Detection theory 101 ELEC-E5410 Signal Processing for Communications

Detection theory 101 ELEC-E5410 Signal Processing for Communications

Binary hypothesis testing Null hypothesis H 0 : e.g. noise only Alternative hypothesis H 1 : signal + noise p(x;h 0 ) γ p(x;h 1 ) Trade-off between Probability of false alarm (decide H 1 when H 0 is true) Miss Both depend on threshold γ P FA Inspired by S.M. Kay, Fundamentals of statistical signal processing. Detection theory, Prentice-Hall, 1998

Likelihood ratio/neyman-pearson test For a given false alarm rate P FA, the probability of detection P D is maximized with the likelihood ratio test where the threshold γ is Assumes that the PDFs p(.) are completely known R.W. 3

Detection of constant signal level in white Gaussian noise (WGN) using N samples Formulas to calculate detection and false alarm probabilities R.W. 4

Receiver operation characteristics (ROC) Way to measure the performance of detectors The performance of a detector is always above the diagonal, because the diagonal is the same as guessing ROC for a constant signal in WGN R.W. 5

Bayesian detector Another, probabilistic, approach for detection Assign prior probabilities to events (if possible) and minimize the probability of error Bayesian detector becomes R.W. 6

Bayesian detector Assign costs to the decisions and minimize Bayes risk where C ij is the cost of deciding H i when H j is the true event Bayesian detector Assumes perfect knowledge of PDFs, probabilities and risks 7

Generalized likelihood ratio test (GLRT) Decide between H 1 and H 0 when parameters of the PDFs depend on the set of unknown parameters θ i Unknown parameters are estimated by maximum-likelihood estimation and substituted in the generalized likelihood ratio There is no optimality associated to GLRT, but it works often well in practise 8

Generalized likelihood ratio test (GLRT) Another form of GLRT is Special case when the PDF under H 0 is completely known (e.g. H 0 is noise only hypothesis and noise power is known) L G (x) =max 1 p(x; ˆ 1, H 1 ) p(x; 0, H 0 ) > 9

Uniformly most powerful test (UMP) Uniformly most powerful test (UMP) yields the highest probability of detection for any signal parameter, say, A For UMP, test statistic and threshold must be found without the knowledge of A For UMP to exist the parameter test must be one-sided One-sided: Two-sided: 10

Bayesian approach Unknown parameters are considered as realizations of random variables and prior PDFs are assigned to them Requires prior knowledge of unknown parameters whereas GLRT does not Requires multidimensional integration over unknown parameters 11

Example: unknown signal in WGN GLRT approach: MLE estimate of the unknown signal s[n] is clearly x[n] The test becomes This is simply the energy detector 12

Example: known signal in WGN with unknown amplitude The detector becomes The detector is simply a correlator 13

Example: known in signal in WGN with unknown amplitude Threshold seems to depend on the amplitude so th test does not look UMP However H 0 is noise only and therefore Given P FA the threshold is independent of A If the sign of the amplitude is known, the test is UMP If the sign is unknown the test is GLRT and MLE of A is 14

Example: sinusoidal signal in WGN Unknown amplitude, known frequency and phase: Unknown amplitude, known frequency and unknown phase: P P Unknown arrival time: Determine MLE of the timing NX 1 max s[n]x[n ] n=0 Unknown frequency: Estimate frequency as the max. peak of the periodogram 15