Re: Detection Question Reformulated

Hi Fred,

Thanks for your response and helping me out. See
my responses to you below.

Fred Marshall wrote:
"Randy Yates" <yates@xxxxxxxx> wrote in message
news:1153872973.044858.313200@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Hi Folks,

Suppose we have a signal y(t) that is composed of possibly colored
noise of unknown but bounded variance n(t) and potentially a sinusoid
of known power s(t) and known frequency F, so that

y(t) = s(t) + n(t).

The object is to detect the presence (or not) of s(t).

Which technique would produce a better detector?

1. Filter the signal y(t) as narrowly as possible and set a
detection threshold.

2. Compare the power in some off-band bandwidth of y(t) about F_e
(the estimation frequency) to the power about F and make a
determination?

It seems that approach 2 is susceptable to the presence of localized
noise sources around F_e. For example, what if there were a cell
phone call at F_e?

Also, it seems that, even if you had a good estimate of the noise power
Pn, it's a good idea idea to measure the power of the signal Ps in
as narrow a bandwidth as possible since there is a variance in the
*estimate* Pn, and in order to minimize the effect of this variance we
need to make Ps >> Pn as much as is practical. Thus it seems like you
might as well simply measure the power around F (i.e, to estimate Ps +
Pn) and not bother in measuring Pn at all.

Does any of this make sense? Is this sort of heuristic reasoning
valid, or does formal estimation theory come to different conclusions?

Randy,

Maybe one would consider doing it both ways. Consider this:

If you look in the narrowest bandwidth possible do you have the opportunity
to observe "signal" and "no signal" conditions? If so, you might set a
threshold at "no signal" + 0.5("signal" - "no signal).

We don't have that opportunity.

If you look in some adjacent band then you're simply trying to estimate "no
signal", right?

Precisely.

So, then one question comes how close to F is F_e? And, how many different
F_e points might you use?

Practically speaking, F_e may be +/- 200 kHz since the channels are
200 kHz wide and on-channel is centered at baseband. Essentially we're
looking for another on-band transmitter before deciding to transmit in
that band.

Some systems, where F isn't known, use spectral normalization by computing a
bunch of frequency bins and then applying a FIR filter in frequency to some
number of adjacent bins. Something like:
-1/6 -1/6 -1/6 1.0 -1/6 -1/6 -1/6

Notice that this method subtracts the average of the off-center bin values
from the center bin value. The average of the off-center bins is the noise
estimate.
You still have to set a threshold of course.

Yes, I see. That's a good idea, if you have the time to do
all that (i.e., measure several other off-band frequencies).
I'm not sure if we do or don't.

In an ideal situation and with a positive SNR you will get:

With "no signal" at the center tap, the ouput is zero.
With "signal" at the center tap, the output is something positive.
With "signal" off the center tap, the output is negative.
What happens with this then is that the output will have reduced values
surrounding the maximum - which is noticeable with a strong signal because
it messes up the noise estimates at off-signal frequencies.

Back to your terminology:
If you don't measure or estimate Pn then what do you use for comparison?

You mean what do we use for a threshold? I would guess some fraction of
the signal power, Ps (probably Ps/2).

Motivated by the idea later in your post, let me make a strawman
example using some real numbers.

Let's say the actual signal power Ps = 1. So we establish a threshold
of 0.5. Assume the actual SNR is 6 dB, so that the actual noise power
Pn = 0.25. Say we measure the total power

Phatt = Ps + Pn,

and it comes out to 0.97 (there is error in it - it should be 1.25 -
because there is a variance associated with any power estimate). So
what? As long as the variance in the estimate does not exceed the
"margin" of the threshold, we're fine, and the margin of the threshold
can be established relatively high by ensuring the SNR is high enough,
e.g., that we do enough filtering.

--Randy

.

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