Randy Yates wrote:
> Stan Pawlukiewicz <spam@spam.mitre.org> writes:
> 
>>[...]
>>A DFT is a matrix vector multiplication.  I can express a filter in
>>precisely the same terms.
> 
> 
> Stan, humor me for a moment. 
> 
> A vector in the sense you refer to regarding the DFT is finite. A
> vector for a "signal" may be in-finite. How *do* you go from one to
> the other without trying to consider things like operating on a subset
> of the signal, end effects due to truncation of the signal, etc.,
> which are all very pertinent to this discussion and not so necessarily
> straight-forward to resolve?

An image is a finite object. Is there some reason that we can't filter 
one of these?

  Wrt to time, we can in theory have signals that extend to infinity in 
both directions but they are finite in practice.  If you do the 
decimation correctly you can neglect start up and end transients in a 
decimated filter bank. If I have finite sequence of time samples stored 
away some place in memory, there is nothing that says you have to 
process it sequentially, particularly for non-recursive filter. There is 
nothing that requires a filter the same number of outputs as inputs. It 
can be a single point, as an example, the classic textbook replica 
correlator.

There have been numerous posts in the past where people have taken the 
liberty of extending the DFT in the limit to infinite duration 
sequences.  I don't recall anyone saying, "Can't do that, its not a DFT 
then, has to be finite".  A matrix is an abstraction, there isn't 
anything that says you can't consider one, as its dimension approaches 
infinity.

Stan Pawlukiewicz <spam@spam.mitre.org> writes:
> [...]
> A DFT is a matrix vector multiplication.  I can express a filter in
> precisely the same terms.

Stan, humor me for a moment. 

A vector in the sense you refer to regarding the DFT is finite. A
vector for a "signal" may be in-finite. How *do* you go from one to
the other without trying to consider things like operating on a subset
of the signal, end effects due to truncation of the signal, etc.,
which are all very pertinent to this discussion and not so necessarily
straight-forward to resolve?
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

Stan Pawlukiewicz wrote:

   ...

> A DFT is a matrix vector multiplication.  I can express a filter in 
> precisely the same terms.

And the motion of a planet in its orbit, and the polymerization of 
isoprene, and ...   Lot's of food for thought there!

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Bob Cain wrote:
> 
> 
> Stan Pawlukiewicz wrote:
> 
>>> But, nonetheless, there is a rate involved.  In the same sense, what 
>>> is the rate of a DFT?
>>
>>
>>
>> Depends on how often you use it.  
> 
> 
> How about once?
> 
>> What's the rate on filtering a single block of data in memory?  
>> Fundamentally this is semantics. 
> 
> 
> Not at all!  The most common use is the one-off because it contains the 
> information relevant to many analyses.  Of what use is one cycle of what 
> is usually called a filter?
> 
> While you can construct a filter bank of sorts from sequential DFTs if 
> you are so inclined, the DFT operation itself is not a filter, it's a 
> domain transformation.  Do filters transform domains? 

A DFT is a matrix vector multiplication.  I can express a filter in 
precisely the same terms.


What's the distinction between an invertible mapping of a metric space 
and a transform?  Isn't a matched filter a projection of an input onto a 
particular waveform?
  None that I've
> seen do that; what comes in is time domain and what comes out is time 
> domain.

Perhaps you should equate yourself with Bacon's idols.

> 
>> Do you actually have a mathematical argument?  
> 
> 
> For what?

The distinction between a mapping and a transform,  What else?
> 
>> Can you find a book that disagrees with Strang's?
> 
> 
> Why?  I can think for myself.

So you disagree with Strang.
> 
> 
> Bob

Stan Pawlukiewicz wrote:

>> But, nonetheless, there is a rate involved.  In the same sense, what 
>> is the rate of a DFT?
> 
> 
> Depends on how often you use it.  

How about once?

> What's the rate on filtering a single 
> block of data in memory?  Fundamentally this is semantics. 

Not at all!  The most common use is the one-off because it 
contains the information relevant to many analyses.  Of what 
use is one cycle of what is usually called a filter?

While you can construct a filter bank of sorts from 
sequential DFTs if you are so inclined, the DFT operation 
itself is not a filter, it's a domain transformation.  Do 
filters transform domains?  None that I've seen do that; 
what comes in is time domain and what comes out is time domain.

> Do you 
> actually have a mathematical argument?  

For what?

> Can you find a book that 
> disagrees with Strang's?

Why?  I can think for myself.


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

Bob Cain wrote:
> 
> 
> Stan Pawlukiewicz wrote:
> 
>> Multirate filters generalize filters so that the number of input 
>> points don't have to equal the number of output points.  
> 
> 
> But, nonetheless, there is a rate involved.  In the same sense, what is 
> the rate of a DFT?

Depends on how often you use it.  What's the rate on filtering a single 
block of data in memory?  Fundamentally this is semantics. Do you 
actually have a mathematical argument?  Can you find a book that 
disagrees with Strang's?
> 
>> I don't think Bob asked any question.  
> 
> 
> We agree on that much.  :-)
> 
> 
> Bob

Stan Pawlukiewicz wrote:

> Multirate filters generalize filters so that the number of input points 
> don't have to equal the number of output points.  

But, nonetheless, there is a rate involved.  In the same 
sense, what is the rate of a DFT?

> I don't think Bob asked any question.  

We agree on that much.  :-)


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

Randy Yates wrote:
> Stan Pawlukiewicz <spam@spam.mitre.org> writes:
> 
> 
>>Bob Cain wrote:
>>
>>>Stan Pawlukiewicz wrote:
>>
>>>>Beg to differ, but it has been shown in the literature that the DFT
>>>>is indeed a special case of a multirate filter bank.
>>
>>>We've had this argument here before and I still can't agree.  Each
>>>point of a DFT is an inner product of some number of cycles of a
>>>sinusoid with a sequence of data points.  It is a static computation
>>>on a sequence of data points.  A filter produces output continuously
>>>in response to input presented continuously.  They could hardly be
>>>different so far as I can see.
>>
>>>OTOH if the DFT was computed anew for each new sample throwing away
>>>the oldest in the sequence, then each point of the DFT is in fact a
>>>filter output with the coeficients being a sinusoid.  But to say
>>>that a one shot DFT is intrinsically a filter seems totally wrong to
>>>me.
>>
>>>I'm curious as to what kind of filter an FIR with coeficients taken
>>>from a sinuoid actually is.  Never heard of one being used per se.
>>
>>>Bob
>>
>>
>>Consider the title of chapter 6 of Elliot and Rao's "Fast Transforms,
>>Algorithms, Analysis,and Applications", Academic Press, 1982,  "DFT
>>FIlter shapes and Shaping".  You can look at Gilbert Strang's book on
>>wavelets where in the introduction, he points out that filters can be
>>expressed as Matrix vector products.  A waveneumber filter can have a
>>single output.  I can make a replica correlator out of a matched
>>filter sampled at a single point.  Does sampling the matched filter at
>>a single point unmake it as a matched-filter?
>>
>>
>>You're free to believe what you want but I can show you books and
>>articles that interperet the DFT a special sort of multirate filter
>>bank.  Show me books or articles that say a DFT is in no circumstance
>>a filter.
> 
> 
> Hi Stan,
> 
> I agree with you, but I think you're leaving out a few details. Namely,
> how DOES one process an indefinite stream of input data with the N-point DFT to provide
> N indefinite streams of output data? Is the DFT taken every sample? Every N/2
> samples? Every N samples? What? I think this is the question Bob is asking,
> and as I've only "heard" the "DFT is a filterbank" argument and not seen it
> operate in action, I'd like to know the answer as well.

Multirate filters generalize filters so that the number of input points 
don't have to equal the number of output points.  I don't know what you 
mean by "indefinite".  I don't think Bob asked any question.  IMHO, he 
voiced the opinion that filters necessarily involve streaming sequences 
of data.  I think filters are a bit more general.  In image processing, 
people use filters without the concept of a sequential stream.  You make 
a output pixel by operating on a set of input pixels. I don't see any 
stream or natural ordering being necessary on the input side.

   Consider the FILTFILT function in matlab.  You put M points in and 
you get M points out.  They use a trick so that the causal and 
anti-causal passes cancel out the startup and ending transients.  Is 
FILTFILT a filter?, after all since filtering is convolution and the 
convolution of an IIR filter with a finite sequence is necessarily an 
infinite sequence, it cannot be, or do we note that we don't really care 
about the zeros appended the to the front and back of the sequence that 
don't matter.

i asked about a comparison (differences/similarities/pros/cons) between
two methods: 1. amdf/comb filtering, 2. fft/fourier transforms, for
achieving the same goal: splitting sound into various frequency bands
throughout the sound (like a graphic equaliser in hifi's).

i think a reasonably correct comparison/comment is: there's not a lot
of difference between the results that the two methods can provide,
assuming a good implemention of both. the results from carrying out
either method to accomplish the goal aren't destined to be inherently
different by choosing one of the mentioned methods as opposed to the
other? (from an end results point of view that is).

i know Benry asnwered that (thanks Benry), and sorry to go on, but it'd
be great to get confirmation of that if possible?

thanks very much, ben.


p.s. i now realise that amdf isn't different or more than comb
filtering at all (which is i thought and said earlier) -- they're the
same thing.

Stan Pawlukiewicz <spam@spam.mitre.org> writes:

> Bob Cain wrote:
> > Stan Pawlukiewicz wrote:
> 
> >
> 
> >> Beg to differ, but it has been shown in the literature that the DFT
> >> is indeed a special case of a multirate filter bank.
> 
> > We've had this argument here before and I still can't agree.  Each
> > point of a DFT is an inner product of some number of cycles of a
> > sinusoid with a sequence of data points.  It is a static computation
> > on a sequence of data points.  A filter produces output continuously
> > in response to input presented continuously.  They could hardly be
> > different so far as I can see.
> 
> > OTOH if the DFT was computed anew for each new sample throwing away
> > the oldest in the sequence, then each point of the DFT is in fact a
> > filter output with the coeficients being a sinusoid.  But to say
> > that a one shot DFT is intrinsically a filter seems totally wrong to
> > me.
> 
> > I'm curious as to what kind of filter an FIR with coeficients taken
> > from a sinuoid actually is.  Never heard of one being used per se.
> 
> > Bob
> 
> 
> Consider the title of chapter 6 of Elliot and Rao's "Fast Transforms,
> Algorithms, Analysis,and Applications", Academic Press, 1982,  "DFT
> FIlter shapes and Shaping".  You can look at Gilbert Strang's book on
> wavelets where in the introduction, he points out that filters can be
> expressed as Matrix vector products.  A waveneumber filter can have a
> single output.  I can make a replica correlator out of a matched
> filter sampled at a single point.  Does sampling the matched filter at
> a single point unmake it as a matched-filter?
> 
> 
> You're free to believe what you want but I can show you books and
> articles that interperet the DFT a special sort of multirate filter
> bank.  Show me books or articles that say a DFT is in no circumstance
> a filter.

Hi Stan,

I agree with you, but I think you're leaving out a few details. Namely,
how DOES one process an indefinite stream of input data with the N-point DFT to provide
N indefinite streams of output data? Is the DFT taken every sample? Every N/2
samples? Every N samples? What? I think this is the question Bob is asking,
and as I've only "heard" the "DFT is a filterbank" argument and not seen it
operate in action, I'd like to know the answer as well.
-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124