parks mcclellan

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Parks-McClellan FIR Filter Design

Islamic University-GazaFaculty Of EngineeringElectrical and Computer dep.

Done By:

Eman R.El-TaweelMaysoon A. Abu Shamla

Submitted to:

Dr.Hatem El-Aydi

2nd May 2007.


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Contents

Introduction. Parks- McClellan.

must there be a transition band using P-MC.

Parks- McClellan Method.

P-Mc design of FIR using Matlab.

Remez exchange algorithm. Simulation.

Approximation Errors.

Minimax Design.

Formal Statement of the L- (Minimax) Design Problem

Alternation Theorem. L- Optimal Lowpass Filter Design Lemma

The Method.

Comments . Conclusion.


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Introduction

Kaiser filters are not guaranteed to be the

minimum length filter which meets the design

constraints.

Kaiser filters do not allow passband and stopband

ripple to be varied independently.

Minimizing filter length is important.


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Parks-McClellan filter


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Parks- McClellan

Often called the Remez exchange method.

This method designs an optimal linear phase filter.

This is the standard method for FIR filter design.

This methodology for designing symmetric filters that

minimize filter length for a particular set of design

constraints {p, s, p, s}.


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Continue

The computational effort is linearly proportional tothe length of the filter.

In Matlab, this method is available as remez().


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Now the

question is

must there bea transition

band using P-

MC ???


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Yes, when the desired response is discontinues.

Since the frequency response of a finite length filter

must be continuous. Without a transition band theworst-case error could be no less than half the

discontinuity.

The answer


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Parks- McClellan Method

The resulting filters minimize the maximum error

between the desired frequency response and the

actual frequency response by spreading the

approximation error uniformly over each band.

Such filters that exhibit equiripple behavior in

both the passband and the stopband, and aresometimes called equiripple filters.


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P-Mc design of FIR using Matlab

1. Use the (remezord) command to estimate the order of the optimal P-Mc FIR filter.

The syntax of the command is as follows:

[n,fo,mo,w]=remezord(f,m,dev)

f:the vector of band frequencies.m:the vector of desired magnitude.

dev:max. devotion of the magnitude response.

2. b= remez(n,fo,mo)

H(z) = b(1) + b(2)z-1 + b(3)z-2 + + b(n + 1)z-n


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Simulation


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Graph the desired and actual frequency responses of a 17th-

order Parks-MC bandpass filter


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Approximation Errors

From the theory of the Fourier series, the rectangularwindow design method gives the best mean square (L 2)

approximation to a desired frequency response for a given

filter length M.


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Minimax Design

simple truncation leads to adverse behavior

near discontinuity's and in the stop band.

Better filters generally result from

minimization of the maximum error (L ) or

a frequency weighed error criterion.


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Formal Statement of the L- (Minimax) Design

Problem

For a given filter length (M) and type (odd length,

symmetric, linear phase, a relative error weighting

function W ()


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The polynomial of degree L that minimizes the maximumerror will have at least L+2 extrema.

The optimal frequency response will just touch

the maximum ripple bounds.

Extrema must occur at the pass and stop band edges and

at either =0 or or both.

The derivative of a polynomial of degree L is a polynomial ofdegree L-1, which can be zero in at most L-1 places. So themaximum number of local extrema is the L-1 local extremaplus the 4 band edges. That is L+3.

Alternation Theorem


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Continue

The alternation theorem doesnt directly suggest a

method for computing the optimal filter.

What we need is an intelligent way of guessing theoptimal filters coefficients.


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L- Optimal Lowpass Filter Design Lemma

The maximum possible number of alternations for alowpass filter is L + 3.

There must be an alternation at either = 0

or =

Alternations must occur at p and s.

The filter must be equiripple except at possibly

= 0 or =.


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The Method

Boundary points are from the band edge specifications.At least 3 of these points must be extreme.

We know how many local extrema there are from theestimated filter length (Harris formula or similar) but

we dont know their positions.

Guess the positions of the extrema are evenly spaced inthe pass and stop bands.

Perform polynomial interpolation and reestimatepositions of local extrema.

Move extrema to new positions and iterate until the

extrema stop shifting.


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Remez exchange algorithm


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Comments Given the positions of the extrema, there exists a

formula for the optimum . However we dont know theoptimum nor the exact positions of the extrema.

Thus we need to iterate. Assume the positions of theextrema, calculate , move the extrema, recalculate ,

until stops changing.

The algorithm generally converges in about 12 iteration.


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Conclusion

Disadvantages of Kaizer window.

The parks McClellan method is the best method to

achieve the desired impulse response with least

error .

we achieved L- Optimal Lowpass Filter Design.

Simulation using Matlab for optimal filter design .


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parks mcclellan

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