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Islamic University-Gaza Faculty Of Engineering Electrical and Computer dep . Parks-McClellan FIR Filter Design. Done By: Eman R.El-Taweel Maysoon A. Abu Shamla Submitted to: Dr.Hatem El-Aydi 2 nd May 2007. C ontents . Introduction.
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Islamic University-Gaza Faculty Of Engineering Electrical and Computer dep. Parks-McClellan FIR Filter Design 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 • 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 • 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. • b= remez(n,fo,mo) • H(z) = b(1) + b(2)z-1 + b(3)z-2 + · · · + b(n + 1)z-n
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)
Alternation Theorem • 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.
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.
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 .
