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Matched filtering
6.011, Spring 2018 Lec 24
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Matched filtering for detecting known signal in white Gaussian noise
g[n] g[0] r[n] LTI, h[∙] Threshold g n = 0 ‘H1 ’ ‘H0 ’ 7 6
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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 - - 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 )
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Matched filter performance
f(g|H0) f(g|H1) g = a r[n]s[n] g sU E E
n
PM PFA
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− − −
− − β γ β
γ − − β
γ − β
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: Useful bounds: For a Gaussian random variable of mean value α and standard deviation , the area under the PDF to the right of some value is
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Q(x) = 1 √ 2π Z ∞
x
e−v2/2 dv x (1 + x2) e−x2/2 √ 2π < Q(x) < x 1 e−x2/2 √ 2π , x > 0 1 β √ 2π Z ∞
γ
e−(w−α)2/(2β2) dw = Q ⇣γ − α β ⌘
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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
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On-off signaling in noise
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d Gen. p(t) h(t) n(t) Dec. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 200 400 600 800 1000 1200
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2 200 400 600 800 1000 1200 2 200 400 600 800 1000 1200
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2 1 200 400 600 800 1000 1200 10
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Antipodal signaling
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d Gen. Dec. 200 400 600 800 1000 1200
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2 200 400 600 800 1000 1200 2 200 400 600 800 1000 1200
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2 200 400 600 800 1000 1200 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 p(t) n(t) h(t)
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Pulse compression for radar
Read the simulation example from https://en.wikipedia.org/wiki/Pulse_compression
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6.011 Signals, Systems and Inference
Spring 2018 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.
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