Re: Questions regarding polyphase filter

On Mar 3, 7:56 pm, "bcarmaint" <bing_carma...@xxxxxxxxx> wrote:
On Mar 3, 6:56=A0am, "bcarmaint" <bing_carma...@xxxxxxxxx> wrote:
On Mar 2, 7:11=3DA0pm, "bcarmaint" <bing_carma...@xxxxxxxxx> wrote:
[snip]

Thanks for the response. I understand the relationship between
Farrow
structure and polyphase filter.

My questions was the relationship between different interpolation
method,
such as cubin bspline interpolation and polyphase filter.

Thanks, Ben

I'm with Jerry, I think you are confusing an efficient implementation
method
(polyphase filter) with a mathematical tool for approximation (cubic
bspline
interpolation). =A0For example, you can implement a cubic bspline
interpolator
as a polyphase filter. =A0So I don't understand what your question
is,
sorry.

Reading your original question again, I think that where you are
confused
is between:
* the formula for computing the coefficients of a filter to apply to
a
signal,
=A0for example for interpolation.
* the efficient implementation of a filter, in fact any filter.

The first one includes the cubic spline formula, etc etc etc. =A0The
second
one includes polyphase structures, lattice filters, etc.

As for your second question, yes, if your original filter has
bandwidth
<=3D3D 1/32 as a fraction of the digital bandwidth, then you can use
it
for
variable rate interpolation, as long as that bandwidth is not so
narrow
that your signal is distorted.

Hope this helps,
Julius

Julius, Jerry

Thanks for the clarifications about the second question. About my
first
question, I still have some confusion... I can't seem to relate a
cubic
spline coefficients with a 32 phase polyphase filter. Do you by any
chanc=
e
have some example in this regard?

Ben

Ben,

A generic explanation of the resampling process is as follows:
1. Upsample to a higher rate (i.e., insert zeros between existing
samples)
2. Use an interpolator to figure out what the values of the samples
should be at the new points (where you inserted zeros)
3. Possibly downsample to achieve the desired rate

Now you can implement step (2) of the process in one of several ways.
One way is to use a linear combination of samples around the point you
want - eg. x(t) =3D 0.5(x(t0)+x(t1)). Basically, the linear combination
of samples with different weights is what constitutes a filter - so
what you have is an "interpolating filter".

Now, you could also choose not to be restricted to a linear technique
for interpolation, i.e., you could use square, cubic, and higher order
terms of the samples to do the interpolation. For example,

x(t) =3D a0.x(t0)^2 + b0.x(t0) + a1.x(t1)^2 + b1.x(t1) + c.

What you have here is a polynomial interpolation technique. In a
sense, the linear interpolating technique is a specific case of the
polynomial technique, with the order of the polynomial restricted to
one.

So when do you choose one over the other? The linear filtering
technique is a low computational complexity technique and you can use
a number of filter design techniques to design your interpolating
filter. On the other hand, in situations where the spacing between
your samples is not constant, or, if the spacing between your
interpolating instant and the sampling instant changes, then the
filter needs to change. The filtering technique falls short because
your filter was designed assuming a particular fixed spacing of
samples on the time axis and does not hold if that changes.

The greatest advantage of polynomial interpolation is that it can be
implemented in a way that factors in the time interval between sampled
and interpolated points - for example,

x(t) =3D x(t0) + (x(t1)-x(t0)).(t-t0)/(t1-t0)

This has the advantage that the interpolation holds even if the
relative spacing between the interpolating instant, t, and sampling
instants t0 and t1, changes. In the case of a communications receiver
for example, the receiver clock can be at a slightly different
frequency from the transmitter clock - so assuming that the two clocks
are in sync at the first sample, the second sample is off by dT, the
third by 2dT, etc. If you can estimate this difference, you could use
a polynomial interpolator to interpolate existing received samples to
the transmitter rate.

And as others have said, a polyphase filter is simply a structure that
allows for efficient implementation of the filter.

- Ravi

Ravi,

Thanks a lot. I think I confused "interpolation filter" and the polyphae
implementation of the anti-aliasing filter used after the interpolation.

Here is my understanding of a overall interpolation procedure:

1) First I will need interpolate the signal (may need to use both
upsampling and downsampling to achieve the required factor); in this
process, I may choose different interpolation techniques you described
above.

2) Implement anti-aliasing/anti-imaging lowpass filter to reduce
artifects. In this step, I can choose the cutoff bandwidth to be the
smallest possible bandwidth I need to support (determined by the highest
interpolation factor, for example, if the factor is 32, then the cutoff
bandwidth should be 1/32); A typical filter like this may have several
hundre taps, and a good choise to implement this filter is polyphase
filter. With this structure, I can support the filtering of interpolation
of 2,3,4...32 discrete steps.

Does this looks correct to you?

Thanks for the help from all of you,

Regards, Ben

You need to be a little careful here. You need to put your anti-
aliasing filter BEFORE downsampling. Looked at in the frequency
domain, the downsampling process causes the frequency response to
stretch and causes aliasing. So basically, you can combine the
interpolating filter and the anti-aliasing filter, provided you choose
your filter bandwidth properly.
.

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