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 ....
inverse of an FIR filter
Started by ●January 18, 2005
Reply by ●January 20, 20052005-01-20
Reply by ●January 20, 20052005-01-20
glen herrmannsfeldt <gah@ugcs.caltech.edu> 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
Reply by ●January 20, 20052005-01-20
Randy Yates wrote:> glen herrmannsfeldt <gah@ugcs.caltech.edu> 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. �����������������������������������������������������������������������
Reply by ●January 20, 20052005-01-20
Jerry Avins <jya@ieee.org> 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
Reply by ●January 20, 20052005-01-20
"Randy Yates" <randy.yates@sonyericsson.com> wrote in message news:xxp3bww2sop.fsf@usrts005.corpusers.net...> glen herrmannsfeldt <gah@ugcs.caltech.edu> 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).
Reply by ●January 20, 20052005-01-20
Jon Harris wrote:> "Randy Yates" <randy.yates@sonyericsson.com> wrote in message > news:xxp3bww2sop.fsf@usrts005.corpusers.net... > >>glen herrmannsfeldt <gah@ugcs.caltech.edu> 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. �����������������������������������������������������������������������
Reply by ●January 20, 20052005-01-20
"Jon Harris" <goldentully@hotmail.com> writes:> "Randy Yates" <randy.yates@sonyericsson.com> wrote in message > news:xxp3bww2sop.fsf@usrts005.corpusers.net... > > glen herrmannsfeldt <gah@ugcs.caltech.edu> 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
Reply by ●January 20, 20052005-01-20
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
Reply by ●January 20, 20052005-01-20
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."
Reply by ●January 20, 20052005-01-20
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. > > -- glenIf 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. �����������������������������������������������������������������������
