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