AWGN The effective noise content of bandpass noise is BNo Experimental measurements show that the pdf of the noise samples can be modeled as zero mean gaussian random variable f(r) x2/2 e 2丌o AKA Normal r V, N(0, 04) E BN The cdf of a gaussian rv F(a)=PX≤]=f(xtx e ∞y2兀o This integral requires numerical evaluation Available in tables
σ σ f x () AWGN • The effective noise content of bandpass noise is BNo – Experimental measurements show that the pdf of the noise samples can be modeled as zero mean gaussian random variable x () = 2πσ 1 e− x 2 / 2σ2 – AKA Normal r.v., N(0,σ2) – σ2 = Px = BNo • The CDF of a Gaussian R.V., α α Fx α = P[X ≤ α] = ∫−∞ fx (x)dx = ∫−∞ 2πσ 1 e− x 2 / 2σ2 dx • This integral requires numerical evaluation – Available in tables Eytan Modiano Slide 6
aWGN continued X()~N(0,02) X(t), X(t2 are independent unless t=t2 EX(t+)X(t)≠0 R,(t)=elX(t+tX(tI E[X()]t=0 0τ≠0 02=0 R2(0) E BN
σ σ E X AWGN, continued • X(t) ~ N(0,σ2) • X(t1), X(t2) are independent unless t1 = t2 • [ (t + τ )]E[ ( X t)] τ ≠ 0 Rx () τ = E[ X(t + τ )X t( )] = E X2 [ (t)] τ = 0 0 τ ≠ 0 = σ2 τ = 0 • Rx(0) = σ2 = Px = BNo Eytan Modiano Slide 7
Detection of signals in AWGN Observe: r(t)=S(t)+n(t), tE O,T Decide which of,, ..,Sm was sent Receiver filter Designed to maximize signal-to-noise power ratio ( sNr) t) y(t) Sample at t=T decide Goal: find h(t that maximized SNR
Detection of signals in AWGN Observe: r(t) = S(t) + n(t), t ∈ [0,T] Decide which of S1, …, S m was sent • Receiver filter – Designed to maximize signal-to-noise power ratio (SNR) h(t) y(t) filter r(t) “sample at t=T” decide • Goal: find h(t) that maximized SNR Eytan Modiano Slide 8
Receiver filter y()=r(1)*h)=|r()(-lr Sampling at t=t=yT)=r(t)h(T-t)dr r(T)=(T)+n()→ 0)JW7-t+m==()+x(T SNR= y(7) EYD] No h(T-tdt
y t y T y T T Receiver filter t () = r t ( ) = ∫ ( ) * h t r(τ )h(t − τ )dτ 0 T Sampling at t = T ⇒ () = ∫ r(τ )h(T − τ )dτ 0 r() τ = s() τ + n() τ ⇒ T T () = τ ∫ s(τ )h(T − τ )dτ + ∫ n( )h(T − τ )dτ = Ys(T) + Yn (T) 0 0 T 2 T s( )h(T − τ )dτ ∫ h( )s(T − τ )dτ ∫ τ τ Y T SNR= s2 () = 0 = 0 [ (T)] T T E Yn2 N0 ∫ h T − t)dt N0 ∫ h T − t)dt 2 ( 2 ( 2 2 0 0 Eytan Modiano Slide 9 2
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 10
0 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