the number of ones (b = z o): The addtons-only soluton requres o 1 addtons. The subtractons-only soluton requres z 1 subtractons. There s always a soluton wth at most b/2 O(1) addtons or subtractons (just take the cheapest of the two solutons). The average cost s also b/2 O(1). Booth encodng has also the same cost. Can t be done better?

9 33 SIGNED POWER-OF-TWO REPRESENTATION Uses three-valued dgts nstead of bnary dgts: A at poston means a contrbuton of to the fnal value (as usual). A at poston means a contrbuton of to the fnal value. 34 CANONICAL SIGNED-DIGIT (CSD) Specal case of sgned-dgt power-of-two, wth mnmal number of non-zero dgts. Canoncal = unque encodng. When used to mnmze addtons n constant multplcaton, reduces number of operatons to b/3 O(1) n average, but stll b/2 O(1) n worst case. Example: Example: 35 TWO S COMPLEMENT TO CSD CONVERSION (1) Two s complement number: 36 2 S COMPLEMENT TO CSD CONVERSION (2) Target: Start from LSB and proceed to MSB usng table on next slde Dummy value (sgn extenson): Carry-n, ntalzed to. Hewltt & Swarzlander, Table 2

10 37 CSD NOT OPTIMAL CSD has mnmal number of non-zeros, but s stll not optmal for the sngle constant multplcaton problem. How come? 38 SINGLE-CONSTANT MULTIPLICATION Number of operatons can be reduced by allowng shftng and addng ntermedate results Example, goal s to multply by Voronenko & Pueschel, Fgure 2 3x add/sub 2x add/sub 39 MULTIPLE-CONSTANT MULTIPLICATION Even more opportuntes for optmzaton occur when multple constants can be optmzed at the same tme (thnk of the transposed form of a FIR flter). Example: Voronenko & Pueschel, Fgure 5 4 COMPUTATIONAL COMPLEXITY The optmzaton of the mplementaton for both the sngleconstant and multple-constant multplcaton problems s NPcomplete. Powerful heurstcs are avalable. Try SPIRAL on-lne applcaton:

11 41 CONSTANT MATRIX-VECTOR MULT. (1) Applcatons n hybrd mplementatons of FIR flters y 1 = 11*x 1 17*x 2 y 2 = 19*x 1 33*x 2 42 CONSTANT MATRIX-VECTOR MULT. (2) Optmzed wth depth constrant of 3: 7 add/sub Unoptmzed: 8 add/sub Aksoy et al., Fgure 3 Optmzed: 5 add/sub Aksoy et al., Fgure 5 43 CHOOSING THE COEFFICIENTS Untl now, the dscusson was about mplementng flters wth gven constant coeffcents as effcently as possble. Classcal approach starts from floatng-pont coeffcents as e.g. computed n Matlab and a blnd fxed-pont converson. It s even more nterestng to take cheap mplementaton as a crteron durng flter desgn. A problem descrpton could e.g. be: Gven a number T, construct a flter wth at most T non-zero bts n ts set of coeffcents whle at the same tme satsfyng the usual crtera such as bandwdth, pass band rpple, etc. See e.g. tools at: