can we just see the Fourier Spectrum of a filter and know if it is a FIR or a IIR?

I have a unit response function h(n), and its DFT is H(n). I am
wondering if we just see the H(n) and know if it is a FIR or a IIR? I
think we cannot since just look at the H(n) I cannot see any noticeable
difference.

Thanks

VijaKhara wrote:
> I have a unit response function h(n), and its DFT is H(n). I am
> wondering if we just see the H(n) and know if it is a FIR or a IIR? I
> think we cannot since just look at the H(n) I cannot see any noticeable
> difference.
> 
> Thanks

Generally the term "discrete Fourier transform" implies that you are 
taking the transform over a finite amount of time.  Because the 
transform is finite there is no way to distinguish a finite sequence 
from an infinite one.

In theory if you took a _full_ Fourier transform of a sequence, 
extending to infinity, you would be able to tell if it could be 
generated by a linear IIR filter vs. an FIR filter.

In practice you could take a DFT of a response function and approximate 
it with a linear IIR filter, then check your approximation to see how 
accurate it would be.

Probably the easiest thing to do, however, would be to do an inverse DFT 
on your H(n), look at the resulting h(n) and see if it looks finite or 
infinite.  All fancy math aside, I think that's what I would do.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"VijaKhara" <VijaKhara@gmail.com> wrote in message
news:1142959978.601223.284610@z34g2000cwc.googlegroups.com...
> I have a unit response function h(n), and its DFT is H(n). I am
> wondering if we just see the H(n) and know if it is a FIR or a IIR? I
> think we cannot since just look at the H(n) I cannot see any noticeable
> difference.
>
> Thanks

I'd say you are correct - in general.
BTW - I presume you mean H(f) (not H(n))
H(f) can have a magnitude vs freq and a phase vs freq. You might be able to
guess a little better if you looked at the phase vs freq response. A
non-linear phase response makes it likely that your filter is a non-FIR
filter.

A few more notes on terminology. I could have an impulse response h(n) which
describes the transfer function of a black box (say a room). I could choose
to use an FIR or IIR implementation that approximates this impulse response
h(n) (call it h(n)'). Studying h(n) and its DFT will give you close to
nothing. Studying h(n)' and its DFT might give a few clues on the filter
(which we've already talked about).

Cheers
Bhaskar

in article 1142959978.601223.284610@z34g2000cwc.googlegroups.com, VijaKhara
at VijaKhara@gmail.com wrote on 03/21/2006 11:52:

> I have a unit response function h(n), and its DFT is H(n). I am
> wondering if we just see the H(n) and know if it is a FIR or a IIR? I
> think we cannot since just look at the H(n) I cannot see any noticeable
> difference.

i would recommend using the notational convention common in DSP texts of
indicating a discrete-time (or discrete-frequency) sequence with brackets
instead of parenths.  use "h[n]" instead of "h(n)".  and leave the parenths
for continuous-time functions such as the implulse response "h(t)" of an
analog filter.

anyway, there are a class of FIR filters sometimes known as "Truncated IIR
filters" (TIIR) which can be efficiently implemented with an IIR filter and
a delay line to subtract out the "tail" of the remaining IIR.

anyway, with TIIR, i can make an FIR approach the IIR to whatever level of
precision i want.  the spectrums will likewise get just as close to each
other.

so i think the answer to your question is simply "no".

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

VijaKhara wrote:

> I have a unit response function h(n), and its DFT is H(n). I am
> wondering if we just see the H(n) and know if it is a FIR or a IIR?

Your filter is FIR (otherwise you couldn't take the DFT of its impulse
response).

"Andor" <andor.bariska@gmail.com> wrote in message 
news:1142968534.670045.177150@t31g2000cwb.googlegroups.com...
> VijaKhara wrote:
>
>> I have a unit response function h(n), and its DFT is H(n). I am
>> wondering if we just see the H(n) and know if it is a FIR or a IIR?
>
> Your filter is FIR (otherwise you couldn't take the DFT of its impulse
> response).
>

Or, said another way, when you take the DFT of a unit sample response then 
you are forced into finite time.  That *makes* it a FIR filter: thus Andor's 
comment "your filter is a FIR".

Now, if the original was an IIR filter and you truncate its unit sample 
response into a FIR, that doesn't mean you might not still implement an IIR 
to get similar results.  Otherwise, it *is* a (possibly long) FIR that 
you're analyzing.

Fred 

Andor wrote:
> VijaKhara wrote:
>
> > I have a unit response function h(n), and its DFT is H(n). I am
> > wondering if we just see the H(n) and know if it is a FIR or a IIR?

Even if it originally was a truncated IIR, you could use the DFT
results to construct a FIR which would the produce identical DFT
results.  Thus you couldn't tell the difference in that direction.

But perhaps there are FIR responses that are impossible to
duplicate via a truncated IIR under certain restrictions (degree,
numeric precision, latency, etc.)  So the answer may be asymmetric
once any details are filled in.

> Your filter is FIR (otherwise you couldn't take the DFT of its impulse
> response).

So is it your opinion that every digitally-represented IIR with
a decay that takes the filter response tail well below the
quantization floor of the filter output and state representation
in finite time is actually a FIR, since you could take a numeric
DFT of the impulse response in that situation?


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
...
> > Your filter is FIR (otherwise you couldn't take the DFT of its impulse
> > response).
>
> So is it your opinion that every digitally-represented IIR with
> a decay that takes the filter response tail well below the
> quantization floor of the filter output and state representation
> in finite time is actually a FIR, since you could take a numeric
> DFT of the impulse response in that situation?

My opinion is every filter with a finite duration impulse response is
an FIR filter.

Andor wrote:
> Ron N. wrote:
> ...
> > > Your filter is FIR (otherwise you couldn't take the DFT of its impulse
> > > response).
> >
> > So is it your opinion that every digitally-represented IIR with
> > a decay that takes the filter response tail well below the
> > quantization floor of the filter output and state representation
> > in finite time is actually a FIR, since you could take a numeric
> > DFT of the impulse response in that situation?
>
> My opinion is every filter with a finite duration impulse response is
> an FIR filter.

I might add that the problem of order reduction is somewhat related,
but much more interesting.

Assume that a measured impulse response has at some time N decayed
below a threshold where we feel that truncation introduces acceptable
error. This truncated impulse response can be perfectly implemented
using an N-th order FIR. How do we go about designing an M-th order IIR
system (if possible sparse) with an impulse response that minimizes
some kind of cost function on the difference vector of the two impulse
resonses (the FIR impulse response can be extrapolated with 0 to
measure the error)?

VijaKhara wrote:
> I have a unit response function h(n), and its DFT is H(n). I am
> wondering if we just see the H(n) and know if it is a FIR or a IIR? I
> think we cannot since just look at the H(n) I cannot see any noticeable
> difference.

No, you can not tell the difference between an IIR and an FIR system by

looking at the spectrum. There are two main arguments:

1) As others already have commented, the DFT works on a finite
   data sequence, so any spectrum is generetaed from the impulse
   response of an FIR filter.

2) Even if you have the N-length spectrum of a "true" FIR filter of
length
    M, M<N, a test for trailing zeros would not work due to numerical
    inaccuracies.

So the two possible tests are inconclusive: You can not test for
infinite
duration since you can only work with finite data sequences, and you
can not test for vanishing coefficients since the impulse response of
a stable IIR filter vanishes "sufficiently far" into the transient.

Rune