     # VLSI Signal Processing - .VLSI Signal Processing Spring 2015 In-Cheol Park Dept. of EE, KAIST - 2

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• VLSI Signal Processing Spring 2015

In-Cheol Park Dept. of EE, KAIST

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Main Textbook:

Keshab K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementations, John Wiley & Sons, 1999.

A: 40 ~ 60%

B: 40 ~ 60%

C: 0 ~ 10%

Homework

Projects

Presentations

Attendance

• - 4 -

Graph Representations

Iteration Bound

Pipelining & Parallel Processing

Retiming

Unfolding

Folding

Systolic Architecture

Fast convolution

Algorithmic Strength Reduction

Low-Power Design

Numerical Strength Reduction

Bit-level Arithmetic Architecture

Redundant Arithmetic

Etc.

• DSP Representations

In-Cheol Park Dept. of EE, KAIST

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• Shortest Path Algorithms

2015. 03

In-Cheol Park Dept. of EE, KAIST

• - 17 -

,

otherwise ,

exists if , edge theoflength the)(

VUVUVUw

ee

e

• - 18 -

A single-source shortest path algorithm for a graph with non-negative edge path costs

Published in 1959 by Edsger Dijkstra

O(|V|2) , where |V| is the number of vertices

O(|E|+|V|log|V|) with a min-priority queue, where |E| is the number of edges,

• - 19 -

1. Assign a tentative distance value to every node .

zero for the initial node and infinity for all other nodes.

2. Set the initial node as current. Mark all other nodes unvisited. Create the unvisited set.

3. For the current node, consider its unvisited neighbors and calculate their tentative distances. Compare the newly calculated tentative distance to the current value and assign the smaller one.

For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B (through A) will be 6 + 2 = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, keep the current value.

4. Mark the current node as visited and remove it from the unvisited set.

• - 20 -

5. If the destination node has been marked visited, or if the smallest tentative distance among the nodes in the unvisited set is infinity (which occurs when there is no connection between the initial node and remaining unvisited nodes), then stop.

6. Select the unvisited node with the smallest tentative distance, and set it as the new "current node. Go back to step 3.

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- A Single-point shortest path algorithm that finds the shortest path from a chosen node to each node in the graph. - Finds the shortest path if no negative cycles exist. - O(n3) complexity

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Shortest paths from U=2;

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Shortest paths from U=2;

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- An all-points shortest path algorithm that finds the shortest path Between any pair of nodes in the graph. - Finds the shortest path if no negative cycles exist. - O(n3)

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• Euclidean GCD Algorithm

In-Cheol Park Dept. of EE, KAIST

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.032060

360163

60632186

631865993

GCD of (993, 186)

GCD of (993, 186) = 3

• - 29 -

GCD (a, 0) = a

GCD (a, a) = a

GCD (a, b) = GCD (b, a-b) when a > b

GCD (a, b) = GCD (b, a mod b) when a > b

GCD (a, b) LCM (a, b) = a b

• - 30 -

Theorem (Euclidean GCD Algorithm)

Given 2 positive integers s and r, their GCD can be computed by iteratively

applying the division algorithm. Suppose r < s, the algorithm is continued

iteratively as shown below:

,

,

,

,

,

11

12

3231

212

11

nnn

nnnn

rQr

rrQr

rrQr

rrQr

rrQs

and the process strops whenever the remainder is zero. The last nonzero

remainder rn is the GCD.

• - 31 -

.),( brasrsGCD

321

2132

12

)()1(

)(

),(

nnnnn

nnnnn

nnn

n

rQrQQ

rQrQr

rQr

rrsGCD

.186169933

1861)1865993(3

1861633

)632186(163

60163

3)186,993(

GCD

Corollary: For any positive integers s and r,

there exist integers a and b such that

Example,

2 1i i i ir r Q r

• - 32 -

)()()()( xrxdxQxc

)()()(

),()()()(

),()()()(

),()()()(

),()()()(

11

12

3231

212

11

xrxQxr

xrxrxQxr

xrxrxQxr

xrxrxQxr

xrxrxQxs

nnn

nnnn

),())(),(( xrxrxsGCD n

Theorem (Division algorithm for polynomials)

For 2 polynomials c(x) and nonzero d(x), there is a unique pair of polynomials

Q(x) and r(x) such that

Theorem (Euclidean GCD algorithm for polynomials)

Given the polynomials s(x) and r(x) over certain field, their GCD can be

computed by applying iteratively the division algorithm. Suppose deg s(x) >=

deg r(x) >= 0, then this computation is carried out iteratively as

and the process stops when the remainder is zero. Then

where is a scalar.

• - 33 -

).()()()())(),(( xrxbxsxaxrxsGCD

3 2 2( ) 1, ( ) 2 1.s x x x x r x x x

( ( ), ( )) 2( 1)GCD s x r x x

Corollary : For any polynomials s(x) and r(x),

there exist polynomials a(x) and b(x) such that

Example,

).22)(2

1

2

1()(

),22()()1()(

xxxr

xxrxxs

)()1()())(),(( xrxxsxrxsGCD

( ) 1 and ( ) ( 1).a x b x x

• Scheduling and Allocation Techniques

In-Cheol Park Dept. of EE, KAIST

• - 35 -

High-Level Synthesis

is believed to be NP-hard

• - 36 -

Scheduling: assigning every node of the DFG to control time steps Resource Allocation: assigning operations to hardware to

minimize the amount of hardware Resourceconstrained synthesis (area-constrained) Time-constrained synthesis (performance-constrained) High-level synthesis Low-level optimization

• - 37 -

yxuyyxdx

ydy

dx

du

dx

dyyuyyxy

3333

033

2

2

}

;;;

;*

);**3()***3(

;

{while

uluylyxlx

dxuyyl

dxydxuxuul

dxxxl

a) (x

Add the nodes of the DFG to the schedule one node at a time Two simple methods: As Soon as Possible (ASAP), As Late as Possible (ALAP)

Example: Second-order differential equation

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• - 39 -

Assumption: Unlimited Resources

• - 40 -

Assumption: Unlimited Resources

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j

j

L

Si

ixiDGjForceSelf )](*)([)(_

25.0

))5.0(*333.2())5.0(*833.2(

))5.00(*333.2())5.01(*833.2(

))2(*)2(())1(*)1((

)2()1()1(_

44

444

xDGxDG

ForceForceForceSelf

MM

25.0

))5.0(*333.2())5.0(*833.2(

))2(*)2(())1(*)1((

)2()1()2(_

44

444

xDGxDG

ForceForceForceSelf

MM

• - 45 -

75.0

))5.0(*833.0())5.0(*333.2(

))5.01(*833.0())5.00(*333.2(

))3(*)3(())2(*)2((

)3(_)2(_)2(_

88

884

xDGxDG

ForceSelfForceSelfForceSucc

MM

4 4 4(2) _ (2) _ (2)

0.25 0.75 1.00

Force Self Force Succ Force

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1

minimize ( * )k k

m

t t

k

c M

kti

k

FUo

tji mkTjforMx .1 ,1 ,0,

.1 ,1, niforxi

i

L

Sj

ji

. ,1)*()*( ,, ki

L

Sj

L

Sj

jkji ooxjxji

i

i

i

such that

; Resource Constraints

; Allocation Constraints

; Dependence Constraints

• - 52 -

u.t. d c u.t. an cAssume alumult 15

)*1*5( alumult MMMinimize

.0

0

0:

3,53,8

2,52,82,42,3

1,51,41,21,1

mult

mult

mult

Mxx

Mxxxx

MxxxxMultiplier

.0

0

0

0 :

4,104,94,11

3,103,63,93,7

2,102,62,9

1,6

alu

alu

alu

alu

Mxxx

Mxxxx

Mxxx

MxALU

• - 53 -

.1 ,1

1 ,1

1 ,1

1 ,1 ,1 ,1 1,

10,410,310,26,36,26,1

9,49,39,25,35,25,1

8,38,24,24,1

11,47,33,22,11,1

xxxxxx

xxxxxx

xxxx

xxxxx

.143232

143232

1322

10,410,310,26,36,26,1

9,49,39,25,35,25,1

8,38,24,24,1

xxxxxx

xxxxxx

xxxx

.1,1,1

,1,1,1,1

,1,1,1,1

11,410,29,4

8,37,36,15,3

4,23,22,11,1

xxx

xxxx

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