Parks-McClellan FIR Filter Design

Islamic University-GazaFaculty Of EngineeringElectrical and Computer dep.

Done By:

Eman R.El-Taweel Maysoon A. Abu Shamla

Submitted to:

Dr.Hatem El-Aydi

2nd May 2007.

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.

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.

Parks-McClellan filterParks-McClellan filter

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 {ωp, ωs, δ p, δ s}.

Continue …

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

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

Now the question is must there

be a transition

band using P-MC ???

Yes, when the desired response is

discontinues. Since the frequency response of a finite length filter must be continuous. Without a transition band the worst-case error could be no less than half the discontinuity.

The answer …

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 are sometimes called equiripple filters.

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

Simulation …

Graph the desired and actual frequency responses of a 17th-order Parks-MC bandpass filter

Approximation Errors

From the theory of the Fourier series, the rectangular window design method gives the best mean square (L 2) approximation to a desired frequency response for a given filter length M.

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.

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 (ω)

The polynomial of degree L that minimizes the maximum error 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 ω=0 or π or both.

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

Alternation Theorem

Continue…

The alternation theorem doesn’t directly suggest a method for computing the optimal filter.

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

L-∞ Optimal Lowpass Filter Design Lemma

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

There must be an alternation at either ω = 0 or ω=π

Alternations must occur at ωp and ωs.

The filter must be equiripple except at possibly

ω = 0 or ω=π.

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 the estimated filter length (Harris formula or similar) but we don’t know their positions.

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

Perform polynomial interpolation and reestimate positions of local extrema.

Move extrema to new positions and iterate until the extrema stop shifting.

الرسمة

Remez exchange algorithm …

Comments

Given the positions of the extrema, there exists a

formula for the optimum δ. However we don’t know the optimum δ nor the exact positions of the extrema.

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

until δ stops changing.

The algorithm generally converges in about 12 iteration.

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 .