Applications of Distributed Arithmetic to Digital Signal Processing: A Tutorial Review

Transcription

1 pplicatios of Distriuted rithmetic to Digital Sigal Processig: Tutorial Review Ref: Staley. White, pplicatios of Distriuted rithmetic to Digital Sigal Processig: Tutorial Review, IEEE SSP Magazie, July, 989 VSP Lecture Distriuted rithmetic -

2 Distriuted rithmetic D, 974 The most-ofte ecoutered form of computatio i DSP: Sum of product Ier-product Executed most efficietly y D VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -

3 Derivatio of D Techique Sum of product: where x is a s-complemet iary umer scaled such that x <, ad is fixed coefficiets x : {,,, - }, wordlegth where is the sig it Express each x as: Su ito > y x VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw y x 3-3

4 VSP Lecture Distriuted rithmetic -4 Derivatio of D Techique -cotiued I Critical step where is the umer of iputs or taps ad is the wordlegth of Data y 4

5 Derivatio of D Techique -cotiued II ow cosider the equatio 4 y has oly possile values has oly possile values We ca store it i a looup-talerom: size VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -5

6 Techical Overview of D dvatage of D: Efficiecy of computig mechaizatio frequetly argued: Slowess ecause of its iheret it-serial ature ot true Some modificatios to icrease the speed y employig techiques: Plus more arithmetic operatios expese of expoetially icreased memory VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -6

7 Derivatio of D Techique -cotiued III Example Let umer of iputs 4 The fixed coefficiets are.7, -.3, 3.95, 4. y We eed 3-word ROM 4 VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -7

8 VSP Lecture Distriuted rithmetic -8 Example Ufoldig , 4 3, 3,, 4

9 Example -cotiued I Hardware architecture x x x 3 x 4 y VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -9

10 VSP Lecture Distriuted rithmetic - Example -cotiued II Shorte the tale Eq y 5

11 Example -cotiued II VSP Lecture Distriuted rithmetic Oly 6 words of ROM are required,ow. -

12 Offset-Biary Codig OBC Chage Iput data from iary to siged-digit x [ x x ] {,,... } 6 x x s-complemet x 7 VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -

13 Offset-Biary Codig OBC cot d x c, { where c {,}, x c 代入 y x y c Q Q Where Q c ad Q VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw Costat -3

14 Offset-Biary Codig OBC Hardware architecture cot d 3 4 VSP Lecture Q Distriuted Q rithmetic cwliu@twis.ee.ctu.edu.tw -4

15 Speed up of D multiplicatio Way I: Plus more arithmetic operatios y Q Q Iitial coditio Q Q... Q Q Q Eve part Odd part VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -5

16 Speedup of D multiplicatio Way I: at the expese of liearly icreased memory & arithmetic operatio Odd part sig} Eve part Iitial Coditio /*Q VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw y Q Q -6

17 Speed up of D Multiplicatio Way II: at the expese of expoetially icreased memory ROM : *7 words *8 words VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -7

18 Coclusios D is a very efficiet mechaism for computatios that are domiated y ier products covolutio good way to trade comiatioal logic with memory for high-performace computatio. Whe a may computig methods are compared, D should e cosidered. It is ot always ut ofte est, ad ever poorly: save gate cout aroud 5% to 8%. pplicatio: VLSI implemetatio of a 6*6 discrete cosie trasform, y M.-T. Su, T.-C. Che,. M. Gottlie, IEEE Trasactios o Circuits ad Systems, Volume: 36 Issue: 4, pril 989, Pages: 6 67, ad may other trasforms ad DSP erels. VSP Lecture Distriuted rithmetic cwliu@twis.ee.ctu.edu.tw -8