Is anyone familiar with the following paper? Preuss, K, "On the Design of FIR Filters by Complex Chebyshev Approximation," IEEE Trans. Acoust., Speach, Signal Processing, vol. 37, pp. 702-712, 1989. The author uses a notation I am not familiar with. Specifically, see equation (8): | E(Omega_nu) | = delta nu = 1(1)m, where n+2 <= m <= 2n+3 What does this 1(1) denote? and how does that apply to equations (13a) and (13b) ? Thanks, -Brandon

# Complex FIR Filter Design - Notation in Preuss' IEEE Trans. Paper

Started by ●September 11, 2006

Reply by ●September 11, 20062006-09-11

Brandon skrev:> Is anyone familiar with the following paper? > > Preuss, K, "On the Design of FIR Filters by Complex Chebyshev > Approximation," IEEE Trans. Acoust., Speach, Signal Processing, vol. > 37, pp. 702-712, 1989. > > The author uses a notation I am not familiar with. Specifically, see > equation (8): > > | E(Omega_nu) | = delta > nu = 1(1)m, where n+2 <= m <= 2n+3After having spent literally half a minute bowsing the first couple of pages, it seems as if the algorithm is based on the Remez exchange algorithm known from the Patks-McClellan FIR filter design method. The Remez is an equiripple method, which means that there is an upper bound on the error in the transfer function. The Omega_nu above seems to be the frequencies where the error is maximum. There is a theorem from th etheory of Chebyshev functions that states some upper and lower bounds on how many such extrema there may be in the error function. The author of the paper seems to make a point of that one can not tell beforehand exactly how many extrema there actually are in any given approximation. This certainly has some significance when implementing the Parks-McClellan method for FIR filter design, it may have some further theoretical importance when discussing the complex- valued filters.> What does this 1(1) denote?I would guess that nu=1(1)m is a way to express what we would write today as nu = 1:m or "let nu take on all integer vaues fro 1 to m, inclusive."> and how does that apply to equations (13a) > and (13b) ?Once you implement the algorithm to estimate the filter function, you need some initial estimates for the frequencies Omega_nu where the extrema in the errors occur. Finding those initial estimates is no trivial task. Here the fact that you don't even know how many extrema are present, jumps up to bite you. Rune

Reply by ●September 11, 20062006-09-11

Rune Allnor wrote:> Brandon skrev: > > Is anyone familiar with the following paper? > > > > Preuss, K, "On the Design of FIR Filters by Complex Chebyshev > > Approximation," IEEE Trans. Acoust., Speach, Signal Processing, vol. > > 37, pp. 702-712, 1989. > > > > The author uses a notation I am not familiar with. Specifically, see > > equation (8): > > > > | E(Omega_nu) | = delta > > nu = 1(1)m, where n+2 <= m <= 2n+3 > > After having spent literally half a minute bowsing the first couple of > pages, it seems as if the algorithm is based on the Remez exchange > algorithm known from the Patks-McClellan FIR filter design method.>From memory (it was a while ago), it's not really a Remez-stylealgorithm, though it does use some of the same theoretcial underpinnings. Also from memory, while the Preuss algorithm guarantees convergence, it doesn't necessarily guarantee convergence to the minimax optimal... or rather it wasn't necessarily solving the minimax problem I thought it was solving. :-) <snip>> Once you implement the algorithm to estimate the filter function, > you need some initial estimates for the frequencies Omega_nu > where the extrema in the errors occur. Finding those initial > estimates is no trivial task. Here the fact that you don't even > know how many extrema are present, jumps up to bite you.Yup! Ciao, Peter K.

Reply by ●September 12, 20062006-09-12

I see. Do either of you have any other papers to recommend on complex FIR filter design? I need to learn how to approximate a complex frequency response with non-linear phase for an equalizer design. My first approach was to window the impulse response, but this just didn't cut it. Now I'm realizing I'll have to use some other sophisticated methods... Thanks, -Brandon Peter K. wrote:> Rune Allnor wrote: > > > Brandon skrev: > > > Is anyone familiar with the following paper? > > > > > > Preuss, K, "On the Design of FIR Filters by Complex Chebyshev > > > Approximation," IEEE Trans. Acoust., Speach, Signal Processing, vol. > > > 37, pp. 702-712, 1989. > > > > > > The author uses a notation I am not familiar with. Specifically, see > > > equation (8): > > > > > > | E(Omega_nu) | = delta > > > nu = 1(1)m, where n+2 <= m <= 2n+3 > > > > After having spent literally half a minute bowsing the first couple of > > pages, it seems as if the algorithm is based on the Remez exchange > > algorithm known from the Patks-McClellan FIR filter design method. > > >From memory (it was a while ago), it's not really a Remez-style > algorithm, though it does use some of the same theoretcial > underpinnings. > > Also from memory, while the Preuss algorithm guarantees convergence, it > doesn't necessarily guarantee convergence to the minimax optimal... or > rather it wasn't necessarily solving the minimax problem I thought it > was solving. :-) > > <snip> > > > Once you implement the algorithm to estimate the filter function, > > you need some initial estimates for the frequencies Omega_nu > > where the extrema in the errors occur. Finding those initial > > estimates is no trivial task. Here the fact that you don't even > > know how many extrema are present, jumps up to bite you. > > Yup! > > Ciao, > > Peter K.

Reply by ●September 12, 20062006-09-12

Brandon wrote:> I see. > > Do either of you have any other papers to recommend on complex FIR > filter design? I need to learn how to approximate a complex frequency > response with non-linear phase for an equalizer design. My first > approach was to window the impulse response, but this just didn't cut > it. Now I'm realizing I'll have to use some other sophisticated > methods... > > Thanks, > -BrandonYou could try Eric's suggestion: http://www.dspguru.com/comp.dsp/tricks/dsn/nlp_fir.htm Ciao, Peter K.

Reply by ●September 13, 20062006-09-13

Brandon skrev:> Now I'm realizing I'll have to use some other sophisticated > methods...That's taking things it a bit too far. The Remez algorithm works, you just need to make an effort to keep track of all the details. That's all there is to it. Rune

Reply by ●September 19, 20062006-09-19

I'm not sure I see how that link helps me. I not only need to design a FIR filter with non-linear phase, but I need to be able to specify an approximation to the magnitude and non-linear phase for an equalization problem. I've been looking through a dissertation by Mathias Lang, which has an algorithm for complex Chebyshev filter design using Lawson's algorithm, but I'm having a hard time getting the algorithm to work well. Here is the desired response I'd like to design: http://img101.imageshack.us/my.php?image=deseqrespwk8.jpg Any thoughts? Thanks, -Brandon Peter K. wrote:> Brandon wrote: > > I see. > > > > Do either of you have any other papers to recommend on complex FIR > > filter design? I need to learn how to approximate a complex frequency > > response with non-linear phase for an equalizer design. My first > > approach was to window the impulse response, but this just didn't cut > > it. Now I'm realizing I'll have to use some other sophisticated > > methods... > > > > Thanks, > > -Brandon > > You could try Eric's suggestion: > > http://www.dspguru.com/comp.dsp/tricks/dsn/nlp_fir.htm > > Ciao, > > Peter K.