Communication Systems Engineering Episode 1 Part 7 pps

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) 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 ττ τ ττ τ ττ τ ττ τ ττ τ 00 0 ττ τ Random Processes • The auto-correlation of a random process x(t) is defined as – R xx (t 1 ,t 2 ) = E[x(t 1 )x(t 2 )] • A random process is Wide-sense-stationary (WSS) if its mean and auto-correlation are not a function of time. That is – m x (t) = E[x(t)] = m – R xx (t 1 ,t 2 ) = R x ( τ ), where τ = t 1 -t 2 • If x(t) is WSS then: – R x ( τ ) = R x (- τ ) – | R x ( τ )| <= |R x ( 0 )| (max is achieved at τ = 0) • The power content of a WSS process is: 1 T / 2 1 T / 2 P x = E[lim 2 ( ) t →∞ T ∫ −T / 2 R x (0)dt =R x (0) t →∞ T ∫ −T / 2 xtdt = lim Eytan Modiano Slide 3 ττ τ Power Spectrum of a random process • If x(t) is WSS then the power spectral density function is given by: S x (f) = F[R x ( τ )] • The total power in the process is also given by: ∞ ∞  ∞  P x = ∫ S x () te − jft dt   dff df = ∫   ∫ R x ( ) 2 π −∞ −∞  −∞  ∞  ∞  x () 2 π = ∫   ∫ Rte − jft df   dt −∞  −∞  ∞  ∞  ∞ = ∫ Rt 2 π t t dt = R x (0) x ()   ∫ e − jft df   dt = ∫ R x () δ () −∞  −∞  −∞ Eytan Modiano Slide 4 White noise • The noise spectrum is flat over all relevant frequencies – White light contains all frequencies S n (f) N o /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 S BP (f) N o /2 f c N o /2 -f c Slide 5 B B σσ σ σσ σ fx () AWGN • The effective noise content of bandpass noise is BN o – Experimental measurements show that the pdf of the noise samples can be modeled as zero mean gaussian random variable x () = 2 πσ 1 e − x 2 / 2 σ 2 – AKA Normal r.v., N(0, σ 2 ) – σ 2 = P x = BN o • The CDF of a Gaussian R.V., α α F x α = P[X ≤ α ] = ∫ −∞ f x (x)dx = ∫ −∞ πσ 2 1 e − x 2 / 2 σ 2 dx • This integral requires numerical evaluation – Available in tables Eytan Modiano Slide 6 σσ σ σσ σ EX AWGN, continued • X(t) ~ N(0, σ 2 ) • X(t 1 ), X(t 2 ) are independent unless t 1 = t 2 • E X t  [(t + τ )] [()] τ ≠ 0 τ ( )] =  R x () = E[ X(t + τ )Xt  EX 2 (t)] τ = 0[  0 τ ≠ 0 =   σ 2 τ = 0 • R x (0) = σ 2 = P x = BN o Eytan Modiano Slide 7 Detection of signals in AWGN Observe: r(t) = S(t) + n(t), t ∈ [0,T] Decide which of S 1 , …, 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 yt yT yT T Receiver filter t () = r t ( ) = ∫ r( τ )h(t − τ )d τ ()* ht 0 T Sampling at t = T ⇒ () = ∫ r( τ )h(T − τ )d τ 0 r() = s() + n() ⇒ τ τ τ T T τ () = ∫ s( τ )h(T − τ )d τ + ∫ n()h(T − τ )d τ = Y s (T ) + Y n (T ) 0 0  T  2  T   s()h(T − τ )d τ     ∫ h()s(T − τ )d τ    ∫ τ τ YT SNR = s 2 () =  0  =  0  [(T )] T T EY n 2 N 0 ∫ hT − t)dt N 0 ∫ hT − t)dt 2 ( 2 ( 2 2 0 0 Eytan Modiano Slide 9 2 0 Matched filter: maximizes SNR Caushy -Schwartz Inequality: 2 ∞ ∞  ∞ gtg 2 ()  ∫ −∞ 1 ( )) 2 (g 2 (t)) 2   ∫ −∞ 1 () t dt   ≤ (gt ∫ −∞ Above holds with equality iff: gt t 1 () = cg 2 () for arbitrary constant c 2  T  T T  s()h(T − τ )d τ   ∫ (( τ )) 2 d τ ∫ hT − ττ T  ∫ τ s 2 ( ) d s SNR =  0 T  ≤ 0 T 0 = 2 ∫ (( τ )) 2 d τ = 2 E s N 0 ∫ hT − t)dt N 0 ∫ hT − t ) dt N 0 0 N 0 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 [...]... distance between signal points 10 -1 orthogonal Pe antipodal 3dB 10 -5 Eb/N0 (dB) Eytan Modiano Slide 25 12 14 Probability of error for M-PAM S1 Si S2 SM = AM Eg , AM = (2m − 1 − M) SM τi dij = 2 Eg for | i − j |= 1 Decision rule : Choose si such that d(r,si ) is minimized P[error | si ] = P[decode si 1 | si ] + P[decode si +1 | si ] = 2P[decode si +1 | si ]  d2   2 Eg  Pe i , i +1  = 2Q  , Peb = Pe =... – • S1 => r > 0 decide S1 => r < 0 decide S2 Probability of error – When S2 was sent the probability of error is the probability that noise exceeds (Eb )1/ 2 similarly when S1 was sent the probability of error is the probability that noise exceeds - (Eb )1/ 2 – Eytan Modiano Slide 20 P(e|S1) = P(e|S2) = P[r “antipodal” signaling Antipodal signals with energy Eb can be represented geometrically as S2 − Eb • • S1 Eb If S1 was sent then the received signal r = S1 + n If S2 was sent then the received signal r = S2 + n 1 − (r − fr | s (r | s1) = e πN0 1 − (r + fr | s (r | s2) = e πN0 Eytan Modiano Slide 19 Eb ) 2 / N 0 Eb ) 2 / N 0 Detection... 4 .1 of text) More on Q function • Notes on Q(x) – – – – • Q(0) = 1/ 2 Q(-x) = 1- Q(x) ∞) =1 Q(∞) = 0, Q(- ∞ ∞ σ σ If X is N(m,σ2) Then P(X>x) = Q((x-m)/ σ) Example: Pe = P[r m = Eb , σ = N0 / 2 Pe = 1 − P[r > 0 | s1] = 1 − Q( Eytan Modiano Slide 22 − Eb N0 / 2 ) = 1 − Q(− 2Eb / N0 ) = Q( 2Eb / N0 ) Error analysis continued • In general, the probability... 2 e 1 − (r − e πN0 −∞ −r 2 / N 0 − 2 Eb / N 0 −∞ ∞ ∫ 0 Eb ) 2 / N 0 dr dr −r 2 / 2 dr dr 2 Eb / N 0 = Q( 2 Eb / N0 ) where, 1 Q( x ) ∆ 2π • • Eytan Modiano Slide 21 ∫ ∞ e −r − 2 /2 dr x Q(x) = P(X>x) for X Gaussian with zero mean and σ2 = 1 Q(x) requires numerical evaluation and is tabulated in many math books (Table 4 .1 of text) More on Q function • Notes on Q(x) – – – – • Q(0) = 1/ 2 Q(-x) = 1- Q(x)...   Notes: 1) the probability of error for s1 and sM is lower because error only occur in one direction 2) With Gray coding the bit error rate is Pe/log2(M) Eytan Modiano Slide 26 Probability of error for M-PAM Eav M2 1 M2 1 = Eg => Ebav = Eg 3 3Log2 ( M ) 3Log2 ( M ) Eg = Ebav 2 M 1   6 Log2 ( M ) Pe Pe = 2Q  Ebav , Peb = 2 Log2 ( M )   ( M − 1) N0   accounting for effect of S1 and SM we... filter receiver Sample at t=kT U(t) rx(t) rx(kT) g(T-t) 2Cos(2πfct) Sample at t=kT U(t) ry(t) 2Sin(2πfct) Eytan Modiano Slide 13 g(T-t) ry(kT) Binary PAM example, continued g(t) 0 => S1 = g(t) 1 => S2 = -g(t) A S(t) T “S1(t)” “S2(t)” Y(t) “Y1(t)” 2T T 3T T T Eytan Modiano Slide 14 2T “Y2(t)” 2T Alternative implementation: correlator receiver r(t) = S(t) + n(t) Sample at t=kT T ∫ () r(t) Y(kT) 0 S(t)... Sm ) (AKA Maximum Likelihood (ML) decision rule S1 S M Eytan Modiano Slide 17 ) Detection in AWGN (Single dimensional constellations) 1 − (r − S m ) 2 / N 0 f (r | Sm ) = e πN0 (r − Sm )2 ln( f (r | Sm )) = − ln( πN0 ) − N0 drSm = (r − Sm )2 Maximum Likelihood decoding amounts to minimizing • Also known as minimum distance decoding – Eytan Modiano Slide 18 Similar expression for multidimensional constellations... 16 MAP detector • MAP detector : max P(Sm | r) Notes: – S1 S M – P(Sm ,r) P(r | Sm )P(Sm ) = P(Sm | r) = P(r ) P(r ) P( Sm | r ) = M fr ( r ) = ∑f fr | s (r | Sm )P(Sm ) fr ( r ) – – MAP rule requires prior probabilities MAP minimizes the probability of a decision error ML rule assumes equally likely symbols With equally likely symbols MAP and ML are the same r|s (r | Sm )P(Sm ) m =1 When P(Sm ) = 1. .. Binary PAM Am ∈ {0 ,1} Matched filter is matched to g(t) g(t) g(T-t) A A T Eytan Modiano Slide 11 T “matched filter” Example, continued t ∫ Ys (t ) = S(τ )h(t − τ )dτ , h(t© = g(T − t© ⇒ h(t − τ ) = g(T +τ − t) ) ) 0 t t ∫ ∫ Ys (t ) = g(τ )g(T + τ − t)dτ = g(τ )g(T − t + τ )dτ 0 0 T ∫ Ys (T ) = g 2 (τ )dτ 0 Ys(t) A2T • Sample at t=T to obtain maximum value t T Eytan Modiano Slide 12 Matched filter receiver . continued g(t) 0 => S 1 = g(t) 1 => S 2 = -g(t) A Eytan Modiano S(t) Y(t) T 2T 3T “S 1 (t)” T “S 2 (t)” T T “Y 1 (t)” “Y 2 (t)” 2T 2T Slide 14 YT Alternative implementation:. (r | S m )P(S m ) m =1 1 When P(S m ) = Map rule becomes : M (max fr | S m ) ( AKA Maximum Likelihood (ML) decision rule ) S 1 S M Eytan Modiano Slide 17 Detection in AWGN (Single. probability that noise exceeds (Eb) 1/ 2 similarly when S1 was sent the probability of error is the probability that noise exceeds - (Eb) 1/ 2 – P(e|S1) = P(e|S2) = P[r<0|S1) Eytan Modiano Slide 20