Signal Detection. Outline. Detection Theory. Example Applications of Detection Theory

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1 Outline Signal Detection M. Sami Fadali Professor of lectrical ngineering University of Nevada, Reno Hypothesis testing. Neyman-Pearson (NP) detector for a known signal in white Gaussian noise (WGN). Matched filter implementation. Detection performance. 1 2 Detection Theory Detection Problem: Decide based on noisy measurements with a measure of confidence 1. If an event occurred or not. 2. Which of a number of possible outcomes has occurred? Statistics: called decision theory. xample Applications of Detection Theory Radar/sonar. Communications. Speech/image processing. Biomedicine. Fault detection. Seismology. 3 4

2 Radar/Sonar Detect the presence of a target. Transmit an electromagnetic pulse. Test received noisy signal: If a pulse is detected, it was reflected from the target and a target is present. If no pulse is detected then no target is present. Classes of Detection Problems Detect signals from noisy measurements 1. Known signals in additive noise. 2. Deterministic signals with unknown parameters in additive noise. 3. Random signals in additive noise. Typically, assume additive Gaussian noise. 5 6 Known Signals Using one or more noisy measurements decide if the known signal is present. Additive noise: Binary Hypothesis: H 0 or H 1 Statistical Decision: D 0 or D 1 Probability density functions: Possible Outcomes of Binary xperiment 1. D 0 H 0 : correct decision. 2. D 1 H 0 : Type I rror, false alarm. 3. D 1 H 1 : correct decision. 4. D 0 H 1 : Type II rror, miss. 7 8

3 Radar/ Sonar Terminology Probability of Detection Decompose observation space into two disjoint subspaces: Define the following probabilities: 1. False alarm Density P D 2. Proper dismissal 0.05 P FA Critical Value 9 10 Likelihood Ratio (LR) Log-likelihood Ratio (LLR) = threshold Different detectors yield different thresholds. Use the likelihood ratio to decide. Both sides of the inequality are positive. Log is an order preserving transformation. Often more convenient (exponential pdfs)

4 Neyman-Pearson Criterion Select the threshold level NP Criterion: Maximize the probability of detection such that the probability of false alarm remains below a specified level. Maximize s.t. the constraint 13 Matched Filters Known deterministic signal in white Gaussian noise NP criterion: upper bound on probability of false alarm. Detection: distinguish between two hypotheses N 14 Likelihood Ratio Simplify Likelihood Ratio / / / / 15 16

5 Log-likelihood Ratio NP Detection: Choose to maximize and satisfy the false alarm rate constraint Replica Correlator is the correlation of the data and the known signal (or its replica). weights the data samples with the values of the signal. The larger the signal amplitude the higher the weight. Negative amplitudes have negative weights Replica Correlator Block Diagram Matched Filter x(n) s(n) + N 1 n0 x ( n) s( n) T(x) D 1 ' D = = Match filter impulse response to the signal. Obtain impulse response by i. Flipping the signal. ii. Right shifting the signal by 1. 20

6 Matched Filter Block Diagram Matched Filter Remarks Derivation requires known arrival time. x(n) h(n) Close at n = N1 T(x) D 1 ' D 0 Filter output is maximum at N Signal energy the square of the norm of the signal vector. For no additive noise, = signal energy Output of Filter: Mean & Variance For any filter with impulse response Signal to Noise Ratio (SNR) SNR ratio of the signal : ratio of signal energy to noise energy

7 Matched Filter Maximizes SNR Consider all possible FIR filters. Maximum signal energy at the output of an FIR filter if and are colinear. Performance of matched filter increases monotonically with the maximum SNR. 25 Matched Filter Performance Obtain for a given. Gaussian: linear combination of Gaussian Mean of Variance of 26 Distribution of Threshold Calculation N N N N Performance improves as increases(snr). Densities move farther apart but have the same shape. 27 = right tail probability for N Solve for 28

8 NP Detection Probability Normalize: standard normal N 29 Non-Gaussian Noise Known deterministic signal in white Gaussian noise: NP criterion and the max SNR criterion give the matched filter detector. Non-Gaussian noise 1. Matched filter detector is not NP optimal but still maximizes SNR. 2. NP filter is nonlinear but the linear filter works well for moderate deviations from Gaussian. 30 References R. D. Hippenstiel, Detection Theory: Application and Digital Signal Processing, CRC Press, Boca Raton, S. M. Kay, Fundamentals of Statistical Signal Processing, Volume II Detection Theory, Prentice-Hall, Upper Saddle River, NJ,