Matched filtering 6.011, Spring 2018 Lec 24 1 Matched filtering - PowerPoint PPT Presentation

Matched filtering 6.011, Spring 2018 Lec 24 1

Matched filtering for detecting known signal in white Gaussian noise ’ ‘H 1 r [ n ] g [ n ] g [0] 7 LTI, h [ ∙ ] Threshold g 6 n = 0 ’ ‘H 0 2

Matched filter performance f ( g | H 1 ) f ( g | H 0 ) s U E g E g = a r [ n ] s [ n ] n P M P FA 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 e − v 2 / 2 dv Q ( x ) = − √ 2 π x Useful bounds: e − x 2 / 2 1 e − x 2 / 2 x − − < Q ( x ) < x x > 0 , √ √ (1 + x 2 ) 2 π 2 π − − For a Gaussian random variable of mean value α and standard deviation , the β area under the PDF to the right of some value is γ Z ∞ 1 ⇣ γ − α γ − ⌘ e − ( w − α ) 2 / (2 β 2 ) dw = Q β − − √ β β β 2 π β γ γ 4

Matched filter properties Matched filter output in noise-free case (and before • sampling) is the deterministic autocorrelation of the signal: g [ n ] = R ss [ 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 0 200 400 600 800 1000 1200 p ( t ) 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 1 1 0 0 1 0 0 1 6

Antipodal signaling 1 0 1 0 0 1 1 0 1 1 0 0 1 d Gen. 2 0 - 2 0 200 400 600 800 1000 1200 p ( t ) 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 - 2 Dec. 0 200 400 600 800 1000 1200 1 0 1 0 0 1 1 0 1 1 0 0 1 7

Pulse compression for radar Read the simulation example from https://en.wikipedia.org/wiki/Pulse_compression 8

MIT OpenCourseWare https://ocw.mit.edu 6.011 Signals, Systems and Inference Spring 201 8 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. 12