IIR Filter Realizations

In most books, block diagram realizations of IIR filters are shown -
these include Direct form1  and Direct form II also, transposed
realization and so on. I was wondering what the point is. Essentially
these are all state-space type realizations but in implmenting a filter
we just use the difference equation. I can see that if you were
implmenting in hardware then you would gain by for example only having
one set of delays rather than II but is this of importance? I expect
FPGA realization may benefit.


Naebad

naebad wrote:
> In most books, block diagram realizations of IIR filters are shown -
> these include Direct form1  and Direct form II also, transposed
> realization and so on. I was wondering what the point is. Essentially
> these are all state-space type realizations but in implmenting a filter
> we just use the difference equation. I can see that if you were
> implmenting in hardware then you would gain by for example only having
> one set of delays rather than II but is this of importance? I expect
> FPGA realization may benefit.

Two sets of delays take twice as long to process and consume twice as 
much variable space. With hardware, the delays run in parallel, but in 
software they are sequential. Is that enough? There can be numerical 
issues as well; some forms are better behaved than others.

Jerry
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naebad wrote:
> In most books, block diagram realizations of IIR filters are shown -
> these include Direct form1  and Direct form II also, transposed
> realization and so on. I was wondering what the point is. Essentially
> these are all state-space type realizations but in implmenting a filter
> we just use the difference equation.

there are different difference equations for the DF1 vs. DF2 vs.
transposed DF1/2 or the lattice or ladder filters.  when you do this
with finite precision words, there are practical differences.

> I can see that if you were
> implmenting in hardware then you would gain by for example only having
> one set of delays rather than II but is this of importance? I expect
> FPGA realization may benefit.

doesn't matter whether it's FPGA or software.  in both cases you have
finite precision words.  we've discussed this before multiple times on
comp.dsp.  suppose you have a high Q filter but one where the output
gain is still 0 dB or less.  (then you have a pair of poles very close
to the unit circle, but in between the poles and the unit circle are
zeros that are even closer to the unit circle) if you have fixed-point
words, the DF2 will essentially boost your signal first (with the
poles) and then cut your signal.  but there will be clipping between
the boost and the cut.  there are two adders and two quantization
points for the DF2.  if it's DF1 and your one-and-only adder is double
precision all 5 feedback and feedforward paths can be added before the
single quantization.  there is no clipping because your zeros happened
before the poles.

r b-j

"naebad" <minnaebad@yahoo.co.uk> wrote in message 
news:1159751598.411833.97300@m73g2000cwd.googlegroups.com...
> In most books, block diagram realizations of IIR filters 
> are shown -
> these include Direct form1  and Direct form II also, 
> transposed
> realization and so on. I was wondering what the point is. 
> Essentially
> these are all state-space type realizations but in 
> implmenting a filter
> we just use the difference equation. I can see that if you 
> were
> implmenting in hardware then you would gain by for example 
> only having
> one set of delays rather than II but is this of 
> importance? I expect
> FPGA realization may benefit.
>

I'm not sure what your "Direct Form" I and II look like but 
my reference shows two diagrams: "Direct" and "Canonic" 
implementations.

Looking quickly at the "Direct" diagram, one could be 
forgiven for concluding that the delays corresponding to the 
zeroes are added to the delays corresponding to the poles. 
On closer inspection, it becomes clear that, from the 
incoming signal's point of view, the zeroes and poles are 
processed in parallel.  Consequently, the delay in the 
"direct" form is governed by the greater of the number of 
poles or zeroes, just as in the "canonic" form.

The main reason for preferring canonic over direct is higher 
stability and lower internal generation of noise caused by 
round-off error in finite precision implementations.  There 
may also be a minor performance difference since one has to 
manage only half as many delay elements in the canonic form.

One of the reasons for the improved noise performance of the 
canonic form is that the signal is first processed by the 
poles, then by the zeroes.  In the direct form, noise 
generated by the zeroes is introduced to the poles, which 
recirculate it over and over (IIR) through their inherent 
feedback mechanism.

naebad wrote:

> In most books, block diagram realizations of IIR filters are shown -
> these include Direct form1  and Direct form II also, transposed
> realization and so on. I was wondering what the point is.

There are the different requirements for the storage memory and for the 
arithmetic precision. Also, the different IIR structures may be more or 
less convenient depending on the specific hardware.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

John E. Hadstate wrote:
>
> I'm not sure what your "Direct Form" I and II look like but
> my reference shows two diagrams: "Direct" and "Canonic"
> implementations.

the word "canonic" had been used for architectures where the number of
states is equal to the order of the filter.  DF2 is canonic, the DF1 is
not.

> Looking quickly at the "Direct" diagram, one could be
> forgiven for concluding that the delays corresponding to the
> zeroes are added to the delays corresponding to the poles.
> On closer inspection, it becomes clear that, from the
> incoming signal's point of view, the zeroes and poles are
> processed in parallel.

DF1:

    y[n] = b0*x[n] + b1*x[n-1] + b2*x[n-2] - a1*y[n-1] - a2*y[n-2]

which can be rewritten as:

    v[n] = b0*x[n] + b1*x[n-1] + b2*x[n-2]
    y[n] = v[n] - a1*y[n-1] - a2*y[n-2]

the sub-filter with zeros processes x[n] to v[n] and that is followed
by a sub-filter with poles that processes v[n] to y[n].  if there is no
quantization imposed upon v[n] (and that is trivial with a typical
fixed-point DSP like the 56K or the fixed-point portion of the SHArC,
all you need to do is keep your double precision word until y[n] is
computed), then the boost from the 2nd sub-filter (the one with the
poles) will boost no quantization noise.

in contrast, the DF2 is:

    v[n] = x[n]    - a1*v[n-1] - a2*v[n-2]
    y[n] = b0*v[n] + b1*v[n-1] + b2*v[n-2]

the poles come first, then the zeros.

> The main reason for preferring canonic over direct is higher
> stability and lower internal generation of noise caused by
> round-off error in finite precision implementations.

John, it's the opposite.  you do a DF2 in fixed-point and, if you have
enough Q in your biquad to make it useful for anything interesting, the
poles will result in a lot of gain which will cause saturation at
frequencies close to the resonant frequency and even though the
following zeros will beat that gain back down, the damage of the
saturation remains.  then, too avoid such saturation you would have to
crank down the input gain which will decrease S/N.  it's the DF1 (the
non-canonical form) that is preferred in a fixed-point context.

>  There
> may also be a minor performance difference since one has to
> manage only half as many delay elements in the canonic form.
>
> One of the reasons for the improved noise performance of the
> canonic form is that the signal is first processed by the
> poles, then by the zeroes.

it's the opposite, John.  doing the polse first causes saturation.  to
avoid the saturation you have to crank down the input.  but the
quantization noise is not going down with it.  that decreases S/N.

>  In the direct form, noise
> generated by the zeroes is introduced to the poles, which
> recirculate it over and over (IIR) through their inherent
> feedback mechanism.

quantizers generate noise, zeros and poles do not.

r b-j

"robert bristow-johnson" <rbj@audioimagination.com> wrote in 
message 
news:1159821032.691362.12240@m73g2000cwd.googlegroups.com...

> quantizers generate noise, zeros and poles do not.
>

I'm sorry.  You are mistaken, and all of your conclusions 
about canonic vs. direct-form IIR's are wrong.  Moreover, I 
proved this more than thirty years ago.

> r b-j
> 

"John E. Hadstate" <jh113355@hotmail.com> writes:
> [...]
> I'm sorry.  You are mistaken, and all of your conclusions 
> about canonic vs. direct-form IIR's are wrong.  Moreover, I 
> proved this more than thirty years ago.

Reference?
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http://home.earthlink.net/~yatescr

robert bristow-johnson wrote:
> John E. Hadstate wrote:
>> I'm not sure what your "Direct Form" I and II look like but
>> my reference shows two diagrams: "Direct" and "Canonic"
>> implementations.
> 
> the word "canonic" had been used for architectures where the number of
> states is equal to the order of the filter.  DF2 is canonic, the DF1 is
> not.

Sadly you can use a term like canonic to mean pretty much anything you 
want. It lacks a well standardised definition. Most books just say 
something is a canonic form, without justifying the term at all. I 
remember at college when we first heard the term applied to something. 
We were wondering how religion got into electronics. :-) When we asked 
our very capable lecturer about the exact meaning of the term, he fudged 
this issue.

Steve

Steve Underwood skrev:
> robert bristow-johnson wrote:
> > John E. Hadstate wrote:
> >> I'm not sure what your "Direct Form" I and II look like but
> >> my reference shows two diagrams: "Direct" and "Canonic"
> >> implementations.
> >
> > the word "canonic" had been used for architectures where the number of
> > states is equal to the order of the filter.  DF2 is canonic, the DF1 is
> > not.
>
> Sadly you can use a term like canonic to mean pretty much anything you
> want. It lacks a well standardised definition. Most books just say
> something is a canonic form, without justifying the term at all. I
> remember at college when we first heard the term applied to something.
> We were wondering how religion got into electronics. :-) When we asked
> our very capable lecturer about the exact meaning of the term, he fudged
> this issue.

>From http://www.thefreedictionary.com/canonic, item 3:

canonic - reduced to the simplest and most significant form possible
without loss of generality;

Rune