Fixed and adaptive - optimal filtering, Electrical Engineering

Assignment Help:

Optimal Filtering

1389_Fixed and Adaptive.png

A system identi?cation structure is shown in Figure 1. The discrete-time signal [ ] forms the input to an unknown system represented by a moving average, FIR, ?lter with parameters, {ω01 ωn} . The output of the unknown system is given by

466_Fixed and Adaptive1.png

This can be represented more concisely in vector notation as

1891_Fixed and Adaptive2.png

where ω and x[k] are Nw + 1 element column vectors, underlines are used to denote vector quantities and the superscript (·)T represents a vector transpose.

A noise signal n[k] is added at the output of the system to yield a realistic measured output signal z[k]. An optimum ?lter is to be designed which operates on the input signal x[k] and measured output signal z[k] to yield the closest match to the unknown system. The optimum ?lter is assumed to be a moving average ?lter with the same number of parameters as the unknown system, i.e. opt.

This problem arises in many applications, for example control of some machinery plant, or of acoustic and electrical echoes in telecommunications. The Wiener solution is found by minimising the mean-square error between z[k] and y[k], where y[k] is the output of the optimum ?lter driven by x[k]. The circum?ex is used to denote that the output of the optimum ?lter is an estimate of the output of the unknown system.

If x[k]and n[k] are assumed to be uncorrelated stationary random discrete signals then the mean square error is de?ned as

2274_Fixed and Adaptive3.png

where {·} is the statistical expectation operation, which to a ?rst approximation can be considered to be an average. The right hand side of equation (1) contains a vector of crosscorrelation components and a matrix of autocorrelation components, i.e.

555_Fixed and Adaptive4.png

Differenting equation (1) with respect to opt and setting the result to zero yields the optimum Wiener ?lter solution which is given by

596_Fixed and Adaptive5.png


Related Discussions:- Fixed and adaptive - optimal filtering

Determine the kva rating of the synchronous motor, An induction motor takes...

An induction motor takes 350 kW at 0.8 power factor lagging while driving a load. When an overexcited synchronous motor taking 150 kW is connected in parallel with the induction mo

Sinusoidal steady-state phasor analysis, Q. Sinusoidal steady-state phasor ...

Q. Sinusoidal steady-state phasor analysis? The kind of response of a physical system in an applied excitation depends in general on the type of excitation, the elements in the

Dtl and ttl logic circuits, Q. DTL AND TTL LOGIC CIRCUITs? Bipolar tran...

Q. DTL AND TTL LOGIC CIRCUITs? Bipolar transistorswere the first solid-state switching devices commonly used to implement digital logic circuits in the 1950s and 1960s. These c

Types of field-effect transistors, Types of Field-Effect Transistors T...

Types of Field-Effect Transistors The channel of a FET (field-effect transistor) is doped to produce either an N-type semiconductor or a P-type semiconductor. The drain and so

Sketch the circuit with all the relevant labelling, Question: (a) Consi...

Question: (a) Consider the subsequent LC impedance function: (i) If the function is to be realised using the Foster I Form, evaluate the values of the different compon

Use delta-wye transformation for network reduction, 1555_Use delta-wye tran...

1555_Use delta-wye transformation for network reduction.png what is the solution to this particular problem?

Architecture of 8085 microprocessor, Architecture of 8085 Microprocessor ...

Architecture of 8085 Microprocessor The 8085  microprocessor  is an 8 bit  general purpose microprocessor. 8 bit  processor  means it can  read write  and process 8 bit  data

Illustrate electric potential and voltage, Q. Illustrate Electric Potential...

Q. Illustrate Electric Potential and Voltage? When electrical forces act on a particle, it will possess potential energy. In order to describe the potential energy that a parti

Test, current and voltage

current and voltage

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd