# Multistage interpolation question

Started by January 27, 2008
```Vladimir Vassilevsky wrote:
>
>
> Jerry Avins wrote:
>
>> A systematic approach to optimizing two stages is a first step to
>> finding a more general procedure.
>
> Unfortunately, no.
>
> The FIR filters are optimized for the passband ripple/stopband
> attenuation rather then cutoff/slope. When designing a FIR with many
> stages it is important to know how the ripples of the different stages
> are added altogether. If we assume the worst case the design may not be
> very optimal.
>
> For the multirate FIR, the transfer function is something like:
>
> P(Z)P(Z^n1)P(Z^n2)...
>
> This results in the nastiness and I don't know if Parks-Mcclellan can be
> applied to this problem. If it is applicable, I would just search
> through all ratios.

The number of taps might be so chosen that the ripple peaks of one
filter fill in the valleys of the other. For three stages, one would
want a three-phase offset. I can't guess the best approach, but I'm
confident that exploring the solution space would provide insights we
don't now have.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```On Jan 28, 10:45&#4294967295;pm, Jerry Avins <j...@ieee.org> wrote:
> DSPGURU wrote:
> > On Jan 28, 9:45 pm, Jerry Avins <j...@ieee.org> wrote:
> >> DSPGURU wrote:
> >>> On Jan 28, 9:29 pm, "bharat pathak" <bha...@arithos.com> wrote:
> >>>>> do any of you know an equation that gives us the
> >>>>> optimum value for I1 (from which we can obtain I2)
> >>>>> that minimizes the total number of taps in the two
> >>>>> interpolation filters?
> >>>>> Thanks,
> >>>>> [-Rick-]
> >>>> Rick,
> >>>> &#4294967295; &#4294967295; For the equation see my other post on Two stage interpolation-Optimum
> >>>> order.
> >>>> Bharat Pathak
> >>>> Arithos Designswww.arithos.com
> >>> Hello my DSP Brothers,
> >>> Why is focus at two stages? Why not more?
> >>> To optimise all reddy under-optimal approach make any sense?
> >> A systematic approach to optimizing two stages is a first step to
> >> finding a more general procedure.
>
> >> Jerry
> >> --
> >> Engineering is the art of making what you want from things you can get.
> >> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;- Hide quoted text -
>
> >> - Show quoted text -
>
> > Dr Avin,
>
> > Might be first step down path that go nowhere. &#4294967295;If real application
> > should just do it design starting closer to answer not as great a trip
> > away as can be. &#4294967295;Close form not required, maybe not possible as get
> > much bigger than too stages.
>
> > Ok, if only gymnastics of brain.
>
> Closed form will not be needed if an effective partitioning procedure
> can be developed. For example, if it becomes evident that the optimum
> two-way split is I1 = I2 = sqrt(I), then a good three-way split is
> likely to be I1 = I2 = I3 = cube-root(I). If, on the other hand, the
> best two-way split involves unequal I's (possibly because the filters
> operate at different rates), then the formula that works there will
> likely also lead to an efficient three-way split.
>
> Once efficient splits can be generated easily, the way is open to
> explore the relation between the number of splits and the size of the
> interpolation.
>
> Almost having a diploma makes you an apprentice, not a guru. You are
> accustomed to learning neat formulas out of textbooks, but you haven't
> yet realized that the messy problems aren't discussed in those texts.
> You have the privilege of witnessing here a process by which those messy
> problems come to be understood, then simplified in practice, and finally
> reduced to formulas suitable for student "gurus".
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;- Hide quoted text -
>
> - Show quoted text -

Dr Jerry,

It seems Vlad and I both not agree with you.  You make lot of

Regards,

Kamar Ruptan
DSP Guru
```
```On Jan 29, 12:11&#4294967295;am, Jerry Avins <j...@ieee.org> wrote:
>
> > Jerry Avins wrote:
>
> >> A systematic approach to optimizing two stages is a first step to
> >> finding a more general procedure.
>
> > Unfortunately, no.
>
> > The FIR filters are optimized for the passband ripple/stopband
> > attenuation rather then cutoff/slope. When designing a FIR with many
> > stages it is important to know how the ripples of the different stages
> > are added altogether. If we assume the worst case the design may not be
> > very optimal.
>
> > For the multirate FIR, the transfer function is something like:
>
> > P(Z)P(Z^n1)P(Z^n2)...
>
> > This results in the nastiness and I don't know if Parks-Mcclellan can be
> > applied to this problem. If it is applicable, I would just search
> > through all ratios.
>
> The number of taps might be so chosen that the ripple peaks of one
> filter fill in the valleys of the other. For three stages, one would
> want a three-phase offset. I can't guess the best approach, but I'm
> confident that exploring the solution space would provide insights we
> don't now have.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;- Hide quoted text -
>
> - Show quoted text -

Yes Jerry,

It might prove insight if you have never done.  I have done.

Regards,

Kamar Ruptan
DSP Guru
```
```On 29 Jan, 06:11, Jerry Avins <j...@ieee.org> wrote:
>
> > Jerry Avins wrote:
>
> >> A systematic approach to optimizing two stages is a first step to
> >> finding a more general procedure.
>
> > Unfortunately, no.
>
> > The FIR filters are optimized for the passband ripple/stopband
> > attenuation rather then cutoff/slope. When designing a FIR with many
> > stages it is important to know how the ripples of the different stages
> > are added altogether. If we assume the worst case the design may not be
> > very optimal.
>
> > For the multirate FIR, the transfer function is something like:
>
> > P(Z)P(Z^n1)P(Z^n2)...
>
> > This results in the nastiness and I don't know if Parks-Mcclellan can be
> > applied to this problem. If it is applicable, I would just search
> > through all ratios.
>
> The number of taps might be so chosen that the ripple peaks of one
> filter fill in the valleys of the other.

Based on vague reminiscence of long-lost memories, this resembles
a perfect reconstruction filter bank? Are you suggesting a
multirate filter with extra decimation/interpolation steps
interleaved?

Rune
```
```"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> On 29 Jan, 06:11, Jerry Avins <j...@ieee.org> wrote:

> > > The FIR filters are optimized for the passband ripple/stopband
> > > attenuation rather then cutoff/slope. When designing a FIR with many
> > > stages it is important to know how the ripples of the different stages
> > > are added altogether. If we assume the worst case the design may not
be
> > > very optimal.
> >
> > > For the multirate FIR, the transfer function is something like:
> >
> > > P(Z)P(Z^n1)P(Z^n2)...
> >
> > > This results in the nastiness and I don't know if Parks-Mcclellan can
be
> > > applied to this problem. If it is applicable, I would just search
> > > through all ratios.
> >
> > The number of taps might be so chosen that the ripple peaks of one
> > filter fill in the valleys of the other.

The periods don't match unless specifically designed for that.

> Based on vague reminiscence of long-lost memories, this resembles
> a perfect reconstruction filter bank? Are you suggesting a
> multirate filter with extra decimation/interpolation steps
> interleaved?

All that I said is if the goal is a decimation/interpolation by the high
factor with the tough requirements to the passband flatness and to the
stopband attenuation, then the filter stages should be jointly optimized.
The formulae based on the cutoff slopes only give the rough idea of where
could be the decimation/interpolation ratios.