Skript HWSW Codesign

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HW/SW Codesign for DSP

G. Fettweis, E. Matus, E. Fischer

July 30, 2014

Skript

Vo d a f o n e C h a i r M o b i l e C o m mu n i c at i o n s S y s t em s

Technische Universitt Dresden

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Contents

1 Introduction 4

2 Pipelining of Non-Recursive Systems 42.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Cut- Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Critical Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Pure Pip elining . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Example: FIR Fi lter . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Latency and Cl o ck Rate . . . . . . . . . . . . . . . . . . . . . . . 13

3 Pipelining Recursive Systems 143.1 Basic Lo ops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Linear Fedback Lo op . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Logarithmic l o ok- ahead . . . . . . . . . . . . . . . . . . . . . . . 203.4 Z- Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Impact of Commutati ve Law . . . . . . . . . . . . . . . . . . . . 24

4 Time Variant Feedback Loops 244.1 M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Dot Op erator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Generalizations 285.1 Al gebrai c Op erator General izations . . . . . . . . . . . . . . . . . 285.2 M ore Than 2 Op erators . . . . . . . . . . . . . . . . . . . . . . . 315.3 M ul ti - Span Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Block Processing 356.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Pi p eli ne Interleavi ng . . . . . . . . . . . . . . . . . . . . . . . . . 376.3 Bl o ck Pro cessing with Paral lel Feedback . . . . . . . . . . . . . . 416.4 Bl o ck Pro cessing with One Feedback . . . . . . . . . . . . . . . . 44

7 On-Chip Communication 467.1 Bi t- level communicati on . . . . . . . . . . . . . . . . . . . . . . . 467.2 Perf ect Shuffl e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3 De Bruij n Networks & Shuffle- Exchange . . . . . . . . . . . . . . 57

8 Measures 61

8.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.2 M easure of Complexi ty . . . . . . . . . . . . . . . . . . . . . . . . 638.3 Wordlength Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 658.4 M - Step Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.5 ATE M easure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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9 Basic Processor Principles 749.1 Hardware Reuse . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9.1.1 Exam pl e: FIR . . . . . . . . . . . . . . . . . . . . . . . . 759.1.2 Exam pl e: Transp osed Form FIR . . . . . . . . . . . . . . 779 .1 .3 I nt er me di ate Ve ry Im por tant Co nc lu si on . . . . . . . . . . 79

9.2 Our Fi rst Di gi tal Signal Pro cessor . . . . . . . . . . . . . . . . . 799.2.1 Implementing Direct Form FIR . . . . . . . . . . . . . . . 819.2.2 Instructions Needed . . . . . . . . . . . . . . . . . . . . . 81

9.3 Instruction Set Architecture (ISA) . . . . . . . . . . . . . . . . . 849.3.1 VLIW Very Long Instruction Word . . . . . . . . . . . . . 849.3.2 RISC Reduced Instruction Set Computer . . . . . . . . . 869.3.3 CISC Complex Instructi on Set Computer . . . . . . . . . 89

9.4 Pi peli ning of O ur Basi c Pro cessor Bl ock Di agram . . . . . . . . . 909.4.1 Pip eli ni ng . . . . . . . . . . . . . . . . . . . . . . . . . . . 909.4.2 VLIW Pi p el ining . . . . . . . . . . . . . . . . . . . . . . . 919.4.3 CISC Pip elini ng . . . . . . . . . . . . . . . . . . . . . . . 93

10 Hazards 9410.1 Control Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . 9410.2 Data Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9910.3 Structural Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . 102

11 Vector Processing 10311.1 Vector & Data Flow Pro cessors . . . . . . . . . . . . . . . . . . . 10311.2 SIM D: Si ngl e Instructi on M utl iple Data . . . . . . . . . . . . . . 105

11.2.1 Exam pl e FIR & Zurich Zip . . . . . . . . . . . . . . . . . 10611.2.2 Transp osed FIR, partial . . . . . . . . . . . . . . . . . . . 11111.2.3 General ization . . . . . . . . . . . . . . . . . . . . . . . . 113

12 Scheduling for Synchronous Data Flow 11612.1 Kahn Pro cess Networks . . . . . . . . . . . . . . . . . . . . . . . 11712.2 SDF: Synchronous Data Flow . . . . . . . . . . . . . . . . . . . . 11812.3 M ul ti Pro cessor Scheduli ng . . . . . . . . . . . . . . . . . . . . . 123

13 MPSoC: Multi Processor System-on-Chip 12813.1 Programming Mo del . . . . . . . . . . . . . . . . . . . . . . . . . 12913.2 Task Schedul ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13413.3 Network- on- Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . 13813.4 Homogeneous / H eterogeneous M PSoC . . . . . . . . . . . . . . . 13913.5 Hi erarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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1 Introduction

T h e c o u r s e H a r d w ar e / S of t wa r e - C o d e s i gn f o r d i g i t a l s i g n a l p r o c e ss o r s i s m a i n l yb as ed o n t he b o ok : [ 1] ( ge rm an v er si o n) / [ 2] ( en gl i sh v er si on ). O th er r ec om -m e nd e d b o o k s c a n b e f o un d a t t h e r e f e re n ce s a t t h e e n d o f t h is s c ri p t: [ 3 ][ 4 ] [5 ] [ 6] .M o re s p e ci fi c s c ie nt i fic p u bl i c at i on s w i l l b e r e fe r en c ed a t t h e b e g in n in g o f t h eaccording sections.

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2 Pipelining of Non-Recursive Systems

Fu r t h er r e a di n g : [ 7 ]

2.1 Basic Principles

Fo r t he p ur p os e o f p i pe li n in g a s ys te m w e c an m ake u se o f a g ra ph ic al r ep re -s e nt a ti o n ( fl ow g r ap h ). F l ow g r ap h s b a si c al l y c o ns i s t o f t wo t yp e s o f e l em e nt s :o p e ra t or s ( i . e. c o mp u ta t io n al o r l o gi c f u nc t io n s) a n d d e la y e l em e nt s ( i . e. r e gi s -t e r s , fl i p - fl o ps , m e m o ri e s , e t c . ) , a s fi g u r e 1 s h o w s . T h e o p e r a t o r i s t i m e - i n va r i a nti f i ts t ra ns fe r f u nc ti o n i s n ot d ep e nd en t t im e. I n o th er w or ds , i f t he i np ut s t oo p er at or a re t im e s hi ft ed t he n t he o ut pu ts o f t he o p er at or s ta y t he s am e a nda re t im e s hi f te d b y t he s am e a mo un t a s t he i np ut s. Fo r e xa mp le , a n o p er at orw i t h i n p u t s ak , bk a n d o u t p ut yk r e a l i z i n g f u n c ti o n

yk = ak bk (1)

i s t i m e - i nv ar i a nt i f a l s o t i m e - s h i f t ed e q u a ti o n h o l d s i . e .

yk+1 = ak+1 bk+1

O n ce t h e t i me - i nva r ia n ce p r op e r ty o f o p e ra t or i s d e fin e d, r e ti m i ng b a se d o nc u t- s et r u le c a n b e i nt r o du c ed .

F i gu r e 1 : G r ap h ic a l i n te r pr e ta t io n o f b a si c o p e ra t or a n d d e la y.

R e gi s te r s c a n e i th e r b e m ov ed f r om t h e o u t p ut s o f a n o p e ra t or t o t h e i n p u ts . . .

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F i gu r e 2 : R e gi s te r b a la n ci n g: M ov i ng o u tp u t r e g i st e r t o t h e i n p u ts

. . . o r t he o th er w ay a ro un d . . .

F i gu r e 3 : R e gi s te r b a la n ci n g: M ov i ng i n pu t r e gi s te r t o t h e o u t p ut s

T h e c u t - se t m e th o d i s a m o re f o rm a l w ay t o e a si l y r e al i z e r e g i st e rb a l a n ci n g / p ip e l i n i n g f o r m o r e c o m p l ex n o n - re c u r si v e s y s t em s .

2.2 Cut-Sets

Assume:

T i m e c l o c ke d s i g n a l- fl o w g r a p h ( S FG )

T i m e i n va r i a nt n o d e o p e r a to r s

D i s c r e t e t i m e d e l a y s ( u s u a l l y a d e l a y i s g i v e n a s m u l t i p l e o f a c l o c k c y c l e )

Def: Cut o f a g r a p h i s p a r t i t i o n i n g o f t h e g r a p h n o d e s i n t o t w o d i s j o i n s u b s e t s .

Def: Cut-set o f t he c ut i s t he s et o f g ra ph e dg es w ho se e nd p oi nt s b e lo ng s t od i ff e re n t g r a p h p a r t it i o n s.

Def: Cut-set rule:1. C o ns t ru c t c l o s ed c u t . . .

2. M o ve d i sc r et e d e la ys f r om b r an ch e s e nt e ri n g c u t- s et s a n d v i ce v er s a b yi n s e r i n g d e l a y s ( r e g i s t e r s ) o n t h e o n e s i d e a n d c o r r s p o n d i n g n e g a t i v e d e l a y s( s p ee d -u p s) o n t h e o t he r s i de

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3. C o m bi n e c o mp l em e nt a ry d e la y e l em e nt s ( r eg i st e r/ s p ee d -u p ) o n c u t- s ete d ge s r e su l ti n g i n z e ro d e la y

A s su m e f o r e x am p l e s y s te m i n F i g. 4 w i th i n pu t sak ,bk a n d o u t p ut sxk and yk .C u t i s c o n s tr u c t ed a r o u n d t h e n o d e . D e l ay a n d s p e e d u p e l e m en t s a r e i n s e r t ed o ni n pu t a n d o u tp u t e d ge s , r e sp e c ti ve l y. S p e ed u p a n d d e la y e l em e nt o n t h e o u tp u te dg e a re c om bi ne d r es ul ti n g i n z er o d el ay. I t i s e vi de nt , t ha t t hi s t ech ni qu ep r es e rv es t i mi n g o f c u t- s et i n pu t a n d o u tp u t s i gn a ls .

F i gu r e 4 : G r ap h ic a l i nt e rp r et a ti o n o f r e ti m i ng b a se d o n c u t- s et r u le .

B e a wa r e t h a t s p e e d - u p e l e m en t s a r e o n l y t h e or e t i ca l ( a u xi l i a r y ) c o n s t r uc t st h at c a n t b e r e al i ze d i n h a rd wa r e. T hu s , a l l s p e ed - up e l em e nt s m us t b er em ov ed ( at t he c os t o f a n i nc re as ed l at en cy ) t o b e a bl e t o i mp l em en t i t a s ar e al c i rc u it . I n t h e e x am p le o f fi g ur e 4 w e c o u ld r e mo ve t h e s p e e d- u p f r o m t h eoutputyk a n d c o n s eq u e nt l y p r ov i d i n g t h e d e l ay e d v e rs i o nyk1a t t h is o u tp u tport.

F i gu r e 5 : G e ne r al i za t io n o f c u t- s et r u le .

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M o re g e ne r al e x am p l e i s i n F i g. 5 . H e re t h e c u t -s e t i s c o ns t ru c te d a r o u nd t h en o d e a s fi r st . Tw o r e al i za t io n o f r e ti m i ng s ch e me s a r e d e pi c te d t h er e .

2.3 Critical Path

Def: Critical Path ( C P ) i s t h e l o n g es t d e la y/ l at e nc y p a th a m on g a l l p a th s i nt h e S F G d e t e r m i n i ng m a x i mu m s y s t em c l o c k f r e q u e n c y.

C P p o s e s a s t h e l i m i t a t i o n o n t h e e x e c u t i o n t i m e o f S F G . I n t h e c a s e o f d i s c r e t e -t i m e ( c l o c k e d ) s y s t e m s , t h e C P d e t e r m i n e s m a x i m u m c l o c k f r e q u e n c y o f s y s t e ma s t h e d e l ay o f C P i s i nv er s e p r o p o rt i on a l t o t h e c l o c k f r e q ue n cy. T hu s , i n m a nyc as es , i t m ay b e o f i nt er es t t o s p ee du p t he s ys te m. I n g en er al , t he s p ee du p o f t h e s y s t e m c a n b e a c h i e v e d b y s p e n d i n g a d d i t i o n a l ( c o m p u t a t i o n a l ) r e s o u r c e s o rb y i n c r e a s i n g c l o c k f r e q u en c y. T h e fi r s t a p p r o a ch r e p re s e nt s s p a t i al p a r a ll e l i s mw h i l e s e c o n d o n e i s a s s o c i at e d w i t h t e m p o r a l p a r a l l el i s m ( p i p e l i n i n g) . N o t e t h a t

i n t he r es t o f t hi s w or k t he t er m p ar al l el p ro c es si ng i s a ss o ci at ed w it h s pa ti alp a ra l le l is m w h il e t h e t e rm p i p el i n in g i s a s so c i at e d w i th t e mp o r al p a ra l le l i sm .I n t h e f o l lo wi n g s e ct i on , p i p el i n in g t e ch n iq u es a r e a d dr e ss e d a s fi r st .

P i p el i n in g r e du c es C P b y i n se r ti n g a d di t i on a l d e la y e l em e nt s ( r eg i st e rs )a lo ng t he C P. H ow ev er , t he t ra ns f er f un ct io n o f t he s ys te m a ft er p ip e li ni ngs h al l b e p r es e rv ed . B a si c al l y, t wo m e th o d s c a n b e u s ed f o r p i p el i ni n g:

R e ti m i ng - i s t e ch n iq u e f o r s t ru c tu r al r e lo c a ti o n o f d e la y e l em e nt s w i th i nS FG s u ch t h at S FG f u nc t io n i s p r es e rv ed .

L o o p t r an s fo r ma t io n - g r ap h s t ru c tu r e i s c ha n ge d w i th o ut a ff ec t in g t h et r an s fe r f u nc t io n r e su l ti n g i n i n cr e as e d n um b er o f d e la y e l em e nt s a n d o p -e ra to rs . N ew ly c re at ed d el ay e le me nt s c an b e m ove d i nt o t he C P b e r e-

timing.I n t he e xa mp le i n F ig . 6 , a ll n od es h ave s am e d el ay T. T hu s, C P p as ses 4n o de s r es ul ti ng i n t ot al d el ay o f 4 T. I n o r d er t o r e d uc e t h e C P, p ip e li ni ngu si ng c ut -s et r ul e i s a pp li ed a lo ng C P t o s ho rt en t he C P b y 1 /2 . Tw o c ut -s et sh ave b e en u se d f or t hi s p ur p os e. F i gu re 7 s ho ws t ha t t he se t wo c ut -s et s c ane ve n b e m e rg e d i n t o a s i ng l e c u t - se t ( d as h ed g r ee n c u t- s et ) . B y i n se r ti n g t w oa d di t io n al c u t- s et s w e c a n p i p el i n e t h e c i rc u it e ve n f u rt h er u nt i l f u ll p i p el i ni n gis achieved. Full pipelining i s a s s o c i a t e d w i t h s u c h g r a p h s t r u c t u r e t h a t a n y t w on o de s i n t he g ra ph a re i so la te d by d el ay e le me nt . I n t he F ig . 7 , t hr ee c ut -s et sa r e c o n s tr u c t ed .

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F i gu r e 6 : C r it i ca l p a t h i n S F G .

F i gu re 7 : G ra ph r et im i ng u si ng c ut -s et p ip e li ni ng . T hr ee c ut -s et s h as b e enc o ns t ru c te d i n o r de r t o f u ll y p i p el i ne t h e g r ap h .

F i gu r e 8 : R e gi s te r a r ra n ge m en t i n S FG a f te r c u t- s et p i p el i ni n g o f S D F i n F i g. 6 .

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I nt e re s ti n g o b se r va t io n : C u t- s et r u le c a n b e s e en a s a p pl i c at i on o f d i st r ib u ti vel aw . E . g. t a ke t h e e x a mp l e i n fi g ur e 9 w h er e 2 d i ffe r en t o p e r at o rs

( m ul t i p l i ca t i o n w i t h c o n s ta n t a a n d a d d i t i o n ) a r e c o n ne c t e d a c c o r di n g t ot h e f o l l ow i n g f o r mu l a : xa = a ya+ a ya . T hi s c an b e t ra ns fo rm ed b y t h ea p p l i ca t i o n o f t h e d i s t ri b u t i ve l a w i n t o : xa = a (ya+ ya), i . e. m ov in g t h em ul t ip l i ca t io n f r om t h e i n p u ts t o t h e o u t p ut . .. C o mp a re i t w i th t h e c u t - se trule.

F i gu r e 9 : R e la t io n b e tw ee n c u t- s et r u le a n d d i st r ib u ti ve l aw

A m or e c om pl ex e xa mp le i s g ive n i n fi gu re 1 0. T he l en gt h o f t he c ri ti ca l p at hi n t hi s e xa mp le i s 5 o p er at io ns . T hu s, 4 p ip e li ni ng s ta ge s s ho ul d b e s uffi ci en tf o r f u l l y p i p el i n in g t h e c i r c ui t . T h e c r i t ic a l p a t h i s c u t o n e s b e tw ee n e ve r y t w oo p e ra t io n s f o r i n se r ti n g r e g i st e rs o n t h e i n p u t e d ge s o f t h e c u t - se t . T h e o u t p ute d g e s o f t h e c u t - s e t w h e r e t h e s p e e d - u p s a r e i n s e r t e d a r e a l s o t h e o u t p u t e d g e s o f t h e c i r c ui t . T h i s a l l ow s a n e a s y r e mo va l o f t h e s p e e d- u ps l a te r o n . T h e l a t e nc yo f t h e c i r c ui t a f te r f u l l y p i p el i n ed i s 4 c l o ck c y cl e s: yk4.

F i gu r e 1 0 : P i p el i ni n g a l on g t h e c r it i ca l p a th

2.4 Pure Pipelining

A s d i sc u ss e d i n p r ev i ou s s e ct i on , c u t- s et c o ns t ru c ti o n b a se d o n " c lo s ed " c u tc u rv es g u ar a nt i es i nva r ia n ce o f s y st e m t r an s fe r f u nc t io n . H ow ev er , d u e t o t h ec u t- s et r u le p r op e r ty, c o mp l em e nt t i me - sh i f ts a r e p l ac e d o n i n pu t a n d o u tp u te d g e s. T h e r ef o r e , n o t p h ys i c a l ly r e a l i za b l e s p e e d u p e l e m e nt s a r e p l a c ed o n i n p u t

o r o ut pu t e dg es o f t he c ut -s et . E ve n i f t he s ys te m t ra ns fe r f un ct io n i s m at h-e ma ti ca ll y c or re ct i n t hi s c as e, i t i s n ot r ea li za bl e d ue t o p re se nc e o f s p ee du pe le me nt . Fo r t ha t r ea so n, i n s om e c as es , p ur e p ip e li ni ng i s u se d. I n t hi s c as e,c u t - s et s a r e c o n s t r u c te d s u ch t h a t t h e y c o nt a i n o n l y p h ys i c a l l y r e a l i z a bl e r e g i s -t e r e l e m en ts ( p o si t i ve d e la y) . A n e x am p le o f p u re p i p el i ni n g f o r s y s t em i n F i g.

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6 i s i n F i g. 1 1 . O n l y r e gi s te r s a r e p l a c ed a t c r o s si n gs o f c u ts a n d e d g es r e su l ti n gi n t he s am e s t r uc tu re a s t h a t i n F ig . 7 .

F i gu r e 1 1 : T h e p r i n ci p le o f p u re p i p el i n in g .

T h e i d e a b e h in d i s t h at t h e c u t s c r o s si n g i n p u t e d g e s o r o u tp u t e d g e s o n l y. I f t hi s i s t he c as e, w e c a n i ns er t a r eg is te r o n e ve ry c ut te d e dg e a nd c lo se t he c utv i a a l l i n pu t s o r o u tp u ts o f t h e c i rc u it , r e sp e c ti ve l y, w h er e t h e s p e ed - up s a r ei n se r te d . T h e s p e e d- u ps a t t h e c i r cu i t i n p u ts a n d o u t p ut s c a n s i mp l y b ed r op p e d a f te r p i p el i n in g a n d o n ly t h e i nt e rn a ll y i n se r te d r e gi s te r s r e ma i n.F i gu r e 1 2 g i ve s a m o re c o mp l ex e x am p le f o r p u re p i p el i ni n g w i th a c r it i ca lp at h o f l en gt h 5 .

F i gu r e 1 2 : M o re c o mp l ex e x am p le f o r p u re p i p el i ni n g

2.5 Example: FIR Filter

A n F I R fi l te r i s a t yp i ca l e l em e nt o f t e le c om mu n ic a ti o ns a n d s i gn a l p r o ce s si n g.I t i s r e al i ze d b y a c o nv ol u ti o na l s u m a s g i ve n b y e q u a ti o n 2 .

yk =L1i=0

aixki (2)

A c o rr e sp o n di n g h a rd wa r e i m pl e me nt a ti o n s h ow s fi g ur e 1 3 . T h e c r it i ca l p a thc o ns i s ts o f 1 c o ns t an t m u l ti p l ie r a n d 3 a d de r s. B y p u re p i p el i n in g 8 a d di t i on a lr eg is te rs a re i ns er te d ( in a dd i ti on t o t he a lr ea dy e xi s ti ng 3 r eg is te rs ). A ft er

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p ip e li ni n g t he C P o nl y c on ta in s a s i ng le a dd er o r c on st an t m ul ti pl i er b ut t hel a te n cy o f t h e c i r c ui t w as i n cr e as e d t o 3 c l o ck c y cl e s.

F i gu r e 1 3 : E x am p l e f o r a p pl i c at i on o f p u re p i p el i ni n g o n F I R fi l t e r

T hi s i s s ti ll n ot t he e nd o f t he s to ry. T he q ue st io n i s: C an w e d o t he p ip e li ni ngi n a m or e e ffic ie nt w ay, i . e. s p en di ng l es s ( th an t he 1 1 u se d) r eg is te rs ? T hi s i sp o s si b le b y e x pl o it i ng t h e a s so c i at i vi ty p r op e r ty o f t h e ( a d d i t i o n) o p e r a to r .T h is i s s h ow n i n fi g ur e 1 4 . B y s l i gh tl y r e ar r an g in g t h e F I R s t ru c tu r e a c c o rd i ngt o t h e a s s o c i a t i v i t y r u l e f r o m (((a0xk+ a1xk1) + a2xk2) + a3xk3) to(a0xk+(a1xk1+ (a2xk2 + a3xk3))), w e a r e n ow a bl e t o d efi ne c ut -s et s t ha t m ak eu s e ( i .e . r e us e ) t h e a l r e ad y e x is t in g F I R s h if t r e gi s te r s f o r t h e p i p el i n in g .

F i g u re 1 4 : E x p l oi t i n g a s s o c i a t i vi t y o f o p e ra t or i n F I R fi l te r p i p el i ni n g.

Fo r t h e p i p e l i ni n g o f t h e r e a r ra n ge d F I R s t r u ct u re w e t h u s o n ly n e ed 4 ( L =4 )a d d i ti o n a l r e g i st e r s ( i n s te a d o f 2 L b e f or e) f or a t ot al o f 7 r eg is te rs . A f te rr e dr aw i ng t h e g r a p h a l i tt l e b i t w e g e t t h e w e l l -k n ow n t r a n s p o se d f o r m o f t h eF IR fi lt er , s ho wn i n fi gu re 1 5, w hi ch h as a l at en cy o f o nl y 1 c l oc k c y c le .

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F i gu r e 1 5 : P i p el i n in g t r an s p os e d F I R fi l te r .

I f w e d on t s hi f t t he r eg is te rs f ro m t he u pp e r b ra nc h t o t he l ow er b ra nch ( inb e t w e e n t h e a d d e r s ) t h e c i r c u i t i s n o t f u l l y p i p e l i n e d , s i n c e w e h a v e s t i l l 3 a d d e r si n t he c ri ti ca l p at h. T hi s f or m o f t he F IR i s c al le d d i r ec t f o r m a nd s ho wn i nfigure.

F i gu r e 1 6 : D i re c t f o r m F I R fi l t e r.

W ha t d i d w e d o :

w e h a d a s h if t r e g i s te r

a n d n o n - pi p e l i n e d l o g i c

w e c r e a te d p i p el i n ed l o gi c b y u s i n g t h e d e l ay r e gi s te r s o f t h e s h i f t r e gi s te r

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F i na ll y, t ak e a l o ok a t fi gu re 1 7 w he re w e c om pa re t he c ri ti ca l p at h o f b o thF I R fi l t e r v e r si o ns . We c l ea r ly fi n d t h a t t h e t r an s p os e d f o r m F I R fi l t e r i s b e t te r

s u it e d f o r t h e h a r d wa r e d e s i gn . H ow ev er , a s w e w i l l s e e l a t e r o n t h is i s a c tu a ll ya b a d i d e a t o b e u s ed f o r s o ft wa r e r e a l i za t io n .

F i gu r e 1 7 : C o mp a ri s on o f c . p. b e tw ee n d i re c t a n d t r an s p os e d f o r m F I R fi l t e r

2.6 Latency and Clock Rate

C l o c k R a t e 1/T i s d e te r mi n ed b y t h e C . P.

1

T > 1 ) t i m e s .

11.2 SIMD: Single Instruction Mutliple Data

N ow , l e t u s m a ke a b o t to m -u p a n al y si s o f t h e d i ff e r en t h i er a rc hy l e ve l s o f p a r-a l l el i s m, s t ar t in g w i th t h e f u l l s e ri a l ( b it - se r ia l ) p r o ce s si n g a n d fi n al l y e n di n gw i th t h e m a xi mu m l e ve l o f p a ra l l el i sm ( S IM D ).

B i t s : b i t - s er i a l fi g u r e 1 1 1 s h o w s a n e x a m p l e o f a b i t - s e r i a l a d d e r t h a t r e c e i v e st w o i n p u t b i t s ai and bi a n d c o m p u t e s t h e s u m b i t si a n d t h e c a r r y o u t p u t couta c co r di n gl y. T h er e by, t h e c a rr y b i t cout i s f ed b ack t o t he i np ut cin s o t ha tm ul t ip l e b i t s o f a p a ra l le l w or d c a n b e c o mp u te d i n a s e qu e nt i al l o o p.

F i gu r e 1 1 1: E x am p le : b i t- s er i al a d de r

Wo r ds : b i t- p ar a ll e l - o p e ra t e t h e s a me o p e ra t or o n a w or d -p a ra l le l A L U ( e .g .m ul t ip l e 1 - bi t a d de r c o nn e ct e d i n a r ow , i . e. c a rr y -r i pp l e a d de r )

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Wo r ds : o n e o p e r at i on o n m ul t ip l e i n p u t w o r ds ( e .g . a d d, m ul t , O R , . . . )

I L P: ( I ns t ru c ti o n l e ve l p a ra l le l i sm ) m a ny o p s o n m ul t ip l e i n pu t s ( e .g . M A C,s a t ( ad d ) , . .. ) , s u p e r s ca l a r i s s u e

I L P: o n e c o m p l ex I L P s e t o f o p e ra t io n s o n s c al a r e l e m e nt sS IM D: o ne c om pl ex I LP s et o f o ps o n ve ct or s

11.2.1 Example FIR & Zurich Zip

To g iv e a n e xa mp le f or S IM D v ec to r p ro c es si ng w e t ak e t he F IR e qu at io n a sbasis:

yk =L1

i=0ai xki

We c an d ec om p os e t he s um t o o d d a nd e ve n p ar ts , w hi ch c an b e c om pu te d i np a r a l le l , i . e . , i n d e p e n d e nt o f e a ch o t h e r.

yk =

L/21i=0

a2i xk2i even

+

L/21i=0

a2i+1 xk2i1 odd

=oddi

ai xki+

even i

ai xki

E xc ep t f or t he fi na l a dd it io n o f t he two s um s, we u se t he s am e i ns tr uc ti on( m ul ti pl y a a nd x a nd a cc um ul at e t he r es ul t ) o n t wo p ar al l el s et s o f d at a

SIMD: S i ng l e I n st r uc t io n M u lt i pl e D a ta ! T h e r e su l ti n g h a rd wa r e s t ru c tu r e i sd e pi c te d i n fi g ur e 1 1 2 . T h e fi n a l a d d i ti o n o f t h e t wo s u ms i s t h er e in r e al i ze d b ya m ul t ip l ex e r a t t h e i n p u t o f t h e s e c o nd a c cu mu l at o r t h a t i s f e d b y t h e o u t p uto f t h e fi r s t a c c um u l at o r .

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F i gu r e 1 1 2 : F I R S I M D E x am p l e

P l ea se n ot e t ha t t he S IM D c on ce pt w it h tw o p ar al le l A LU s d iff er s f ro m t hec l as s ic a l V P m a ch i ne i m p le m en ta t io n , w h er e w e h a v e o n l y a s i ng l e A L U t h at i sN - ( in t hi s c as e 2 -) f ol d p ip e li ne d a s fi gu re 1 13 s ho ws .

F i g u r e 1 1 3 : F I R I m p l e m e nt a t i on a s Ve c t o r P r o c e s s i ng M a ch i n e

T he p ro bl em o f t he S IM D s ol u ti on f ro m fi gu re 1 12 i s t ha t t he n ec es sa ry m em -o ry b an dw id th h as a ls o b e en d ou bl ed c om pa re d t o t he n on - pa ra ll el s ol ut io nt o f ee d a ll t he i np ut s. We n ee d t o p as s 4 i np ut s p er c yc le : aieven ,xkieven ,aiodd ,xkiodd , i n s t e a d o f 2 i n p u t s p e r c y c l e f o r t h e n o n - p a r a l l e l s o l u t i o n : ai ,xki .

T h is e m ph a si z es fi g ur e 1 1 4 b y d i re c t c o mp a ri s on o f t h e b a nd w i dt h r e qu i re d f o ri n pu t s/ o ut p ut s o f t h e S I MD p r o ce s so r a n d p u re h a rd wa r e s o lu t io n .

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F i gu r e 1 1 4 : C o mp a ri s on o f S I MD p r o ce s so r & H W s o lu t io n w . r . t . b a nd w id t h

o f i n p u ts / o u tp u t s

A m o re e ffi ci e nt a p pr o ac h i s t h e f o l l ow i ng : l e t u s c o mp u te t h e s u m s e qu e nt i al l ya s b e fo re , b ut i ns te ad c om pu te m ul ti pl e o ut pu ts (yk and yk+1 f or N =2 ) i nparallel:

yk =L1i=0

aixki

yk1 =L1i=0

aixki1

T h e r e su l ti n g h a rd wa r e s t ru c tu r e i s d e pi c te d i n fi g ur e 1 1 5. We c a n c o mp u te ykand yk1 i n c o mp l et e ly s e pa r at e d ( i . e . p a ra l l el ) D P Us . T h ey p e r fo r m t h e s a meo p e r a t i o n o n t w o d i ff e r e n t d a t a s e t s ( S I M D ) . N o v e c t o r m e r g i n g b e t w e e n t h e2 s ta ge s i s n ec es sa ry a ft er t he l as t i te ra ti on a s i n t he a pp ro ach b e fo re . T he aii np ut s n ee d o nl y t o b e r ea d o nc e f or b o th D PU s. M or eov er , t he xki a l s o c a nb e r eu se d f or b o th D PU s b y i nt ro d uc in g a s im pl e r eg is te r a s d ep ic te d i n t he

fi g ur e . F i na l l y, t h e m e m o ry b a nd w id t h f o r r e ad i ng t h e i n pu t s d i d n o t i n cr e as e( 2 r e ad o p s / c y cl e ) u s i n g t h is s e co n d a p p r oa ch .

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F i gu r e 1 1 5: F I R I m pl e me nt a ti o n a s Z u ri ch Z i p

S o, w ha t d id we g ai n?

N - f o l d p a r a ll e l i z a ti o n

N s e t s o f d a t a

1 s i n g l e i n s t r u ct i o n

N s e t s o f D P s ( w i t h I L P )

butWe a re n ow a bl e t o g en er at e N s et s o f d at a, o ut o f s ca l ar s et o f i np ut d at a!i . e. w e g a i n ed N - fo l d p r o ce s si n g p o we r w / o N - f o l d m e mo r y b a n d wi d th ( e x-p l oi t in g t h e s o c a ll e d d a ta l o c al i ty p r in c ip l e)

A c t u a l l y , t h i s i d e a h a s o r i g i n a l l y b e e n i n v e n t e d f o r s c a l a r p r o c e s s i n g ( w i t h s i n g l e -p o rt m em or y) b y I BM R es ea rch /Z ur ich a nd w as t he re fo re c al l ed Z ur ic h Z ip( Z ip p in g b e tw ee n 2 A CC U s) . T h e o r ig i na l c o nc e pt f o r t h e ( s ca l ar ) F I R c o mp u -t at io n w i th a 1 -p o rt m em or y s ho ws fi gu re 1 16 . T he m ai n i de a o f t hi s c on ce pti s t h a t o n l y o n e o f t h e t w o i n p u t r e g i s t e r s ai andxki+1 h a d t o b e u p d a t e d p e rc l o ck c y cl e , i . e . b o t h r e g i st e rs o n ly n e ed a n u p d a t e e v e ry s e co n d c l o c k c y c le . Fo rt h i s p u r p o s e , t wo o u t p ut syk and yk+1 a r e c o mp u te d i n a n a l te r na t in g m a nn e ri n t wo p a ra l le l a c cu mu l at o rs ( s im i l ar t o t h e S I MD e x am p le f r om fi g ur e 1 1 5) :

yk + =ai xkiyk+1 + =ai xki+1yk + =ai+1 xki+1yk+1 + =ai+1 xki+2...

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H e nc e , w e r e co g ni z e t h at o n ly a s i ng l e i n pu t c ha n ge s b e tw ee n e ve r y t wo c o m-putations!

F i gu r e 1 1 6 : G e ne r al i d ea o f Z u ri ch Z i p

F i na l l y, w e g a in e d N - f ol d p r o ce s si n g p o we r b y a p pl y i ng S I MD w i th Z u ri ch Z i p,b ut w e n ee d a s ke wi ng t ri an gl e c on si de ra ti on h er e! A s fi gu re 1 17 s ho ws , w eg e t N - 1 s e tu p c y cl e s a n d N - 1 c o mp l et i on c y cl e s f o r a n N - f ol d p a ra l le l i za t io n t ofi l l /fl u sh t h e r e gi s te r c ha i n. D u ri n g t h e i n it i al i za t io n a n d fl u sh i ng p h as e , w e c a nn o t r e a l ly d o f u l l p a r a ll e l S I MD p r o ce s si n g. H e nc e , t h e p a r a ll e li z at i on i s o n lye ffe ct iv e, i f t he p ar al le li za ti on f ac to r i s m uch l ow er t ha n t he n um be r o f l o opc y cl e s ( L ), s o t h at t h e o ve r he a d b y t h e i n i t/ fl us h p h as e c a rr i es n o w ei g ht : N

L. T h e e ff ec t iv e p a ra l l el i z at i on f a ct o r, t a ki n g t h e o ve r he a d o f t h e i n i ti a li z at i ona n d fl u sh i ng p h as e i nt o a c co u nt , c a n b e d e te r mi n ed a s f o l lo ws :

P = L

L + 2N 2 N

F i g u r e 1 1 7 : S k ew i n g t r i a n g l e c o n s i d e ra t i o n f o r S I M D p r o c e s s i n g

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Fo l l ow i ng t a bl e s u mm a ri z es b o t h a p pr o ac he s , e m ph a si z i ng t h ei r a d va nt a ge s( g re e n) . T h er e in , L co rr es p on ds t o t he n um be r o f l o op i te ra ti on s ( i. e. # o f

F I R t a p s ) a n d N i s t h e v e c to r p r o ce s si n g s i z e ( i .e . # o f p a ra l l el D P Us ) .

v ec t or p ar a ll e l pr o c . z i pp e d p ro c e ss i ng

latency L/N+ N 1 L + N 1s p e ed u p P P=N LL+N1 P =N

LL+N1

reads/cycle 2L LL+N1 2 L

L+N1

writes/cycle PLPL

C on se qu en tl y, i f l at en cy i s n ot c ri ti ca l i t i s u su al ly a g o od i de a t o g o f or t hez i p p e d p r o c e ss i n g a p p r oa c h.

11.2.2 Transposed FIR, partial

T h e m e t h o d s p r e s e n t e d i n t h e p r e v i o u s s e c t i o n a r e n o t t h e o n l y o p t i o n s t o r e a l i z eS I MD p r o ce s si n g. A d i ffe r en t a p p r oa ch f o r t h e S I MD F I R i m p l e me nt a ti o n i sp rov i de d b y t he s o c al l ed p ar ti a l t ra ns p os ed f or m ( i. e. a m i xt ur e o f t he d i re ctf o rm a n d t h e t r an s p os e d f o r m F I R fi l t e r) . T h e s t r u ct u re i s i l l us t ra t ed i n fi g ur e1 18 . I t i s e as y t o s ee t ha t t he c om pu ta ti on o f t hi s 6 -t ap fi lt er c an b e r ea li ze db y 3 p ar al le l d at a p at h u ni ts w it h i nt er na l I LP. A l l D PU s e xh ib it t he s am es t ru c tu r e, a s d e pi c te d i n fi g ur e 1 1 9, s o t h at h a rd wa r e r e us e i s e n ab l ed .T hi s s ma ll e xa mp le s ho ul d d em on st ra te t ha t t he re i s o ft en a l ar ge n um be r o f a l te r na t iv es o f h ow t o h a nd l e d a ta .

F i na l l y, w e c a n a l so g e ne r al i z e t h e c o nc e pt o f t h e p a r t ia l t r an s p os e d f o rm F I Rf o r d i ff e re n t a l g o r i t h ms , s i n c e :

o p e r a t o r d o e s n t m a t t er o p e ra t or e ve n d o e sn t h av e t o b e d i st r ib u ti ve

yk = 2L1

i=0 ai 1xki

Fo r e x am p le t h e p a rt i al t r an s p os e d f o rm i s a l so p o s si b le f o r:

yk =

|ai xki|

|ai xki|2

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F i gu r e 1 1 8: F I R fi l te r i n P ar t ia l Tr a ns p o se d Fo r m

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F i gu r e 1 1 9: R e us a bl e D P U f o r P ar a ll e l P r o ce s si n g o f P ar t ia l Tr a ns p o se d F I R

11.2.3 Generalization

F i na l l y, w e c o nc l ud e t h e S I M D t o pi c w i th a g e ne r al c o ns i de r at i on . F i gu r e 1 2 0s ho ws a g en er al v ie w o nt o S IM D p ro c es si ng . T he re in , t wo ( or m or e g en er al :m u l t i p l e ) N - e l e m e n t v e c t o r s a r e p a s s e d t o N p a r a l l e l d a t a p r o c e s s i n g p a t h s ( D P ) .

F i gu r e 1 2 0: G e ne r al V i ew O nt o S I MD P r o ce s si n g

T h e r eb y, t wo p r o b le m s c o n c er n i n g t h e i n p u t v e c t o r h a v e t o b e c o n s i de r e d :

1 . I np ut ve ct or i s n ot a li g ne d i n m em or y

2 . I np ut ve ct or d o es n ot m at ch N ( # o f p ar al le l D PU s)

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F i g u r e 1 2 4 : C o m bi n e d a p p r o a ch f o r Z - Z i p a n d v e c t o r -p a r a ll e l p r o c e s si n g

12 Scheduling for Synchronous Data Flow

Fu r t h er r e a di n g : [ 1 5 ] [ 1 6]

U p t o n ow , w e d i sc u ss e d e ffi ci e nt s o lu t io n s f o r d i ffe r en t s e le c te d a l go r th i m s a n da r ch i te c tu r es i n h a rd wa r e o r s o ft wa r e. A p r ob l em t h at c o nn e ct s b o t h w or l ds i s

t h e ( t a s k) s ch e du l in g . T h is m e an s t h at w e h a ve a p i ec e o f s o ft wa r e t h a t c a n b ed e vi d ed i nt o ( s ub - ) t a s k s o n t h e o n e h a nd ( fi gu r e 1 2 5 , l e f t ) . O n t h e o t h er h a nd ,w e h a v e a S i l ic o n- C hi p t h at c o ns i s ts o f a b u nc h o f s o c a ll e d p r o ce s si n g e l em e nt s( P E s ): e . g . D S P s , g e n e r a l p u r p o s e p r o c e s s o r s, d e d i ca t e d h a r d w ar e a c c el e r a to r s ,e tc . N ow , w e h ave t o a ll o ca te t he s of tw ar e t as ks o nt o t he a va i la bl e P Es , i . e. ,d es ig n a t em p or al ( wh en ?) a nd s pa ci al ( w h i ch P E ? ) s c he d u l e , t h e r eb yc o ns i de r in g d a ta d e p en d en c ie s b e tw ee n t h e s o ft wa r e t a sk s ( e .g . t as k 2 n e ed so u tp u t o f t a sk 1 a s i n pu t ) .

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F i gu r e 1 2 5: G e ne r al v i ew o nt o s ch e du l in g p r ob l em

T h e r i g h t s i d e o f fi g u r e 1 2 5 s h o w s t h e 3 m a i n c h a l l e n g e s o f t h e s c h e d u l i n g p r o b -l e m, d e pi c te d a s 3 D m a pp i ng s p ac e :

1 . M P c h a ll e ng e : m a pp i ng t a s k s o n P E s

2 . M e mo r y a l l o ca t io n & c o he r en c e: W h ic h d a t a t o s t or e w h e r e t o a vo i d l o n gd at a t ra ns f er s b e tw ee n g lo ba l a nd l o ca l ( e. g. P E c ach e) m em or y? H owt o ke ep t he d at a c oh er en t, i . e. , h ow m us t t he r es ul ts b e m er ge d, i f t as ksw o rk o n d i ff e r en t m e m o ri e s ?

3 . O n -c hi p i nt e rc o nn e ct ( N et wo r k- o n- C h ip , N o C) : H ow t o s ch e du l e t a sk s t om a ke d a ta / pr o gr a m t r an s fe r s a s f a st a s p o s si b le ( s ho r t r o ut e s, u n if o rml o a d , e t c . )?

12.1 Kahn Process Networks

To h an dl e t he p ro bl em o f s ch ed ul i ng f or ma l t o ol s c an b e u se d t o m o de l t hep r ob l em a n d fi n d a s o lu t io n . K a hn P r o ce s s N e t wo r ks ( K PN ) a r e a v er y s i mp l eg r ap h b a se d m o d el t h at c a n b e u s ed t o m o d el s o ft wa r e t a s k s ( o r p r o ce s se s ) a n dt h ei r i nt e rd e p en d en c ie s . T h er i n, t a sk s /p r o ce s se s a r e m o d el e d b y g r ap h n o d es ,c a ll e d a c t or s , a s d e p ec i te d i n fi g ur e 1 2 6 . T h e a r c s b e t we e n t h e n o d es a r e u s e dt o m o d el t h e i nt e rd e p en d en c ie s b e tw ee n t h e t a sk s o r t h e t a sk i n pu t s/ o ut p ut s ,r e sp e c ti ve l y. A l so f e ed b ac k l o o ps a r e a l lo we d . A n a ct or c an b e e xe cu te d (

fired ) as soon as all input conditions are satisfied. A va l id o u tp u t i s c r ea t edo n e ve r y o u t p ut e d ge w h en a n a c to r i s fi r ed ( n ot e t h at e mp ty o u tp u ts a r e a l s op o s si b le ) . A l m os t a ny s o ft wa r e c a n b e d e sc r ib e d b y a K P N!

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F i gu r e 1 2 6 : K P N g r a ph m o d el

F i gu r e 1 2 7 s u mm a ri z es s o me p r ob l em s t h at c a n b e a d dr e ss e d a n d e x am i ne d b ym e a n s o f t h e K P N m o d e l .

F i gu r e 1 2 7 : S e le c ti o n o f d e si g n p r o b le m s t o b e m o d el e d a n d e x am i ne d b y K P N

12.2 SDF: Synchronous Data Flow

A s u b - c l a s s o f t h e g r a p h i c a l K P N m o d e l i s t h e s o c a l l e d S y n c h r o n o u s D a t a F l o w( SD F) . I n c ont ra st t o t he m or e g en er al K PN a ll i np ut s a re a ss um ed t o s wi tcht i m e s y n c h r on o u s , i . e . , w e h a v e d i s c r e te p r o c e s si n g t i m e s a n d a d i s c r et e a m o u nto f d a ta t h at fl ow s b e t we e n t h e a c t o rs . I . e. e ve r y t i m e a n a c to r i s fi r ed :

a g iv en # o f e le me nt s i s c on su me d a nd

a g i ve n # o f e l em e nt s i s p r o du c ed .

T h e n um b er o f c o ns u me d /p r o du c ed e l em e nt s i s t h er e by t i me i n de p e nd e nt ( i .e .c o ns t an t a t r u n- t i me ) ! T h is m o d el s u it s p e r fe c tl y t o m o d el s y nc hr o no u s ( i .e .c lo cke d) d ig it al s ys te ms t ha t a re c om mo nl y u se d t o da y! F ig ur e 1 28 s ho ws av er y s i mp l e S D F w i th o n e i n p u t a c t o r a n d t w o o u t pu t a c t or s and , w h e r e and d e p en d o n t he o ut pu t o f .

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F i gu r e 1 3 0: S D F a n no t at i on s

T h er e by, t h e fi r in g r a te o f d e t er m i ne s t h e s a mp l e r a te a t i t s o u tp u ts :

R 3 = F c

R 4 = F d

To m a ke t h e S D F f r om fi g ur e 1 3 0 a va l id ( i .e . e x ec u ta b le ) s ch e du l e w e h av e t oe n su r e t h a t n o b u ffe r o ve r flo w o c c u r s a t t h e i n p u ts . T h is i s t r ue i f t h e f o l l ow i ngc o n d i ti o n i s f u l fi l l ed :

F =R 1

a =R 2

b .

F i gu r e 1 3 1 g i ve s a s l ig ht l y m o r e c o mp l ex S D F e x a m pl e w i th 3 a c to r s a n d d a taamount annotations.

F i gu r e 1 3 1 : S D F s a m p l e r a t e c o nc e pt

T h e q u e st i on t h at n ow a r is e s i s : D o e s t h i s w o r k? / I s t h i s S D F e x ec u ta b le ? Wec a nn o t a n sw er t h is q u es t io n , s i nc e i t d e p en d s o n t h e c o n c re t e a m o u nt o f d a ta

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c on su me d a nd p ro d uc ed , w hi ch h as o nl y b e en a nn ot at ed i n a bs tr ac t f or m i nfi gu re 1 31 ! L et u s e xa mi n e a c on cr et e i ns ta nc e o f t hi s S DF e xa mp le , s ho wn i n

fi gu re 1 32 . T he ri n, w e s e e t ha t p r o du ce s 2 t ok en s o n i ts o ut pu t t o o n ce i ti s fi re d. O n t he o ut pu t e d ge t o 1 t ok en w il l b e p ro d uc ed b y. N ow , canb e fi r ed o n ce , c o ns u mi n g t h e 1 t o ke n f r om a t i t s i n p u t a n d p r o d u c i n g 1 t o k e na t i t s o u t p ut . F i na l ly, c a n b e fi re d o nc es , c on su mi ng 1 t ok en f r om a nd 1t o ke n f r o m. N ow , we fi ni sh ed a s in gl e p e ri o d a nd s ta rt f ro m t he b e gi nn in g.T he p ro bl em , h ow ev er , i s t ha t t he re i s s ti l l 1 t ok en l ef t a t t he i np ut b uff er o f , s in ce h as p ro du ce d 2 t oke ns o n t hi s a rc b ef or e! I f we r ep ea t t hi s g am ec ont in uo us l y, w e e nd u p w it h a b uff er o ve rfl ow ( as su mi ng a r ea l s ys te m w i thl i m it e d b u ff e r s i z e ) w h i ch i n t u rn l e ad s t o a m e mo r y d e a d lo c k, s i nc e c a n n o tb e fi re d a ny mo re i n t hi s c as e. T hi s m ea ns t ha t t he S DF i s n ot e xe cu ta bl e d uet o a s a mp l e r a te i n co n si s te n cy !

F i gu r e 1 3 2: E x am p l e f o r S D F w i th s a mp l e r a te i n co n si s te n cy

T h e f o l l ow i n g t wo q u e s ti o n s r e m a in :

H ow t o c he ck f o r s u ch m e mo r y d e ad l o ck s ( d ue t o b u ffe r o ve r flo ws ) ?

H o w t o c r ea t e a s ch e du l e f r o m a g i ve n S D F ?

T h e s e c o nd q u es t io n i s s u b j ec t o f t h e f o l l ow i ng s u b s e c t io n . To c he ck f o r s a m p l er a te c o ns i st e nc y w e c a n s u cc e ss i ve l y c o mp u te t h e fi r in g r a te s o f t h e a c t o rs a n dt he d at a r at e b e tw ee n t he a ct or s, a s d ep ic te d i n fi gu re 1 33 . T he fi ri ng r at e o f t h e i n pu t a c to r c an b e c om pu te d b y t he i np ut r at eRin a n d t he a mo un t o f d a ta t h at i s c o ns u me d b y (namely b ): F = Rin/b. Fo ll ow i ng , w e a r e a bl e t oc o mp u te t h e d a t a r a t e s a t t h e o u t pu t e d ge s o f , s in ce w e k n ow t he a mo un t o f d a t a p r o d u c e d w h e n i s fi r e d (c and d resp.):

Rce= c F = Rin c/b,

Rdf=d F = Rin d/b.

We c o nt i nu e w i th t h is u nt i l a l l fi r in g a n d d a t a r a te s h av e b e e n d e t e rm i n ed .

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F i gu re 1 34 : S DF g ra ph s ho wi ng t he n o de a nd a rc n um be ri ng . T he i np ut a nd

o u tp u t a r cs a r e i g no r ed f o r n ow .

w he re s i s t he n um be r o f a ct or s. T ha t m ea ns t he m at ri x m us t b e r an k d efi -c i en t i . e . h as t o h av e l e s s l i ne a r i n d e p en d en t r o w s t h an c o lu m ns ( under-d e te r mi n ed l i ne a r s y s t em o f e q ua t io n s) . C h ec k i t f o r y o u rs e lf f o r t h e e x a m pl ef r om fi g ur e 1 3 2 . Yo u w i l l fi n d t h a t a l l 3 r ow s a r e l i n ea r i n d e p en d en t i n t h is c a se !T h e r a n k d e fic i en c y c o n d it i on i s b a se d o n t h e f o l l ow i ng i d ea : A s su m e t h a t o u r3 a ct or s a re fi re dF = (F, F , F)

T t i me s d u ri n g a s i ng l e p e r i o d o f a s ch e du l e.

T h an t h e f o l lo wi n g e q ua t io n y i el d s t h e fi n al b u ffe r s t at eB = (B, B , B)T

att he e nd o f t he s ch ed ul e p e ri o d: F = B . We a lr ea dy k no w t h a tB m us t b e0 a t t he e nd o f a s ch ed ul e p er io d f or t he S DF t o b e r at e c on si st ent . H en ce : F = 01s . We w il l a lway s b e ab le to fi nd a s ol uti on f or th is s ys te m o f

e q u a ti o n s , n a m e ly F = 0. H ow ev er , w e r e qu i re Fi > 0 t o r e su l t i n a r e as o na b les ch e du l e. C o ns e qu e nt l y, t o fi n d a n F = 0 w e n ee d a t l ea st a s ec on d s ol ut io n.H e nc e , t h e s y st e m o f e q ua t io n s m us t b e u n de r -d e te r mi n ed a n d t h er e fo r i t n e ed st o b e r a nk d e fic i en t.We c o ul d g e ne r al i z e t h e S D F c o nc e pt t o s t o ch a st i c q u eu e s w i th va r ia b le i n pu tr a te s . I n t h is c a se w e w o u ld s i m pl y t a ke t h e a v er a ge r a te a s o u r c o n st a nt d a tar at e i n t he S DF , e .g ., a= E [a] and b= E [b]. T hi s a pp ro ac h w i l l w o rk a s l on ga s t h e v a ri a nc e o f t h e s t o c ha s ti c i n pu t p r o ce s s i s s m al l c o mp a re d t o t h e q u e u es i ze , s u ch t h at n o o ve r flo ws o c c ur s .

12.3 Multi Processor Scheduling

N ow , t h at w e d i sc u ss e d t h e q u es t io n o f c he ck i ng t h e S D F f o r f e as i bi l i ty ( r at e

c o ns i s te n cy ) , w e w i ll a d dr e ss t h e q u e s ti o n i n t h is s u b s e c t io n o f h ow t o a c tu a ll yc r ea t e a s ch e du l e f r om t h e S D F f o r a s i ng l e- o r h o mo g en e ou s m ul t i -p r o ce s so ra rch it ec tu re . L et s s ta rt o ff w i th t he s m al l S DF e xa mp l e p re se nt ed i n fi gu re1 35 ( le ft ). I t c on si st s o f 3 a ct or s w i th d ep e nd en ci es i n a f ee db ack l o op . A fi rs tc he ck c o nfi r ms t h at t h er e a r e n o r a te i n co n si s te n ci e s i n t h is g r ap h , s o t h at w e

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c an c on ti nu e w it h i t. P le as e n ot e t ha t w e n ee d b uff er i ni ti al i za ti on i n c as e o f f e ed b ac k l o o p s . O t he r wi s e, w e w o u ld g e t a d e ad l o ck r i gh t a t t h e b e g i nn i ng . T h is

i s b e c au s e e v e ry a c to r i s d e p en d en t o n t h e o u t p ut o f a n ot h er a c to r , s u ch t h atn o a c to r c a n s t a r t fi r i n g ! B y t h e b u ff e r i n i t i al i z at i on w e a v oi d t h is d e ad l o ck a n da l l ow a c e rt a in a c to r (s ) t o s t ar t . We a l so n e ed t o c o ns i de r t h e a c t o r p r o c e ss i ngd ur at io n t ha t h as b e en c ho se n t o b e e qu al t o t he n o de n um be r o f t he a ct or f ors i m p l i ci t y ( fi g u re 1 3 5 , r i g ht ) .

F i gu r e 1 3 5 : L e ft : E x am p l e S D F w i t h l o o p f o r M P s ch e du l in g ; R i g ht : p r o ce s si n gdelays

N ow , w e c an c ont in ue t o g en er at e a s ch ed ul e by c on st ru ct in g t he s o c al l edp re ce de nc e g ra ph . T hi s g ra ph c on ta in s 1 n o de f or e ach a ct or i n a p o te nt ia ls ch ed ul e. I t c on ta in s 1 e dg e f ro m n o de a to b, if a h as t o b e fi re d b ef or e b(i.e. a precedes b) . I t s h ow s a c o nc r et e c hr o no l og i ca l fi r in g ( e xe c ut i on ) o r de r ,i n c on tr as t t o t he S DF t ha t o nl y s ho ws t he g en er al d ep e nd en ci es ! F ig ur e 1 36s h o w s t h e p r e c e d e n c e g r a p h f o r o u r s m a l l S D F e x a m p l e f o r J = 1 , i . e . , e v e r y a c t o rh a s t o b e fi r e d a t l e a s t o n c e .

F i gu r e 1 3 6 : P r ec e nd e nc e g r ap h ( J =1 ) f o r S D F e x am p le

T he t wo a ct or s # 1 a nd # 3 c an d ir ec tl y b e fi re d a t t he b e gi nn i ng ( du e t o t heb u ffe r i n i ti a l iz a ti o n) . A ct o r # 1 c a n e v e n b e fi r ed t wi c e r i g ht a t t h e b e g i n ni n g.A f te r a ct or # 1 h as b e en fi re d t wi ce , a ct or # 2 c an b e fi re d, s in ce i t c on su me st wo t ok en s f ro m t he o ut pu t o f a ct or # 1. N ow , w e a r e a lr ea dy d on e w it h t hec o ns t ru c ti o n o f t h e p r ec e de n ce g r ap h , s i nc e e ve r y a c to r w as fi r ed a t l e as t o n ce !

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Fr o m t h e p r ec e de n ce g r ap h w e c a n e a si l y g e ne r at e d i ffe r en t s i ng l e p r o ce s so rs c h ed u l i n g a l t e r n a ti v es , a s d e p i ct e d i n fi g u r e 1 3 7 .

F i gu r e 1 3 7: P os s i bl e s ch e du l es f o r P = 1 p r o ce s so r s

A l l t h es e o p ti o ns a r e r e a so n ab l e s c h ed u le s . T h e q u e s ti o n i s : A r e s o m e o f t h es es ch ed ul es b et te r t ha n o th er s? I n f ac t, t hi s i s t he c as e s in ce t he two g re en

e n ci r cl e d a l te r na t iv es m a ke u s e o f t h e d a ta l o c al i ty p r in c ip l e, a s s h ow n i n fi g ur e1 38 . T hi s m ea ns t ha t t he a ct or s a re d ir ec tl y e xe cu te d i n t he o rd er o f t he ird e p en d en c ie s . R e me mb e r : A d a ta d e p en d en c y f r o m a c to r to m e a ns t h at n e ed s t h e o u t p ut d a ta f r om a s i n pu t. I f i s d i re c tl y e x ec u te d a f te r ont he s am e P E, i t c an r eu se t he r es ul ts i n t he l o ca l m em or y. I f a d i ffe re nt a ct or i se x e c u t e d o n t h e P E , t h e r e s u l t s f r o m m u s t b e m o v e d t o a g l o b a l m e m o r y a n dt r an s fe r ed b a ck t o t h e l o c al m e mo r y b e f or e i s e x e c ut e d two unnecessaryd a t a t r a n sf e r s !

F i gu r e 1 3 8: B e st s ch e du l i ng o p ti o ns t h at e x pl o i t d a ta l o c al i ty

L e t s c o nt i nu e w i th t h e s ch e du l in g e x am p le f o r t wo p r o ce s so r s. We c a n a g ai ne a si l y fi n d va l i d s ch e du l es f r om o u r p r ec e de n ce g r ap h . F i gu r e 1 3 9 p r es e nt s t woa l te r na t iv es . A l th o ug h , t h es e a r e r e as o na b le s ch e du l es , w e r e co g ni z e t h at t h eya r e n o t e nt i re l y o p t i m al , s i nc e w e h a ve t o s ch e du l e n o p s i n b e tw ee n a n d t h u sn o t f u l l y u t il i z e t h e t wo p r o ce s so r s. Fr o m t h e e ffi ci e nc y p e r sp e c ti ve i t l o o ks a sf ol l ow s: We k no w t ha t E 1AT . I f we n orm al iz e A t o t he a rea o f a s ing lep r o ce s so r , w e g e tA = P = 2. I f w e f u rt he r n or ma li ze t he p ro c es si ng r at e ( i. e.t he l en gt h o f a s in gl e s ch ed ul e p e ri o d) t o t ha t o f a s in gl e p ro c es so r s ol ut io n,

we g et t he s pe ed up S = 1T = 74

= Scheduleperiodf or P=1Schedule periodf or P=2

. F in al ly, we g et

E 74 12 =

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F i gu r e 1 3 9: P os s i bl e s ch e du l es f o r P = 2 p r o ce s so r s

To b e a bl e t o fi ll t he g ap s i n t he s ch ed ul es o f fi gu re 1 39 , w e h a ve t o a lt er na tet h e p r o c e ss o rs b e tw ee n a d ja c en t p e r i o d s. Fo r t h is p u rp o s e w e n e ed t o e x te n dt he p er io d o f o ur p re ce de nc e g ra ph t o J =2 ( i. e. e ve ry a ct or i s fi re d a t l ea st

t w i ce ) . T h e r e s u l t i ng g r a p h i s p r e s e n te d i n fi g u r e 1 4 0 ( t h e b r o w n i n d i c e s p r o vi d ei n f o r ma t i o n o n t h e c o r r e s p o n d in g s c h ed u l e p e r i o d ) .

F i gu r e 1 4 0 : P r ec e nd e nc e g r ap h ( J =2 ) f o r S D F e x am p le

Fr o m t h e e x te n de d p r ec e de n ce g r ap h w e c a n n ow a g ai n e a si l y d e ri ve a s ch e du l ef or P =2 p ro c es so rs . A s fi gu re 1 41 s how s, t he g ap s c ou ld b e r em ov ed f ro m t heschedule.

F i gu r e 1 4 1: S ch e du l e f o r P = 2 p r o ce s so r s c o ns i de r in g J = 2 p e r io d s

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F i na ll y, w e w er e a bl e t o fi nd a n o pt im um s ch ed ul e f or t he P =2 c as e w i th a ne ffi c i e nc y c o m p ar a b l e t o t h a t o f t h e o p t i mu m s i n g l e- p r o c e ss o r s o l u t io n :

E= S 1P

= 147

12

= 1.

W h a t a b o u t t h e c a s e P = 3 ? F i g u r e 1 4 2 s h o w s t h e c o r r e s p o nd i n g s c h e du l e ( J = 1 ) .T h e e ffi ci e nc y i s f u rt h er d e cr e as e d c o mp a re d t o P = 2:

E= S 1

P =

7

3

1

3=

7

9 2, s i nc e w e a r e n o t a b l e t o i n it i al l y s c h ed u le m o ret h an t h e t h r e e a c t o rs ( # 1, # 1 ,# 3 ) i n p a ra l l el a n d t h u s w i ll n o t a c h ie ve a h i gh e rd e g r ee o f p a r a l le l i s m .

F i gu r e 1 4 2: S ch e du l e f o r P = 3 p r o ce s so r s

I n st e ad , l e t u s s l i gh tl y m o d if y o u r S D F e x am p l e b y s i mp l y i nt r o du c in g a n a d -d it io na l i ni ti al i za ti on b uff er a t t he i np ut o f a ct or # 3. N ow , a ct or # 3 c an b efi r ed t wo t i me s i n i ti a ll y ! T h i s h a s q u it e a n i m p a c t o n o u r s c h ed u le , s i nc e i t n owi nd ee d m ak es s en se t o g o f or t he J =3 p re ce de nc e g ra ph , a s fi gu re 1 43 ( ri gh t)s h ow s ! N ow , w e fi n d t h re e i n de p e nd e nt p r ec e de n ce t r ee s i n t h e g r a p h a n d c a ns i m pl y b u il d a n o p ti m al s ch e du l e f o r P = 3 p r o c e ss o rs o u t o f i t (E= S/P= 1 ).

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3 . O ut pu t i s c re at ed a t t he e nd o f a n a ct or /t as k.

N o i nt e r a c to r /t a sk c o nt r ol o n ce fi r ed , i . e. e ve r y a c to r /t a sk c o ll e ct s a l l n e c-e s sa r y c o n tr o l i n f or m at i on ( a s p a rt o f t h e i n p u t d a ta ) b e f or e i t s t ar t s. I f a P E ( pr o ce ss i ng e le me nt ) h as i ts i np ut f or a n a ct or ( lo c al l y) ava il ab le ,t h en t h e a c to r c a n c o mp l et e .

S o w ha t a re t he f ea tu re s o f D DF n ow ?

1 . N o p r ed e fin e d r u n - ti m e s

2 . A ct or s c a n b e c re at ed by a ct or s ( e . g. d y na m i c t a s k s ch e du l er w h ic h i sa n a c to r i t se l f ) .

3 . I nt e rd e p en d en c y o f a c to r s ( a rc s ) c a n b e c r ea t ed d y na m ic a ll y ( n ec e ss i tyt h at f o ll ow s f r om 2 . ).

F i gu r e 1 4 5 g i ve s a s m al l e x am p le t o m o ti va t e t h e n e ce s si ty o f d y na m ic s ch e du l -i n g. T h er e in , w e c o n s i de r t wo a c to r s t h a t p e r fo r m m o t i on e s ti m at i on o n a 2 Di m a ge t h at i s s t or e d i n a g l ob a l m e m o ry. We c a n d i s t i ng u is h t h re e c a se s :

1. 1& 2 ( a c t or s w o r k o n i n d e p e nd e nt m e m o r y a r e a s ) o n e g l ob a l m e m-o r y o . k.

2. 1& 3 ( a c to r s w o rk o n o ve r l a pp i n g m e m o ry a r e a s) sequential accesst o m e mo r y o . k.

3. 1& 3 w i t h w r i t e b a c k m o d i f y p r o b l e m t o w o r k o n t h e s a m e m e m o r y !

i n c a se a c to r 2 3 b u t i n p u t /o u t p ut a r r ay c h os e n d y n a mi c a l l y ( i m a g eprocessing)

F i gu r e 1 4 5: I m ag e p r o ce s si n g a s e x am p le f o r d y na m ic s ch e du l i ng

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T hi s s ma ll e xa mp le s ho ul d d em on st ra te a c as e w he re w e a re n ot a bl e t o a p-p l y s t at i c s ch e du l in g , s i n ce t h e d e p en d en c ie s a n d e x ec u ti o n o r de r a n d e ve n t h e

m em or y m ap pi n g d ep e nd s o n w hi ch a re a o f t he i m ag e a ct ua ll y h as t o b e p ro -c e ss e d b y t h e t w o a c t o rs w h i ch i n t u rn i s d y na m ic a ll y s e le c te d a t r u n- t i me . T h ise m ph a si z es t h e n e ed f o r a d y na m ic ( r un - ti m e) s ch e du l er f o r:

h a n d l i n g h a z a r d c o n fl i c t s i n m e m o ry

m a p p i n g t a s ks o n to c o r e s ( t i m e / s pa c e )

r e s e rv i n g i n te r c o nn e c ts ( N o C )

T o a l l o w d y n a m i c s c h e d u l i n g a t r u n - t i m e w e n e e d a s p e c i a l p r o g r a m m i n g m o d e lt ha t a ll ow s u s t o d iv id e t he p ro gr am i nt o s ub t as ks w hi ch c an b e i ss ue d i na r b i tr a r y e x e c u t i o n o r d e r ( c o ns i d e r in g t h e d a t a d e p e n d en c i e s ) b y t h e r u n - ti m es ch e du l er o nt o a p pr o pr i at e P E s. Fo r t h is p u rp o s e w e h av e t o d e sc r ib e t h e a c -

t o rs / ta s ks a s s h ow n i n t h e f o l lo wi n g:

Algorithm 6 P r o g ra m mi n g m o d el f o r M P So C w i th r u n- t i me s ch e du l er

c a l l a c t o r ( i n 0 , i n 1 , . . . ; o u t 0 , o u t 1 , . . . )actor...e n d o f a c t o r

T he m em or y a dd re ss es o f t he i np ut s a nd o ut pu ts c an b e u se d b y t he r un -t im es c h ed u l e r t o i d e nt i f y d a t a d e p e n de n c i es ( i . e . o ve r l a pp i n g m e m o r y a r e a s ) . B a s e do n t h e d e p e n d e n c y a n a l y s i s t h e s c h e d u l e r i s a b l e t o fi n d a n a d m i s s i b l e e x e c u t i o no r de r w i th r e sp e c t t o t h e t a rg e te d o p ti m i za t io n s t ra t eg y ( s p ee d , p o we r , e t c. ) .

T h e a c t u a l p r o g r a m c o d e o f t h e a c t o r i s s t o r e d i n t h e g l o b a l m e m o r y a n d t r a n s -f er ed t o t he l oc al m em or y o f t he s el ec te d P E w he n t he a ct or i s r ea dy t o b ee x ec u te d . F i gu r e 1 4 6 s h ow s t h e M P S o C p r o g ra m mi n g m o d e l f r om t h e P E p o i nto f v i ew . T h e p r o c es s or r e ce i ve s t h e i n p u t d a t a a n d c o nt r ol f r om a g l ob a l m e m -o r y, a s w el l a s t h e p r o gr a m c o d e t h at m a tc he s t h e P E t yp e . C e rt a in a c to r s m a yb e s u it e d t o b e e x ec u te d o n d i ff e r en t P E t yp e s . I n t h i s c a se d i ffe r en t v e rs i on s o f t h e s a m e p r og r am c o d e m u s t b e l o c at e d i n t h e g l o b al m e mo r y ( t h e a p pr o pr i at ev er s io n o f t h e p r o gr a m c o d e i s s e le c te d b y t h e s ch e du l er ) . A l o c al s c ra t ch p a dm e mo r y i s u s ed b y t h e P E t o s t o r e i n t er m ed i at e r e s u l ts . T h e a c t ua l o u tp u ts o f t h e c o m p ut a ti o n a r e t r an s fe r ed b a ck t o a g l ob a l m e m o r y ( u n l es s t h e d a t a c a nb e r e us e d b y t h e s u bs e qu e nt l y s ch e du l ed a c to r d a t a l o c a l i ty ) .

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F i gu r e 1 4 6: M P So C p r og r am m i ng m o d el

L e t u s e x am i n e t h e p r o g ra m mi n g m o d e l f o r o u r F I R e x a m pl e a l i t tl e b i t c l o s ert o u n d e rs t a n d t h e c o n s eq u e n ce s o f t h e i n p u t/ o u t pu t p r o c e du r e :

yk =L1i=0

aixki

A s s u m e t h a t o u r a c t o r c o m p u t e s t h e F I R i n d i r e c t f o r m f o r M s t e p s i n p a r a l l e l ,a s d ep ic te d i n fi gu re 1 47 f or t he c as e L =4 a nd M = 3.

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F i gu r e 1 4 7 : F I R e x a mp l e f o r L = 4, M = 3

I n t h is c a se w e m u s t p a s s t h e f o l l ow i ng i n pu t s:

x= (xkL+1,...,xk+M1) = xkL+1,...,xk1 ; xk,...,xk+M1 state space input

a= (a0,...,al1)

I n c a s e o f M = 3 a n d L = 4 o u r s t a t e s p a c e c o n s i s t s o f (xk3,...,xk1) a n d t h e i n -p u t s f o rx are(xk,...,xk+2) ( r ef e r t o fi g ur e 1 4 7) . T h e o u t p ut s t h at a r e p r o d uc e db y t h e a c to r a r e t h e f o l l ow i ng :

y= (yk,...,yk+M1)

xs = (xk+ML+1,...,xk+M1) ( t he s t at e s p ac e f o r t h e n e xt c o mp u ta t io ncycle)

Fo r t h e M = 3 , L = 4 c a s e w e g e t a c co r di n gl y(yk,...,yk+2) a s a c tu a l o u tp u ts . Fo r

t h e n e x t c o m p u ta t i o n c y c l e ( i . e . t o c o m p u t e(yk+3,...,yk+5)) w e n e e d t o s a v e t h ea c co r di n g s t at e s p ac e w h ic h i s(xk+2,...,xk) f or t he c as e i n fi gu re 1 47 . Fo r t hi sp u r p o s e , t h e s t a t e s p a c e xs m u s t a l s o b e p a s s e d t o t h e o u t p u t t o b e t r a n s f e r e d t ot h e g l o b al m e mo r y. T h i s i s n e ce s sa r y, s i n ce t h e P E c o ul d b e u s ed b y a d i ffe r en ta c to r i n t h e m e a nw h il e t h at w ou l d o v er w ri t e t h e s t at e o f o u r F I R a c to r . T h is i sa n i m p o r t a n t c o n s e q u e n c e t h a t c o m e s w i t h o u r d y n a m i c p r o g r a m m i n g m o d e l f o r

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M P So C s a n d d i ff e rs f r om t h e a s s u mp t io n t h at w e m a de b e f or e t h at o u r c u r r en ts t at e i s a l wa ys l o c al l y a va i la b le i n t h e P E . F i gu r e 1 4 8 b r i e fl y s u m m ar i ze s o u r

o b s e rv at i o n s f o r t h e F I R fi l t e r.

F i gu r e 1 4 8 : M - st e p F I R e x am p le

I n c as e o f S DF /D DF ( xD F) t he m ai n p ro gr am i s m er el y a s eq ue nc e o f a ct orcalls:

Algorithm 7 M a i n ( c on tr o l) p r og r am s e qu e nc e i n c a se o f x D F

m a i n c o d ec a l l a c t o rc a l l a c t o rc a l l a c t o r

case...

d e p e nd e n c y f o r a c t u al e x e c ut i o n o r d e r i s d e t e rm i n e d b y i n / o ut p u t s

I n c a s e w e h a v e P P E s (P >1) t h is l e ad s t o t h e f o l lo wi n g c o ns e qu e nc e s: s c he d u l i ng a c t or s a c c o rd i n g d e p e n de n c i es a n d n o t n e c es s a r i l y i n - o r d e r out- of- order # t a s k s r e ad i n m a i n c on tr o ll e r c a n b e > P, w i t h P b e i ng i s s ue d i n p a ra l l el a t m o st S u p e r S c a l a r Ta s k - L ev e l P r o c e s s i n g ( s e e s e c t i o n 9 . 3 . 2 )

13.2 Task Scheduling

O ne o f t he m ai n c ha l le ng es o f M PS oC s i s t he t as k s ch ed ul i ng t ha t i s b ri efl ya dd re ss ed i n t hi s s ec ti o n. A s a lr ea dy m en ti on ed b e fo re , w e d is ti ng ui s h t woc a se s : s t at i c ( 1 ) a n d d y na m ic ( 2 ) s ch e du l in g .

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1 . S t at i c T as k S ch e du l i ng : T h is c a se i s m o re e a sy t o h a nd l e, s i nc e w e c a n c o m-p ut e t he s ch ed ul e a t co mp il e t im e u si n g t he c om m on S DF m o de l t ha t h as

b e en p re se nt ed i n s ec ti on 1 2. F i gu re 1 49 s ho ws a s im pl e e xa mp le t o c re at e a nA S AP ( as s o o n a s p o s si b le ) o r A L AP ( as l a te a s p o s si b le ) s c h ed u le g i ve n a nS D F g r a p h w i th a n no t at e d e x e c ut i on t i me s a s i n pu t . U s i ng t h e A S A P p o l i cy,actor a c an s ta rt i m me di at el y a t t =0 . A ct or b c an b e fi re d a t t =2 00 a nd i ss ch e du l ed t o t h e s t il l i d le P E # 2 . A ct o rc c a n d i r e c t l y b e fi r e d a t t = 3 0 0 a f t e r ah as b e en c om pl et ed a nd i s e xe cu te d o n P E # 1. D ue t o i ts l on g r u n -t im e o f 8 00c y c l es , a c t o rb o c c u p i e s P E # 2 n e a r l y u p t o t h e e n d o f t h e s c h e d u l e , s o t h a t t h es u bs e qu e nt a c to r s m us t b e e x ec u te d s e qu e nt i al l y o n P E # 1 : d a t t =4 00 a nd ea t t = 70 0 . F i na l ly, a f te rb h a s fi n is h ed , t h e o u tp u t a c to rf c a n b e s ch e du l ed o nP E # 2 a t t =9 00 . T he s ch ed ul e p e ri o d i s c om pl et ed a t t =1 00 0. Fo r t he A LA Ps ch e du l e w e s i mp l e a p pl y t h e s a m e p r o ce d ur e i n r e ve r se o r de r , i . e. s t ar t in g w i thactorf b e ....

F i gu r e 1 4 9 : E x am p le f o r s t at i c M P So C s ch e du l e

2 . D y na m ic Ta s k S ch e du l i ng ( C or e Ma n ag e r) : Fo r t h e c a se o f d y na m ic s ch e du l -i n g w e n e ed a r u n- t i me s ch e du l er t h at w e c a l l C or e M a na g er . I t i s r e sp o n si b l ef o r f e t ch i ng a b u nc h o f t a sk s ( a ct o rs ) i nt o i t s l o c a l b u ffe r . N ow , t h e C o r e M a n -a g er h a s t o d i st i ng u is h t wo c a se s :

1 . t as k i s r ea dy t o b e fi re d t he r e a r e n o d e p en d en c ie s o n o t he r r u nn i ngtasks fi r e a s s o on a s p o ss ib l e ( i . e. w he n t he n ex t P E i s a va il ab l e)

2 . t as k i s al m os t r e ad y t o b e fi re d a l mo s t m e an s t h at t h e a c t o r i n p u tsa r e b e i ng c a lc u la t ed b y o t he r a c to r s ( c ur r en tl y r u nn i ng t a sk o r i n b u ffe rt o b e e x ec u te d ); C o re M a na g er m us t c he ck t h e d e p e n de n ci e s t o fi n d a

va l i d e x e c ut i o n o r d e r

T he C or e M an ag er c on ce pt i s d ep ic te d o n fi gu re 1 50 . T he re in , t he m ir co -p r o ce s so r (P) c ont ro ls t he t op -l ev el p ro gr am fl ow ( i. e. e xe cu te s t he m ai n r o ut i ne ) a n d p u sh e s t a sk d e sc r ip t io n s ( f or t a sk s t o b e s ch e du l ed o nt o P E s) i nt o

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t h e t a s k q u e u e o f t h e C o r e M a n a ge r . T h e C o r e M a n a g er a n al y ze s t h e d e p e n de n -c i es b e tw ee n s ch e du l ed t a sk s a n d f o r w ar d s t h em i n a n a d mi s si b le o r de r t o t h e

p r o ce s si n g e l e m e nt s t o b e e x ec u te d . Fo r t h is p u rp o s e, t h e p r o g ra m c o d e a s w el la s t he i np ut d at a i s t ra ns f er ed f ro m t he g lo ba l m em or y t o t he l o ca l m em or y o f t h e c o r re s p on d in g P E ( u su a ll y b y m e a ns o f a D M A) . A f te r t h e P E fi n is h ed t h et a sk e x ec u ti o n, i t i n f or m s t h e C o r e M a na g er t h at i t i s a va i l ab l e f o r n e w t a s k s.T h e r e s u l t o f t h e c o m p ut a ti o n i s t r an s fe r ed b a ck t o t h e g l o b al m e mo r y, o r k ep ti n l o ca l m em or y t o b e u se d f ur th er b y a s ub se qu en t t a s k ( d a t a l o c a l i ty ) .

F i gu r e 1 5 0: C o re M a na g er c o nc e pt f o r d y na m ic s ch e du l in g

We m a ke f o l l ow i n g a s s u m pt i o n s:

H W P E s c a n e x e c u t e a t a s k , i f

e n ou g h m e mo r y f o r d a ta & p r og r am i s a va i la b le ,

t h e I S A o f t h e s e l e ct e d P E t yp e m a tc he s t h e p r o gr a m c o d e ,

t h e N o C a l l o w s t o d e l i v e r a n d r e t ri e v e i n p u t s /o u t p ut s .

T h e f o l l ow i n g a d va n ta g e s o ff e r s d y n a mi c s c he d u l i ng :

r e ac t o n t h e n u mb e r o f P E s c h a ng i ng i n t i me ( e .g . d y na m ic s w it ch - off d u et o t e m p e ra t u r e c o n t r ol , f a i l u r es , e t c . )

p o we r u p /d ow n P E s a s n e ed e d

s u p p o r t o f v o l ta g e / cl o c k s c a l i n g h av e m ul ti pl e C or e M an ag er s f or d i ffe re nt t hr ea ds fi gh t f o r t he s am e

sea- of- PEs

d yn am ic al ly d ec id e w hi ch k in d o f P Es t o a ll o ca te g o od s up p or t o f h e t er o g e ne o u s M P S o C a r ch i t e ct u r es ( s e e fi g u r e 1 5 1 b e l o w)

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F i g u r e 1 5 1 : H e t e ro g e n eo u s a c t o r/ P E t y p e s

r e ac t o n p r io r it y a c to r s ( in te r ru p ts ) [ i . e. a n i nt e rr u pt i ng a c to r t h at h a sp r i o ri t y o v e r a l l c u r r en t l y r u n n i n g a c t o rs ] ;a ss um e a n M P So C w it h a C or e M an ag er ( CM ) a nd 4 b us y P Es :

C M n e ed s t o r e ac t t o i nt e rr u pt p r io r it y a c to r k i ll r u nn i ng a c to r o n a ny P E t o s e rv i ce i nt e rr u pt r e s ta r t k i l le d a c to r w i th i n it i al i n pu t w h i l e s e r v i n g t h e i n t e r r u p t t h e a c t u a l p a r a l l e l i z a t i o n d e g r e e i s d e c r e a s e dt o P =3 i n c a s e o f s t a t i c s c h e d u l i n g w e w o u l d h a v e P = 0 d u r i n g i n t e r r u p t s e r v i c e ,s i nc e t h e w h ol e s ch e du l e m us t b e r e st a rt e d a g ai n a f te r wa r ds !

D D F s c h ed u l i n g p o s s i b l e

U s i n g d y n a m i c s c h e d u l i n g t h e C o r e M a n a g e r h a s a p o o l ( b u ff e r ) o f f e t c h e d a c t o r sl o c a l l y a v a i l a b l e a n d h a s t o m a k e a d e c i s i o n o n w h i c h P E t o s c h e d u l e w h i c h t a s ki n w h ic h o r de r w i th r e sp e c t t o t h e t a sk i nt e r- d ep e n de n ci e s.

T h er e by, i t c a n p u rs u e d i ffe r en t o p ti m i za t io n s t ra t eg i es - t h e i s s ue d e ci s i on c a nb e c o nt r o l l ed f o r :

s p e e d o f c o m p le t i o n

power consumption

l o c a l i t y o f d a t a /p r o g ra m

L et s g iv e a s ma ll e xa mp l e t o m ot iva te t hi s . A ss um e a P E w he re c ur re nt ly a na c to r i s r u nn i ng . T h e C o r e M a n a ge r n o w h a s t o d e ci d e w h i ch o f t h e a c t or s t h at

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a r e a v a i l a b l e i n i t s b u ff e r s h o u l d b e i s s u e d n e x t t o t h a t P E . W i t h r e s p e c t t o t h ed a t a l o c a l i t y p r i n c i p l e, t h e f o l l o w i n g f a c t s i n fl u e n c e t h i s d e c i si o n :

C a n i n pu t d a ta b e r e us e d?

C a n o u tp u t d a t a b e r e us e d?

C a n p r og r am c o d e b e r e us e d?

I n c o n s e q u e nc e , m e m o ry t r a n sf e r s a r e m i n i m i ze d , . . . N o C l oa d a nd m em or y R /W a cc es s i s r ed uc ed a nd fi na ll y . . . e n e rg y c o n s um p t i on i s m i n i m i ze d !

13.3 Network-on-Chip

T h e o n - c hi p i n te r co n ne c ti o n o f a l a rg e n um b er o f P E s i s a c ha l l en g e t h a t g e ts

m or e a nd m or e i mp o rt ant f or m o de rn M P So Cs . Si m pl e b us se s o r c ro ss -b ars w i tc he s d o n o t o ff er t h e n e c e ss a ry b a nd w id t h a n d fl e xi b i li ty a n d d o n o t s c al ew i th t he nu mb e r o f P Es . N et wo rk -o n- Ch ip ( No C) o ffe rs a s ol u ti on f or t hi sp ro bl em . I t c on si st s o f c on ne ct or n o de s ( ro ut er s) a nd w ir es ( li nk s) t ha t a reu s ed t o a r ra n ge t h e P E s, c o nt r ol p r o ce s so r s, C o re M an a ge r s, m e mo r ie s , e t c. , i na rb it ra ry t op o lo gi es : e .g . m es h, r in g, s ta rt , e tc . T hi s i s a na lo go us t o o ff- ch ipn etw or ks , l i ke e th er ne t, b ut u si ng a m uc h t hi n ne r p ro to c ol s ta ck . N oC s a rep a ck e t- s w i t ch e d n e tw o rk s w h e r e t h e a t o m ic n e tw o rk p a ck e t i s c a l l ed F l i t ( fl owc o nt r o l d i g i t ) . B a s i ca l l y, w e c a n d i s t i ng u i s h t w o t r a ffi c ( s e rv i c e ) c l a s s e s :

B es t E ffo rt ( BE ) t ra ffic : U si ng t hi s t ra ffic c la ss , t he p acke t i s r ou te dt h r o u g h t h e n e t w o r k a s f a s t a s p o s s i b l e , b u t w i t h o u t g i v i n g a n y l a t e n c y o rt h ro u gh p ut g u ar a nt e es . T h is m e an s t h at c e rt a in l i nk s m ay e ve n s t ar ve , i f

t h e n e t wo r k i s u n de r v er y h i gh l o ad .

G u a r an te e d S e r v ic e ( G S) t r affi c : W i th G S t r affi c a fi x r o ut e t h ro u gh t h en e tw or k m us t b e r e se r ve d a t fi r st b e tw ee n t h e t wo c o mm un i ca t io n p a rt -n e r s . T h e c o m m u n i ca t i o n p a r t n er s h a ve n o w h i g h e s t p r i o r i ty o n t h i s r o u t e,s o t ha t t he l in k c an no t b e i nt er f er ed b y o t h er t ra ffic . W it h t hi s k in d o f c i rc u it - sw i tc hi n g t h e N o C i s a b le t o p r ov i de g u ar a nt e es f o r l a te n cy a n dt h r o ug h p u t w h i c h i s e s p e c i al l y i n te r e s ti n g f o r r e a l -t i m e a p p l i c a t i on s . Wec a n f u r t he r s u b- d iv i de t h is c l as s a c co r di n g t o t h e m e t h o d t h a t i s u s ed f o rt h e G S r o ut e a l lo c a ti o n:

D i s t r ib u t e d a l l o c a t io n : B y s e n d i n g a s p e c i a l s e t u p /t e a rd o w n fl i t f r o mt h e s e n d e r t o t h e r e c e i v e r ( u s i n g c o m m o n B E r o u t i n g t e c h n i q u e s s u c h

a s X Y - ro u ti n g o r fl o o d in g ), t h e G S r o ut e i s e s ta b li s he d /r e le a se d . C e nt ra l a ll o ca ti on : A c ent ra l c on tr ol u ni t ( he re : c al l ed N oC M an -

a g er ) i s r e sp o n si b le f o r e s ta b li s hi n g/ r el e as i n g t h e G S r o ut e s. T h eN oC M an ag er h as a g l ob al v ie w o f t he N oC a nd a cc es s t o t he r es er -v at i o n c o n t r o l m e c h a ni s m s o f a l l r o u te r s .

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F in al ly, l et s t ake a l oo k a t t he s ma ll e xa mp le i n fi gu re 1 52 . We s ee a N oCa r ra n ge d i n a 2 D -m e sh t o p ol o gy o f s i ze 4 x 4. We r e co g ni z e t h a t t h i s i s a h e te r o-

g e n e ou s M P S o C, s i n c e d i ff e r e nt P E t y p e s ( i l l u s t r at e d b y c i r c l es a n d r e c t a n g l es )a r e c o n ne c te d t o t h e r o ut e r n o d e s ( s m a ll b u bb l es ) . E . g. o n e o f t h e n o d e s c o u l db e u s ed a s c e nt r al m i cr o -p r o ce s so r t o r u n t h e m a in c o d e. O t he r P E n o d es c o ul db e u s e d f o r c e n t r a l m a n a g e r s , s u c h a s t h e C o r e M a n a g e r o r t h e N o C m a n a g e r ( ac e nt r a l c o n t r ol i n s t an c e f o r t h e l i n k a l l o c a ti o n i n t h e n e t w or k ) . Fu r t h er m o r e, w efi n d t w o s u cc e ss f ul l y e s ta b li s he d r o ut e s i n t h e N o C ( g re e n l i n e s ). A t h ir d c o n-n e ct i on r e qu e st w as n o t s u cc e ss f ul ( r ed l i n e) , s i nc e t h e r o ut e i s a l re a dy b l o ck edb y a n o t he r c r o s si n g r o u t e.

F i gu r e 1 5 2: 4 x 4 2 D - Me s h e x am p le f o r N e tw or k -o n -C h i p

T h is b r ie f i n si g ht s h ou l d g i ve a s m al l i m p re s si o n o f t h e o n -c hi p i nt e rc o nn e ct i onchallenge.

13.4 Homogeneous / Heterogeneous MPSoC

We c a n d i st i ng u is h M P So C s a c co r di n g t o P E t yp e s u s ed .

Homogeneous MPSoC

c a n h a v e a n i r re g ul a r s t ru c tu r e ( i .e . t o p ol o gy )

b ut : a ll P Es & r ou te rs a re o f t h e s am e ty p e. . .

e x c e p t f o r :

C o r e M a n a ge r ,

N o C M a n a ge r ,

M i c r o -P r o ce s so r c a n b e o f a d i ffe r en t t y p e

A n e x am p le f o r a h o mo g en e ou s M P So C i s t h e w el l - kn ow n I B M C e ll p r o ce s so r .A c o ar s e s ch e ma t ic s ke t ch s h ow s fi g ur e 1 5 3.

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F i gu r e 1 5 3 : I B M C e l l p r o ce s so r a s e x am p le f o r a h o mo g en e ou s M P So C

Heterogeneous MPSoC

T hi s M PS oC ty pe c an c on si st o f m ul ti pl e d iff er ent P Es ( e. g. x 86 s &ARM s) r ef e r t o fi g ur e 1 5 1 i n s e ct i on 1 3 .2 .

O n e e x am p le w i th m ul t ip l e D S Ps a n d A R M p r o ce s so r s i s t h e Q u a l co m mM S M, a c e ll u la r m o d em c hi p .

A t yp i ca l a p pl i ca t io n e x am p le t h at c a n e x pl o i t h e te r og e ne o us M P So C i st h e r e c e nt m o b i l e c o m m u n i ca t i o n s t a n da r d LT E ( L o n g T e r m E v o lu t i o n) .I n c o rr e sp o n di n g m o d em s f o r t h is s t an d ar d w e t yp i ca l l y fi n d:

~ 1 0 A R M s ( 3 k i nd s )~ 1 0 H W a c ce l er a to r s ( e .g . fi l te r s, d e co d e rs , e t c. )~ 1 0 D S Ps~ 5 0 m e m o ri e s

13.5 Hierarchy

N ew ch al l en ge s c om e u p w it h t he i nc re as in g s i ze o f M PS oC s. I f w e t a ke a l o oka t s ta te -o f -t he -a rt M PS oC s, w e u su al l y fi nd a f ew d oz en o r m ay b e a h un dr edo f p ro c es so rs . I n n ea r f ut ur e, w e w il l a lr ea dy fi nd m ul ti - pr o ce ss or c hi ps w i tht h ou s an d s o f P E s, u s ua l l y r e fe r re d t o a s m a ny - co r e s y st e ms - on - ch i p o r F P PA( fi e l d p r o g ra m m a b le p r o c e ss o r a r r ay p r o c e ss o r a r ra ys w i th p r og r am m ab l ei nt e rc o nn e ct s , a n al o go u s t o t o d ay s F P GA s [ fi el d p r og r am m ab l e g a te a r ra y] ) .

A s su m i ng s u ch l a rg e n um b er o f P E s, c o nv en ti o na l t o p ol o gi e s w i l l n o t b e s u ffi-c i e nt a n y m o r e . H i e r ar c hi c a l n e t wo r k t o p o l o g i e s w i t h P E s p o o l e d i n c l u s te r s a n dd y na m ic a ll y c o nt r ol l a bl e c l us t er s i ze s m i gh t b e a s o lu t io n t o h a nd l e s u ch l a rg eS o C s ( fi g u r e 1 5 4 ).

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F i g u re 1 5 4 : A d va n c e t o c l u s te r e d m a n y - c o re a r ch i t e ct u r e s

F i na ll y, w e c an a ls o s ta ck m ul ti pl e M P So Cs t og et he r o n a h ig he r l ev el o f h i-e ra rc hy. T hi s c ou ld e it he r b e d on e o n o ne p l an e ( 2D ) o r by s ta ck in g m ul ti pl eM P So Cs o n t op o f e ach o th er ( 3D ).

F i gu r e 1 5 5: H i er a rc hy o f m ul t ip l e M P So C

Design-Space Exploration

A no th er b ig c ha ll en ge i s t he d es i gn o f a n e ffic ie nt M P So C f or a s p ec ifi c p ro b-l e m. T h is t a sk i s s u b j ec t o f t h e s o c a ll e d D e s i gn S p ac e E x p l o ra t io n ( D S E ) t h a ts y st e ma t ic a ll y e va l ua t es a l a rg e n um b er o f d i ffe r en t a r ch i te c tu r e a n d m a pp i nga l te r na t iv es w i th r e sp e c t t o t h e s u it a bi l i ty f o r a g i ve n c o mp u ta t io n p r ob l em( e. g . s i gn al p ro c es si n g c h ai n f or U MT S m o de m) . B ro ad ly s p ea ki ng , w e c a nd i st i ng u is h t h re e d i ffe r en t a p pr o ac he s t o s t ar t o ff w i th a D S E:

1 . g re e nfi e ld a p pr o ac h: g i ve n a s y st e m p r o b le m fi nd a H W/ SW s ol u-tion f r ee c h oi c e o f h a r d wa r e a r ch i t e ct u r e , i n t er c o nn e c t , e t c . , s o f tw a r ep a r t it i o n i ng a n d H W / S W m a p p i n g

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2 . g i ve n s y s t em p r ob l em a l so g i ve n H W /S W s o lu t io n a s s t ar t in g p o i nt fi n d a n o p ti m al H W /S W m a pp i ng o r d y na m i c s ch e du l in g a l go r it h m f o r

t h e s y s t e m p r o b le m

3 . g i ve n H W / SW s o l u t io n C a n a s y s t e m p r o b l e m w i t h a d i ff e r e n t H W / S Ws o lu t io n b e fi t te d f o r t h e n e w H W / S W s o l ut i on ? ( p o r t a bi l i t y p r o b l em )

Outlook 2020

F i na l l y, w e w an t t o g i ve a b r ie f o u tl o o k c o nc e rn i ng t h e d e v el o pm e nt t r en d s o f M P So C s w i t h i n t h i s s e ct i on . To d ay s c hi p s c o n s is t o f a f e w d o ze n o r h un d re d so f p r o ce s so r s. H e re a r e t wo e x am p l e:

1 . nV i di a 5 0 0 P E s o n a c h i p i n 4 0 n m

2 . To m a h aw k c h i p 1 4 c o r e s i n 1 3 0 n m

A s t ro n g g r o w th o f t h e p a r a ll e l is m d e gr e e i s p r ed i ct e d a l r e ad y f o r n e ar f u tu r em a n y- c o r e c h i ps :

2 02 0: ~ 0 .1 M - 1 M c o re s 2 03 0 : ~ 1 B c o re s (! )

T h i s m e a n t h a t t h e p r e d i c t i o n f o r t h e n u m b e r o f c o r e s i n 2 0 3 0 i s c o m p a r a b l e t ot h e n um b er o f t r an s is t or s i n t o d ay s c hi p s!

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References

[ 1 ] P. P i r s ch ,Architekturen der digitalen Signalverarbeitung. S t ut t ga r t: Te u b-n e r , 1 9 9 6 .

[ 2] , Architectures for Digital Signal Processing. N ew Yo rk , NY , US A:J o hn W i le y & S o ns , I n c. , 1 9 98 .

[ 3] K . K . P a rh i,VLSI digital signal processing systems / design and implemen-tation. N ew Yo rk , N Y ; We in he im [ u. a. ] : W il ey, 1 99 9.

[ 4 ] V . P. N e l s on ,Digital logic circuit analysis and design. U pp e r S ad dl e R i ve r,N J : P r en ti c e H a ll , 1 9 95 .

[ 5 ] V . M a d i se t ti ,VLSI digital signal processors / an in

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