presentation on 2-d digital filters

7/30/2019 Presentation on 2-d Digital filters

1/37

8 - 1

Under the Guidance of

Dr. Haranath Kar

Professor

Presented By

Rohan A. Borgalli

2011el21

Department of Electronics & Communication

Engineering

Motilal Nehru National Institute of Technology

Allahabad-211004

Summer Training PresentationOn

Study of a criteria for elimination of overflowoscillation in fixed-point 2-D state-space digital

filter employing 2s complement overflow arithmetic


2/37

Outline Introduction

State space representation

Representation of numbers Finite Wordlength Effects

Linear Matrix Inequalities (LMI)

Standard Models of 2-D Linear Systems

Existing Criteria Results

Conclusions

8 - 2


3/37

Introduction

Digital filter

1-D and 2-D Digital filter Stability of Filter

8 - 3


4/37

State space representation

8 - 4


5/37

Representation of numbers In theoretically, we were considering implementations of discrete-time

systems without any considerations of finite-word-length effects thatare inherent in any digital realization, whether in hardware or software.

Let us consider first two different representations of numbers.

Fixed-point representation

A real numberXis represented as:

8 - 5


6/37

Floating-Point Representation

8 - 6

The one-bit sign field is the sign of the stored value.

The size of the exponent field, determines the range of values

that can be represented.

The size of the significand determines the precision of the

representation.

Computer representation of a floating-point number consists ofthree fixed-size fields:


7/37

Floating point Vs Fixed point Cost

Power consumption

Speed

Complexity

Dynamic range

Relaxation in Design

Tread-off between

We consider fixed-pointfilter implementations, with a `short word-

length

8 - 7


8/37

Finite Wordlength EffectsThe finite word-length problem : We assumed number representation can be performed to an infinite

precision.

In practice, numbers can be represented only to a finite precision, and

arithmetic operations are subject to errors (truncation/rounding/...)

Issues:- quantization of filter coefficients- quantization & overflow in arithmetic operations

8 - 8


9/37

Cont

8 - 9

Finite word-length effects in arithmetic operations: Inlinear filters, have to consider additions & multiplications

Addition:

if, two B-bit numbers are added, the result has (B+1) bits.

Multiplication:

if a B1-bit number is multiplied by a B2-bit number, the

result has (B1+B2-1) bits.

For instance, two B-bit numbers yield a (2B-1)-bit product

Typically (especially so in an IIR (feedback) filter), the result of anaddition/multiplication has to be represented again as a B-bit number(e.g. B=B).


10/37

Cont

8 - 10

Option-1: Most significant bits (MSBs)

If the result is known to be upper bounded such that 1 or more MSBsare always redundant, these MSBs can be dropped without loss ofaccuracy.

better usage of available word-length

better SNR.

Option-2 : Least significant bits (LSBs)

Rounding/truncation/ to B bits introduces quantization noise.Quantization, however, is a deterministic non-linear effect, which may

give rise to limit cycle oscillations.


11/37

Optimum Wordlength Longer wordlength

May improve applicationperformance

Increases hardware cost Shorter wordlength

May increase quantization errorsand overflows

Reduces hardware cost

Optimum wordlength

Maximize application performanceor minimize quantization error

Minimize hardware cost

8 - 11

Cost c(w)

Distortion d(w)

[1/performance]

Optimum

wordlength

Wordlength (w)


12/37

Quantization Quantization is an interpretation of a continuous

quantity by a finite set of discrete values

8 - 12

(a) Roundoff. (b) Magnitude truncation. (c) Value truncation


13/37

limit cycle

8 - 13

zero-input limit cycle oscillations :

Example:

y[k] = -0.625.y[k-1]+u[k]4-bit rounding arithmetic

input u[k]=0, y[0]=3/8

output y[k] = 3/8, -1/4, 1/8, -1/8, 1/8, -1/8, 1/8, -1/8,

1/8,..

=oscillations in the absence of input (u[k]=0)


14/37

Cont

8 - 14

Example: y[k] = -0.625.y[k-1]+u[k]

4-bit truncation (instead of rounding)

input u[k]=0, y[0]=3/8

output y[k] = 3/8, -1/4, 1/8, 0, 0, 0,.. (no limit cycle!)Example: y[k] = 0.625.y[k-1]+u[k]

4-bit rounding

input u[k]=0, y[0]=3/8

output y[k] = 3/8, 1/4, 1/8, 1/8, 1/8, 1/8,..

Example: y[k] = 0.625.y[k-1]+u[k]

4-bit truncation

input u[k]=0, y[0]=-3/8

output y[k] = -3/8, -1/4, -1/8, -1/8, -1/8, -1/8,..


15/37

Overflow Characteristics

8 - 15

( )f x

( )f x

( )f x

( )f x

p p

pp

-p

-p -p

-p

-p

-p p p

2p 2p-2p-2p

( )a ( )b

( )c ( )d

x

x

x

x

(a) Saturation. (b) Zeroing. (c) Twos compliment. (d) Triangular.


16/37

Effect of 2s Complement Overflow Circular representation

Intermediate overflows do not alter the final result

This is not the case for saturation

Example of N = 3 bits: Calculate x = 3+2-4, the theoretical result is 1

With 2s complement overflow:

Calculate first y=(3+2)= 011+010 =101 =-3 overflow

Then (y-4)=101+100=1 001 = 1 and carry =1 correctresult

With saturation:

Calculate first y=(3+2)=3 saturation

Then (y-4) = 011+100=111=-1 wrong result 8 - 16


17/37

8 - 17

tastxif

xxAxx

0)(

)0( 0

Consider a system represent in state space:

All the eigenvalues of the system have negative real

parts (i.e. in the LHP)

Asymptotically stable

Asymptotically Stability condition


18/37

Lyapunov stability A state of an autonomous system is called an equilibrium state,

if starting at that state the system will not move from it in the absence

of the forcing input.

8 - 18

)),(),(( ttutxfx

0,0),0,( tttxf e )(

1

1

32

10tuxx

1x

Equilibrium point

0

00

32

10

0)(

2

1

2

1

e

e

e

e

x

x

x

x

tulet

example

2x


19/37

Lyapunovs method

8 - 19

0 AifAxx

pxxxVT)(

)( PAPAQT

Qxx

xPAPAx

PAxxPxAx

xPxPxxxV

T

TT

TTT

TT

)(

)(

Consider linear autonomous system

If Q is p.d. then is n.d.)(xV

0x is asymptotically stable

Let Lyapunov function

..0 SEx


20/37

LMI editor

8 - 20


21/37

System under consideration

8 - 21

tv

n

vvvh

m

hhhlkylkylkylkylkylkylkylkylky )],()....,(),(),(|),().....,(),,(),,([),( 321321

),(

),(

2221

1211

lkx

lkx

AA

AAv

h

Ax(k,l),==


22/37

Standard Models of 2-D Linear Systems Roesser Model(RM)

8 - 22

jDuijixjix

CCjyi

juiB

B

jix

jix

AA

AA

jix

jix

v

h

v

h

v

h

,),(

),(

21,

,2

1

),(

),(

2221

1211

)1,(

),1(

where, Rn2 is the vertical state vector at the point (i, j ) Z +

Rn1

is the horizontal state vector at the point (i, j )

Z+ui,j Rm is the input vector

yi,j Rp is the output vector

Z + is the set of nonnegative integers

),( jix v

),( jix

h


23/37

Cont

Second 2-D Fornasini-Marchesini model (SF-MM)

8 - 23

),(),(),(

,

)1,(),1()1,(2),1()1,1( 211

jiDujiCxjiy

Zji

jiuBjiuBjixAjixAjix

First 2-D Fornasini-Marchesini model (FF-MM)

),(),(),(

,

),()1,(2),1(),()1,1( 10

jiDujiCxjiy

Zji

jiBujixAjixAjixAjix


24/37

Existing criterion P-DTPD > o

Modified criterion G-ATGA>0

8 - 24

v

v

vvv

h

h

hhhvh

M

DMDGand

M

DMDGGGGwhere

0

0,

0

0,,


25/37

Stability Analysis using LMI For choosing A matrix to satisfy condition we get

unknown matrices Using LMI Toolbox

8 - 25

Thus, present system is globally asymptotically stable.


26/37

LMI editor window

8 - 26


27/37

Cont

8 - 27


28/37

MATLAB program algorithm Matrix Initialization

A=

Initial State of System

8 - 28

2221

1211

AA

AA=

300,1

1),0('

1

1

1

),0(

300,1

1)0,('

1

11

)0,(

,0,1,30,0),(,0),(

,0,1,30,0),(,0),(

jjxjx

iixix

ijjixjix

jijixjix

vh

vh

vh

vh


29/37

Cont Output of the System

2s Complement Overflowarithmetic

Repeat above Steps till

i or j or (i+j)

8 - 29

),(

),(

2221

1211

jix

jix

AA

AAv

h

Y(i,j)=

)},({

)},({

)1,(

),1(

jiyR

jiyR

jix

jixv

h

v

h


30/37

8 - 30

x1h(i,j)

Cont

3-D Mesh Plot


31/37

8 - 31

x2h(i,j)


32/37

8 - 32

x3h(i,j)


33/37

8 - 33

x1v(i,j)


34/37

8 - 34

x2v(i,j)


35/37

Conclusion The stability Analysis has been carried out by solving

an LMI

The Modified criterion is less restrictive thanPrevious.

2-D fixed point state space Roesser model theoscillations are tends to zero as i or j or

(i+j)

8 - 35


36/37

References R.P. Roesser, A discrete state-space model for linear image processing,

IEEE Trans. Automat. Control 20 (1) (1975) 110.

N.G. El- Agizi, M.M. Fahmy, Two-dimensional digital filters with no

overflow oscillations, IEEE Transactions on Acoustics, Speech andSignal Processing 27 (1979) 465469.

D. Liu and A. N. Michel, Asymptotic stability of discrete-time systems

with saturation nonlinearities with application to digital filters, IEEE

Trans. Circuits Syst. I, vol. 39, no. 10, pp. 798-807, Oct. 1992

Haranath Kar: A new criterion for the global asymptotic stability of 2-D

state-space digital filters with two's complement overflow

arithmetic. Signal Processing 92(9): 2322-2326 (2012) 8 - 36


37/37

Thank You

presentation on 2-d digital filters

Documents