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