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

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

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.

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

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.

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.

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.

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

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.