SIGNAL MODELING

BY

DEBANGI GOSWAMI

CONTENTS Introduction 1.Title description 2.Need and importance of signal modeling Theory 1.Least mean square direct method. 1.1. Brief Overview 1.2. Disadvantages 2.Pade Approximation 3.Prony’s Approximation 4.Shanks Method 5.Stochastic process-ARMA,MA,AR Application Least Mean Square Inverse FIR filter Conclusion References

WHAT IS MODELING:

Modelling of signal is basically mathematical representation of signal.

Fourier series, Fourier transform are kind of signal models.

WHY MODELLINGNEED OF MODELING: 1.EFFICIENCY OF TRANSMISSION 2.PREDICTION

SIGNAL MODELING IN EFFICIENCY OF TRANSMISSION

Data compression x(n)Two approaches:

A. Point by point basis i.e x(0) followed by x(1)B. Record model parameters amplitude ,the frequency

and phaseSignal model

x(n)

SIGNAL MODELING IN PREDICTION

Signal interpolation & prediction(extrapolation)

PROBLEMS:

A. x(n) may be known for n=0,1…N-1 and goal is to predict the next value(N)

B. One of consecutive values may be missing or severely distorted over interval[N1,N2],recover x(n)..

SOLUTION

Suppose x(n) can be modeled with recursion of form:

x(n)

Given x(n) for nthe next value x(N),could be estimated

x(n)

STEPS IN MODELLING

PARAMETRIC FORM OF MODEL

MODEL PARAMETER THAT BEST

APPROXIMATE THE GIVEN SIGNAL

PARAMETRIC FORM OF MODEL

There are various approches for forming a model

1. Signal as sum of sinosoid2. Signal as sum of damped sinosoid x(n)3. Here signal is represented as the output of a

causal linear shift invariant filter that has a rational system function of the form:

H(z)== Signal model includes filter coefficient ap(k) and bq(k) and input signal.

TYPES OF SIGNAL TO BE MODELLED

SIGNALS

DETERMINISTIC

FIXED SIGNAL SUCH AS UNIT SAMPLE

SIGNALSREPRESENTED BY A FEW PARAMETER

RANDOMINPUT WILL BE STOCHASTIC

PROCESS,WHITE NOISE

MODEL PARAMETER

Models must be computationally efficient procedure for deriving the model parameters.

Various approaches to signal modeling THE LEAST SQUARE DIRECT METHOD THE PADE APPROXIMATION PRONY’S METHOD

1. Pole-Zero modeling

2. Shanks method

3. All-Pole Modeling DETERMINISTIC

4. Linear Prediction ITERATIVE PREFILTERING FINITE DATA RECORDS

1.The Autocorrelation Method

2.The Covariance Method THE STOCHASTIC MODELS-ARMA,AR,MA RANDOM

LEAST SQUARE (DIRECT) METHOD:

Models deterministic signal (n),as sample response of h(n)

x(n)=0 for n<0 and filter h(n) is causal

Modeling error by e’(n)

The error measure that is to be minimized is square error

h(n) and x(n) are both assumed to be zero for n<0, e’(n)=0 n<0

A necessary condition for the filter cofficient ap(k) and bq(k) to minimize the squared error is that partial derivative w.r.t ap(k)

And bq(k) vanish,i.e.

LEAST SQUARE (CONTINUED)

Using Parseval writing in frequency domain

Setting the partial derivative w.r.t ap*(k) equal to zero we have

Treating ap(k) and ap*(k) as independent variable

For k=1,2…..q,differentiating w.r.t bq(k)

LEAST MEAN(DIRECT) METHOD

BLOCK DIAGRAM

DISADVATAGES OF LM(DIRECT):

Not mathematically tracable and not amendable to real time processing.

Optimum set of model parameter are defined implicitely in terms of a set of p+q+1 non linear equation.If there are 10 poles we have to find inverse of (1010) matrix which is tedious job.

PADE APPROXIMATION Pade approximation only requires solving a set linear

equation. In Pade we force the filter output h(n) to be equal to given

signal x(n) for p+q+1 values of n.

In time domain,

Where h(n)=0 for n<0 and n>q.To find the cofficients ap(k) and bq(k) that gives an exact fit of data model in [0,p+q] we set h(n)=x(n)

In matrix form,

For soving the equation we use two step approach first solving for denominator ap(k) and then bq(k).ap(k) last p equations

CONCLUSIONS ON PADE APPROXIMATION

The model formed from Pade approximation will produce an exact fit to data over the interval[0,p+q].But has no guarantee on how accurate the model will be for n>p+q.

Pade approximation will give correct model parameters provided the model order is chosen to be large enough.

Since the Pade approximation forces the model to match the signal only over limited range of values,the model generated is not stable

PRONY’S METHOD The limitation of Pade approximation-Only uses values of the signal x(n) over the interval [0,p+q] to determine model parameter and over this interval, it models the signal without error.

There is no guarantee on how well the model will approximate the signal for n>p+q

POLE ZERO MODELLING:

Similar to pade x(n)=0 for n<0.A least square minimization of e’(n) results in set of non-linear equation for filter cofficient

Multiplying by Ap(z) we have new error

That is linear cofficients.In time domain:

Since bq(n)=0 for n>q,error can be explicitly written as

Instead of setting e(n)=0 for n=0,1,…….p+q as in Pade approximation,Prony’s method begins by finding the cofficient ap(k) that minimizes squared error.

Equivalently,

Prony’s normal equation:

In matrix form normal equation are

Consisely written as

ESTIMATION OF ERROR IN PRONY’S

Minimun value of modeling error:

e(n) and x*(n) are uncorrelated so from orthogonality principle the second term is zero.

Thus the minimum value:

PRONY NORMAL EQUATION

Augmenting the previous equation in this

Using matrix notation written as

COMPARISION LOW PASS FILTER CHARTERISTICS OF PRONY AND PADE

SHANKS APPROXIMATION

In Prony e(n)=0 for n=1, 2,….q.Although this allows the model to be exact over [0,q],this does not take into account data for n>q.

Shanks performs mininization of model error over entire length of data record.

Filter can be viewed as cascade of two filter Ap(z) and Bq(z)

g(n) can be computed using the equation:

To compute cofficient filter Bq(z),which produces the approximation x(n) when input to filter g(n).Instead of forcing e(n) to zero for first q+1 values of n as in Prony,Shank minimizes the squared error.

……..(1)

………….(2)

Substituting (1) in (2)

MSE AND COMPARISION WITH PRONYMinimum squared error

1.Shanks method is more involved than Prony’s method.

2.Extra compution of sequence g(n),autocorretion of g(n),cross correlation of g(n) and x(n).But in shank the mean square error reduces considerably

STOCHASTIC PROCESS:AR,MA,ARMA

Signals whose time evolution is governed by unknown factors like electrocardiogram,unvoiced speech,population statistics,economic data and seismograph

ARMA:

White noise is filtered with causal Linear shift invariant filter having p poles and q zeros.

YULE WALKER

STOCHASTIC PROCESS CONTINUED..

MODIFIED YULE WALKER

Here k=q+1,…..q+p

Comparing above eq with Pade approximation the data consist of a sequence of autocorrelations rx(k) .

AR:

A WSS AR process of order p is a special case of ARMA(p,q) process in which q=0.

STOCHASTIC CONTINUED….

In matrix form

MA: A MA process is generated by filtering unit variance white noise with an FIR filter order q

:

APPLICATION OF SIGNAL MODELLINGFIR LEAST SQUARES INVERSE FILTER Given a FIR filter the inverse filter relation can be written as

NEED: Equalization filter in digital communicatilon.Assume that channeal

transfer function is G(z),the eqalization filter is found by relation.

Procedure:

g(n) is causal filter to be equalized,the problem is to find FIR filter hN(n) of length N such that

The equation is same as shank,the solution of least square inverse filter is

FIR LS INVERSE FILTER CONTINUED

where

Matrix form:

From shank’s it follows the concise form: …………..(1)

PROBLEM ON LEAST SQUARE INVERSE FIR FILTERLet us find the FIR square inverse for system having a unit sample response

The inverse filter is

We first find the least square inverse of length N=2.The autocorrelation of g(n) is

Therefore

PROBLEM CONTINUED

With squared error of

The system function of least square inverse filter is

Which has a zero at

Least square inverse from equation 1 is

For n=1,2….N these equation may be represented in homogeneous form

The general solution to this equation is

………….(2)

PROBLEMc1,c2 are constants and determined from boundary condition at n=0 and n=N-1

i.e first and last equation

…(3)

Substituting (2) in (3)

Which after cancelling common term can be simplified to

PROBLEM CONTINUED

Finally with n=0

It follows that squared error is

Asymtotically seeing at N|

This is the inverse filter.i.e.

CONCLUSION The various methods of signal modelling for

both deterministic and stochastic process are discussed.

PROBLEM IN SIGNAL MODELING MODEL ORDER ESTIMATION:

In the cases assumed so far we have assumed that a model of given order to be found.In the absence of any information about the correct model order ,it becomes necessary to estimate what an appropiate model order should be.Misleading information may result in an inappropiate model order.

REFERENCES

STATISTICAL SIGNAL PROCESSING AND MODELING… MONSON H.HAYES

Signal Modeling Techniques In Speech Recognition by,Joseph Picone

Spectrum Estimation and Modeling by Petar M. Djuri´c State University of New York at Stony Brook Steven M. Kay University of Rhode Island

SOME REMARKS ON PADÉ-APPROXIMATIONS M.Vajta

Comparing Autoregressive Moving Average(ARMA) coefficients determination using Artificial Neural Networks with other techniques Abiodun M. Aibinu, Momoh J. E. Salami, Amir A. Shafie and Athaur Rahman Najeeb

THANKS FOR YOUR ATTENTION