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I am working on a problem which raised what should be a simple question, but I'm stuck on it. The objective is to detect a shifted and scaled version of a known signal in the presence of Additive White Gaussian Noise (AWGN).

Think of a radar system. In a radar system, you know the transmitted signal. The returned system is delayed in time and scaled. I understand that a time-delay, only shifts the location of the matched filter's peak output. No problems there. But if the scaling is unknown, then how does one construct the reference signal?

Using a reference signal with the an amplitude that doesn't match the amplitude of the signal in the noisy data will mean that a threshold tests based on that reference signal will be incorrect.

I'm guessing that in a radar system, the noise power is approximately known, and so the power of the received signal is estimated by taking the power of the received signal and subtracting the noise power. That value could then be used to scale the reference signal. Still, that would only be and approximation, and it would still effect the probability of detection and false alarm in some way. Any help is appreciated. Thanks!

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    $\begingroup$ The scaling of the matched filter really doesn't matter. Thanks to linearity, any scaling you apply to the matched filter will just scale its output as well. Most importantly, the signal and noise at the filter output will be scaled equally, so the output SNR is independent of any scaling that you apply to the filter itself. For a radar receiver, it is common to have a "noise riding" detection threshold, where the noise level is continuously estimated and a threshold is set some distance above that level to achieve the desired $P_d$ and $P_{fa}$. $\endgroup$ – Jason R May 12 '15 at 11:28
  • $\begingroup$ Another option is normalized cross-correlation (the example shown here is 2D, but the same technique applies in 1D). You can normalize the filter's output so that it is always in the range $[-1, 1]$ based on the measured standard deviation of the input signal and matched filter impulse response. This removes any constant scaling from the picture and allows simpler threshold selection (although the normalization may complicate theoretical calculation of $P_d$ and $P_{fa}$ somewhat). $\endgroup$ – Jason R May 12 '15 at 11:32
  • $\begingroup$ @JasonR You are correct that linearity simply scales your output; however, if your receiver output is scaled you should also scale your detection threshold. Thus, the scaling of the received signal does matter with regard to setting your detection threshold. If you use the a value for your reference signal's scaling that doesn't match the one in the actual signal your $P_d$ and $P_{f_a}$ will be incorrect. $\endgroup$ – Stephen Hartzell May 12 '15 at 17:05
  • $\begingroup$ You are correct. Hence my suggestions to use a noise-riding threshold, where the threshold is set relative to the estimated noise level. Thus, the only thing that matters is the signal-to-noise ratio. You select a threshold that gives the desired $P_{fa}$ and at least the desired $P_d$ at a specified minimum SNR. For well-behaved cases (such as detection on the AWGN channel), you can derive closed-form theoretical expressions for each, assuming you accurately estimate the noise level. $\endgroup$ – Jason R May 12 '15 at 18:51
  • $\begingroup$ JasonR's comment that the matched filter reference signal needs no scaling is spot on (+1). Indeed, another important reason (not often appreciated by theoretical minded folks) for preferring antipodal signalis (instead of, say, on-off keying, for example) in digital communications systems is that the threshold is always zero, regardless of signal strength. $\endgroup$ – Dilip Sarwate May 13 '15 at 12:56
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I think the problem is not as bad as you suspect it is.

I wasn't around at the time, but from what I've read, early radar systems essentially connected the matched filter's output to an oscilloscope, and a trained operator would look at the phosphor and decide, from experience and intuition, when the signal raised above the noise ("the grass") indicating a reflection.

In practice, you never know in advance the amplitude of the pulse you're looking for. There is attenuation in the channel that can change over time, even on very short time scales. Sometimes you don't know how much power the transmitter is using.

There are at least two solutions to this problem. One is to use automatic gain control in the receiver, so that you know in advance the power or the peak amplitude of the pulse going into the matched filter. The second is to use a training sequence or a pilot signal, so that the receiver can calibrate the thresholds appropriately.

To add a bit of detail: to simplify the math, assume the signal $p(t)$ is time-symmetric and has energy 1. The transmitter emits signal $kp(t)$, with $k$ unkown at the receiver. The receiver correlates the received signal with a matched filter with impuse response $lp(t)$. When the transmitted signal is present at the receiver, the matched filter's output is $$r=kl+n,$$ which is a Gaussian random variable with mean $kl$ and variance $N_0/2$.

When nothing but noise is present at the receiver, the matched filter's output is just $$r=n,$$ which is a Gaussian random variable with mean 0 and variance $N_0/2$. If the receiver can estimate $l$, then the problem reduces to standard hypothesis testing (see for example these notes).

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  • $\begingroup$ In those cases, your probabilities of detection and false alarm aren't going to be quite right, but probably close enough. I can see those solutions working. If no one gives a more mathematical answer then I'll mark yours as accepted. Thanks! $\endgroup$ – Stephen Hartzell May 11 '15 at 20:03
  • $\begingroup$ @StephenHartzell I added a bit of detail and linked to some MIT notes which go into this in much more depth. I hope it's useful. $\endgroup$ – MBaz May 12 '15 at 0:47
  • $\begingroup$ Okay, so your more detailed answer is similar to what I was supposed one would do to solve the problem. The receiver can estimate $l$, but that estimation itself won't be perfect. The variance in $l$ should be included in the hypothesis testing. However, it seems like most treatments such as the MIT one, just assume that $l$ is perfectly known. $\endgroup$ – Stephen Hartzell May 12 '15 at 17:00
  • $\begingroup$ @StephenHartzell, you may also be interested in this: en.wikipedia.org/wiki/Blind_equalization. $\endgroup$ – MBaz May 12 '15 at 17:09
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I would suggest first estimation of the gain/loss by running a window with the size of your matched filter that will be compared to the value of the "filter" with the same window - this will average the noise impact as well (square root of the window size). Use the scale as the window moves to scale your signal and then run the matched filter.

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  • $\begingroup$ I think that is a perfectly valid approach, but then the question becomes, how does the fact that you are only estimating that signal value effect your probability of detection and false alarm. The reference signal is then not exactly correct. $\endgroup$ – Stephen Hartzell May 12 '15 at 17:09
  • $\begingroup$ In implementation of DSP algorithm the key question is PD and PFAR, as you are asking. Under the proper assumptions regarding the signal behavior, this approach should provide with a quite acceptable results. To understand the specifics on the results require way more involved work that is somewhat need CONSULTANT work:) $\endgroup$ – Moti May 13 '15 at 15:31

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