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

Digital Signal Processing, What is the difference between the unit step fun...

What is the difference between the unit step function u(n+4) and the time-scaled function u(2n+8)

Common-collector amplifier, A common-collector (CC) amplifier is also known...

A common-collector (CC) amplifier is also known as an emitter follower (or a voltage follower) due to the fact that the output voltage "follows" the input by being approximately eq

Show operation on jfet, Q. Show Operation on JFET? The junction in the ...

Q. Show Operation on JFET? The junction in the JFETis reverse-biased for normal operation.No gate current flows because of the reverse bias and all carriers flow from source to

Explain the operation of n-channel fet, Q. Explain the operation of N-chann...

Q. Explain the operation of N-channel FET? Transistor is a linear semiconductor device that controls current with the application of a lower-power electrical signal. Transistor

Explain rotameter and hot wire anemometer, Write down about the following t...

Write down about the following terms: (i) Pirani Gauge (ii) Rotameter (iii) Hot wire Anemometer (iv) Drag Force Flow Meter

Measurement and control systems, Electronic signals are used by almost all ...

Electronic signals are used by almost all measurement and control systems to transfer information. Much of this data is in the form of analogue voltages, which are prone to the ef

Give the basic structure of a basic dma module, a) Give four features that ...

a) Give four features that were traditionally reserved to RISC architectures? b) Determine the overall throughput for 21 sequential instructions assuming a four-stage pipeline a

Gis for distribution network management, Gis for Distribution Network Manag...

Gis for Distribution Network Management: You are well aware of the problems facing the distribution utilities to meet the day-to-day requirements of the power consumers, like

Calculate the pu voltage and line to line voltage, The one line diagram of ...

The one line diagram of a simple power system is shown below. The data of the system are given in the table below. (a) Draw the impedance diagram of the system and mark all impe

Network theorems, what is the limitations of maximum power transfer thorem ...

what is the limitations of maximum power transfer thorem ?

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