Matched filter: maximizes snr Caushy-Schwartz Inequality 11t≤1g8(0)2(g2() Above holds with equality iff: g,(t=cg2(t)for arbitrary constant c 2E SNR= (s(t))'di |h2(7-t h(T-t)dt 0 Above maximum is obtained iff: hT-t)=cs(t) (t)=csTt)=st-t) h(t is said to be "matched? to the signal S(t) Slide 100 Matched filter: maximizes SNR Caushy -Schwartz Inequality : 2 ∞ ∞  ∞ g t g2 ()  ∫−∞ 1( ))2 (g2 (t))2 ∫−∞1() t dt ≤ (g t ∫−∞ Above holds with equality iff: g t t 1() = cg2 () for arbitrary constant c 2 T  T T  s( )h(T − τ )dτ  ∫ ( (τ ))2 dτ ∫ h T − ττ T ∫ τ s 2 ( )d SNR= s  0 T  ≤ 0 T 0 = 2 ∫ ( (τ ))2 dτ = 2Es N0 ∫ h T − t)dt N0 ∫ h T − t)dt N0 0 N0 2 ( 2 ( 2 2 0 0 Above maximum is obtained iff: h(T-τ) = cS(τ) => h(t) = cS(T-t) = S(T-t) Eytan Modiano h(t) is said to be “matched” to the signal S(t) Slide 10