Computer Architecture: A Constructive Approach

Sequential Circuits

ArvindComputer Science & Artificial Intelligence Lab.Massachusetts Institute of Technology

Revised February 21, 2012 (Slides from #16 onwards)

February 15, 2012 L3-1http://csg.csail.mit.edu/

6.S078

Combinational circuits

OpSelect - Add, Sub, ... - And, Or, Xor, Not, ... - GT, LT, EQ, Zero, ...

ResultComp?

A

BALU

Sel

OA0A1

An-1

Mux...

lg(n)Sel

O0

O1

On-1

A

Dem

ux ...

lg(n)

A

Deco

der ...

O0

O1

On-1

lg(n)

Such circuits have no cycles (feedback) or state

February 15, 2012 L3-2http://csg.csail.mit.edu/

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A simple synchronous state element

ff QDC

CDQ

Metastability

Data is sampled at the rising edge of the clock

Edge-Triggered Flip-flop

February 15, 2012 L3-3http://csg.csail.mit.edu/

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Flip-flops with Write Enables

ff QDC

EN

C

DQ

ENff QD

C

EN

01

ff QD

CEN

dangerous!

Data is captured only if EN is onFebruary 15, 2012 L3-4

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Registers

Register: A group of flip-flops with a common clock and enable

Register file: A group of registers with a common clock, input and output port(s)

ff

D

ff

D

ff

D

ff

D

ff

D

ff

D

ff

D

ff

QQQQQQQQ

D

CEn

February 15, 2012 L3-5http://csg.csail.mit.edu/

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Register FilesReadData1ReadSel1

ReadSel2WriteSel

Register file

2R + 1WReadData2

WriteData

WEClock

No timing issues in reading a selected register

register 1

WSel C

register 0

WData

WE RSel2

RData2

RSel1

RData1

February 15, 2012 L3-6http://csg.csail.mit.edu/

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Register Files and Ports

Ports were expensive multiplex a port for read & write

ReadData1ReadSel1ReadSel2WriteSel

Register file

2R + 1WReadData2

WriteData

WE

ReadDataReadSelR/WSel Register

file1R + 1R/W

R/WData

WE

February 15, 2012 L3-7http://csg.csail.mit.edu/

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We can build useful and compact circuits using registersExample: Multiplication by repeated addition

February 15, 2012 L3-8http://csg.csail.mit.edu/

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Multiplication by repeated addition

1101 (13)1011 (11)1101

+ 1101 + 0000 + 1101 10001111 (143)

b Multiplicanda Muliplier *

add4

add4

add4

a0

a1

a2

a3

m0

m1

m2

m3

0

mi = (a[i]==0)? 0 : b;

February 15, 2012 L3-9http://csg.csail.mit.edu/

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Combinational 32-bit multiplyfunction Bit#(64) mul32(Bit#(32) a, Bit#(32) b);

Bit#(32) prod = 0; Bit#(32) tp = 0;

  for(Integer i = 0; i < 32; i = i+1)  begin     Bit#(32) m = (a[i]==0)? 0 : b;     Bit#(33) sum = add32(m,tp,0);     prod[i] = sum[0];     tp = truncateLSB(sum);  end  return {tp,prod};endfunction

February 15, 2012 L3-10http://csg.csail.mit.edu/

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Design issues with combinational multiply

Lot of hardware 32-bit multiply uses 31 addN circuits

Long chains of gates 32-bit ripple carry adder has a 31 long chain

of gates 32-bit multiply has 31 ripple carry adders in

sequence!

The speed of a combinational circuit is determined by its longest input-to-output path

February 15, 2012 L3-11http://csg.csail.mit.edu/

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Expressing a loop using registersfunction Bit#(64) mul32(Bit#(32) a, Bit#(32) b);

Bit#(32) prod = 0; Bit#(32) tp = 0;

  for(Integer i = 0; i < 32; i = i+1)  begin     Bit#(32) m = (a[i]==0)? 0 : b;     Bit#(33) sum = add32(m,tp,0);     prod[i] = sum[0];     tp = truncateLSB(sum);  end  return {tp,prod};endfunction

February 15, 2012 L3-12http://csg.csail.mit.edu/

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Need registers to hold a, b, tp prod and i

Update the registers every cycle until we are done

Expressing a loop using registers

February 15, 2012 L3-13http://csg.csail.mit.edu/

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for(Integer i = 0; i < 32; i = i+1)  begin     let s = f(s);       endreturn s;

s

fs0

i (init 0)

+1

<32

Sequential multiply Reg#(Bit#(32)) a <- mkRegU(); Reg#(Bit#(32)) b <- mkRegU();

Reg#(Bit#(32)) prod <-mkRegU();Reg#(Bit#(32)) tp <- mkRegU();Reg#(Bit#(6)) i <- mkReg(32);

  rule mulStep if (i < 32);     Bit#(32) m = (a[i]==0)? 0 : b;     Bit#(33) sum = add32(m,tp,0);     prod[i] <= sum[0];     tp <= sum[32:1]; i <= i+1;  endrule

state elements

a rule to describe dynamic behavior

February 15, 2012 L3-14http://csg.csail.mit.edu/

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So that the rule won’t fire until i is set to some other value

Dynamic selection requires a mux

February 15, 2012 L3-15http://csg.csail.mit.edu/

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a[i]a

i

a[0],a[1],a[2],…a

>>

0

Replacing repeated selections by shifts Reg#(Bit#(32)) a <- mkRegU(); Reg#(Bit#(32)) b <- mkRegU();

Reg#(Bit#(32)) prod <-mkRegU();Reg#(Bit#(32)) tp <- mkRegU();Reg#(Bit#(6)) i <- mkReg(32);

  rule mulStep if (i < 32);     Bit#(32) m = (a[0]==0)? 0 : b; a <= a >> 1;     Bit#(33) sum = add32(m,tp,0);    prod <= {sum[0], (prod >> 1)[30:0]};     tp <= sum[32:1]; i <= i+1;  endrule

February 15, 2012 L3-16http://csg.csail.mit.edu/

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Sequential MultiplybIn

b

ai

February 15, 2012 L3-17http://csg.csail.mit.edu/

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== 32

0

done

+1

prod

result (low)

<<

[30:0]

aIn

<<

31:0

tp

result (high)

add

&……………&0

310

0

s1 s1

s1

s2 s2 s2 s2

s1

s1 = start_ens2 = start_en | !done

32:1

rdyenInt#(32)

Int#(64)rdy

star

tre

sult M

ultip

lym

odul

e

Int#(32)

implicit conditions

interface Multiply; method Action start (Int#(32) a, Int#(32) b); method Int#(64) result();endinterface

Multiply Module

Many different implementations can provide the same interface: module mkMultiply (Multiply)

February 15, 2012 L3-18http://csg.csail.mit.edu/

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Multiply Modulemodule mkMultiply32 (Multiply32); Reg#(Bit#(32)) a <- mkRegU(); Reg#(Bit#(32)) b <- mkRegU();

Reg#(Bit#(32)) prod <-mkRegU();Reg#(Bit#(32)) tp <- mkRegU();Reg#(Bit#(6)) i <- mkReg(32);

  rule mulStep if (i < 32); Bit#(32) m = (a[0]==0)? 0 : b;     Bit#(33) sum = add32(m,tp,0);    prod <= {sum[0], (prod >> 1)[30:0]};     tp <= sum[32:1]; a <= a >> 1; i <= i+1;   endrule method Action start(Bit#(32) aIn, Bit#(32) bIn) if (i == 32); a <= aIn; b <= bIn; i <= 0; tp <= 0; prod <= 0; endmethod method Bit#(64) result() if (i == 32); return {tp,prod}; endmethod endmodule

February 15, 2012 L3-19http://csg.csail.mit.edu/

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State

Internalbehavior

Exte

rnal

inte

rfac

e

Module: Method Interface

February 15, 2012 L3-20http://csg.csail.mit.edu/

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s1 = start_ens2 = start_en | !done

bInb

ai

== 32

0

done

+1

prod

result (low)

<<

[30:0]

aIn

<<

31:0

tp

result (high)

add

&……………&0

310

0

s1 s1

s1

s2 s2 s2 s2

s1

32:1

aInbIn

enrdy

resultrdy

star

tre

sult

s1

ORs1 s2

rdyenabInt#(32)

Int#(64)rdy

star

tre

sult M

ultip

lym

odul

e

Int#(32)

implicit conditions

interface Multiply; method Action start (Int#(32) a, Int#(32) b); method Int#(64) result();endinterface

Polymorphic Multiply Module

n

#(Numeric type n)

n

Tadd(n+n)

n nTAdd#(n,n)

n could beInt#(32),Int#(13), ...

The module can easily be made polymorphic

http://csg.csail.mit.edu/SNU L02-21January 11, 2012

Sequential n-bit multiplymodule mkMultiplyN (MultiplyN); Reg#(Bit#(n)) a <- mkRegU(); Reg#(Bit#(n)) b <- mkRegU();

Reg#(Bit#(n)) prod <-mkRegU();Reg#(Bit#(n)) tp <- mkRegU();

Reg#(Bit#(Add#(Tlog(n),1)) i <- mkReg(n); nv = valueOf(n);  rule mulStep if (i < nv); Bit#(n) m = (a[0]==0)? 0 : b;     Bit#(TAdd#(n,1)) sum = addn(m,tp,0);    prod <= {sum[0], (prod >> 1)[(nv-2):0]};     tp <= sum[n:1]; a <= a >> 1; i <= i+1;   endrule method Action start(Bit#(n) aIn, Bit#(n) bIn) if (i == nv); a <= aIn; b <= bIn; i <= 0; tp <= 0; prod <= 0; endmethod method Bit#(TAdd#(n,n)) result() if (i == nv); return {tp,prod}; endmethod endmoduleFebruary 15, 2012 L3-22

http://csg.csail.mit.edu/6.S078