ECE 407 Computer Aided Design for Electronic Systems. Instructor: Maria K. Michael. Overview. CAD tools for multi-level logic synthesis:

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1 407 Computr Aidd Dsign for Elctronic Systms Multi-lvl Logic Synthsis Instructor: Maria K. Michal 1 Ovrviw Major Synthsis Phass Logic Synthsis: 2-lvl Multi-lvl FSM CAD tools for multi-lvl logic synthsis: Factord forms & rprsntations Oprations MIS 2 407: Computr-Aidd Dsign for VLSI 1

2 2-lvl vs multi-lvl 2-lvl logic: ü Fast signal propagation (pro) o Larg gat fanins à rducd gat prformanc (con) o Larg ara (con) 2-lvl optimization Ø min # of products Ø min # of litrals multi-lvl optimization Ø not clar what nds to b optimizd! Ø # of gats? Ø fwr litrals? Ø mor of a synthsis problm (rasonabl & valid solution) than an optimization problm 3 Logic Synthsis Procss Two stps Tchnology indpndnt stp: factor out common sub-logic to rduc gat fanins à incrass circuit lvls Tchnology dpndnt stp: map rsults of prvious stp into particular implmntation using th givn tchnology library Ex. no 4OR or 3OR 4 407: Computr-Aidd Dsign for VLSI 2

3 Factord Forms In multi-lvl synthsis, Boolan xprssions can b rprsntd in a tr-lik structur à factord form: altrnats btwn OR and AND, using hirarchical parnthsis x. F = (a.d + d.b)(g + c.(+a.b) + a.c) + ( + c )(f.a) + k can rwrit as: F = F1.F2 + F4 + k + F whr, F1 = a.d+d.b F2 = g+c.f3+a.c k F4 F3 = +a.b + F1 + F2 + F5 F6 F4 = F5.F6 F5 = +c g c f a F6 = f.a a d d b c F3 + a c This IS a tr (no circls or rconvrgncis) Dpth indicats critical path lvl 5 a b Multi-lvl Simplification Critrion Ovrall, connctions cost mor! (thy xcd by far th # of gats) à minimiz # of litrals What s th # of litrals in th FF of prvious slid? (Fi also counts sinc rfrncd in th xprssion) à multi-lvl simplification rlis on a sris of oprations that manipulat th logic graph à Typically, known scripts ar usd to gt good rsults 6 407: Computr-Aidd Dsign for VLSI 3

4 Basic Oprations Usd for multi-lvl ntwork manipulation: Dcomposition Extraction Factoring Substitution Collapsing Polynomial Division & Multi-lvl factoring 7 Dcomposition Dcompos a singl Boolan xprssion into a collction of nw xprssions x. F = abc + abd + a c d + b c d Lt x = ab, y = c+d à F = xy + x y 7 gats, 8 litrals, 3 lvls 9 gats, 12 litrals, 2 lvls ( = ab(c+d) + (ab) (c+d) ) ( = abc + abd + (a +b )(c d ) ) ( = abc + abd + a c d +b c d ) 8 407: Computr-Aidd Dsign for VLSI 4

5 9 Extraction Applid to a collction of functions to idntify common subxprssions Hardst on to implmnt (must xprss functions in trms of factors and thn find common factors) x. f = (a+b)cd + g = (a+b) h = cd Lt x = a+b (common to f and g) c y = cd (common to f and h) d (x and y ar calld primary divisors/krnls/cubs) à f = xy + g = x h = y x = a + b y = cd 9 gats, 11 litrals, 3 lvls 7 gats, 11 litrals, 3 lvls c d a b c d a b a b f g f g h h Factoring R-xprss 2-lvl to multi-lvl, without any nw sub-functions Usually usd bfor xtraction, to find common subxprssions 6 gats, x. f = ac + ad + bc + bd + 9 litrals, 2 lvls à f = a(c+d) + b(c+d) + = (a+b)(c+d) + 4 gats, 5 litrals, 3 lvls Lt x = a+b, y = c+d Dcomposition will thn giv f = xy : Computr-Aidd Dsign for VLSI 5

6 Substitution R-xprss a function f with rspct to som othr function g Usd aftr factoring & dcomposition/xtraction x. f = AB + C, G = AB à f = G + C x. f = A + BC, G = A+B à f = G(A + C) 11 Collapsing Rvrs of substitution (usd to rduc th # of lvls to mt timing rquirmnts) x. f = (x + y(a + B)).C, x=cd, y=ad à f = (CD + AD(A + B)).C = (CD + AD + ABD).C = (CD + AD).C = CD + ACD = CD 1 gat, 2 litrals, 1 lvl 4 gats, 5 litrals, 4 lvls : Computr-Aidd Dsign for VLSI 6

Polynomial Division Most of th prvious oprations hav strong rlationship with multiplication/division of polynomials Lt F our function (διαιρετέος) D divisor (διαιρέτης) Q quotint (πηλίκο) R rmaindr

F = ac + ad + bc + bd + and D = a+b à F = D(c+d) +, with Q = c+d, R = ( F = ac+ad+bc+bd+ = a(c+d)+b(c+d)+ = (a+b)(c+d)+) ) 13 14 Polynomial Division (cont) Not: Rmmbr to think Boolan x.

7 Polynomial Division Most of th prvious oprations hav strong rlationship with multiplication/division of polynomials Lt F our function (διαιρετέος) D divisor (διαιρέτης) Q quotint (πηλίκο) R rmaindr (υπόλοιπο) à F = D.Q + R x. F = ac + ad + bc + bd + and D = a+b à F = D(c+d) +, with Q = c+d, R = ( F = ac+ad+bc+bd+ = a(c+d)+b(c+d)+ = (a+b)(c+d)+) ) Polynomial Division (cont) Not: Rmmbr to think Boolan x. F = ad + bcd + Challng in multi-lvl is d a divisor? synthsis: YES is a+bc a divisor? find YES good divisors à givs factord xprssionsà is a+b a divisor? YES Boolan (NO Algbraic) dcomposition/xtraction à F = d(a+bc) + substitutionà F = (a+bc)d + F = (a+b)q + if Algbraic, Q dos not xists; ls, Q = (a+c)d, sinc F = (a+b)(a+c)d+ = (aa+ab+ac+bc)d+ = (a+bc)d+ = ad+bcd+ 407: Computr-Aidd Dsign for VLSI 7

8 Multi-lvl Synthsis MIS: Brkly multi-lvl logic synthsis tool (now part of SIS), rad nots Also rad, Tutorial: Dsign of a Logic Synthsis Systm by Richard Rudll, DAC : Computr-Aidd Dsign for VLSI 8

Slide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS

Slide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt