how to filter this optimally?

hi,

I wonder how you guys would proceed to solve the following (probably rather
standard) problem:

- given a system that has an average sample rate of approx 200 Hz
- a space-time signal needs to be smoothed, i.e. find x_hat[k] given x[k],
y_hat[k] given y[k] where x_hat, y_hat are the estimates of the unknown
truth
- noise is present in the signal. the noise seems to be additive and follow
a normal distribution with 0 mean and a variance that is not constant but
which, in average is constant and rather small compared to average step
size and magnitude of the signal of interest
- afaik, usually this problem is known as a stochastic filtering or
smoothing problem. here however the noise is pretty well known, but the
underlying signal cannot be modelled easily: curvilinear motion model may
hold, however, its parameters are totally random and time-varying.

the problem solving constraints are the following:
- it's a real time problem, no decimation is allowed
- filter as much of the noise as possible, but no specific design goal
- no overshoot
- pass band must have unity gain, no error tolerance
- maximum group delay in pass band shall not exceed 4 samples
- implementation complexity does not matter, FPU is available if necessary

my approach was the following:
- given the nature of the problem it'll be hard/impossible to use a
stochastic filter like Kalman filter, because the process model is not
known
- therefore, simple low-pass filtering is my choice: however, which
filter?
  - 1st order IIR LPF, e.g. the exponential smoother?
  - simple averager on 8 samples?
  - Gaussian FIR LPF on 8 samples?

anything better?

thanks for all your advices and tips...

Andy
	 

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On Wed, 26 Mar 2014 05:42:52 -0500, ombz wrote:

> hi,
> 
> I wonder how you guys would proceed to solve the following (probably
> rather standard) problem:
> 
> - given a system that has an average sample rate of approx 200 Hz - a
> space-time signal needs to be smoothed, i.e. find x_hat[k] given x[k],
> y_hat[k] given y[k] where x_hat, y_hat are the estimates of the unknown
> truth - noise is present in the signal. the noise seems to be additive
> and follow a normal distribution with 0 mean and a variance that is not
> constant but which, in average is constant and rather small compared to
> average step size and magnitude of the signal of interest - afaik,
> usually this problem is known as a stochastic filtering or smoothing
> problem. here however the noise is pretty well known, but the underlying
> signal cannot be modelled easily: curvilinear motion model may hold,
> however, its parameters are totally random and time-varying.
> 
> the problem solving constraints are the following:
> - it's a real time problem, no decimation is allowed - filter as much of
> the noise as possible, but no specific design goal - no overshoot - pass
> band must have unity gain, no error tolerance - maximum group delay in
> pass band shall not exceed 4 samples - implementation complexity does
> not matter, FPU is available if necessary
> 
> my approach was the following:
> - given the nature of the problem it'll be hard/impossible to use a
> stochastic filter like Kalman filter, because the process model is not
> known - therefore, simple low-pass filtering is my choice: however,
> which filter?
>   - 1st order IIR LPF, e.g. the exponential smoother?
>   - simple averager on 8 samples?
>   - Gaussian FIR LPF on 8 samples?

If you frame your problem right, the Kalman filter that you get will be a 
first order IIR LPF with a varying pole that tends toward a constant.

If delay is an issue then your best bet is an IIR filter, probably a 
first or second order.  Given your "no overshoot" requirement, that's 
pointing to a first-order.  If you do use a second-order, then start with 
a continuous-time Bessel prototype and design from there.

I think I'd just see what I could do with a 1st-order or 2nd-order Bessel 
that meets your group delay requirement, then decide if I wanted to tell 
my boss that I'm a hero, or quietly brush up my resume.

-- 

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

Hi,

what I'd do is use a very long FIR filter but enforce a group delay of four
samples in the numeric optimization. It's almost an IIR filter but it has
more design parameters than a conventional implementation with feedback.
You can't meet all the requirements so specifying the best trade-off
becomes the main problem. The numerical optimization itself is usually
robust for a few dozen parameters. For more taps, use an analog filter /
IIR impulse response of your choice as initial value.	 

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mnentwig <24789@dsprelated> wrote:

(previously snipped question)

> what I'd do is use a very long FIR filter but enforce a group delay of four
> samples in the numeric optimization. It's almost an IIR filter but it has
> more design parameters than a conventional implementation with feedback.
> You can't meet all the requirements so specifying the best trade-off
> becomes the main problem. The numerical optimization itself is usually
> robust for a few dozen parameters. For more taps, use an analog filter /
> IIR impulse response of your choice as initial value.    

The four sample group delay seems a little restrictive.

For either the long FIR or IIR, you will have to allow any transients
to die out. The OP didn't mention restrictions on that.

-- glen

On 3/26/14 8:27 PM, glen herrmannsfeldt wrote:
> mnentwig<24789@dsprelated>  wrote:
>
> (previously snipped question)
>
>> what I'd do is use a very long FIR filter but enforce a group delay of four
>> samples in the numeric optimization.

curious what the FIR, h[n], is for n >> 8 ?

>>  It's almost an IIR filter but it has
>> more design parameters than a conventional implementation with feedback.
>> You can't meet all the requirements so specifying the best trade-off
>> becomes the main problem. The numerical optimization itself is usually
>> robust for a few dozen parameters. For more taps, use an analog filter /
>> IIR impulse response of your choice as initial value.
>
> The four sample group delay seems a little restrictive.
>

it is.  it means anything beyond the 9th tap (or "8th" if you count the 
"0th" tap) contributes to a group delay greater than 4 samples.

> For either the long FIR or IIR, you will have to allow any transients
> to die out. The OP didn't mention restrictions on that.

but doesn't the physics (or maths)?

how much energy do you dare put into h[n] after n=8?  (i know we can 
sorta PWM a low frequency component, but i think we have to "enforce a 
group delay of four samples" on a restricted range of frequencies.)


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Dear all,

Thanks for your answers and hints!

So it's definitely either a 1st order IIR or an 8-tap FIR, as I thought.
Not sure about that more-than-8-tap FIR. Of course when you simply add very
weak taps you don't really add much of a group delay, but doesn't filter
more neither essentially... But I try to experiment a little.

The group delay spec actually is my translation from a 20 ms "latency" spec
(@fs=200 Hz). What is "latency" in my system, how it is measured and
perceived that's a whole different story. For me, the term "latency" is
pretty inacurrate, or even inappropriate, and I think of it rather as a
marketing term... But I realized that group delay and "latency" match
pretty well in my case. So that's why: max group delay = 4 samples.

@Tim: I really appreciate your proposals, especially the one on being a
hero, or brushing up my resumé secretly haha. ;)
hmmp, I ever wondered how to get a Bessel filter right in the digital
domain. I never succeeded. Any hints?

Enjoy your week end

Andy


>On 3/26/14 8:27 PM, glen herrmannsfeldt wrote:
>> mnentwig<24789@dsprelated>  wrote:
>>
>> (previously snipped question)
>>
>>> what I'd do is use a very long FIR filter but enforce a group delay of
four
>>> samples in the numeric optimization.
>
>curious what the FIR, h[n], is for n >> 8 ?
>
>>>  It's almost an IIR filter but it has
>>> more design parameters than a conventional implementation with
feedback.
>>> You can't meet all the requirements so specifying the best trade-off
>>> becomes the main problem. The numerical optimization itself is usually
>>> robust for a few dozen parameters. For more taps, use an analog filter
/
>>> IIR impulse response of your choice as initial value.
>>
>> The four sample group delay seems a little restrictive.
>>
>
>it is.  it means anything beyond the 9th tap (or "8th" if you count the 
>"0th" tap) contributes to a group delay greater than 4 samples.
>
>> For either the long FIR or IIR, you will have to allow any transients
>> to die out. The OP didn't mention restrictions on that.
>
>but doesn't the physics (or maths)?
>
>how much energy do you dare put into h[n] after n=8?  (i know we can 
>sorta PWM a low frequency component, but i think we have to "enforce a 
>group delay of four samples" on a restricted range of frequencies.)
>
>
>-- 
>
>r b-j                  rbj@audioimagination.com
>
>"Imagination is more important than knowledge."
>
>
>	 

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On 3/28/14 1:55 PM, ombz wrote:
>
> So it's definitely either a 1st order IIR or an 8-tap FIR, as I thought.


actually a 9-tap FIR.  h[0] is a tap with non-zero coefficient and h[8] 
is another (and equal valued, if it's phase-linear) tap.

a 1-tap FIR is really just a static gain, and has group delay of 0 samples.


> Not sure about that more-than-8-tap FIR.

it cannot have constant group delay of 4 samples (assuming you mean 
"9-tap FIR").

> Of course when you simply add very
> weak taps you don't really add much of a group delay, but doesn't filter
> more neither essentially...

yup.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Fri, 28 Mar 2014 12:55:13 -0500, ombz wrote:

> Dear all,
> 
> Thanks for your answers and hints!
> 
> So it's definitely either a 1st order IIR or an 8-tap FIR, as I thought.
> Not sure about that more-than-8-tap FIR. Of course when you simply add very
> weak taps you don't really add much of a group delay, but doesn't filter
> more neither essentially... But I try to experiment a little.
> 
> The group delay spec actually is my translation from a 20 ms "latency" spec
> (@fs=200 Hz). What is "latency" in my system, how it is measured and
> perceived that's a whole different story. For me, the term "latency" is
> pretty inacurrate, or even inappropriate, and I think of it rather as a
> marketing term... But I realized that group delay and "latency" match
> pretty well in my case. So that's why: max group delay = 4 samples.
> 
> @Tim: I really appreciate your proposals, especially the one on being a
> hero, or brushing up my resumé secretly haha. ;)
> hmmp, I ever wondered how to get a Bessel filter right in the digital
> domain. I never succeeded. Any hints?
> 
> Enjoy your week end
> 
> Andy

Have you tried a bilinear transform of the Bessel filter?  I haven't checked
the responses in great detail but they seem to work.  Of course, if you're
sure you are limited to 1st order there isn't much point to it!

>> it cannot have constant group delay of 4 samples (assuming you mean 
"9-tap FIR").

not strictly constant, as the impulse response is asymmetric. But that's no
different from any recursive filter (exception: special cases like CIC).
It's a trade-off between latency, group delay variation and magnitude
response / filtering. 

The bad news here is, no linear filter can do a better job. I could simply
take its impulse response and sample it into a long FIR.
But you can use a numeric optimizer and _design_ the FIR for your own
trade-off that is better suited to your specific requirements than one of
the well-known textbook prototypes.

BTW when designing this with numerical optimization, there is usually a bit
of "cooking" involved, the requirements evolve with the filter design.	 

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When you SAR "no overshoot" do you mean no overshoot with a step input or do you mean that if the input is bounded, the output must also be bounded within the same limits? I think it makes a difference. 

Bob