Hi Folks,

I read that the inverse of an FIR filter is always an IIR filter. I
have no idea how to prove this mathematically. If anybody could tell me
where to start, I'd be more than happy.

I'm a DSP newbie, please do excuse me if the question is a little
stupid.

Thanks,
Karthik.

From:	Andor
Subject:	Re: inverse of an FIR filter
Date:	20 Jan 2005 00:59:53 -0800

glen herrmannsfeldt wrote:
.....
> Consider the problem of the reciprocals of integers.
> 1/3 is a repeating decimal, 1/5 is not.
Why not? 1/5 = 0.2000000 ....

From:	bryant_j_j at yahoo.com
Subject:	Re: inverse of an FIR filter
Date:	19 Jan 2005 04:01:17 -0800

easy. let U(z),Y(Z) and H(z) denote the input, output Z-transforms, and
the transfer function of the FIR filter, respectively. Then
Y(z)=H(z)U(z) and equivalently U(z)=1/H(z) Y(z). hence the inverse of
the FIR filter is IIR.

From:	glen herrmannsfeldt
Subject:	Re: inverse of an FIR filter
Date:	Wed, 19 Jan 2005 09:58:45 -0800

bryant_j_j@yahoo.com wrote:

> easy. let U(z),Y(Z) and H(z) denote the input, output Z-transforms, and
> the transfer function of the FIR filter, respectively. Then
> Y(z)=H(z)U(z) and equivalently U(z)=1/H(z) Y(z). hence the inverse of
> the FIR filter is IIR.

Continuing the discussion, there isn't a requirement that an IIR
filter actually have an infinite impulse response, only that it
may have one. That is, FIR are a subset of IIR.

For an example, consider H(z)=1.

-- glen

From:	Rune Allnor
Subject:	Re: inverse of an FIR filter
Date:	19 Jan 2005 04:08:18 -0800

Karthik wrote:
> Hi Folks,
>
> I read that the inverse of an FIR filter is always an IIR filter. I
> have no idea how to prove this mathematically. If anybody could tell
me
> where to start, I'd be more than happy.

Step 1: Derive the frequency-domain expression for the transfer
function of a FIR filter, given the filter coefficients.
In other words, use the coefficients of the time-domain
difference equation to find the frequency domain function.

Step 2: Find the filter inverse to the FIR in frequency domain

Step 3: Find a general expression for the digital filter in
frequency domain, given both feed-forward coefficients
and feedback coefficients (both the a's and b's in the
difference equation)

Step 4: Compare the filter you found in step 2 with the general
expression you found in step 3. How do the non-zero
coefficients from step 2 fit into the expression of
step 3?

Rune

From:	Karthik
Subject:	Re: inverse of an FIR filter
Date:	19 Jan 2005 18:51:54 -0800

Thank you, Thanks a ton, Everybody :-). I'm going to spend some time
with my notebook and a pencil this evening, and all these responses
will be plenty of food for thought.

Thanks again :-)
Karthik.

From:	Randy Yates
Subject:	Re: inverse of an FIR filter
Date:	19 Jan 2005 09:15:31 -0500

"Karthik" writes:

> Hi Folks,
>
> I read that the inverse of an FIR filter is always an IIR filter. I
> have no idea how to prove this mathematically.

That's good, because it's false. Here's a counterexample: Consider
the FIR H(z) = z^{-1}. The inverse filter is G(z) = z. The impulse
response of G(z) is g[n] = d[n+1], where d[] is the unit impulse
function. This is a finite impulse response, hence the claim is
disproven.
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

From:	Jerry Avins
Subject:	Re: inverse of an FIR filter
Date:	Wed, 19 Jan 2005 13:48:12 -0500

Randy Yates wrote:

> "Karthik" writes:
>
>
>>Hi Folks,
>>
>>I read that the inverse of an FIR filter is always an IIR filter. I
>>have no idea how to prove this mathematically.
>
>
> That's good, because it's false. Here's a counterexample: Consider
> the FIR H(z) = z^{-1}. The inverse filter is G(z) = z. The impulse
> response of G(z) is g[n] = d[n+1], where d[] is the unit impulse
> function. This is a finite impulse response, hence the claim is
> disproven.

That's neat! But I think it's fair to say that the original assertion is
true PROVIDED THAT both impulse responses are causal.

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

From:	Jerry Avins
Subject:	Re: inverse of an FIR filter
Date:	Wed, 19 Jan 2005 14:14:41 -0500

Jerry Avins wrote:

> Randy Yates wrote:
>
>
>>"Karthik" writes:
>>
>>
>>
>>>Hi Folks,
>>>
>>>I read that the inverse of an FIR filter is always an IIR filter. I
>>>have no idea how to prove this mathematically.
>>
>>
>>That's good, because it's false. Here's a counterexample: Consider
>>the FIR H(z) = z^{-1}. The inverse filter is G(z) = z. The impulse
>>response of G(z) is g[n] = d[n+1], where d[] is the unit impulse
>>function. This is a finite impulse response, hence the claim is
>>disproven.
>
>
> That's neat! But I think it's fair to say that the original assertion is
> true PROVIDED THAT both impulse responses are causal.
>
> Jerry

Glen's H(z) = 1 is a counterexample. Are there others?

jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

From:	glen herrmannsfeldt
Subject:	Re: inverse of an FIR filter
Date:	Wed, 19 Jan 2005 12:29:37 -0800

Jerry Avins wrote:
(someone wrote)

>>>>I read that the inverse of an FIR filter is always an IIR filter. I
>>>>have no idea how to prove this mathematically.

(snip)

> Glen's H(z) = 1 is a counterexample. Are there others?

Yes, but I also claimed that FIRs are a subset of IIRs.

Consider the problem of the reciprocals of integers.
1/3 is a repeating decimal, 1/5 is not. I used to know
some math people who claimed that you could write
1/5 instead of 0.2, as 0.199999999999999999999...., that is,
as a repeating decimal.

Of all possible FIRs only a very small fraction have an
FIR as their inverse, but you might get lucky.

-- glen

From:	Randy Yates
Subject:	Re: inverse of an FIR filter
Date:	20 Jan 2005 07:25:42 -0500

glen herrmannsfeldt writes:

> Jerry Avins wrote:
> (someone wrote)
>
> >>>>I read that the inverse of an FIR filter is always an IIR filter. I
> >>>> have no idea how to prove this mathematically.
>
>
> (snip)
>
> > Glen's H(z) = 1 is a counterexample. Are there others?
>
> Yes, but I also claimed that FIRs are a subset of IIRs.

That doesn't make sense to me, Glen. If the total class of digital
filters can be partitioned into two sets, those with finite impulse
response (i.e., "FIR" filters), and those with infinite impulse
response (not including finite impulse responses), then including FIRs
in the set of IIRs causes the set of IIRs to be equivalent to the
total class of digital filters. If that is the case, then the original
claim is trivially true, "I read that the inverse of a [digital] FIR
filter is always a digital filter."
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

From:	Jon Harris
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 10:33:02 -0800

"Randy Yates" wrote in message
news:xxp3bww2sop.fsf@usrts005.corpusers.net...
> glen herrmannsfeldt writes:
>
> > Jerry Avins wrote:
> > (someone wrote)
> >
> > >>>>I read that the inverse of an FIR filter is always an IIR filter. I
> > >>>> have no idea how to prove this mathematically.
> >
> >
> > (snip)
> >
> > > Glen's H(z) = 1 is a counterexample. Are there others?
> >
> > Yes, but I also claimed that FIRs are a subset of IIRs.
>
> That doesn't make sense to me, Glen. If the total class of digital
> filters can be partitioned into two sets, those with finite impulse
> response (i.e., "FIR" filters), and those with infinite impulse
> response (not including finite impulse responses), then including FIRs
> in the set of IIRs causes the set of IIRs to be equivalent to the
> total class of digital filters. If that is the case, then the original
> claim is trivially true, "I read that the inverse of a [digital] FIR
> filter is always a digital filter."

Glen in I had a similar discussion recently in my little IIR puzzle thread. The
terms FIR and IIR certainly do imply a nice neat distinction. However, if you
consider IIR filters to be filters with both poles and zeros and FIR filters to
be filters with only (non-trivial) zeros, then it makes more sense that one is a
sub-set of the other. I know there are boundary cases that muddy the waters,
but from that perspective, it makes some sense that what we like to call FIR
filters (zero only) are really a sub-set of a larger class of filters that we
tend to call IIR filters (pole and zero).

From:	robert bristow-johnson
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 15:26:52 -0500

in article 35abq6F4j58phU1@individual.net, Jon Harris at
goldentully@hotmail.com wrote on 01/20/2005 13:33:

> it makes some sense that what we like to call FIR
> filters (zero only) are really a sub-set of a larger class of filters that we
> tend to call IIR filters (pole and zero).

remember that FIR filters *do* have just as many poles as they have zeros.
it's just that all of the poles of an FIR filter are at z=0.

--

r b-j rbj@audioimagination.com

"Imagination is more important than knowledge."

From:	Jon Harris
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 14:27:34 -0800

"robert bristow-johnson" wrote in message
news:BE157E3C.3F3C%rbj@audioimagination.com...
> in article 35abq6F4j58phU1@individual.net, Jon Harris at
> goldentully@hotmail.com wrote on 01/20/2005 13:33:
>
> > it makes some sense that what we like to call FIR
> > filters (zero only) are really a sub-set of a larger class of filters that
we
> > tend to call IIR filters (pole and zero).
>
> remember that FIR filters *do* have just as many poles as they have zeros.
> it's just that all of the poles of an FIR filter are at z=0.

You snipped the part where I said "non-trivial", which was my way of
acknowledging that issue. I was considering the z=0 poles of an FIR filter to
be "trivial poles".

From:	robert bristow-johnson
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 19:16:18 -0500

in article 35aphtF4j6ut4U1@individual.net, Jon Harris at
goldentully@hotmail.com wrote on 01/20/2005 17:27:

> "robert bristow-johnson" wrote in message
> news:BE157E3C.3F3C%rbj@audioimagination.com...
>> in article 35abq6F4j58phU1@individual.net, Jon Harris at
>> goldentully@hotmail.com wrote on 01/20/2005 13:33:
>>
>>> it makes some sense that what we like to call FIR
>>> filters (zero only) are really a sub-set of a larger class of filters that
>>> we tend to call IIR filters (pole and zero).
>>
>> remember that FIR filters *do* have just as many poles as they have zeros.
>> it's just that all of the poles of an FIR filter are at z=0.
>
> You snipped the part where I said "non-trivial", which was my way of
> acknowledging that issue. I was considering the z=0 poles of an FIR filter to
> be "trivial poles".

ok, fine. i didn't know that's what you meant.

it's just that the kernal of the answer to the OP question is that when you
make the inverse of and FIR or IIR filter, the poles become zeros and the
zeros become poles which means that only minimum-phase filters can be
inverted to be a stable filter. whether that be FIR or IIR. those poles at
the origin become zeros.

--

r b-j rbj@audioimagination.com

"Imagination is more important than knowledge."

From:	Jerry Avins
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 13:55:51 -0500

Jon Harris wrote:

> "Randy Yates" wrote in message
> news:xxp3bww2sop.fsf@usrts005.corpusers.net...
>
>>glen herrmannsfeldt writes:
>>
>>
>>>Jerry Avins wrote:
>>>(someone wrote)
>>>
>>>
>>>>>>>I read that the inverse of an FIR filter is always an IIR filter. I
>>>>>>>have no idea how to prove this mathematically.
>>>
>>>
>>>(snip)
>>>
>>>
>>>>Glen's H(z) = 1 is a counterexample. Are there others?
>>>
>>>Yes, but I also claimed that FIRs are a subset of IIRs.
>>
>>That doesn't make sense to me, Glen. If the total class of digital
>>filters can be partitioned into two sets, those with finite impulse
>>response (i.e., "FIR" filters), and those with infinite impulse
>>response (not including finite impulse responses), then including FIRs
>>in the set of IIRs causes the set of IIRs to be equivalent to the
>>total class of digital filters. If that is the case, then the original
>>claim is trivially true, "I read that the inverse of a [digital] FIR
>>filter is always a digital filter."
>
>
> Glen in I had a similar discussion recently in my little IIR puzzle thread. The
> terms FIR and IIR certainly do imply a nice neat distinction. However, if you
> consider IIR filters to be filters with both poles and zeros and FIR filters to
> be filters with only (non-trivial) zeros, then it makes more sense that one is a
> sub-set of the other. I know there are boundary cases that muddy the waters,
> but from that perspective, it makes some sense that what we like to call FIR
> filters (zero only) are really a sub-set of a larger class of filters that we
> tend to call IIR filters (pole and zero).

Poles-and-zeros vs. zeros-only is one dichotomy, but another -- closer
to the names -- is an impulse response that may decay but whose end
point depends only on the precision of the calculation vs. an impulse
response with a definite end. Integers are a subset of real numbers, but
integer vs. non-integer is a true dichotomy.

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

From:	glen herrmannsfeldt
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 11:45:54 -0800

Jerry Avins wrote:
> Jon Harris wrote:

(snip)

>>Glen in I had a similar discussion recently in my little IIR puzzle thread. The
>>terms FIR and IIR certainly do imply a nice neat distinction. However, if you
>>consider IIR filters to be filters with both poles and zeros and FIR filters to
>>be filters with only (non-trivial) zeros, then it makes more sense that one is a
>>sub-set of the other. I know there are boundary cases that muddy the waters,
>>but from that perspective, it makes some sense that what we like to call FIR
>>filters (zero only) are really a sub-set of a larger class of filters that we
>>tend to call IIR filters (pole and zero).

> Poles-and-zeros vs. zeros-only is one dichotomy, but another -- closer
> to the names -- is an impulse response that may decay but whose end
> point depends only on the precision of the calculation vs. an impulse
> response with a definite end. Integers are a subset of real numbers, but
> integer vs. non-integer is a true dichotomy.

Yes, that is the question. In the previous discussion we had:

y(n) = A*y(n-1) + B*x(n) + C*x(n-1)

As a first order IIR filter, or, as you say recursive.

If you implement this filter in hardware or software you say
that it is an implementation of an IIR filter. The hardware or
software doesn't change if A happens to be zero. Every cycles
y(n-1) is multiplied by zero.

Consider a Fortran programmer: 2 is an integer, 2.0 is real.

(In C, 2.0 is double, the analogy doesn't work.)

Some might consider it the difference between theoretical
science and experimental science. That is, whether you actually
build it or just discuss it.

-- glen

From:	Jerry Avins
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 15:34:54 -0500

glen herrmannsfeldt wrote:

> Jerry Avins wrote:
>
>> Jon Harris wrote:
>
>
> (snip)
>
>>> Glen in I had a similar discussion recently in my little IIR puzzle
>>> thread. The
>>> terms FIR and IIR certainly do imply a nice neat distinction.
>>> However, if you
>>> consider IIR filters to be filters with both poles and zeros and FIR
>>> filters to
>>> be filters with only (non-trivial) zeros, then it makes more sense
>>> that one is a
>>> sub-set of the other. I know there are boundary cases that muddy the
>>> waters,
>>> but from that perspective, it makes some sense that what we like to
>>> call FIR
>>> filters (zero only) are really a sub-set of a larger class of filters
>>> that we
>>> tend to call IIR filters (pole and zero).
>
>
>> Poles-and-zeros vs. zeros-only is one dichotomy, but another -- closer
>> to the names -- is an impulse response that may decay but whose end
>> point depends only on the precision of the calculation vs. an impulse
>> response with a definite end. Integers are a subset of real numbers, but
>> integer vs. non-integer is a true dichotomy.
>
>
> Yes, that is the question. In the previous discussion we had:
>
> y(n) = A*y(n-1) + B*x(n) + C*x(n-1)
>
> As a first order IIR filter, or, as you say recursive.
>
> If you implement this filter in hardware or software you say that it is
> an implementation of an IIR filter. The hardware or software doesn't
> change if A happens to be zero. Every cycles y(n-1) is multiplied by
> zero.
>
> Consider a Fortran programmer: 2 is an integer, 2.0 is real.
>
> (In C, 2.0 is double, the analogy doesn't work.)
>
> Some might consider it the difference between theoretical science and
> experimental science. That is, whether you actually build it or just
> discuss it.
>
> -- glen

If one equates IIR to recursive, then he must accept your categories. I
prefer to think that y(n) = A*y(n-1) + B*x(n) + C*x(n-1) is recursive,
but that for some values of A, B, and C, it is FIR.

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

From:	Randy Yates
Subject:	Re: inverse of an FIR filter
Date:	20 Jan 2005 14:28:51 -0500

"Jon Harris" writes:

> "Randy Yates" wrote in message
> news:xxp3bww2sop.fsf@usrts005.corpusers.net...
> > glen herrmannsfeldt writes:
> >
> > > Jerry Avins wrote:
> > > (someone wrote)
> > >
> > > >>>>I read that the inverse of an FIR filter is always an IIR filter. I
> > > >>>> have no idea how to prove this mathematically.
> > >
> > >
> > > (snip)
> > >
> > > > Glen's H(z) = 1 is a counterexample. Are there others?
> > >
> > > Yes, but I also claimed that FIRs are a subset of IIRs.
> >
> > That doesn't make sense to me, Glen. If the total class of digital
> > filters can be partitioned into two sets, those with finite impulse
> > response (i.e., "FIR" filters), and those with infinite impulse
> > response (not including finite impulse responses), then including FIRs
> > in the set of IIRs causes the set of IIRs to be equivalent to the
> > total class of digital filters. If that is the case, then the original
> > claim is trivially true, "I read that the inverse of a [digital] FIR
> > filter is always a digital filter."
>
> Glen in I had a similar discussion recently in my little IIR puzzle thread. The
> terms FIR and IIR certainly do imply a nice neat distinction. However, if you
> consider IIR filters to be filters with both poles and zeros and FIR filters to
> be filters with only (non-trivial) zeros, then it makes more sense that one is a
> sub-set of the other.

Essentially you're postulating that an IIR is a filter with a rational transfer
function, while an FIR is a filter with a polynomial transfer function with a
possible exception of poles at z = 0.

Yeah, I'll buy that Jon (i.e., that one is a subset of the other, defined this
way).
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

From:	Jon Harris
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 14:29:47 -0800

"Randy Yates" wrote in message
news:xxpsm4vzyq4.fsf@usrts005.corpusers.net...
> "Jon Harris" writes:
>
> > "Randy Yates" wrote in message
> > news:xxp3bww2sop.fsf@usrts005.corpusers.net...
> > > glen herrmannsfeldt writes:
> > >
> > > >
> > > > Yes, but I also claimed that FIRs are a subset of IIRs.
> > >
> > > That doesn't make sense to me, Glen. If the total class of digital
> > > filters can be partitioned into two sets, those with finite impulse
> > > response (i.e., "FIR" filters), and those with infinite impulse
> > > response (not including finite impulse responses), then including FIRs
> > > in the set of IIRs causes the set of IIRs to be equivalent to the
> > > total class of digital filters. If that is the case, then the original
> > > claim is trivially true, "I read that the inverse of a [digital] FIR
> > > filter is always a digital filter."
> >
> > Glen in I had a similar discussion recently in my little IIR puzzle thread.
The
> > terms FIR and IIR certainly do imply a nice neat distinction. However, if
you
> > consider IIR filters to be filters with both poles and zeros and FIR filters
to
> > be filters with only (non-trivial) zeros, then it makes more sense that one
is a
> > sub-set of the other.
>
> Essentially you're postulating that an IIR is a filter with a rational
transfer
> function, while an FIR is a filter with a polynomial transfer function with a
> possible exception of poles at z = 0.
>
> Yeah, I'll buy that Jon (i.e., that one is a subset of the other, defined this
> way).

Yes, and thanks for "rigourizing" it (taking my descriptive text and adding more
mathematical meat)!

From:	Jerry Avins
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 10:39:18 -0500

Randy Yates wrote:
> glen herrmannsfeldt writes:
>
>
>>Jerry Avins wrote:
>>(someone wrote)
>>
>>
>>>>>>I read that the inverse of an FIR filter is always an IIR filter. I
>>>>>>have no idea how to prove this mathematically.
>>
>>
>>(snip)
>>
>>
>>>Glen's H(z) = 1 is a counterexample. Are there others?
>>
>>Yes, but I also claimed that FIRs are a subset of IIRs.
>
>
> That doesn't make sense to me, Glen. If the total class of digital
> filters can be partitioned into two sets, those with finite impulse
> response (i.e., "FIR" filters), and those with infinite impulse
> response (not including finite impulse responses), then including FIRs
> in the set of IIRs causes the set of IIRs to be equivalent to the
> total class of digital filters. If that is the case, then the original
> claim is trivially true, "I read that the inverse of a [digital] FIR
> filter is always a digital filter."

That's what we get for writing "IIR" when we mean "recursive".

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

From:	Randy Yates
Subject:	Re: inverse of an FIR filter
Date:	20 Jan 2005 11:15:27 -0500

Jerry Avins writes:
> [...]
> That's what we get for writing "IIR" when we mean "recursive".

I did not take Glen's definition of IIR to include "only the subset of
finite impulse responses that can be realized recursively" but instead
interpreted it as "ALL finite impulse responses."
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124

From:	Jerry Avins
Subject:	Re: inverse of an FIR filter
Date:	Wed, 19 Jan 2005 16:39:06 -0500

glen herrmannsfeldt wrote:

> Jerry Avins wrote:
> (someone wrote)
>
>>>>> I read that the inverse of an FIR filter is always an IIR filter. I
>>>>> have no idea how to prove this mathematically.
>
>
> (snip)
>
>> Glen's H(z) = 1 is a counterexample. Are there others?
>
>
> Yes, but I also claimed that FIRs are a subset of IIRs.
>
> Consider the problem of the reciprocals of integers.
> 1/3 is a repeating decimal, 1/5 is not. I used to know

That seems artificial to me, as it depends on the representation base.
A number is divisible by 7 if the sum of its digits are, provided you
work in octal.

> some math people who claimed that you could write
> 1/5 instead of 0.2, as 0.199999999999999999999...., that is,
> as a repeating decimal.
>
> Of all possible FIRs only a very small fraction have an
> FIR as their inverse, but you might get lucky.
>
> -- glen
>

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

From:	Bob Cain
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 16:44:55 -0800

glen herrmannsfeldt wrote:

> Of all possible FIRs only a very small fraction have an
> FIR as their inverse, but you might get lucky.

OTOH, in world of audio impulse responses I have yet to see
an inverse that isn't decaying to zero such that it
eventually becomes finite just due to digital representation
limits.

Most inverses aren't much longer than the original for all
practical purposes.

Bob
--

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

A. Einstein

From:	glen herrmannsfeldt
Subject:	Re: inverse of an FIR filter
Date:	Fri, 21 Jan 2005 09:37:53 -0800

Bob Cain wrote:

(I wrote)

>> Of all possible FIRs only a very small fraction have an
>> FIR as their inverse, but you might get lucky.

> OTOH, in world of audio impulse responses I have yet to see an inverse
> that isn't decaying to zero such that it eventually becomes finite just
> due to digital representation limits.

> Most inverses aren't much longer than the original for all practical
> purposes.

IIR, or more specifically recursive filters, have the advantage
in the number of terms required for some common filter designs,
but it would seem that in finite precision arithmetic (most
audio systems) FIR could do just as well.

-- glen

From:	Jon Harris
Subject:	Re: inverse of an FIR filter
Date:	Thu, 20 Jan 2005 17:33:35 -0800

"Bob Cain" wrote in message
news:cspja301527@enews3.newsguy.com...
>
>
> glen herrmannsfeldt wrote:
>
> > Of all possible FIRs only a very small fraction have an
> > FIR as their inverse, but you might get lucky.
>
> OTOH, in world of audio impulse responses I have yet to see
> an inverse that isn't decaying to zero such that it
> eventually becomes finite just due to digital representation
> limits.

Worst case is probably something like an IIR reverb, which decays into the
digital noise floor typically after "seconds".