Why one can show erfc sqrt where recall pfd exp snr for given snr

References

 “Design and analysis of Modern Tracking Systems,” Samuel Blackman and Robert Popoli, Artech House, 1999.

Electrical Background

Simulating Noise and Signals in

% Plot random series
clf
plot(Time, NoSig, 'r', Time, Sig, 'b', [0 N], ...

[Threshold Threshold], 'g',[0 N],[0 0],[0 N], [Signal Signal]) legend('Noise', 'Noise+Signal', 'Threshold')
xlabel('Time')
ylabel('Voltage')
title('Voltage vs. Time')

		Going beyond this over simplification is beyond the scope of this course.  It can be shown that:
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	Note: maximum of RhoR occurs near r = σ
	Note: one may show E(Npower) = σ2 / Ro = kTeBw
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			The Noise distribution may be approximated by a Rayleigh Distribution
	Note: maximum of RhoN occurs at r = A
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			Derivation is beyond scope of this course

MATLAB Code: sDetection

% Script Name: sDetect
% Version: 1.3
% Date: 3/22/08
% Author: John MacCarthy
% Description: This script generates gaussian curve for the Noise case
% and for the Signal in noise case. For a given SNR Threshold,
% the program determines the Ptd, Ptnd, Pfd, & Pmd

False Detections and Missed Detections


			
				We report a detection (we get it right):
					Characterized by the probability of (true) detection, Pd
			

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	We do not report a detection (we get it right):
		Characterized by the probability of (true) non-detection, Pnd	
			

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			Pmd = 1-Pd  Question: So what should we select as Vt that minimizes Pfd and Pmd?
		Characterized by the probability of false detection, Pfd
		Characterized by the probability of false detection, Pfd	
			
			Curve(s)
*Pfd = exp(-Vt2/(2 σ2 )) = exp(-SNRt)**			Curve(s)

 We are unable to get all these. Why?

 One can show:

 Note: Integration over multiple pulses can increase the Pd and decrease the Pfd for a given SNR. But a discussion of this is beyond the scope of this course.

Radio/Radar Design Implications

 Requirements for a radar will generally take the form:

Summary
 Signals in the presence of Noise:

	White (Thermal) Noise exhibits a Gaussian Distribution about a mean value of 0 and SigmaNoise = sqrt(kTB)
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 Signal Present: properly recorded, recorded a miss  Each has a probability of occurrence that is a function of SNRt, Ao, & SigmaNoise:
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	Again, Decision Theory can help here.
 This analysis applies to both Communications and Sensor Systems		11