CSE4210 Architecture and Hardware for DSP

Transcription

1 4210 Archtecture and Hardware for DSP Lecture 1 Introducton & Number systems

2 Admnstratve Stuff 4210 Archtecture and Hardware for DSP Text: VLSI Dgtal Sgnal Processng Systems: Desgn and Implementaton. K. Parh. Wley Interscence Posted artcles

3 Admnstratve Stuff Offce hours: Monday 1-2pm TR 3-4pm Room 2026 B x40607 HW 10% Quzes 10% Mdterm 25% Project 15% Fnal 40%

4 Topcs Number systems Fast arthmetc Algorthm representaton Transformaton (retmng, unfoldng, foldng) Systolc arrays and mappng algorthms nto hardware Low power desgn

5 Introducton Introducton to DSP algorthms Non-termnatng programs n real tme. Speed depends on applcatons (audo, vdeo, 2-D, 3-D, ) Need to desgn famles of archtectures for specfed algorthm complexty and speed constrants

6 Typcal DSP Programs nt 3T 2T T 0 Input DSP System Output 3-Dmensonal optmzaton: Area, Speed, Power) Usually, speed s a requrement, area-power tradeoff P=C V 2 F

7 Examples FIR flter, x(n) s the nput, y(n) output J 1 yn ( ) = h( jxn ) ( j) j= 0 IIR flter P = + yn ( ) ayn ( ) ( ) bkxn ( ) ( k) Q = 1 k= 0

8 Examples Convoluton M 1 N 1 = = yn ( ) xhn ( ) ( ) h( jxn ) ( j) = 0 j= 0 For n=1 to M+N-2, y(n)=0, For =0:M-1, y(n)=y(n)+x()*h(n-) end end MAC operaton

9 More Complex Examples Moton Estmaton Image (frame) s dvded nto macroblocks Each macroblock s compared to a macroblock n the reference frame usng some error measure. The search s conducted over a predetermned search area. A vector denotng the dsplacement of the (moton) s sent.

10 More Complex Examples Moton Estmaton Many measures of errors could be used. The dsplaced block dfference s(m,n) usng MAD (Mean Absolute Dfference) s defned as s( m, n) N = 1 N 1 x(, j) y( + m, j + = 0 j= 0 n) m,n are n s the search area, N s the macroblock sze. The one wth the mnmum error s chosen

11 More Complex Examples Vector Quantzaton Used n compresson A group of samples (vector) are quantzed together For example consder k pxels, wth W bts. That vector s compared to a group of N codewords, choose the one wth the mn. dstorton. We transmt the ndex of that codeword

12 More Complex Examples Vector Quantzaton Compresson rato = KW/log 2 N Eucldean dstance s used as a measure of dstorton. d( x,c = x 2 j ) = x c 2( x c j j 2 + e = j ), k 1 ( x = 0 e j c = j 1 2 ) 2 c j 2 = 1 2 k 1 c = 0 2 j

13 Dscrete Cosne Transform The 1-D DCT s defned as X ( K) = e( k) N 1 n= 0 (2n + 1) kπ x( n)cos 2N, k = 0,1,2,, N 1 e( k) = 1 f k = otherwse

14 More Complex Examples Vterb Decodng FFT Wavelets and Flter banks See the book for detals

15 Requrements Consder block matchng algorthm, the computatonal requrement s as follows 3(2p+1) 2 NMF 3*(2*7+1)*288*352*30=2GOP Much hgher for hgher resoluton and bgger frames How to acheve these requrements?

16 hardware Mcroprocessors Mcroprocessors wth DSP extenson DSP FPGA ASIC

17 Number System Numbers and ther representaton Bnary numbers Negatve numbers Unconventonal numbers

18 Bnary Numbers An ordered sequence The value of the number s x n {0,1} The range [X mn, X max ] s the range of the numbers to be represented, n the prevous case [0,2 n -1]

19 Bnary numbers The prevous representaton s nonredundant and weghted (w ) The n-dgt number can be parttoned nto a fracton part (n-k bts) and an ntegral part(k bts)

20 Bnary numbers Gven the length of the operand, n, the weght r -m of the least sgnfcant dgt ndcates the poston of the radx pont. Unt n Last Poston ulp=r -m Smplfes the dscusson and there s no need to partton the number nto fractonal and ntegral parts.

21 Converson Convert nto bnary 36/2 Dvson by 2 Quotent Remander Multplcaton by 2 Integer Fracton

22 Negatve Numbers Sgned magntude Complement Dmnshed radx complement (1 s complement for bnary) Radx complement (2 s complement n bnary)

23 Sgned magntude The n th bt (dgt) s the sgn n-1 dgts for magntude (k-1 ntegral and m fractonal). Largest value = X max =r k-1 -ulp Smallest negatve value (r k-1 -ulp) Two representaton for zero

24 Sgned magntude Operatons may be more complcated than usng complement. For example, addng 2 numbers, a postve number X, and a ve number Y, the result depends on f X>Y or not If X > Y the result s X+(-Y) If Y>X swtch the 2 numbers, subtract, attach mnus sgn (Y-X)

25 Complement representaton Postve numbers are represented just as sgned-magntude Negatve numbers are represented as R- number, where R s a constant Note that (-y)=r-(r-y))=y The choce of R must satsfy 2 condtons Calculatng the complement s easy Smplfyng or elmnatng correcton

26 Complement representaton For radx complement R=r k For the dmnshed radx complement R=r k -ulp

27 r=2, k=n=4, m=0, ulp=2 0 =1

28 2 s complement Example 6= = = = Carry n = carry out Ignore carry out Carry n carry out Overflow

29 1 s complement Example 6= = = = Carry n = carry out Add to LSB Carry n carry out Overflow

30 Arthmetc Shft Consder the number Fnte extenson of sgned magntude s 0,0{ x, x,.,.,. }0,0, n 2 n 3 x0 2 s complement 1 s complement { xn 2, xn 3,.,.,. x0}... xn 1, xn 1{ xn 2, xn 3,.,.,. x0}0,0, xn 1, xn 1{ xn 2, xn 3,.,.,. x0} xn 1, x n 1

31 Arthmetc Shft 2 s Comp 1 s Comp

32 Unconventonal Number System Negatve radx number system A general class of of fxed-radx number system Sgned-dgt number system Resdue number system

33 Negatve radx Number System The radx could be negatve, r=-β, β s a postve number. Dgt set 0,1,, β-1 Value of X n = 1 x = 0 ( xn 1, xn 2,..., x0 ) ( β ) = = 127 Range = [090,909] -10, or [-9,909] = -8-2= -10 Range=[1010,0101] -2 = [-10,5] 10

34 Negatve radx Number System Algorthms do exst for basc operatons. Not better than 2 s complement systems

35 General Class of Fxed Radx Number System Characterzed by (n,β,λ), β s a postve radx, dgt set 0,1, β-1, and a vector of length n Λ=(λ n-1, λ n-2, λ 0 ) λ = {-1,1} X n 1 = = 0 λ x β 2 s Complement Λ={-1,1,1,.1}

36 General Class of Fxed Radx Number System P p P = { p = n 1 1/ 2( λ + 1)( β 1) β = 0 = 1/ 2 n 1, p [ n ] Q + ( β 1) Where Q s n 2,..., 0 } β 1 f λ = + 1 = 0 otherwse p the value of = Max. postve number 1 2 ( λ + 1)( β 1) n 1 n 1 = 1/ 2 λ ( β 1) β + ( β 1) β = 0 = 0 the tuple ( β 1, β 1,, β 1) Fnd the smallest representable number

37 Sgned Dgt Number System The dgts could be postve or negatve Redundant (more than one representaton for the same number) For a radx β, x {β-1, β-2,,1,0,1 β-1} To reduce redundancy, X r -1 = 2 r { a, a 1,...,1,0,1,... a}, where a 1

38 Sgned Dgt Number System Example: β=10,a=6,

39 Breakng the Carry Chan Add two numbers X,Y 0, f ) ( f ) ( f = + = < = + = + + t t w s a y x a y x a y x t rt y x w Example, a=6,r=

40 Breakng the Carry Chan

41 Breakng the carry chan For no carry s = w +t a, -> w a (1) ) ( w (lower bound) a y Case (1) 2 w (upper bound) 2a y Case1 + + = = = = + = = = + r a r a r a a r a r w r a r a x r a r a r a x

42 Breakng the carry chan For bnary SD, r=2,a=1 t s mpossble for the prevous condton to hold. We do have carry n SD addton

43 Breakng the carry chan X,y 00, X -1,y -1 Neth er s 1 At least one s 1 Neth er s 1 At least one s 1 t w

44 Breakng the carry chan Example

45 Breakng the carry chan Example

46 Breakng the carry chan