Digital Communications

Digital Communications Chapter 5 Carrier and Symbol Synchronization Po-Ning Chen, Professor Institute of Communications Engineering National Chiao-Tung University, Taiwan Digital Communications Ver 218.7.26 Po-Ning Chen 1 / 69

5.1 Signal parameters estimation Digital Communications Ver 218.7.26 Po-Ning Chen 2 / 69

Models Channel delay τ exists between Tx and Rx. For AWGN + channel delay τ + carrier phase mismatch φ, given s(t) transmitted, one receives As lowpass equivalent r(t) = s(t τ) + n(t). s(t) = Re [s l (t)e ı 2πfct ] but r(t) = Re [r l (t)e ı 2πfct+ ı φ ] we have r(t) = Re [r l (t)e ı 2πfct+ ı φ ] = Re [s l (t τ)e ı 2πfc(t τ) ] + Re [n l (t)e ı 2πfct+ ı φ ] = Re {[s l (t τ)e ı φ + n l (t)] e ı 2πfct+ ı φ } where φ = 2πf c τ φ. Digital Communications Ver 218.7.26 Po-Ning Chen 3 / 69

φ = 2πf c τ φ In general, τ T, but is far from since f c is large. 2πf c τ mod 2π So, we should treat τ and φ as different random variables r l (t) = s l (t; φ, τ) + n l (t). Note that since the passband signal s(t τ) is real, there does not appear a phase mismatch φ for the real passband signals. In other words, φ appears due to the imperfect down-conversion at the receiver. Digital Communications Ver 218.7.26 Po-Ning Chen 4 / 69

Let Θ = (φ, τ), and set r l (t) = s l (t; φ, τ) + n l (t) r l (t) = s l (t; Θ) + n l (t). Let {φ n,l (t), 1 n N} be a set of orthonormal functions over [, T ), where T T, such that r j,l = r l (t), φ n,l (t) and we have a vector representation r l = s l (Θ) + n l. Digital Communications Ver 218.7.26 Po-Ning Chen 5 / 69

Assuming Θ has a joint pdf f (Θ), the MAP estimate of Θ is ˆΘ = arg max Θ f (Θ r l) = arg max f (r l Θ) f (Θ) Θ f (r l ) = arg max f (r l Θ) f (Θ). Θ Assume Θ is uniform and holds constant for an observation period of T T (slow variation), ˆΘ = arg max Θ The latter is the ML estimate of Θ. f (r l Θ) f (Θ) = arg max f (r l Θ) Θ Note that f (r l Θ) is the likelihood function. Digital Communications Ver 218.7.26 Po-Ning Chen 6 / 69

r l = s l (Θ) + n l and ˆΘ = arg maxθ f (r l Θ) For a fixed {φ j,l (t)} N j=1 and E[ n j,l 2 ] = σ 2 l = 2N, f (r l Θ) = ( 1 N ) πσl 2 N r n,l s n,l (Θ) 2 exp { }. Text (5.1-5) uses bandpass analysis and yields 1 f (r Θ) = ( 2πσ 2 )N exp { N r n s n(θ) 2 n=1 } with σ 2 = N 2σ 2 2. We will show later that both analyses yield identical result! Assume that {φ n,l (t)} N j=1 is a complete orthonormal basis. Then n=1 σ 2 l N n=1 r n,l s n,l (Θ) 2 = T r l (t) s l (t; Θ) 2 dt. Digital Communications Ver 218.7.26 Po-Ning Chen 7 / 69

The likelihood function Given s l (t) known to both Tx and Rx, the ML estimate of Θ is and the term ˆΘ = arg max Λ(Θ) Θ Λ(Θ) = exp { 1 σ 2 l T r l (t) s l (t; Θ) 2 dt} will be referred to as the likelihood function. Digital Communications Ver 218.7.26 Po-Ning Chen 8 / 69

Exemplified block diagrams Digital Communications Ver 218.7.26 Po-Ning Chen 9 / 69

Exemplified block diagrams Digital Communications Ver 218.7.26 Po-Ning Chen 1 / 69

5.2 Carrier phase estimation Digital Communications Ver 218.7.26 Po-Ning Chen 11 / 69

Example 1 (DSB-SC signal where τ = and φ = φ φ) Assume the transmitted signal is s(t) = A(t) cos (2πf c t + φ) Rx uses c(t) with carrier reference φ to demodulate c(t) = cos (2πf c t + φ) So even n l (t) =, the down-covertion Rx gives LPF{s(t)c(t)} = 1 2 A(t) cos (φ φ) Performance is severely degraded due to phase error (φ φ). Hence, signal power is reduced by a factor cos 2 (φ φ): A phase error of 1 leads to.13 db of signal power loss. A phase error of 3 leads to 1.25 db of signal power loss. Digital Communications Ver 218.7.26 Po-Ning Chen 12 / 69

Example 2 (QAM or PSK where τ = and φ = φ φ) For QAM or PSK signals, Tx sends s(t) = x(t) cos (2πf c t + φ) y(t) sin (2πf c t + φ) and Rx uses the c I (t) and c Q (t) to demodulate c I (t) = cos (2πf c t + φ) c Q (t) = sin (2πf c t + φ) Hence r I (t) = LPF{s(t)c I (t)} = x(t) cos(φ φ) y(t) sin(φ φ) 2 r Q (t) = LPF{s(t)c Q (t)} = x(t) sin(φ φ) + y(t) cos(φ φ) 2 Even worse, both power degradation and crosstalk occur. Digital Communications Ver 218.7.26 Po-Ning Chen 13 / 69

5.2-1 ML carrier phase estimation Digital Communications Ver 218.7.26 Po-Ning Chen 14 / 69

Assume τ = (or τ has been perfectly compensated) & estimate φ. So Θ = φ. The likelihood function is Λ(φ) = exp { 1 σ 2 l T r l (t) s l (t; φ) 2 dt} = exp ( 1 σ 2 l T [ r l (t) 2 2Re {r l (t)s l (t; φ)} + s l(t; φ) 2 ] dt) r l (t) 2 is irrelevant to the maximization over φ. Digital Communications Ver 218.7.26 Po-Ning Chen 15 / 69

For the term s l (t; φ) 2, we have s l (t; φ) = s l (t)e ı φ. So, Thus s(t, φ) 2 = s(t) 2 ˆφ = arg max exp { 2 φ σl 2 T Re {r l (t)sl (t; φ)} dt} = arg max exp { 4 φ σl 2 T r(t)s(t; φ) dt} (With σ2 l = 2N, this formula is the same as (5.2-8) obtained based on bandpass analysis!) Note that from Slide 2-24, x(t), y(t) = 1 2 Re { x l(t), y l (t) }. We will use one of the two above criterions (i.e., baseband or passband) to derive ˆφ, depending on whichever is convenient. Digital Communications Ver 218.7.26 Po-Ning Chen 16 / 69

Example Assume s(t) = A cos(2πf c t) and r(t) = A cos (2πf c t + φ) +n(t) s(t;φ) where φ is the unknown phase here. ˆφ = arg max exp { 4 φ σl 2 T T = arg max r(t)s(t; φ) dt φ r(t)s(t; φ) dt} Digital Communications Ver 218.7.26 Po-Ning Chen 17 / 69

So we seek ˆφ that minimizes Λ L (φ) = A T r(t) cos (2πf c t + φ) dt Since φ is continuous, a necessary condition for a minimum is that dλ L (φ) = dφ φ= ˆφ It yields T = cos( ˆφ) r(t) sin(2πf c t + ˆφ) dt = T r(t) sin(2πf c t) dt + sin( ˆφ) T r(t) cos(2πf c t) dt ˆφ = tan 1 T r(t) sin(2πf c t) dt T r(t) cos(2πf c t) dt Digital Communications Ver 218.7.26 Po-Ning Chen 18 / 69

Performance check Recall with unknown phase φ. E [ E [ r(t) = A cos (2πf c t + φ) + n(t) T T Then on the average r(t) sin(2πf c t) dt] = 1 2 AT sin(φ) r(t) cos(2πf c t) dt] = 1 2 AT cos(φ) ˆφ = tan E [ T 1 r(t) sin(2πf c t) dt] E [ T r(t) cos(2πf c t) dt] = φ which is the true phase. Digital Communications Ver 218.7.26 Po-Ning Chen 19 / 69

A one-shot ML phase estimator Digital Communications Ver 218.7.26 Po-Ning Chen 2 / 69

5.2-2 The phase-locked loops Digital Communications Ver 218.7.26 Po-Ning Chen 21 / 69

Instead of one-shot estimate, a phase-locked loop continuously adjusts φ to achieve T r(t) sin(2πf c t + ˆφ) dt = We can then change the sign of sine function to facilitate the follow-up analysis. T r(t) ( sin(2πf c t + ˆφ)) dt = Digital Communications Ver 218.7.26 Po-Ning Chen 22 / 69

The analysis of the PLL can be visioned in a simplified basic diagram: Digital Communications Ver 218.7.26 Po-Ning Chen 23 / 69

If T is a multiple of 1/(2f c ), then T e(t)dt = A T 2 sin(φ ˆφ ML )dt A T 2 sin(4πf c t + φ + ˆφ ML )dt = AT 2 sin(φ ˆφ ML ) The effect of integration is similar to a lowpass filter. Digital Communications Ver 218.7.26 Po-Ning Chen 24 / 69

Digital Communications Ver 218.7.26 Po-Ning Chen 25 / 69

Effective VCO output ˆφ(t) can be modeled as ˆφ(t) = K t where v( ) is the input of the VCO. v(τ)dτ Digital Communications Ver 218.7.26 Po-Ning Chen 26 / 69

The nonlinear sin( ) causes difficulty in analysis. Hence, we may simply it using sin(x) x. In terms of Laplacian transform technique, we then derive the close-loop system transfer function. H(s) = ˆφ(s) φ(s) = = = ˆφ(s) [φ(s) ˆφ(s)] + ˆφ(s) (KG(s)/s)[φ(s) ˆφ(s)] [φ(s) ˆφ(s)] + (KG(s)/s)[φ(s) ˆφ(s)] KG(s)/s 1 + KG(s)/s Digital Communications Ver 218.7.26 Po-Ning Chen 27 / 69

Second-order loop transfer function G(s) = 1 + τ 2s 1 + τ 1 s, where τ 1 τ 2 for a lowpass filter. where H(s) = =τ 2 ω n (2ζ ω n /K)(s/ω n ) + 1 (s/ω n ) 2 + 2ζ(s/ω n ) + 1 ω n = K/τ 1 ζ = ω n (τ 2 + 1/K)/2 natural frequency of the loop loop damping factor Digital Communications Ver 218.7.26 Po-Ning Chen 28 / 69

Assume 2ζ ωn K ωn/k 2ζ (i.e., 2ζ = H(s) 1 Kτ 2 +1 1). Hence, 2ζ(s/ω n ) + 1 (s/ω n ) 2 + 2ζ(s/ω n ) + 1. Damping factor=response speed (for changes) Digital Communications Ver 218.7.26 Po-Ning Chen 29 / 69

Noise-equivalent bandwidth of H(f ) Definition: One-sided noise-equivalent bandwidth of H(f ) 1 B eq = max f H(f ) 2 where we use τ 2 ω n = 2ζ ωn K 2ζ. H(f ) 2 df = 1 + (τ 2ω n ) 2 8ζ/ω n Tradeoff in parameter selection in PLL 1 + 4ζ2 ω n 8ζ It is desirable to have a larger PLL bandwidth ω n in order to track any time variation in the phase of the received carrier. However, with a larger PLL bandwidth, more noise will be passed into the loop; hence, the phase estimate is less accurate. Digital Communications Ver 218.7.26 Po-Ning Chen 3 / 69

5.2-3 Effect of additive noise on the phase estimate Digital Communications Ver 218.7.26 Po-Ning Chen 31 / 69

Assume that r(t) = A c cos(2πf c t + φ(t)) + n(t) where n(t) = n c (t) cos(2πf c t + φ(t)) n s (t) sin(2πf c t + φ(t)) and n c (t) and n s (t) are independent Gaussian random processes. Digital Communications Ver 218.7.26 Po-Ning Chen 32 / 69

For convenience, we abbreviate φ(t) and ˆφ(t) as φ and φ ML in the derivation. Digital Communications Ver 218.7.26 Po-Ning Chen 33 / 69

Digital Communications Ver 218.7.26 Po-Ning Chen 34 / 69

Digital Communications Ver 218.7.26 Po-Ning Chen 35 / 69

As a result, ˆφ(s) φ(s) + n 2 (s) = = = ˆφ(s) [φ(s) ˆφ(s) + n 2 (s)] + ˆφ(s) (KG(s)/s)[(φ(s) ˆφ(s)) + n 2 (s)] [φ(s) ˆφ(s) + n 2 (s)] + (KG(s)/s)[(φ(s) ˆφ(s)) + n 2 (s)] KG(s)/s 1 + KG(s)/s = H(s) ˆφ(s) = [φ(s) + n 2 (s)]h(s) ˆφ(t) = [φ(t) + n 2 (t)] h(t) = φ(t) h(t) + n 2 (t) h(t) noise Digital Communications Ver 218.7.26 Po-Ning Chen 36 / 69

Let s calculate noise variance: σ 2ˆφ = = = N B eq A 2 c Φ n2 (f ) H(f ) 2 df N 2 1 A 2 c H(f ) 2 df max H(f ) 2 f σ 2ˆφ is proportional to B eq; since the signal power is fixed as 1/2, SNR is inversely proportional to B eq. Digital Communications Ver 218.7.26 Po-Ning Chen 37 / 69

Exact PLL model versus liberalized PLL model Digital Communications Ver 218.7.26 Po-Ning Chen 38 / 69

It turns out that when G(s) = 1, the σ of the exact PLL model is 2ˆφ tractable (Vitebi 1966). The linear model gives σ 2ˆφ = 1/γ L. The linear model well approximates the exact model when γ L = A 2 c/(n B eq ) > 3 4.77 db. Digital Communications Ver 218.7.26 Po-Ning Chen 39 / 69

5.2-4 Decision directed loops Digital Communications Ver 218.7.26 Po-Ning Chen 4 / 69

For general modulation scheme, let s l (t) = I n g(t nt ); n= then Hence, by letting T = KT, Λ L (φ) = = = s(t; φ) = Re {s l (t)e ı φ e ı 2πfct } T T T r(t)s(t; φ) dt r(t)re { n= I n g(t nt )e ı φ e ı 2πfct } dt K 1 r(t)re { I n g(t nt )e ı φ e ı 2πfct } dt n= Digital Communications Ver 218.7.26 Po-Ning Chen 41 / 69

Λ L (φ) = T r(t)re { K 1 n= I n g(t nt )e ı φ e ı 2πfct } dt Λ L (φ) = Re = Re e ı φ K 1 I n n= e ı φ K 1 I n n= nt T r(t)g(t nt )e ı 2πfct dt (n+1)t r(t)g(t nt )e ı 2πfct dt y n = Re {e ı φ K 1 I n y n } n= K 1 K 1 = Re { I n y n } cos(φ) Im { I n y n } sin(φ) n= n= Digital Communications Ver 218.7.26 Po-Ning Chen 42 / 69

Now Λ L (φ) = Re { K 1 n= I n y n } cos(φ) Im { K 1 n= I n y n } sin(φ) dλ L (φ) dφ K 1 K 1 = Re { I n y n } sin(φ) Im { I n y n } cos(φ) n= n= and the optimal estimate ˆφ is given by ˆφ = tan Im { K 1 1 n= I n y n } Re { K 1 n= I n y n } This is called decision directed estimation of φ. Digital Communications Ver 218.7.26 Po-Ning Chen 43 / 69

Note that from Slide 2-24, Hence, x(t), y(t) = 1 2 Re { x l(t), y l (t) }. Re{I n y n } = nt (n+1)t = 1 2 Re { nt r(t) Re {I n g(t nt )e ı 2πfct } dt (n+1)t (n+1)t r l (t)i n g (t nt ) dt} = 1 2 Re{I n r l (t)g (t nt ) dt } nt y n,l = 1 2 Re {I n y n,l } Digital Communications Ver 218.7.26 Po-Ning Chen 44 / 69

Im{I n y n } = = (n+1)t nt (n+1)t nt = 1 2 Re { nt r(t) Im {I n g(t nt )e ı 2πfct } dt r(t) Re {( ı )I n g(t nt )e ı 2πfct } dt (n+1)t = 1 2 Im {I n y n,l } r l (t) ı I n g (t nt ) dt} ˆφ = tan 1 Im { K 1 n= In y n,l } Re { K 1 n= In y n,l } Final note: The formula (5.2-38) in text has an extra sign because the text (inconsistently to (5.1-2)) assumes s(t; φ) = Re {s l (t)e ı φ e ı 2πfct }; but we assume s(t; φ) = Re {s l (t)e + ı φ e ı 2πfct } as (5.1-2) did. Digital Communications Ver 218.7.26 Po-Ning Chen 45 / 69

5.2-5 Non-decision-directed loops Digital Communications Ver 218.7.26 Po-Ning Chen 46 / 69

For carrier phase estimation with σ 2 l = 2N, we have shown that ˆφ = arg max φ exp { 2 N T = tan 1 Im { K 1 n= I n y n } Re { K 1 n= I n y n } r(t)s(t; φ) dt} When {I n } K 1 n= is unavailable, we take the expectation with respect to {I n } n= K 1 instead: ˆφ = arg max φ = arg max φ E [exp { 2 N E [exp { 2 N = arg max E [exp { 2 K 1 φ N n= T T r(t)s(t; φ) dt}] K 1 r(t)re { I n g(t nt )e ı φ e ı 2πfct } dt}] I n y n (φ)}] n= where we assume both {I n } and g(t) are real and y n (φ) = (n+1)t nt r(t)g(t nt ) cos(2πf c t + φ)dt. Digital Communications Ver 218.7.26 Po-Ning Chen 47 / 69

If {I n } i.i.d. and equal-probable over { 1, 1}, K 1 ˆφ = arg max φ n= K 1 = arg max φ n= K 1 = arg max φ n= K 1 = arg max φ n= E [exp { 2 N I n y n (φ)}] (exp { 2 N y n (φ)} + exp { 2 N y n (φ)}) cosh ( 2 N y n (φ)) log cosh ( 2 N y n (φ)) We may then determine the optimal ˆφ by deriving K 1 n= log cosh ( 2 N y n (φ)) φ =. Digital Communications Ver 218.7.26 Po-Ning Chen 48 / 69

For x 1 (low SNR), log cosh(x) x2 2 exaponsion). For x 1 (high SNR), log cosh(x) x. (By Taylor K 1 ˆφ = arg max log cosh ( 2 y n (φ)) φ n= N arg max φ K 1 2 n= y N 2 n 2 (φ) N large arg max φ K 1 2 n= N y n (φ) N small arg max φ K 1 n= yn = 2 (φ) N large arg max φ K 1 n= y n (φ) N small Digital Communications Ver 218.7.26 Po-Ning Chen 49 / 69

When x small, log(cosh(x)) = log e x + e x and 2 = log [1 x + 1 2 x 2 1 6 x 3 + O(x 4 )] + [1 + x + 1 2 x 2 + 1 6 x 3 + O(x 4 )] 2 = log (1 + 1 2 x 2 + O(x 4 )) = 1 2 x 2 + O(x 4 ) log(cosh(x)) e x e x lim = lim tanh(x) = lim x x x x e x = 1. + e x Digital Communications Ver 218.7.26 Po-Ning Chen 5 / 69

Digital Communications Ver 218.7.26 Po-Ning Chen 51 / 69

When K = 1, which covers the case of Example 5.2-2 in text, the optimal decision becomes irrelevant to N : arg max φ y ˆφ 2 = (φ) N large arg max φ y (φ) N small = arg max φ = arg max φ = arg max φ where tan(θ) = T make T cos(φ) r(t)g(t) cos(2πf c t + φ)dt T sin(φ) r(t)g(t) cos(2πf c t)dt T r(t)g(t) sin(2πf c t)dt cos(φ) cos(θ) sin(φ) sin(θ) = arg max cos(φ + θ) r(t)g(t) sin(2πfct)dt T φ r(t)g(t) cos(2πfct)dt. So the optimal ˆφ should ˆφ = θ = tan 1 T r(t)g(t) sin(2πf ct) dt T r(t)g(t) cos(2πf ct) dt. Digital Communications Ver 218.7.26 Po-Ning Chen 52 / 69

5.3 Symbol timing estimation Digital Communications Ver 218.7.26 Po-Ning Chen 53 / 69

Assume φ = (or φ has been perfectly compensated) & estimate τ. In such case, r l (t) = s l (t; τ) + n l (t) = s l (t τ) + n l (t). We could rewrite the likelihood function (cf. Slide 5-8 with σ 2 l = 2N ) as Λ(τ) = exp { 1 T r l (t) s l (t; τ) 2 dt} 2N = exp ( 1 T [ r l (t) 2 2Re {r l (t)sl 2N (t; τ)} + s l(t; τ) 2 ] dt) Same as before, the 1st term is independent of τ and can be ignored. But T s l (t; τ) 2 dt = T s l (t τ) 2 dt could be a function of τ. So, we would say when τ T, the 3rd term is nearly independent of τ and can also be ignored. Digital Communications Ver 218.7.26 Po-Ning Chen 54 / 69

This gives: Λ L (τ) = Re { T r l (t)sl (t; τ)dt} Assume Then s l (t) = I n g(t nt ) n= K 1 Λ L (τ) = Re { In n= T K 1 = Re { In ỹ n,l (τ)} n= r l (t)g (t nt τ)dt} where ỹ n,l (τ) = T r l (t)g (t nt τ)dt Digital Communications Ver 218.7.26 Po-Ning Chen 55 / 69

Decision directed estimation The text assumes that both {I n } and g(t) are real; hence r l (t) can be made real by eliminating the complex part, and where y n,l (τ) = Λ L (τ) = T K 1 I n y n,l (τ) n= Re{r l (t)}g(t nt τ)dt. Then the optimal decision is the ˆτ such that dλ L (τ) dτ = K 1 dy n,l (τ) I n n= dτ = Likewise, it is called decision directed estimation. Digital Communications Ver 218.7.26 Po-Ning Chen 56 / 69

Non-decision directed estimation Consider again the case of BPSK, i.e. I n = ±1 equal-probable; then because the complex noise can be excluded, we have Λ(τ) = exp ( 1 N T Re{r l (t)}s l (t; τ)dt) = exp ( 1 K 1 T I n N Re{r l (t)}g(t nt τ) dt) n= K 1 = exp ( 1 I n y n,l (τ)) n= N Λ(τ) = E [Λ(τ)] = K 1 cosh ( 1 y n,l (τ)) n= N Digital Communications Ver 218.7.26 Po-Ning Chen 57 / 69

Thus log Λ(τ) = K 1 log cosh ( 1 y n,l (τ)) n= N For low to moderate SNR, people simplify log cosh(x) to 1 2 x 2, log Λ(τ) 1 2N 2 K 1 yn,l 2 (τ). n= Taking derivative, we see a necessary condition for ˆτ is N 2 d log Λ(τ) = dτ τ=ˆτ K 1 n= y n,l (ˆτ) y n,l K 1 (ˆτ) = yn,l 2 (ˆτ)y n,l (ˆτ) n= y n,l (ˆτ) = Digital Communications Ver 218.7.26 Po-Ning Chen 58 / 69

For high SNR, people simplify log cosh(x) to x, log Λ(τ) 1 K 1 y n,l (τ). N n= Taking derivative, we see a necessary condition for ˆτ is d log Λ(τ) N = dτ τ=ˆτ K 1 sgn(y n,l (ˆτ)) y n,l(ˆτ) = n= K 1 y n,l (ˆτ) y n,l (ˆτ) n= y n,l (ˆτ) = Note that here, we use f (x) x f (x), f (x) > = f (x), f (x) < } = sgn(f (x))f (x). Digital Communications Ver 218.7.26 Po-Ning Chen 59 / 69

5.4 Joint estimation of carrier phase and symbol timing Digital Communications Ver 218.7.26 Po-Ning Chen 6 / 69

The likelihood function is Assuming we have Λ(φ, τ) = exp { 1 2N s l (t) = s l (t; φ, τ) = T r l (t) s l (t; τ, φ) 2 dt} (I n g(t nt τ) + ı J n w(t nt τ)) n= (I n g(t nt τ) + ı J n w(t nt τ)) e ı φ n= Here, I use e ı φ in order to synchronize with the textbook. PAM: I n real and J n = QAM and PSK: I n complex and J n = OQPSK: w(t) = g(t T /2) Digital Communications Ver 218.7.26 Po-Ning Chen 61 / 69

Along similar technique used before, we rewrite Λ(φ, τ) as log Λ(φ, τ) = Re { T = Re {e ı φ K 1 n= r l (t)sl (t; φ, τ) dt} T r l (t) (I n g (t nt τ) ı J n w (t nt τ)) dt} = Re {e ı φ K 1 (In y n,l (τ) ı Jn x n,l (τ))} n= = Re {e ı φ (A(τ) + ı B(τ))} = A(τ) cos(φ) B(τ) sin(φ) where y n,l (τ) = T r l (t)g (t nt τ)dt x n,l (τ) = T r l (t)w (t nt τ)dt A(τ) + ı B(τ) = K 1 n= (I n y n,l (τ) ı Jn x n,l (τ)) Digital Communications Ver 218.7.26 Po-Ning Chen 62 / 69

The necessary conditions for ˆφ and ˆτ are log Λ(φ, τ) τ τ=ˆτ = and log Λ(φ, τ) φ φ= ˆφ = Finally solving jointly the above two equations, we have the optimal estimates given by ˆτ satisfies A(τ) A(τ) τ 1 B(ˆτ) ˆφ = tan A(ˆτ) + B(τ) B(τ) τ = Digital Communications Ver 218.7.26 Po-Ning Chen 63 / 69

5.5 Performance characteristics of ML estimators Digital Communications Ver 218.7.26 Po-Ning Chen 64 / 69

Comparison between decision-directed (DD) and non-decision-directed (NDD) estimators Comparison between symbol timing (i.e., τ) DD and NDD estimates with raisedcosine(-spectrum) pulse shape. β is a parameter of the raised-cosine pulse Digital Communications Ver 218.7.26 Po-Ning Chen 65 / 69

Raise-cosine spectrum T, f 1 β 2T ; T X rc (f ) = 2 {1 + cos [ πt 1 β ( f β )]} 2T, 1 β 1+β 2T f 2T ;, otherwise β [, 1] roll-off factor β/(2t ) bandwidth beyond the Nyquist bandwidth 1/(2T ) is called the excess bandwidth. Digital Communications Ver 218.7.26 Po-Ning Chen 66 / 69

Raise-cosine spectrum Digital Communications Ver 218.7.26 Po-Ning Chen 67 / 69

Comparison between symbol timing (i.e., τ) DD and NDD estimates with raisedcosine(-spectrum) pulse shape. The larger the excess bandwidth, the better the estimate. NDD variance may go without bound when β small. Digital Communications Ver 218.7.26 Po-Ning Chen 68 / 69

What you learn from Chapter 5 MAP/ML estimate of τ and φ based on likelihood ratio function and known signals Phase lock loop Linear model analysis and its transfer function with and without additive noise Decision-directed (or decision-feedback) loop Non-decision-directed loop Take expectation on a quantity, proportional to probability. Digital Communications Ver 218.7.26 Po-Ning Chen 69 / 69