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Lecture Signals, systems & inference – Lecture 24: Matched filtering

The following will be discussed in this chapter: Matched filtering for detecting known signal in white Gaussian noise, matched filter performance, Q(.) function for area in Gaussian tail, matched filter properties, on-off signaling in noise, antipodal signaling,. | Lecture Signals, systems & inference – Lecture 24: Matched filtering Matched filtering 6.011, Spring 2018 Lec 24 1 Matched filtering for detecting known signal in white Gaussian noise r[n] g[n] g[0] ‘H1’ LTI, h[∙] Threshold g 7 6 n=0 ‘H0’ 2 Matched filter performance f(g|H0) f(g|H1) sU E g = a r[n]s[n] g E PM n PFA 3 Q(.) function for area in Gaussian tail The tail area to the right of x under a Gaussian PDF of mean 0 and standard deviation 1 is tabulated as the tail-probability function: Z 1 1 v 2 /2 Q(x) = p e dv 2⇡ x Useful bounds: 2 2 x e x /2 1 e x /2 2 p < Q(x) < p , x>0 (1 + x ) 2⇡ x 2⇡ For a Gaussian random variable of mean value ↵ and standard deviation , the area under the PDF to the right of some value is Z 1 ⇣ 1 (w ↵)2 /(2 2 ) ↵⌘ p e dw = Q 2⇡ 4 Matched filter properties •  Matched filter output in noise-free case (and before sampling) is the deterministic autocorrelation of the signal: g[n] = Rss [n] •  Matched filter frequency response magnitude accentuates frequencies where signal has strength relative to (spectrally flat) noise •  Matched filter frequency response phase cancels signal phase characteristic to allow all components to contribute at sampling time •  Matched filter maximizes “SNR” of sample fed to threshold test 5 On-off signaling in noise 1 0 1 0 0 1 1 0 1 1 0 0 1 d Gen. 2 0 -2 p(t) 0 200 400 600 800 1000 1200 2 1 0 -1 n(t) 0 200 400 600 800 1000 1200 10 0 -10 h(t) 0 200 400 600 800 1000 1200 2 0 -2 Dec. 0 200 400 600 800 1000 1200 1 0 1 1 0 6 1 1 0 0 1 0 0 1 Antipodal signaling 1 0 1 0 0 1 1 0 1 1 0 0 1 d Gen. 2 0 -2 p(t) 0 200 400 600 800 1000 1200 2 0 -2 n(t) 0 200 400 600 800 1000 1200 10 0 -10 h(t) 0 200 400 600 800 1000 1200 2 0 Dec. -2 0 200 400 600 800 1000 1200 1 0 1 0 0 7 1 1 0 1 1 0 0 1 Pulse compression for radar Read the simulation example .