Adv DSPSpring-2015

Lecture#9Optimum Filters (Ch:7)

Wiener Filters

Introduction Estimation of one signal from another is one

of the most important problems in signal processing

In many applications, the desired signal is not available or observed directly. Speech, Radar, EEG etc

Desired signal may be noisy and highly distorted

In very simple and idealized environments, it may be possible to design classical filters such as LP,HP or BP, to restore the desired signal from the measured data.

Introduction These classical filters shall rarely be

optimal in the sense of producing the “best” estimate of the signal.

Class of filters called OPTIMUM DIGITAL FILTERS

Two important Types Digital Wiener Filter Discrete Kalman Filter

Wiener Filter In the 1940’s, Norbert Wiener pioneered

research in the problem of designing a filter that would produce the optimum estimate of a signal from a noisy measurement or observation.

Wiener Filter The Wiener filtering problem, is to design a filter

to recover a signal d[n] from noisy measurement

Assuming that both d[n] and v[n] are wide-sense stationary random process, Wiener considered the problem of designing the filter that would produces the minimum mean square estimate of d[n] (by using x[n])

Wiener Filter Thus the error terms are (Mean Square Error)

Problem is to find the filter (filter coefficients, FIR or IIR) that minimizes ξ (minimum mean square error).

Wiener Filter Depending on how the signals x[n] and

d[n] are related to each other, a number of different and important problems may be cast into Wiener filtering framework.

These problems are Filtering Smoothing Prediction De-convolution

Wiener Filter

Wiener Filter

The FIR Wiener Filter Design of an FIR Wiener Filter

That will produce the minimum mean-square estimate of a given process d[n] by filtering a set of observations of statistically related process x[n]

It is assumed that x[n] and d[n] are jointly wide-sense stationary with known autocorrelations rx[k] and rd[k], and known cross-correlation rdx[k]

The FIR Wiener Filter Denoting the unit sample response of the

Wiener filter by w[n], and assuming (p-1)st order filter, the system function W(z) is

With x[n] as input to this filter, the output is (using DT convolution)

The FIR Wiener Filter Wiener filter design problem requires that

we find the filter coefficients w[k], that minimize the mean-square error:

Using optimization steps, for k=0,1,…,p-1

Error

Taking PartialDerivative forkth value

Orthogonality Principle

After putting e[n] back

The FIR Wiener Filter Since x[n] and d[n] are jointly WSS

Set of ‘p’ linear equations in the ‘p’ unknowns w[k] for k=0,1,….,p-1

The FIR Wiener Filter In matrix form using the fact that

autocorrelation sequence is conjugate symmetric rx[k]=r*x[-k]

Wiener-Hopf Equations

The FIR Wiener Filter Wiener-Hopf Equations

The FIR Wiener Filter The minimum mean square error in the

estimate of d[n] is

Equal to zero bz of Orthognality

The FIR Wiener Filter After taking expected values

In Vector Notation

The FIR Wiener Filter

Filtering In the filtering problem, a signal d[n] is to

be estimated from a noise (v[n]) corrupted observation x[n]

Assuming that noise has a zero mean and it is uncorrelated with d[n]

Filtering The cross-correlation between d[n] and

x[n] becomes

Filtering With v[n] and d[n] uncorrelated, it follows

To simplify these equations, specific information about the statistic of the signal and noise are required Example:7.2.1

Linear Prediction With noise-free observations, linear

prediction is concerned with the estimation (prediction) of x[n+1] in terms of linear combination of the current and previous values of x[n]

Linear Prediction An FIR linear predictor of order ‘p-1’ has

the form

where w[k] for k=0,1,…,p-1 are the coefficients of the prediction filter.

Linear predictor may be cast into Wiener filtering problem by setting d[n]=x[n+1]

Linear Prediction Setting up the Wiener-Hopf equations

Ex:7.2.2

Linear Prediction in noise With noise present, a more realistic model

for linear prediction is the one in which the signal to be predicted is measured in the presence of noise.

Linear Prediction in noise Input to Wiener filter is given by

Goal is to design a filter that will estimate x[n+1] in terms of linear combination of ‘p’ previous values of y[n]

Linear Prediction in noise The Wiener-Hopf equations are

If the noise is uncorrelated with signal x[n], then Ry, the autocorrelation matrix for y[n] is

Linear Prediction in noise The only difference between linear

prediction with and without noise is in the autocorrelation matrix for the input signal.

In the case of noise that is uncorrelated with x[n],

Multi-Step Prediction In one-step linear prediction, x[n+1] is

predicted in terms of linear combination of the current and previous values of x[n]

In multi-step prediction, x[n+δ] is predicted in terms of linear combination of the ‘p’ values x[n],x[n-1],…,x[n-p+1]

Multi-Step Prediction In multi-step prediction

In multi-step prediction, sincePositive Integer

Multi-Step Prediction Wiener-Hopf equations are

Noise Cancellation The goal of noise canceller is to estimate a signal

d[n] from a noise corrupted observation, that is recorded by primary sensor.

Unlike the filtering problem, which requires that the autocorrelation of the noise be known, with noise canceller this information is obtained from a secondary sensor that is placed within the noise field.

Noise CancellationPrimary sensor

Secondary sensor

Noise Cancellation Although the noise measured by secondary

sensor, v2[n], will be correlated with the noise in the primary sensor v1[n], the two will not be same.

Reasons for being not same: Difference in sensor characteristics Difference in the propagation path from noise

source to the two sensors. Since v1[n]≠v2[n], it is not possible to

estimate d[n] by simply subtracting v2[n] from x[n]

Noise Cancellation Noise canceller consists of Wiener filter

that is designed to estimate the noise v1[n] from the signal received by the secondary sensor

This estimate is then subtracted from the primary signal x[n] to form an estimate of d[n] which is given by

Noise Cancellation With v2[n] as the input to Wiener filter, that

is used to estimate the noise v1[n], the Wiener-Hopf equations are

Rv2 is the autocorrelation matrix of v2[n] rv1,v2 is the vector of cross-correlations

between desired signal v1[n] and Wiener filter input v2[n]

Noise Cancellation The cross-correlation between v1[n] and

v2[n] is

If we assume v2[n] is uncorrelated with d[n], then second term is zero, hence

Example:7.2.6

Example:7.2.6