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Subject: Re: Historical question: negative feedback and the op amp
X-Newsreader: Microsoft Outlook Express 5.50.4920.2300
Date: Thu, 14 Nov 2002 15:55:18 GMT
NNTP-Posting-Date: Thu, 14 Nov 2002 10:55:18 EST
Organization: Cox Communications
"Chuck Simmons" wrote in message
> Mike wrote:
> > "Chuck Simmons" wrote in message
> > news:3DD2FD36.8877972F@webaccess.net...
> > > Mike wrote:
> > > >
> > > > "Chuck Simmons" wrote in message
> > > > news:3DD27F3D.667A4D75@webaccess.net...
> > > > > Mike wrote:
> > > > > >
> > > > > > "Jeroen" wrote in message
> > > > > > news:GyuA9.80964$I6.email@example.com...
> > > > > > >
> > > > > > >
> > > > > > > In case of IIR filters, feedback reduces the number of
> > calculations
> > > > > > needed.
> > > > > > > In this case feedback isn't a thing that slows things down.
> > > > > >
> > > > > > Actually, it does slow down the maximum operating frequency. I
> > > > pipeline
> > > > > > an FIR filter almost without limit, and obtain very high clock
> > > > frequencies.
> > > > > > In an IIR filter, the maximum operating frequency is often
> > determined by
> > > > the
> > > > > > feedback loop. In the general case, the feedback term has to be
> > fully
> > > > > > calculated in one clock cycle.
> > > > >
> > > > > An IIR can produce output with one multiply and one add at the
> > > > > of the sample for that iteration. Does anyone not do this? It is
> > basic.
> > > > > This means that there is no difference in output speed of an FIR
> > versus
> > > > > an IIR with fixed sampling and infinite speed arithmetic.
> > > >
> > > > Yes, but if your arithmetic isn't infinite speed, then there's a
> > practical
> > > > limitation due to the feedback loop in the IIR filter. As a result,
> > FIR
> > > > clock rate can be significantly faster than for an IIR filter with
> > > > equivalent word widths.
> > >
> > > I don't see this at all. Forgetting that an FIR filter cannot be used
> > > a compensator for controlling a dynamical system of order two or
> > > greater, exactly the same amount of time is available for computation
> > > fixed sample rate for both. For both the IIR and the FIR, the
> > > computation is a difference equation but the FIR has no output terms
> > > it. In the control situation, the pipeline used for the FIR must be
> > > short because, in most cases, the nth output from the controller
> > > on the nth input. If not, the controller will fail anyway unless made
> > > even more complex than the high order already required to control a
> > > dynamical system because of delay. Delay requires estimation which
> > > simply means the order goes up. Estimators are of limited use because
> > > errors introduced from the outside that are unknown to the estimator
> > > which is designed for perfect dynamics (what else?).
> > Okay, here's what I'm thinking: the FIR output is given by
> > y(k) = a0*x(k-1) + a1*x(k-2) + .... + aN-1*x(k-N)
> > For the sake of simplicity, let's ignore the multiplications for this
> > example. Once the multiplications are completed, I still have to find
> > sum of the products. If I'm running at a fast clock rate, there's not
> > to do all these additions, but I can start pipelining. On the first
> > cycle, I'll add adjacent terms in pairs:
> > b01(k) = a0*x(k-1) + a1*x(k-2)
> > b23(k) = a2*x(k-3) + a3*x(k-4)
> > ...
> > On the next clock cycle, add the b terms in pairs:
> > c02(k) = b01(k) + b23(k)
> > c46(k) = b45(k) + b67(k)
> > ...
> > Keep going until the process is complete. Each level is registered; the
> > computation takes place at the same time that the b(k+1) terms are being
> > calculated.
> > We can go further: if there's inadequate time to complete the additions
> > one clock cycle, they can be pipelined as well. I built an FIR a few
> > ago that used a Booth multiplier with a latency of three clock cycles,
> > tree of adders with latency of two clocks at the top of the tree, and
> > clocks at the bottom (as the word widths increased). The overall latency
> > something like 12 clock cycles from the time that the last data word
> > the filter.
> > In the IIR filter, the y(k) result is needed during the next clock
> > and there's no practical way to delay it in the general case.
> > -- Mike --
> But in typical IIR implementations the data present in the processor at
> each sample is only one data point less than for your FIR. However,
> piplining will usually not be of interest because of the lower order
> required for realization of an IIR filter.
> However, let's take up the control of dynamical systems and see why an
> FIR, in general, will not work.
> Generally, a sampled system modelled by a system of n linear first order
> ODEs (the same as an nth order ODE) with constant coefficients gives
> rise to a quotient of polynomials say p(z)/q(z) where the degree of q(z)
> is n and the degree of p(z)<=n (deg(p)>deg(q) is non-causal). Assume
> that there is an FIR controller we wish to apply which is given by
> r(z)/z^k where r(z) is a polynomial of degree k. Obviously, the closed
> loop system including the FIR is stable if all of the zeros (roots) of
> z^k*q(z)-r(z)*p(z) are inside the unit circle. There will be n+k zeros
> of this polynomial. However, of the coefficients of this polynomial, we
> may select arbitrarily at most k+1 leaving n-1 coefficients to chance.
> In other words, we can force at most k+1 zeros inside the unit circle by
> selecting the coefficients of r(z) while the other n-1 can be anywhere.
> That is, give an arbitrary p(z)/q(z), we cannot, in general, stabilize
> it with an FIR.
> An interesting point is that if n=1, we can always use an FIR. At n=2,
> we have one pole we cannot place.
> It is easy to see that for any p(z) and q(z) there is a stabilizing
> controller of the IIR type which is of at most order n (the polynomials
> are of degree at most n).
> Another FIR problem is that an FIR places k system zeros at the origin
> of the unit circle. This causes extreme parameter sensitivity which
> translates to poor stability margin even assuming an FIR stabilizing
> controller exists (by no means given as shown).
> An advantage in an IIR is that it can be broken into cascades of order
> at most 2. This greatly reduces parameter sensitivity. The penalty for
> cascades is one multiply and add for each cascade section if we require
> that output from a new sample be available with one multiply and add.
> Cleary, there is little reason to even look for a possible but unlikely
> FIR to controller for a dynamical system.
Okay, but I was answering a different question. I can make a faster FIR
filter because I can trade latency (and hardware) for speed, even if the FIR
is higher order. Maybe I missed the point of your original comment - I
thought it was limited to a comparison of IIR and FIR filters, not those
filters contained in in a feedback loop.
On the other hand, the speed of a control loop containing an IIR or FIR
filter is another matter. Assuming that an FIR filter could be used to
control the system under consideration, the additional latency of a
pipelined FIR introduces even more delay in the loop, which decreases
stability even further.
-- Mike --
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