1.0 Probability of Error for non-coherent BFSK

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1 Probability of Error, Digital Sigalig o a Fadig Chael Ad Equalizatio Schemes for ISI Wireless Commuicatios echologies Sprig 5 Lectures & R Departmet of Electrical Egieerig, Rutgers Uiversity, Piscataway, NJ Istructor: Dr. Naraya Madayam Summary by Rady Mitch Abstract his documet discusses the probability of error for o-coheret FSK ad DPSK. Digital sigalig o a frequecy selective chael will be reviewed as well as equalizatio schemes to elimiate ISI.. Probability of Error for o-coheret BFSK Recall from the last lecture that we have Biary FSK orthogoal modulatio. Whe a is selected the S (t) is trasmitted ad whe a is selected the S (t)is trasmitted. S (t) ad S (t) are orthogoal. he received sigal is g (t) if S (t) was trasmitted ad g (t) if S (t) is trasmitted. he received sigals g (t) ad g (t) are also orthogoal. he receiver structure is show i figure. X(t) Filter Matched to S(t) Evelope Detector t = l Compare l > l Filter Matched to S(t) Evelope Detector t = l l > l Figure. No-coheret Receiver for BFSK Let S (t) be trasmitted, the a error occurs if l > l. l = x + x I Q Where X I ad X Q are both Gaussia with zero mea ad psd = N /. l l exp, l fl ( l ) = N N, otherwise

2 l dl { } { λ } ( ) Pr obability of error P = Pr l > l = Pr l > l f l e l Pr{ l > λ l} = fl ( l ) dl = exp N l N where l = xi + x Q, x, I N E ad N xq N, ( xi E ) fx ( x ) exp I I = π N N x Q fx ( x ) exp Q Q = π N N ( error ) Pr obability of error P p x x f dx d x = I Q e I Q xi x Q ( xi E) xq + x I xq Pe = exp exp exp d xi dx Q N N N E xq + x I + x I E + x Q = x I E + x Q + E x Q Pe = exp exp ( x ) exp I E dx I dx π N N N N Rewritig ( ) ( ) Q E Pe = exp N. Probability of error for o-coheret M-ary FSK I geeral for M-ary FSK with o-coheret detectio: ( i ) M i Eb Pe = ( ) exp ( m ) i= in Where = log M A o-coheret receiver for M-ary FSK is show i Figure. As with the biary FSK receiver a compariso of the output of the evelope detectors selects the appropriate brach. Figure 3 illustrates the probability of bit error for various values of M. As M icreases the badwidth efficiecy decreases but the power efficiecy icreases. More costellatios require more chus of orthogoal spaces.

3 Filter Matched to S(t) Evelope Detector t = X(t) Filter Matched to S(t) Evelope Detector t = Compare Filter Matched to Sm(t) Evelope Detector t = Figure. No-coheret Receiver for M-ary FSK 3

4 Figure 3. No-coheret M-ary FSK BER 3. Differetial Phase Shift Keyig If iformatio is eyed oto the phase, PSK, it caot be detected by o-coheret methods. Use phase differece betwee two cosecutive waveforms to carry the iformatio. Assumptio: It wors provided the uow phase itroduced by the chael varies slowly (i.e. slow eough to be cosidered costat over bit itervals). Cosider iput sequece {m }. Geerate differetially ecoded sequece {d } from {m } as follows:. Sum d - ad m Modulo.. Set d to be the complemet of the result from above. 3. Use d to phase shift a carrier as follows: d = θ = d = θ = π Example: if { m } { d } { d} { θ} ππ Observe: Symbol is uchaged from previous symbol if icomig symbol is. d Symbol is toggled from previous symbol if icomig symbol is. d 4. 4 π - DQPSK Exploit the above observatio to derive the probability of error. Let the DPSK sigal i Eb t b be cos( π fct). If the ext bit (i the iterval b t b ) is, the the b phase is uchaged. If the ext bit is, the the phase is shifted by π. 4

5 S S () t () t It is possible to cosider E b cos( π ft c ), t b b = Eb cos( π ft c ), b t b b E b cos( π ft c ), t b b = Eb cos( π ft c π), b t + b b () modulatio over a bit iterval, t b. S t ad S ( t ) as similar to o-coheret orthogoal P e E = exp N Figure 4 compares the BER for DPSK ad FSK. Observe that DPSK BER is approximately 3 db better tha o-coheret FSK. Figure 4 BER Comparisos of DPSK ad FSK 5

6 5. Digital Sigalig o Frequecy Selective Fadig Chaels Modelig: Recall ay modulated sigal is give as vt () = A b( t, x ) ourselves to liear modulatio. ( ) ( ) { } btx, = xh t x complex symbol sequece a. We restrict amplitude samplig pulse he above sigal is trasmitted through a chael c(t) that results i the received sigal w(t). = ( ) wt () = xh t + zt () ht () = ha ( τ ) ct ( τ) dτ. zero-mea AWGN For causal chaels ht ( ) = ha ( τ) ct ( τ) dτ, t. We also assume h(t) = for t ad h(t) = for t L. L is some real positive iteger. x δ ( t ) h () t c( t) a u ( t) h * () t y ( ) ht () z ( t ) t= Figure 5. Matched Filter Receiver i AWGN Chael If we ow h(t), we ca implemet a matched filter as above. yt () = x f( t ) + vt () = Filter oise 6

7 * f () t = h h( + t) d ( ) τ τ τ. f() t is the overall pulse respose ad it accouts for trasiet filter, chael ad receive filters. y = y( t) = x f ( ) + v( ) = = x f + v = x f + x f + v = called itersymbol iterferece(isi) ISI is caused by the chael so if we ca elimiate it the we ca treat the chael as AWGN chael. herefore to achieve the same performace as o a AWGN chael, we require x f =. I order to do this we must mae f =. Alterately, = if i j f = δ f where δij =. if i= j Equally i the frequecy domai F f + = f, F( f) = F{ () } = f t It ca be show that f(t) ca be ay fuctio that has equally spaced zero crossigs. What is the optimum receiver? We must express wt () = lim wφ() t usig the Karhue-Loeve expasio. ( h = h t Φ t dt * ( ) ( ) z = ztφ tdt * () () w= w, w,... w ) is a multivariate Gaussia distributio because z(t) is Gaussia. 7

8 N pwxh (, ) = exp w xh = π N N = where H = h, h, h3,..., h ad h = (..., h 3, h, h, h) herefore the optimum receiver is the Maximum Lielihood receiver, for the case of AWGN it reduces to N arg max ux ( ) = w xh x = = he bottom lie is:. We eed owledge of { f } { c } i order to perfectly elimiate ISI. herefore we eed to estimate the chael i order to equalize it. to uderstad the chael respose ad owledge of. A additioal problem results as well by observig the followig: yt () = x f( t ) + vt () = where the oise fuctio is * vt () = h( ) zt ( ) d τ τ τ is still Gaussia but is ot white. herefore the oise samples at the output are correlated. I this case we ca obtai a discrete time white oise model as follows. x δ ( t ) h () t c( t) a h * ( t) Whiteig Filter v ht () z ( t ) t= Figure 6. Whiteig Filter with Matched Filter Receiver he output of the whiteig filter v L = gx + η = v is give by 8

9 where g is the overall filter for the chael ad the whiteig filter ad η is the white oise process. 3. Aother importat cosideratio is the desig of ISI equalizig filters is extremely sesitive to timig iformatio. We ca solve this sesitivity i two ways. a. Use pulse shapig. Raised cosie pulse shapig allows us to derive the legth of the pulse by samplig at eve poits. b. Use fractioal samplig. Sample the output y(t) at a rate higher tha /. Although we will still have correlated oise ad will eed a whiteig filter the overall pulse shape will be less sesitive to timig errors. 6. Equalizatio Schemes here are two types of equalizatio schemes.. Symbol by symbol equalizers that ca be liear or o-liear.. Sequece estimatio equalizers that are o-liear. 6. Symbol-by-Symbol Equalizers We cosider the discrete time white oise model show i figure 7. Where { a } is the iput sigal, L is the memory of the chael, η is the additive oise ad g = ( g, g..., g ) l is the chael vector that describes the overall chael impulse, respose. { } a... X g X g X g L L r = g a + η = Figure 7. Discrete ime White Noise Model η 9

10 Liear equalizers estimate the th trasmitted symbol aˆ M = cr j j Where { } M j is the equalizer filter of (M+) taps. M is a j= M c + M desig choice ad C is the chose umber of taps to estimate aˆ. 6. Zero Forcig Equalizer he Zero Forcig Equalizer combies the chael ad equalizer impulse respose to force to zero at all but oe of the taps i the DL filter. he tap coefficiets, c, are chose to zero-out the ISI. Give the chael vector, g, we ca select the tap coefficiets, c, to get the desired respose q = (,,,..., q,,...). he Zero Forcig Equalizer zeros out all but the desired result. he tap coefficiets are calculated usig the relatioship q = c g q aˆ. Zero forcig equalizers ca perfectly elimiate ISI as M but it also ehaces the oise. I practice, the receiver does ot ow the chael vector ad fiite legth traiig sequeces are used to choose the tap coefficiets. 6.3 Miimum Mea Square Equalizer (MMSE) he MMSE is aother symbol-by-symbol equalizer but is superior to the zero forcig equalizer i performace. he MSME utilizes the mea square error criterio to adjust the tap coefficiets. We defie a estimatio error: ε = a aˆ as Where a is the symbol set ad aˆ is the estimated symbol. he fuctio to be miimized is: J = mi E ε = mi E ( a a ˆ) c c M = mi E ( a cjr j) c j= M so as to mae the error sequece orthogoal to he error is miimized by choosig { c j } the sigal sequece r, for l M l E ε =, l M. he optimum c is, i.e. [ r ] l obtaied by solvig the followig set of equatios. M ce j r jr l = E[ ar l], l M j= M

11 Defie the followig fuctios. Γ r r E j l, j l, = M,..., M ad b= ( E[ a ] [ ]... r+ m E ar+ m E[ ar m] ) the copt =Γ b o implemet the MSME we eed to us adaptive algorithms because the chael respose is ot readily ow. ypically, we use the steepest descet algorithm. cj( + ) = cj( ) µ E ε ( ) c j, j = M... + M Where µ is a positive umber. Note: E ε ( ) ε ( ) = E ε( ) = E ε( ) r j = er( ) cj c R j j he term R er is the cross correlatio betwee the iput ad the error. he equalizer taps ca be obtaied by implemetig a stochastic gradiet algorithm. cj( + ) = cj( ) µε ( ) r j o evaluate ε ( ) i this algorithm we use a traiig sequece. 6.4 Decisio Feedbac Equalizer (DFE) A DFE is a oliear equalizer that cosists of a feed forward sectio ad a feedbac sectio. DFEs are very effective i frequecy selective chaels because they mitigate the effects of oise ehacemets that degrade the performace of liear equalizers. he DFE uses previous decisios to elimiate ISI caused by previously detected symbols o curret symbols. he output of the equalizer ca be expressed as: ˆ j j a = c r + ca% j= M feed forwad filter M + taps M j= j j feedbac filter M taps Where a%, a%,..., a% M are earlier detected symbols. Both the feed forward loop ad the feedbac loop eed adaptatio to adjust the coefficiets.

12 6.5 Maximum Lielihood Sequece Estimatio (MLSE) Recall the discrete-time white oise chael model. L r = g a + η m m m= Assume symbols are trasmitted over the chael. he after receivig the sequece { r }, the ML receiver decides i favor of the sequece { } = a = lielihood fuctio ( r r, r a, a a ) log,...,,..., that maximizes the Sice the oise samples are idepedet ad trasmitted symbols. r depeds oly o the L most recet ( r r, r a, a a ) ( r a, a a l) ( r r, r a, a a) log,...,,..., = log,., + log,..,,.., his has already bee calculated where a L= for L Sice the d term has bee calculated at the previous time (-), oly the first term eeds to be computed at time for each icomig sigal r. he Veterbi algorithm ca be used to implemet the MLSE equalizer. Adaptive MLSE is used i GSM. Refereces: [] N. Madayam, Wireless Commuicatio echologies, course otes. []. Rappaport, Wireless Commuicatios. Priciples ad Practice. d Editio, Pretice-Hall, Eglewood Cliffs, NJ: 996. [3] J. Proais, Digital Commuicatios, 3 rd Editio, McGraw-Hill, NY: 995. [4] G. Stuber, Priciples of Mobile Commuicatio, d Editio, Kluwer Academic Publishers, Norwell, MA:. [5] B. Slar, Digital Commuicatios. Fudametals ad Applicatios, Pretice-Hall, Eglewood Cliffs, NJ: 988. [6] D. Wu, Wireless Commuicatio echologies, Rutgers Uiversity, Piscataway, NJ:

Lecture 1: Channel Equalization 1 Advanced Digital Communications (EQ2410) 1. Overview. Ming Xiao CommTh/EES/KTH

Lecture 1: Channel Equalization 1 Advanced Digital Communications (EQ2410) 1. Overview. Ming Xiao CommTh/EES/KTH : 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