Lectures 8-9: Signal detection in Noise Eytan Modiano AA Dept
Lectures 8-9: Signal Detection in Noise Eytan Modiano AA Dept. Eytan Modiano Slide 1
Noise in communication systems S(t) Channel r(t)=s(t)+n(t) Noise is additional"unwanted "signal that interferes with the transmitted signal Generated by electronic devices The noise is a random process Each"sample " of n(tis a random variable Typically, the noise process is modeled as"Additive White Gaussian noise”(AWGN) White: Flat frequency spectrur Gaussian noise distribution
Noise in communication systems S(t) Channel r(t) = S(t) + n(t) r(t) n(t) • Noise is additional “unwanted ” signal that interferes with the transmitted signal – Generated by electronic devices • The noise is a random process – Each “sample” of n(t) is a random variable • Typically, the noise process is modeled as “Additive White Gaussian Noise” (AWGN) – White: Flat frequency spectrum – Gaussian: noise distribution Eytan Modiano Slide 2
Random processes The auto-correlation of a random process x(t is defined as Rxx(t, t2=E[x(t x(t2) A random process is wide-sense-stationary (WsS) if its mean and auto-correlation are not a function of time that is m()=E【(=m Rx(1,+2)=R(T, whereτ=t少t2 If x(t)is WSS then: RA(T=R-可 IRx )<=Rx(ol(max is achieved at t=0) The power content of a wSS process is: cT/2 T/2 P= ellit x(tdt= lim R2(0)dt=R2(0) 1→∞TJ-7/2 1→∞TJ-7/2
τ τ τ τ τ 0 τ Random Processes • The auto-correlation of a random process x(t) is defined as – Rxx(t1,t2) = E[x(t1)x(t2)] • A random process is Wide-sense-stationary (WSS) if its mean and auto-correlation are not a function of time. That is – mx(t) = E[x(t)] = m – Rxx(t1,t2) = Rx(τ), where τ = t1-t2 • If x(t) is WSS then: – Rx(τ) = Rx(-τ) – | Rx(τ)| <= |Rx(0)| (max is achieved at τ = 0) • The power content of a WSS process is: 1 T / 2 1 T / 2 Px = E[lim 2 ( ) t→∞ T ∫−T / 2 Rx (0)dt =Rx (0) t→∞ T ∫−T / 2 x t dt = lim Eytan Modiano Slide 3
Power Spectrum of a random process If x(t is wSS then the power spectral density function is given by Sx ( = F[R] The total power in the process is also given by: S, (df=R,(e 72nd df Rr (tJe / df dt R(Je tiu df at=R,(8(tWt=R(O)
τ Power Spectrum of a random process • If x(t) is WSS then the power spectral density function is given by: Sx(f) = F[Rx(τ)] • The total power in the process is also given by: ∞ ∞ ∞ Px = ∫ Sx () t e− j ftdt f df = df ∫ ∫ Rx ( ) 2π −∞ −∞−∞ ∞ ∞ x () 2π = ∫ ∫ R t e− j ftdf dt −∞−∞ ∞ ∞ ∞ = ∫ R t 2π t t dt = Rx (0) x ( ) ∫ e− j ftdf dt = ∫ Rx ( )δ() −∞ −∞ −∞ Eytan Modiano Slide 4
White noise The noise spectrum is flat over all relevant frequencies White light contains all frequencies N/2 Notice that the total power over the entire frequency range is infinite But in practice we only care about the noise content within the signal bandwidth as the rest can be filtered out After filtering the only remaining noise power is that contained within the filter bandwidth B) PBP(f) N/2 B B
White noise • The noise spectrum is flat over all relevant frequencies – White light contains all frequencies Sn(f) No/2 • Notice that the total power over the entire frequency range is infinite – But in practice we only care about the noise content within the signal bandwidth, as the rest can be filtered out • After filtering the only remaining noise power is that contained within the filter bandwidth (B) Eytan Modiano SBP(f) No/2 fc No/2 -fc Slide 5 B B
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
