ELEG 635 Digital Communication Theory. Lecture 10

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1 ELEG 635 Digital Cmmuiati hery Leture

2 Ergdi Radm Presses CPM reeiver Sigal Parameter Estimati Carrier Syhrizati Ageda

3 Ergdi Radm Presses - A radm press is said t be ergdi if time averagig is equivalet t esemble averagig. his implies a sigle sample time sigal f the press tais all pssible statistial variatis f the press. Averages If the radm press is ergdi the a esemble r a time average may be used If the radm press is t ergdi the a time average must be used

4 Ergdi Radm Presses - Autrrelati ime average R X (τ =lim X (t X (t+τdt Esemble average R X (τ =E [ X X ] = x x f X X ( x x dx dx

5 Optimum Reeiver Review f sigal rrelati fr a QAM X (dt Sampler g(t s(π f g (t si (π f t+ϕ t+ϕ Cmpute Distae Metri X (dt Sampler

6 Optimum Reeiver Review f sigal rrelati fr a QAM r (t= A g (ts(π f mi t+ϕ+ A g (tsi (π f mq t+ϕ (t= (ts(π f t + (tsi(π f t r r = A + s(ϕ A si(ϕ mi mq s = A + si(ϕ+ A s(ϕ mq mq s = (t g (tdt= / s= (t g(tdt= s /

7 Ctiuus Phase Mdulati (CPM s(t = E s[ π f t+ϕ(t ; I+ϕ ϕ(t ; I =π k= Where {I k } is I k h k q(t k ±,±3,...,±(M t (+, {h k } is a sequee f mdulati symbls ad q(t is sme rmalized wavefrm shape. t q(t= g(τd τ If g(t = fr t> the sigal is alled full respse CPM If g(t fr t> the sigal is alled partial respse CPM ] LREC g(t = L t L therwise LRC g (t = ( s( π t/ L L t L therwise GMSK Q(π B (t / Q(π B (t+ / g(t = l

8 Sigalig Shemes with Memry Maximum Likelihd Sequee Detetr (MLSD K s r (t s(t Optimal Deteti Rule (ŝ (,ŝ (,..., ŝ K dt= k= ( K = k s (k (s ( s r(t s(t,s ( argmi,..., s dt (K Υ K k= rellis r (k s (k dt (ŝ (,ŝ (,..., ŝ ( K = (s (,s ( argmi,..., s K (K Υ k= D(r (k,s (k

9 rellis fr RZI Sigal Assume we start with state s. / (E b / (E b / (E b s s / (E b / ( E b / (E b / ( E b / (E b / (E b / (E b t = t = t = 3 D D (,= ( r + E b + ( r + E b (,= ( r E b + ( r E b D (,= ( r + E b + ( r E b D (,= ( r E b + ( r + E b here are tw paths t arrive at state s at t = We a alulate the Eulidea distae usig the utput f the demdulatr ad disard the larger distae fr state s here are als tw paths t arrive at state s at t = We a alulate the Eulidea distae usig the utput f the demdulatr ad disard the larger distae fr state s his meas we ly eed t keep tw paths

10 rellis fr RZI Sigal Assume we start with state s. / (E b / (E b / (E b s s / (E b / ( E b / (E b / ( E b / (E b / (E b / (E b t = t = t = 3 D D (,,= D (,+ ( r + E 3 b (,,= D (,+ ( r + E 3 b D (,,= D (,+ ( r E 3 b D (,,= D (,+ ( r E 3 b w sider t = 3

11 ϕ(t ; I = π h k = Optimum Reeiver fr CPM s(t = E s[ π f t+ϕ(t ; I+ϕ I k L q(t k =π h k= q(t= fr t< ad t L t q(t= g(τd τ g (t = fr t< ad t L I k ] + π h k= L+ I k q(t k=θ +θ(t ; I t (+ If h is ratiale, h = m/p where m ad p are relative prime psitive itegers, the CPM sheme a be represeted by a trellis Θ Θ =(, π m s p, π m p =(, π m s p, π m p p π m,...,( p m,...,(p π p m is eve m is dd

12 Optimum Reeiver fr CPM If L > there are additi states due t the partial respse harater θ(t ; I= π h k= L+ I k q(t k + π h I q(t t (+ he mbie phase state ad rrelative state is S = ( θ, I, I,..., I L+ Where L > the umber f states is s = pm L pm L eve m dd m Whe t = (+ the state bemes S += ( θ, I +, I,..., I L+ Where θ =θ +π h I + L+

13 CPM Demdulatr x LPF x (+ (dt s[θ(t ; I +θ ] r(t s( π f si( π f t t Phase Geeratr θ(t;i + θ + Viterbi Deder x LPF x si [θ(t ; I +θ (+ (dt ]

14 Optimum Reeiver fr CPM Metri Calulatis CM (+ (I = =CM r(ts[ π f ( (+ (I + t+ϕ(t ; I]dt r(ts[ π f t+θ(t ; I +θ ]dt Ctributi t the metri frm the sigal betwee t (+ here are M L pssible symbls I = ( I, I, I,..., I L+ here are s = pm L dd m pm L dd m here are s = pm L pm L dd m dd m Survivig sequees

15 rellis fr CPM M =, L =, h = ½, g(t = REC States - States, -3π/4, -π/4, π/4, 3π/4, te: Frm symbl perids away there are **4 = 6 pssible paths t reah the 4 states. Frm symbl perid away there are ** = 8 pssible paths t reah the 4 states

16 Detetig CPM M =, L =, h = ½, g(t = REC Largest metri I = {- - -}

17 distae Distaes d i, j = [ s (t s j E lg M b (t ] dt= j [ s(ϕ(t ; I i ϕ(t ; I ]dt j H = ½ L = MSK Samples

18 -Cheret Deteti f CPM What if we uld rever the phase ad differetiate? MSK diff(phi

19 Sigal Parameter Estimati Start with a mathematial mdel fr a reeived sigal r (t=s(t τ+(t where s(t =R ( s l (t e j π f t ad τ is the prpagati delay r (t=r ( [s l (t τe j ϕ + z(t]e j π f t where ϕ= π f τ r (t=s(t ;ϕ, τ+(t ad we must estimate τ ad φ

20 Sigal Parameter Estimati simplify the tati all f the radm variable we wat t estimate a be represeted by a vetr θ. r (t=s(t ;θ+(t where θ=[ϕ, τ] here are tw methds fr estimatig parameters ML ad MAP Usig the MAP riteri the sigal parameter vetr, θ, is mdeled as a radm variable ad haraterized by a a priri prbability desity futi p(θ. Usig the ML riteri the sigal parameter vetr, θ, is mdeled as determiisti, but ukw

21 ML Estimati We will assume ML fr the remaider f the effrt. p (r θ=( π σ ( = e [ r s ] σ r = r(t ϕ (t dt s (θ= s(t ;θϕ (tdt r (t= r s(t ;θ= ϕ s (t (θϕ (t lim [r (t s (t ;θ] = σ [r (t s(t ;θ] dt Λ(θ=e [r (t s (t ; θ] dt his is equivalet t maximizig p(r θ

22 Carrier Phase Estimati What if yu d t aurately estimate the phase? s(t = A(ts( π f t+ϕ B (tsi( π f t+ϕ I (t=s( π f t+ ϕ (t= si ( π f Q t+ ϕ y I (t= A(ts(ϕ ϕ B(tsi (ϕ ϕ y (t= Q B(ts(ϕ ϕ+ A(tsi (ϕ ϕ tie that t ly is amplitude impated by the phase iauray, but there is rss talk

23 ML Carrier Phase Estimati Λ(θ=e [r (t s (t ; θ] dt ( =exp r (tdt+ r(t s(t ; ϕdt s (t ;ϕdt Λ(θ=C exp( r(ts(t ;ϕdt Λ (θ= L r (t s(t ;ϕdt I rder t maximize the futi abve Csider r (t= A(ts( π f r(t si ( π f t+ ϕ t+ϕ dt= ML d Λ (ϕ L = d ϕ r(t si( π f X t+ ϕ ML (dt VCO

ELEG 635 Digital Communication Theory. Lecture 11

ELEG 635 Digital Communication Theory. Lecture 11 EEG 635 Digital Cmmuiati Thery eture 11 110801 Ageda Carrier Syhrizati Phase k p (P) Deisi Direted ps N-Deisi Direted ps Timig Syhrizati Deisi Direted ps N-Deisi Direted ps frmati Thery Sha's aw iear Blk