1ESPL

Wordlength Optimization withComplexity-and-Distortion Measure andIts Application to Broadband Wireless

Demodulator Design

Kyungtae Han and Brian L. Evans

Embedded Signal Processing LaboratoryWireless Networking and Communications Group

The University of Texas at Austin

2ESPL

Fixed-Point Design

• Digital signal processing algorithms– Often developed in floating point

– Later mapped into fixed point for digital hardware realization

• Fixed-point digital hardware– Lower area

– Lower power

– Lower per unit production cost

Idea

Floating-Point Algorithm

Quantization

Fixed-Point Algorithm

Code Generation

Target System

Algorithm

Level

Implem

entationL

evel

Range Estimation

Introduction

3ESPL

Fixed-Point Design

• Float-to-fixed point conversion required to target– ASIC and fixed-point digital signal processor core

– FPGA and fixed-point microprocessor core

• All variables have to be annotated manually– Avoid overflow

– Minimize quantization effects

– Find optimum wordlength

• Manual process supported by simulation– Time-consuming

– Error prone

Introduction

4ESPL

Optimum Wordlength

• Longer wordlength– May improve application

performance– Increases hardware cost

• Shorter wordlength– May increase quantization errors

and overflows– Reduces hardware cost

• Optimum wordlength– Maximize application performance

or minimize quantization error– Minimize hardware cost

Wordlength (w)

Cost c(w)Distortion d(w)[1/performance]

Optimumwordlength

Background

5ESPL

Wordlength Optimization

• Express wordlengths in digital system as vector

• Wordlength range for kth wordlength

• Cost function c

),,,,( 210 nwwww w

nkwww kkk ,...,1

RI nc :

n

kkk

n wcc1

)()(, wIw

Background

6ESPL

Wordlength Optimization

• Application performance function p

• Wordlength optimization problem

• Iterative update equation

• Good choice of update direction can reduce number of iterations to find optimum wordlength

reqPp )(w

},)(|)({min wwPpc reqI n

wwww

)()()1( hhh ww

Background

7ESPL

Sequential Search [K. Han et al. 2001]

• Greedy search based on sensitivity information (gradient)

• Example– Minimum wordlengths {2,2}

– Direction of sequential search

– Optimum wordlengths {5,5}

– 12 iterations

• Advantage: Fewer trials• Disadvantage: Could miss global optimum point

jjj sww

1

wopt

wb

w1

w2

dw1

dw2

5

5

Search Methods

8ESPL

Measures for Optimum Wordlength

• Complexity measure method [W.Sung and K.Kum 1995]

– Minimize complexity c(w) subject to constraint on distortion d(w)

– Update direction uses complexity sensitivity information• Distortion measure [K. Han et al. 2001]

– Minimize distortion d(w) subject to constraint on complexity c(w)

– Update direction uses distortion sensitivity information

?w0w1

wn

…

Complexity

w

wc

)( ?

w0w1

wn

…

Distortion

w

wd

)(

Measures

9ESPL

• Combine complexity measure with distortion measure by weighting factor (0≤α≤1)

• Tradeoffs between measuresby changing weighting factor

• Update direction uses both sources of sensitivity information

Complexity-and-Distortion Measure

)()1()()( www dcf

)]()1()([ ww dc

},)(,)(|)({min wwCcDdf reqI n

wwwww

Update direction

Objective function

Optimization problem

?w0w1

wn…

α

Complexity

Distortion

Measures

10ESPL

Broadband Wireless Access(IEEE 802.16a) Demodulator

w0: Input wordlength of orthogonal frequency division multiplex (OFDM) demodulator which performs a fast Fourier transform (FFT)w1: Input wordlength of equalizerw2: Input wordlength of channel estimatorw3: Output wordlength of channel estimator

EncoderOFDM

Modulator

WirelessChannelModel

OFDMDemodulator

ChannelEstimator

DecoderBit error

ratetester

DataSource

ChannelEqualizer

w0w1

w2w3

Case Study

11ESPL

Simulations

• Assumptions– Internal wordlengths of blocks have bee

n decided

– Complexity increases linearly as wordlength increases

• Required application performance– Bit error rate of 1.5 x 10-3 (without error

correcting codes)

• Simulation tool– LabVIEW 7.0

Case Study

Input Weight

FFT 1024

Equalizer

(right)

1

Estimator 128

Equalizer (upper)

2

12ESPL

Minimum Wordlengths

• Change one wordlength variable while keeping other variables at high precision{1,16,16,16},{2,16,16,16},...{16,1,16,16},{16,2,16,16},...……{16,16,16,15},{16,16,16,16}

• Minimum wordlength vector is {5,4,4,4}

Case Study

13ESPL

Number of Trials

• Start at {5,4,4,4} wordlength• Next wordlength combinatio

n for complexity measure (α = 1.0)

{5,4,4,4},

{5,5,4,4}, …

• Increase wordlength one-by-one until satisfying required application performance

Case Study

14ESPL

Complexity and Number of Iterations

• Each iteration computes complexity & distortion measures

• Distortion measure: high cost, low iterations

• Complexity-distortion: medium cost, fewer iterations

• Complexity measure: low cost, more iterations• Full search: low cost, more iterations

Method α w Complexity Iterations

Distortion Only

Complexity-Distortion

Complexity Only

Full Search

0

0.5

1

n/a

{10,9,4,10}

{7,10,4,6}

{7,7,4,6}

{7,7,4,6}

10781

7702

7699

7699

16

15

69

210

Case Study

15ESPL

Conclusion

• Summary– Fixed-point conversion requires wordlength optimization

– Develop complexity-and-distortion measure

– Complexity-and-distortion method finds optimal solution in one-third the time that full search takes for case study

• Future extensions for wordlength optimization– Automate selection of wordlength range

– Combine simulation-based and analytical approaches

– Employ genetic algorithms

16ESPL

Fixed-Point Representation

• Fixed point type– Wordlength

– Integer wordlength

• Quantization modes– Round

– Truncation

• Overflow modes– Saturation

– Saturation to zero

– Wrap-around

S X X X X X

Wordlength

Integer wordlength

SystemC formatwww.systemc.org

Introduction

X X X X X

Wordlength

Integer wordlength = 2Back

17ESPL

Full Search [W. Sung and K. Kum 1995]

• Exhaustive search of all possible wordlengths

• Advantages– Does not miss optimum points – Simple algorithm

• Disadvantage– Many trials (=experiments)

• Distance• Expected number of iterations

wb

wopt

w1

w2

dw1

dw2

21

24

22

23

5

5

Direction of full search:minimum wordlengths {2,2}

optimum wordlengths = {5,5}d = 6

trials = 24!

)1)(2)...(1()(

N

dddNddE N

FS

Ndwdwdwd ...21

Search Methods

Back

18ESPL

FFT Cost

1024

256log2

256Cost 2FFT

• N Tap FFT cost

• 256 Tap FFT cost

NN

2FFT log2

Cost

Back

19ESPL

• Uses complexity sensitivity information as direction to search for optimum wordlength

• Advantage: minimizes complexity• Disadvantage: demands large number of iterations

Complexity Measure [W.Sung and K.Kum 1995]

)()( ww cf

)(wc

},)(|)({min max wwDdfnI

wwww

Update direction

Objective function

Optimization problem

Measures

Back

20ESPL

• Applies the application performance information to search for the optimum wordlengths

• Advantage: Fewer number of iterations• Disadvantage: Not guaranteed to yield optimum

wordlength for complexity

Distortion Measure [K. Han et al. 2001]

)()( ww df

)(wd

},)(,)(|)({min maxmax wwCcDdfnI

wwwww

Update direction

Objective function

Optimization problem

Measures

Back

21ESPL

Broadband Wireless Access Demodulator Simulation

Case Study

22ESPL

Top-Level Simulation

Case Study

Back

23ESPL

Tools for Fixed-Point Simulation

• gFix (Seoul National University) – Using C++, operator overloading

• Simulink (Mathworks)– Fixed-point block set 4.0

• SPW (Cadence)– Hardware design system

• CoCentric (Synopsys)– Fixed-point designer

gFix a(12,1);gFix b(12,1);gFix c(13,2);c = a + b;

float a;float b;float c;c = a + b;

Introduction

Wordlengths determined manuallyWordlength optimization tool needed

24ESPL

Wordlength Optimization Methods

• Analytical approach– Quantization error model

– Overestimates signal wordlength

– For feedback systems, instability and limit cycles can occur

– Difficult to develop analytical quantization error model of adaptive or non-linear systems

• Simulation-based approach– Wordlengths chosen while observing error criteria

– Repeated until wordlengths converge

– Long simulation time

Background

25ESPL

Optimum Wordlength Search Methods

• Full search [W. Sung and K. Kum 1995]

• Min + b bit search [W. Sung and K. Kum 1995]

• Max – b bit search [M. Cantin et al. 2002]

• Hybrid search [M. Cantin et al. 2002]

• Sequential search [K. Han et al. 2001]

• Preplanned search [K. Han et al. 2001]

• Branch and bound search [H. Choi and W.P.Burleson 1994]

• Simulated annealing search [P.D. Fiore and L. Lee 1999]

Search Methods

26ESPL

Procedure

1. Range estimation– Find maximum and minimum values for each

2. Find minimum wordlengths– Defines starting wordlength values to use

3. Iterative search– Increase/decrease wordlengths one-by-one until meeting

specification using one of the measures

Case Study