Noise Is 3 Orders of Magnitude Greater Than A Wave Form

Everything is known about the transmitted wave, i.e., the shape & phase angle, except the amplitude.
All that is necessary is to recover is the amplitude of the wave. Can
this be done when the noise is several orders of magnitude greater than the signal?
Bret Cahill
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Bret Cahill wrote:

Narrowing the final detection bandwidth is the only hope. If the noise spectrum is white, narrow bandwidth through averaging works. If you cannot average, there is trouble. The old lockin amplifiers used a modulation or chopping signal and a long time constant in the final filter. I had signals at times that required an hour of integration to detect. It was slow but it worked.
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I forgot to mention that the signal wave form can be anything that can be generated.
For example, if something similar to an AM radio signal, say, f(t) = (sint)(sin10t)(sin10t) was possible and the frequency of the noise was about the same as the sin(t) factor, then f(t) will plot the noise every time f(t) = 0.
In this case that would be ten times as often as the sin(t) factor.
The noise can then be subtracted to recover the wave form.
Bret Cahill
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On Thu, 2 Jul 2009 20:10:41 -0700 (PDT), Bret Cahill

I think what's happening is that you have
1. Defined the noise to be bandlimited to radian frequencies below about 1
and
2. Up-converted (modulated) the signal to have usable (recoverable) components around 10 times that frequency.
So (am I allowed to say 'duh'?) a simple highpass filtering operation will remove the noise from the upconverted signal. You moved the signal to where you knew there was no noise.
It's more interesting when the signal and the noise occupy the same bandwidth.
John
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The noise and the signal are both low and have about the same frequency, ~ x/2pi. There is little that can be done to change this situation. There is no time to wait more than several cycles for the result either.

To plot the noise. Every time sin 10t = 0, f(t) = zero, and the only thing left is the low frequency noise.
Then a high pass filter can smooth out the (sin10t)(sin10t) component so something like the original signal can still be recovered after the noise is subtracted out.
Alternatively traditional filters can be eliminated altogether. The signal + noise as well as the noise alone can be traced out from the high frequency signal.
The noise is then subtracted from the signal + noise to recover the signal.

Plot f(t) and it's easy to see the original signal sill exists, although in a somewhat discontinuous form.
It would be interesting if this has never been done before.
Bret Cahill
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On Fri, 3 Jul 2009 08:04:15 -0700 (PDT), Bret Cahill

You are assuming the ability to high-frequency modulate the signal before the noise is added to it. So you already know what the noiseless signal looks like.
John
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As I said in the OP, "Everything is known about the transmitted wave, i.e., the shape & phase angle, except the amplitude."
This works for the same reason reading a newspaper in a foreign language is easy. You already know what they are going to say.
Bret Cahill
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On Fri, 3 Jul 2009 15:30:36 -0700 (PDT), Bret Cahill

Are you familiar with the way a lockin amplifier works? That sounds maybe like what you are doing. If you know everything about the signal but its amplitude, then you have or can construct a normalized (unity amplitude) version of it. That will positively correlate with the unknown-amplitude version of the signal but have zero correlation to random noise.
Things like IR absorption spectrometers commonly chop (square wave modulate) the light source and recover the signal with a synchronous rectifier. That washes out any noise picked up in the optical path or the detector. Things like this commonly dig signals out from 1000x the noise... but slowly.
If the noise is known to be bandlimited, it's a lot easier... almost cheating.
John
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Many thanks for the tip but phase lag is just a more sophisticated form of filtering which is valuable in many situations where the noise is all over the spectrum.
This is not filtering noise; it's measuring it then subtracting it.
Basically the signal is brought to zero to identify the noise. Then the noise is subtracted from whatever the receiver is putting out.
For an accurate signal measurement both the noise and the noise + signal output from the receiver must be known to a higher accuracy.
Integrating should yield that higher accuracy but it isn't always necessary as it can work over _one single higher frequency wave cycle_.

I ran it by a lawyer and he said it was completely legal.
Bret Cahill
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On Sat, 4 Jul 2009 12:24:15 -0700 (PDT), Bret Cahill

A lockin doesn't work by phase lag; it works by correlation.

That only works if the zero+measure thing is done at or above the noise's Nyquist frequency, and that is in turn only meaningful if the noise is bandlimited.
And you are able somehow to turn the signal on and off at that rate.
So all you need is a highpass filter. But the math algorithm you describe is about equivalent.

I thought you *were* a lawyer.
John
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The noise frequency is limited to about +/- 50% - 75% of the signal frequency.

You'ld lose the signal with a simple filter.

The signal is transformed to something that has a lot of the characteristics of the original signal. For example, the integral is 1/2 the integral of the original curve.
The advantage is the transformed curve plots out the difference between the noise and the transformed signal. Then the noise is subtracted.
That is not a conventional filter.
Bret Cahill
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Bret Cahill wrote:

It's an unconventional way of implementing a filter, but it's still a filter, and the end result will be equivalent to using some conventional filter to separate the carrier from the noise.
I'm not saying your technique won't work. Quite likely it could be made to work. But it won't work any *better* than using conventional AM modulation and filtering. It can't, for fundamental mathematical reasons.
--
Greg

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It would be interesting if it's unconventional.
I came up with it a few minutes after the OP killing time because of a delay.

The system will not respond in the same way as to a conventional AM wave form.
For one thing, higher frequencies are attenuated more than lower frequencies.
The higher frequency here only changes sign as often as the signal wave so the media doesn't "see" a +/- high frequency AM wave but something much like the original low frequency signal.

Bret Cahill
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Bret Cahill wrote:

As long as the medium is linear, this makes no difference. There's nothing special about the zero crossings.
--
Greg

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Can you have the same resonance as the original signal?
Bret Cahill
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Fundamentally, I have a major problem with the discussion. and its not the fact that people are fishing their lines of pursuit, which miss the fundamental.
The info was NEVER posted.
And Jon, your earlier post-given freely had a portion copy of my work pre-last semester which I did give freely. And I already gave you a consideration.
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The sensor receives the loud noise, say, 10sin(0.7x), plus the small signal, say, sin(x ). The signal, however, can be transformed into sin (x)sin^2(100x) so the sensor receives
(10sin(0.7x) + sin(x)sin^2(100x)) x from 2 to 3.4
This can be viewed by pasting the entire line into www.wolframalpha.com
The blue area is the signal to be extracted from the noise. It's over the noise to the left of x = pi and under the noise to the right of pi.
The high frequency curve runs between the signal and the noise and maps out both curves to any precision depending on frequency and regression.
Bret Cahill

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Bret Cahill wrote:

What you are doing is changing the problem. You are now looking at a sin(100x) modulated by sin(x). This says the noise and the signal are independent of one another. If you can do this, make it 1000x and make you life easier.

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As I pointed out above, I forgot to mention in the OP that the designer could change the signal to suit the problem.

Actually it's a sin^2(100x) which is always positive.
Sin100x will not work as the sign of the original signal must be preserved.
For example, try pasting
(10sin(0.7x) + sin(x)sin(100x)) x from 2 to 3.4
into www.wolframalpha.com
How would you know the noise curve?

Bingo!
In some situations it may be difficult to use very high frequencies.
That's why regression was mentioned.
Bret Cahill
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Bret Cahill wrote:

Sin(100x) will work just fine. To detect sin(x) just multiply the signal by sin(100x) again or, to be careful multiply by sin(100x) and also by cos(100x). This does a quadrature detection and the magnitude of the two term is independent of the phase relative to sin(100x)

The point of using the high frequency is to remove your signal from the noise value.

The physics requirements are always the same. You need to have the signal power greater than the noise power in the detection bandwidth. Any fancy processing scheme is just trying to narrow the detection bandwidth.

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