: 1 Advaced Digital Commuicatios (EQ2410) 1 Tuesday, Ja. 20, 2015 8:15-10:00, D42 1 Textbook: U. Madhow, Fudametals of Digital Commuicatios, 2008 1 / 1 Overview 2 / 1
Chael Model Itersymbol iterferece (ISI) Successive symbols iterfere with each other. ISI is caused by Multi-path propagatio Radio commuicatios: sigals are reflected by walls, buildigs, hills, ioosphere,... Uderwater commuicatios: sigals are reflected by the groud, the surface, iterface betwee differet water layers,... Frequecy-selective ad badlimited chaels Cables ad wires are modeled by (liear) LCR circuits. Frequecy divisio multiplexig (FDM) requires limited badwidth per chael. Mathematical model ISI ca be modeled by a liear filter. (implicit assumptio: liearity) I geeral: time-variat liear filter. 3 / 1 Chael Model Trasmitted sigal: u(t) = u(t) g C (t) ISI Chael b[]g T (t T ) = : symbol sequece trasmitted at rate 1/T : impulse respose of the trasmit filter T : duratio of oe symbol Received sigal: = b[]p(t T ) + = p(t) = (g T g C )(t): impulse respose of the cascade of the trasmit ad chael filters 2. g C (t): chael impulse respose : complex additive white Gaussia oise (AWGN) with variace σ 2 = N 0 /2 per dimesio Chael equalizatio: extract from 2 Covolutio of two sigals a(t) ad b(t): q(t) = (a b)(t) = a(u)b(t u)du. 4 / 1
Receiver Frot Ed g C(t) Equivalet pulse p(t) = (g T g C)(t) Matched filter p MF (t) = p ( t) z[] Samplig, t = T Theorem (Optimality of the Matched Filter) The optimal receiver filter is matched to the equivalet pulse p(t) ad is specified i the time ad frequecy domai as follows: g R,opt (t) = p MF (t) = p ( t) G R,opt (f ) = P MF (f ) = P (f ). I terms of a decisio o the symbol sequece, there is o loss of relevat iformatio by restrictig attetio to symbol rate samples of the matched filter output give by z[] = (y p MF )(T ) = p MF (T t)dt = p (t T )dt. [U. Madhow, Fudametals of Dig. Comm., 2008] 5 / 1 Eye Diagrams Visualizatio of the effect of ISI (for the oise-free case) Received sigal (oise free): r(t) = b[]x(t T ) Effective impulse respose: x(t) = (g T g C g R )(t) (icl. trasmit, chael, ad receive filter) Eye diagram superimpose the waveforms {r(t kt ), k = ±1, ±2,...} Example (a) BPSK sigal with ISI free pulse i (ope eye); (b) BPSK sigal with ISI (closed eye). [U. Madhow, Fudametals of Dig. Comm., 2008] 6 / 1
Nyquist Criterio g C(t) Equivalet pulse p(t) = (g T g C)(t) Matched filter p MF (t) = p ( t) z[] Samplig, t = T Theorem (Nyquist 3 Criterio ad Nyquist Rate) The received sigal after samplig (samplig rate 1/T ) is give as z(t ) = b[m] x(t mt ) + (T ) = b[] x(0) + b[m] x(t mt ) + (T ), m= m with the effective impulse respose: x(t) = (g T g C g R )(t). Uder the assumptio that X (f ) = F{x(t)} = G T (f )G C (f )G R (f ) = 0 for f > W, the trasmissio system is ISI free if x(t ) = { 1 for = 0 0 for 0 m X ( f m ) = T. T ISI-free trasmissio at symbol rate R is possible if 0 < R R N where the upper boud R N is give by the Nyquist rate R N = 2W. 3 Harry Nyquist 1928 (Swedish/America ivetor) 7 / 1 Nyquist Criterio (a) 1 α=0 α=0.5 α=1 (b) 1 α=0 α=0.5 α=1 0.8 x(t) 0.6 0.4 0.2 0 0.2 X(f) 0.8 0.6 0.4 0.2 t/t 0.4 4 3 2 1 0 1 2 3 4 0 1.5 1 0.5 0 0.5 1 1.5 Pulse shapig for ISI-free trasmissio Raised-cosie pulses ca be desiged to be ISI free for 0 < R 2W I time domai (see plot (a)) ( ) t cos παt/t x rc (t) = sic T 1 4α 2 t 2 /T 2 ft I frequecy domai (see plot(b)) T ( ( X rc (f ) = T 2 [1 + cos πt α f 1 α ))], for f 1 α, for 1 α < f < 1+α 0, for f > 1+α Desig of the trasmit ad receive filters (matched filters): X rc (f ) 1/2 1/2 X rc (f ) G T (f ) = K 1 G C (f ) 1/2 ad G R (f ) = K 2 G C (f ) 1/2 with K 1 so that g 2 T (t)dt = E b ad K 2 arbitrary. 8 / 1
Maximum Likelihood Sequece Estimatio Based o the cotiuous-time model g C (t) Equivalet pulse p(t) = (g T g C )(t) s b (t) ML Sequece Estimator {ˆb[]} Goal: fid b that maximizes the likelihood fuctio 4 L(y b) = p(y b) p(y) ( ) 1 = exp σ 2 (Re( y, s b ) s b 2 /2) with s b (t) = b[]p(t T ). Or equivaletly: fid b that maximizes the cost fuctio Λ(b) = Re( y, s b ) s b 2 /2 Brute-force detector try out all realizatios of b ot feasible: N symbols with M-ary modulatio lead to M N possible sequeces b. 4 Ier product of two sigals a(t) ad b(t): a, b = a(t)b (t)dt. 9 / 1 Maximum Likelihood Sequece Estimatio Decompositio of Λ(b) Useful defiitio: h[m] = p(t)p (t mt )dt = (p p MF )(mt ) = x(mt ) sampled effective impulse respose (trasmit/chael/receiver filter) useful property: h[ m] = h [m] First term i Λ(b) (see e.g. textbook, p. 205) ( ) Re( y, s b ) = Re b [] p (t T )dt = Re(b []z[]) Secod term i Λ(b) (see e.g. textbook, p. 206) s b 2 = s b, s b = b[]b [m]h[m ] =... m = h(0) b[] 2 + 2Re(b []b[m]h[ m]) m< 10 / 1
Maximum Likelihood Sequece Estimatio Decompositio of Λ(b) Itermediate result Λ(b) = { ( Re(b []z[]) h[0] 2 b[] 2 Re b [] )} b[m]h[ m] m< The cost fuctio is additive i. The -th summad of the sum is a fuctio of the curret symbol b[] ad the past symbols {b[m], m < }. Iterpretatio: the sum over m removes the ISI from previously trasmitted symbols from z[]. 11 / 1 Maximum Likelihood Sequece Estimatio Viterbi Algorithm Assumptio: the system has a limited impulse respose, i.e., h[] = 0, > L, ad we get Λ(b) = {Re(b []z[]) h[0] 1 2 b[] 2 Re b [] b[m]h[ m] } good approximatio for practical systems! oly the previous L symbols cause ISI. m= L State defiitio: s[] = (b[ L],..., b[ 1]), M L states. Brach metric: λ (s[] s[ + 1]) = λ (b[], s[]) = Re(b []z[]) h[0] 2 b[] 2 Re b [] 1 m= L b[m]h[ m] Accumulated metric (AM) at time k k Λ k (b) = λ (s[] s[ + 1]) = λ k (s[k] s[k + 1]) + Λ k 1 (b) = =1 k λ (b[], s[]) = λ k (b[k], s[k]) + Λ k 1 (b) =1 12 / 1
Maximum Likelihood Sequece Estimatio Example Viterbi Algorithm 5 BPSK-modulated sigal, h[0] = 3/2, h[1] = h[ 1] = 1/2, i.e., L = 1. [U. Madhow, Fudametals of Dig. Comm., 2008] Each legth-k path through the trellis is associated with a legth-k symbol sequece ad a AM Λ k (b) At ay give state, oly the icomig path with the best AM Λ k (b) (survivor) has to be kept. Let Λ (1 : k, s ) be the AM of the survivor at state s[k] = s. The AM for the path emergig from s[k] = s ad edig at s[k + 1] = s is give as ad we have Λ 0 (1 : k + 1, s s) = Λ (1 : k, s ) + λ k+1 (s s) Λ (1 : k + 1, s) = max s Λ 0 (1 : k + 1, s s) If the ed of the trellis is reached, the best survivor is the maximum likelihood sequece. 5 See Figure 5.5-6 i [U. Madhow, Fudametals of Dig. Comm., 2008] 13 / 1 Maximum Likelihood Sequece Estimatio Alterative Formulatio Based o the discrete-time model gt (t) gc(t) Equivalet pulse p(t) = (gt gc)(t) Matched filter pmf (t) = p ( t) z[] Samplig, t = T Whiteig Filter v[k] After matched filterig, the additive oise i z[] is colored; a whiteig filter is required. Model for the received sequece: v[k] = L f []b[k ] + η k, =0 with discrete-time impulse respose f [] describig the cascade of trasmit, chael, receive, ad whiteig filter; complex additive white Gaussia oise η k with oise variace σ 2 per dimesio. Cost fuctio to be miimized g(b) = L v[k] f []b[k ] 2 k =0 ML sequece ca be foud with the Viterbi algorithm. 14 / 1