Re: Optimum length for a mean filter

On Nov 15, 4:19 pm, "mermeladeK" <nil.gar...@xxxxxxxxx> wrote:
Hi Jerry,

When I say that the noise increases or decreases I am refering to the
relative value to the signal. Exactly the filter is just a low pass filter
if you ommit the second half which is not really relevant.

I don't want another type of filter since this is the most narrow
frequency filter I can have, that is to say, the one that deletes more
noise. The main question is, when I increase the length of the mean filter
(the low pass filter), there is a point where the SNR stops increasing. Do
you know why?



mermeladeK wrote:
Hi Jerry,

Yes the input signal is a step, as you defined it, plus AWGN. The
impulse
response is AC as you said as well. The differentiator is as you say
not
good for the signal since it decreases the signal more than the noise.
However is before the negative "square" of the impulse response starts
working that it is important me. That is just to make the filter AC.

So before the negative square there is the positive one, that works as
an
integrator, so it improves the SNR. The maximum of the output signal,
the
triangle, is proportional to the amplitude of the input step.

What I was asking is that in theory, in order to have the best
relation
signal to noise, the longer the impulse response the better. But in
practice, it's not true. When I use too long impulse responses the
noise
stops deacreasing. What is more it even increases.

So my question, is there some more theory that explains why there is a
point where longer lengths for the impulse response don't increase the
SNR? And what is the optimum length?
I think I heard something about too much lenght would distort the
signal,
hence stop improving the SNR...

When you say that the noise increases, do you mean in absolute value, or
relative to the signal? The signal, after all, is a linear ramp whose
slope is determined by the height of your square, and whose duration is
equal to the square's. Increasing the duration increases the height in
proportion. Do you observe that the signal-to-noise /ratio/ actually
decreases?

The second half of your filter's impulse response is not material to the
question at hand. (It may have an important purpose; I'd like to know
what.) The first half is a poor low-pass filter; you might do better
with a different one. If that seems to be a possibly reasonable
direction for you, I and others here will be happy to discuss an
implementation.

Jerry
--
Engineering is the art of making what you want from things you can get.

This is how it looks to me:

Input-no signal: zero mean AWGN
Signal: Step function, I'll assume a sign (positive) to make it easier
to talk about
Detection filter impulse response: single cycle square wave, first
positive then negative

Output of noise w/o signal: zero mean AWGN proportional to the square
root of the impulse response duration

Output of signal w/o AWGN: triangle wave, peak proportional to the
impulse response duration, impulse response duration equal to that of
the square wave.

The operations are linear so the output will be the sum of the two
previous items.

Now, what is the optimum length of the impulse response? Well, optimum
for what?

To merely detect the signal, a threshold crossing can be used. For a
fixed false alarm rate, the level will have to be increased as the
square root of the impulse response length of the square wave. As the
impulse response length is increased at constant false alarm rate, the
system will be capable of detecting signals of amplitude inversely
proportional to impulse duration, but the delay between the signal
onset and the detection will increase. Is this what was meant by
distortion?

To detect and measure the amplitude of an input signal, the value of
peaks in the output can be used. If the peak exceeds a threshold, a
detection is called. The amplitude of the peak is proportional to the
height of the input step function and the time of signal onset
precedes the detected peak by half the impulse response length of the
square wave. The threshold can be varied at a fixed impulse length to
tradeoff between false alarm rate and sensitivity. If the sensitivity
is increased by increasing the impulse response length, you have to be
able to wait longer to get your amplitude measurement.

A more complicated detection structure could verify that the threshold
exceedances were due to triangle waves and would add more delay to the
process.

SNR at the peak of the triangle output should continue to increase as
the impulse response length increases. But it will be worse at a given
delay from the onset of the step function until the signal response
has had time to exceed the noise output from longer impulse response
length.

Limits on the real usefulness of increasing the impulse response
length can come from nonstationarity of the additive noise, limits
(such as AC coupling) on how long the input signal really represents a
step function and the tolerable delay in response time.

The square wave is a differentiator. Fortunately it is a poor and
narrow bandwidth differentiator. The usefulness of the negative
portion of the impulse response may come from the multiplierless
ability to set a constant baseline for the detection process just as
the positive portion provides a poor but multiplierless matched filter
for the input signal.

Dale B. Dalrymple
http://dbdimages.com
http://stores.lulu.com/dbd

.

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