(96) J ohn G. Proakis, Digital Communications, 4rd edition, McGraw-Hill International Editions, 2001. 831 (03) 5731670 E-mail: qponing@mail.nctu.edu.tw Office Hours : GCD Office Hours : 10:00am~12:00pm 40% 40% 20% (229)(222) http://shannon.cm.nctu.edu.tw ext. 54570 ED823 shin30@gmail.com Office Hours: Monday 6:30pm~8:30pm (ED823) ext. 54570 ED823 bbnctu@hotmail.com Office Hours: Monday 6:30pm~8:30pm (ED823) ext. 54570 ED823 yaao27757@hotmail.com Office Hours: Monday 6:30pm~8:30pm (ED823) Chapter 1 1.1 Elements of a digital communication system1.2 Communication channels and their characteristics 1.3 Mathematical models for communication channels 1.4 A historical respective in the development of digital communications 1.5 Overview of the Book Chapter 4 4.1 Representation of bandpass signals and systems : Fourier Transform 2.2 Stochastic processes (2.2.1, 2.2.2 & 2.2.4) 4.2 Signal space representation 4.3 Representation of digitally modulated signals 4.4 Spectral characteristics of digitally modulated signals 2.2.6 Cyclostationary processes 4.5 Bibliographical notes and references Chapter 5 5.1 Optimal receiver for signals corrupted by additive white Gaussian noise 5.2 Performance of the optimumreceiver for memoryless modulation : White Noise & 7.1.2 Channel capacity & 7.1.3 Achieving Channel capacity with orthogonal signals Chapter 13 13.1 Model of spread spectrumdigital communications system13.2 Direct sequence spread spectrumsignals (exclude 13.2.4 Excision of narrow band inference in DS spread spectrumsystems) Chapter 14 14.1: Characterization of fading multipath channels 2.1.4 Some useful probability distributions (Rayleigh, Rice and Nakagami m-distributions) 14.2 The effect of signal characteristics on the choice of a channel model 14.3 Frequency-nonselective, slowly fading channel 14.4 Diversity techniques for fading multipath channels 14.5 Digital signaling over a frequency-selective, slowly fading channel (exclude 14.5.4 Receiver structures for channels with intersymbol interference) Chapter 6 6.1 Signal parameter estimation 6.2 Carrier phase estimation (exclude 6.2.5: Non-decision-directed loops, such as Costas loop) 6.3 Symbol timing estimation (exclude the early-late gate synchronizers in subsection 6.3.2) 6.4 J oint estimation of carrier phase and symbol timing 6.5 Performance characteristics of ML estimators Chapter 15 15.1 Introduction to multiple access techniques 15.2 Capacity of multiple access methods 15.3 Code-division multiple access (exclude 15.3.4 Successive interference cancellation) --- Po-Ning Chen --- NCTU/CM1 1- -1 1Digital CommunicationsChapter 1 : Introduction1-2Digital communications What we study in this course is:Theories on transmission of information in digital formfromone or more sourcesto one or more destinations.--- Po-Ning Chen --- NCTU/CM1-31.1 Elements of a digital communication system Functional diagramof a digital communication systemChannelInformationsource andinput transducer Sourceencoder Channelencoder DigitalmodulatorDigitaldemodulatorChanneldecoderSourcedecoderOutputtransducerOutputsignalBasic elements of a digital communication system1-41.2 Communication channels and their characteristics Physical channel mediamagnetic-electrical signaled wire channelmodulated light beamoptical (fiber) channelantenna radiated wireless channelacoustical signaled water channel Virtual channelmagnetic storage media--- Po-Ning Chen --- NCTU/CM1-51.2 Communication channels and their characteristics Noise characteristicthermal noise (additive noise)signal attenuationamplitude and phase distortionmulti-path distortion Limitation of channel usagetransmitter powerreceiver sensitivitychannel capacity (such as bandwidth)1-61.3 Mathematical models for communication channels Additive noise channel (with attenuation)where is the attenuation factor, s(t) is the transmitted signal, and n(t) is the additive randomnoise process. Additive Gaussian noise channel If n(t) is a Gaussian noise process.) ( ) ( ) ( t n t s t r + = n(t)Channels(t) r(t)+--- Po-Ning Chen --- NCTU/CM1-71.3 Mathematical models for communication channels The linear filter channel with additive noiseto ensure the specified bandwidth limitations.Channeln(t)s(t)r(t) + + =+ =) ( ) ( ) () ( ) ( * ) ( ) (t n d t s ct n t c t s t rLineartime-invariant filterc(t)1-8 + =+ =) ( ) ( ) ; () ( ) ; ( * ) ( ) (t n d t s t ct n t c t s t r1.3 Mathematical models for communication channels The linear time-variant filter channel with additive noiseLineartime-variantfilter c( ;t)Channeln(t)s(t)r(t)+c( ;t) : is the argument for filtering;t is the argument for time-dependence.(The time-invariant filter can be viewed as a special case of the time-variant filter. Cf. the next slide.)--- Po-Ning Chen --- NCTU/CM1-91.3 Mathematical models for communication channels = 0 1 2 3c()s(1) = 0 1 2 3c()s(2) = 0 1 2 3c()s(3)r(1)r(2)r(3) = 0 1 2 3s(1) = 0 1 2 3s(2) = 0 1 2 3c(;1)c(;2)c(;3)s(3) Assume that n(t) = 0.1-101.3 Mathematical models for communication channels The linear time-variant filter channel with additive noisec( ;t) usually has the formwhere {ak(t)}represents the possibly time-variant attenuation factor for the L multipath propagation paths, and {k}are the corresponding time delays.Hence,=+ =Lkk kt n t s t a t r1 ) ( ) ( ) ( ) ( = =Lkk kt a t c1 ) ( ) ( ) ; ( --- Po-Ning Chen --- NCTU/CM1-111.3 Mathematical models for communication channels The linear time-variant filter channel with additive noiseTransmitter Receiver( )1 1 ), ( t a( )2 2 ), ( t a( )3 3 ), ( t a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3 32 21 1t nt s t at s t at s t a t r+ + + =1-121.4 A historical respective in the development of digital communications 1830s : Morse codevariable-length code 1875 : Baudot codefixed-length binary code with length 5 1924 : NyquistDetermine the maximumsignaling rate without intersymbol interference over a, e.g., telegraph channel--- Po-Ning Chen --- NCTU/CM1-13 Nyquistdefine basic pulse shape g(t) which is bandlimited to W) (t gone wants to transmit {1,1}signals in terms of g(t),or equivalently, one wants to transmit a0, a1, a2, in {1,1}in terms of s(t) defined asL + + + = ) 2 ( ) ( ) ( ) ( 2 1 0T t g a T t g a t g a t s1.4 A historical respective in the development of digital communications1-141.4 A historical respective in the development of digital communications) (0t g a) (1T t g a ) 2 (2T t g a ) , 1 , 1 , 1 ( ) , , , ( Example. 2 1 0 K K + + = a a aL + + + =) 2 ( ) ( ) ( ) (2 10T t g a T t g at g a t sNo interference--- Po-Ning Chen --- NCTU/CM1-15 Question that Nyquist shoots for:What is the maximumrate that the data can be transmitted under the constraint that g(t) causes no intersymbol interference? Answer : 2W pulses/second.What is one of the g(t)sthat achieves this rate ?Answer : WtWtt g=2 ) 2 sin() (1.4 A historical respective in the development of digital communicationsConclusion : A bandlimited-to-W basic pulse shape signal (or symbol) can convey at most 2W pulses/second (or symbols/second) without introducing inter-pulse (or inter-symbol) interference.1-161.4 A historical respective in the development of digital communications 1948 : Shannonsampling theorem A signal of bandwidth W can be reconstructed fromsamples taken at the Nyquist rate (= 2W samples/second) using the interpolation formula( ) =nW n t WW n t WW n s t s) 2 / ( 2 )] 2 / ( 2 sin[2 / ) (channel capacity of additive white Gaussian noisewhere W is the bandwidth of the bandlimited channel, P is the average transmitted power, and N0is single-sided noise power per hertz.+ =02 1 logWNPW C bits/second--- Po-Ning Chen --- NCTU/CM1-171.4 A historical respective in the development of digital communicationsConclusion of Shannons channel coding theorem Let R be the information rate of the source. Then if R < C, it is theoretically possible to achieve reliable (asymptotically error-free) transmission by appropriate coding; if R > C, reliable transmission is impossible. This gives birth to the new field named I nformation Theory.1-181.4 A historical respective in the development of digital communications Other important contributionsHarvey (1928), based on Nyquists result, conclude that A maximumreliably transmitted data rate exists for a bandlimited channel under maximumtransmitted signal amplitude constraint and minimumtransmitted signal amplitude resolution constraint.Kolmogorov (1939) and Wiener (1942) optimumlinear (Kolmogorov-Wiener) filter whose output is the best mean-square approximation to the desired signal s(t) in presence of additive noise.Hamming (1950) Hamming codes--- Po-Ning Chen --- NCTU/CM1-191.4 A historical respective in the development of digital communications Other important contributionsMuller (1954), Reed(1954), Reed and Solomon (1960), Bose and Ray-Chaudhuri (1960), and Goppa (1970,1971) new block codes, such as Reed-Solomon codes, Bose-Chaudhuri-Hocquenghem(BCH) codes and Goppa codes.Forney (1966) concatenated codesChien (1964), Berlekamp (1968) Berlekamp-Massey BCH-code decoding algorithm1-201.4 A historical respective in the development of digital communications Other important contributionsWozencraft and Reiffen (1961), Fano (1963), Zigangirov (1966), J elinek (1969), Forney (1970,1972, 1974) and Viterbi (1967, 1971) convolusional code and its decodingUngerboeck (1982), Forney et al. (1984), Wei (1987) trellis-coded modulationZiv and Lempel (1977, 1978) and Linde et al (1980) source encoding and decoding algorithms, such as Lempel-Ziv codeBerrou et al (1993) turbo codes and iterative decoding--- Po-Ning Chen --- NCTU/CM1-211.5 Overview of the book We will not coverChapter 3 Source CodingChapter 7 Channel Capacity and Coding (mostly)Chapter 8 Block and Convolutional Channel CodesChapter 9 Signal Design for Band-Limited Channels Chapter 10 Communication Through Band-Limited Linear Filter Channels (Self-Study)Chapter 11 Adaptive EqualizationChapter 12 Multichannel and Multicarrier SystemsChapter 15 Multiuser Communications1-221.5 Overview of the book We will coverChapter 1 I ntroduction Quick and briefChapter 4 Characterization of Communication Signals and Systems I n great detailsChapter 5 Optimum Receivers for the Additive White Gaussian Noise Channel Only optimal receivers for memoryless modulations, including matched filter and the Viterbi demodulator, are covered.Chapter 6 Carrier and Symbol Synchronization Almost all--- Po-Ning Chen --- NCTU/CM1-231.5 Overview of the book We will coverChapter 13 Spread Spectrum Signals for Digital Communications Only direct sequence spreading spectrum is covered; further performance enhancement of DSSS system suffering narrowband interference in subsection 13.2.4 is excluded.Chapter 14 Digital Communications through Fading Multipath Channels 14.6 Coded waveforms for fading channelsis not covered, since it is related to channel coding that we did not cover in this lecture.)(14.7 Multiple-antenna systemsis excluded.1-241.5 Overview of the book We will coverChapter 15Multiuser Communications (if time permitted) 15.4 Random access methodsis not covered, since it is the main subject in courses regarding networking. --- Po-Ning Chen --- NCTU/CM1-251.5 Overview of the book We will coverSome selective content fromChapter 2 Probability and Stochastic Processes will be introduced whenever they are necessary.1-26What you learn from Chapter 1 Mathematical models of time-variant and time-invariant additive noise channelsmultipath channels Nyquist rates and Sampling theorem--- Po-Ning Chen --- NCTU/CM4-1Digital CommunicationsChapter 4 : Characterization of Communication Signals and Systems4-24.1 Representation of bandpass signals and systems Carrier modulation : carrier = Accos(ct+c)signal. baseband the is ) ( wheremodulation Phase : ) ( modulation Frequency : ) ( modulation Amplitude : ) ( ct mt mt mt m Acc The transmitted signal (after carrier modulation) is usually a real-valued bandpass signal. Mathematical model of a real-valued narrowband bandpass signal.| | ) (B cB c B cf ff f f f f f S>> + and for 0--- Po-Ning Chen --- NCTU/CM4-3cfcf 4.1.1 Representation of bandpass signals) ( ) ( ) (21f S f u f S =+) (21f SSymmetric, if s(t) is real-valued.) ( f S For analytical convenience, the real-valued transmitted bandpass signal is usually analyzed in terms of its complex-valued equivalent lowpass signal. Hence, we need to realize the relation between them. * u(f) gives 0, 0.5 or 1 depending on whether f is negative, zero, or positive.4-44.1.1 Representation of bandpass signals| | + + = = + =+ =((

+ = ===dtst stt s t s j t st stj t st stjtf S F f u Fdf e f S f udf e f S t sft jft j) ( 1) (1) ( ), ( ) () (1) () ( ) ()] ( [ )] ( 2 [) ( ) ( 2) ( ) (1 122 where Goal : to develop a mathematical representation (in time domain) of S+(f) and S(f) Analytic signal (or pre-envelope) of a bandpass input signalTransform Fourier Inverse ] [1F--- Po-Ning Chen --- NCTU/CM4-5On Fourier transform Inverse Fourier transform] 1 [ )] [sgn( )] ( 2 [1 1 1 + = F f F f u F)]. ( [ lim )] ( [ | ) ( | ) ( and ) ( lim ) ( i.e., sense, the in ansform Fourier trinverse its derive therefore We sense. the in exist not does) sgn( of ransform Fourier t inverse the , | ) sgn( | Since1 1nf S F f S Fdf f S n f S f Sf df fnnnn = < = =extendedstandard* sgn(f) gives 1, 0 or 1 depending on whether f is negative, zero, or positive.4-6|||.|

\|===((

+ =((

+ + =((

+ = + + .0 ,0 044lim2121limlim ) sgn( lim). sgn( ) sgn(2 2 2000) 2 (0) 2 (02 | |00 | |ttjt ,tjt at jt j a t j adf e df e df e f ef f eaat j a f t j a faft j f aaa f aOn Fourier transform|||.|

\|== + = .0 ,0 ), () ( )] ( 2 [1ttjt tttjf u F--- Po-Ning Chen --- NCTU/CM4-7 + = = + = dtst stt s t s j t s t s) ( 1) (1) ( ), ( ) ( ) ( wherett h1) ( =) (t s ) ( t sHilbert transform,0 ,0 , 00 ,) sgn( ] / 1 [ ) ( = = =f jff jf j t F f H 4.1.1 Representation of bandpass signalsHilbert transform is basically a 90-degree phase shifter. Analytic signal (pre-envelope) of a bandpass input signal< = > = =; 0 for 2 / ) ( ; 0 for 2 / ) (0 for 1 | ) ( |f f f ff f H 4-8On Fourier transform Fourier transform of a Hilbert Filter). sgn( ) sgn( ) / 1 ( ] / 1 [ ) sgn( ] / [ f j f j t F f t j F = = = = = analysis. - real by , undefinedsymmetry; by , 01) 0 ( dttHfunction. indicator set } { where} 0 ) ( { | ) ( | } 0 ) ( { | ) ( | ) ( analysis, real In < = 11 1 dx x f x f dx x f x f dx x f--- Po-Ning Chen --- NCTU/CM4-9On Fourier transform Some useful Fourier transforms). sgn( ] / 1 [ ) (). /( )] [sgn( ) (] [ ) (1 42] [ ) (1 | | |, | 11 | | , 0) ( )] ( sinc [ ) (5 . 0 | | , 15 . 0 | | , 0) ( )] ( sinc [ ) (1 )] ( [ ) (2 22 2| |2f j t F oddf j t F odde e F evenfe F evenf fff t F evenfff t F event F evenf tt = == += < = = < > < > < == > < == 4.2.3 Orthogonal expansions of signals--- Po-Ning Chen --- NCTU/CM4-634.2.3 Orthogonal expansions of signals-1010.5 1 1.5 2 2.5 3 3.5-1010.5 1 1.5 2 2.5 3 3.5-1010.5 1 1.5 2 2.5 3 3.5-1010.5 1 1.5 2 2.5 3 3.5) (1t v) (4t v) (2t v) (3t v Example 4.2-2 : Find a Gram-Schmidt orthonormal basis of the following signals.4-644.2.3 Orthogonal expansions of signals) (|| ) ( ||) () () (otherwise , 0) 3 2 ( , 1) ( ) ( ) ( 2 ) ( 0 ) () (2) () ( ) (2) () ( ) () ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ) (2) (|| ) ( ||) () ( ), ( ) ( 0 ) () ( ) ( ) ( ) () ( ) ( ), ( ) ( ) ( ) (2) (|| ) ( ||) () ( ) ('3 '3'331 3 1 2 313013 23023 31 1 3 2 2 3 3'32'2'22 2 1 21301 2 21 1 2 2'21111t ut ut ut utt v t v t u t u t vt u dtt vt v t u dtt vt v t vt u t u t v t u t u t v t v t u iiit vt ut ut u t v t u t vt u dt t u t v t vt u t u t v t v t u iit vt vt vt u i= = < = = =|.|

\||.|

\| => < > < == = = =|.|

\| => < == = --- Po-Ning Chen --- NCTU/CM4-654.2.3 Orthogonal expansions of signals( ) ( ) ( ). 0 ) ( 2 ) ( ) () ( 2 ) ( ) () ( 2 ) ( 0 ) ( 1 ) () ( ) ( ) ( ) ( ) ( ) () ( )] ( ) ( )[ ( ) () ( ) ( ), () ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ) (1 3 41 3 41 2 3 41301 4 2302 43301 3 4 41 1 42 2 4 3 3 4 4'4= + = + = =|.|

\| |.|

\||.|

\| => < > < > < = t v t v t vt u t u t vt u t u t u t vt u dt t u t v t u dt t u t vt u dt t v t v t v t vt u t u t vt u t u t v t u t u t v t v t u iv( ). ) ( ), ( ), ( basis l orthonorma ) (3 2 1t u t u t u v =4-664.2.3 Orthogonal expansions of signals Example 4.2.3: Representing the signals in Example 4.2-2 in terms of the orthonormal basis obtained in the same example. ) 1 , 0 , 2 ( ) () 1 , 0 , 2 () ) ( ) ( , ) ( ) ( , ) ( ) ( ( ) () 0 , 2 , 0 ( ) () 0 , 0 , 2 ( ) ( 0 ) ( 0 ) ( 2 ) (4303 3302 3301 3 323 2 1 1 ==== = + + = t vdt t u t v dt t u t v dt t u t v t vt vt u t u t u t v--- Po-Ning Chen --- NCTU/CM4-674.2.3 Orthogonal expansions of signals The orthonormal basis is not unique.) 1 , 1 , 1 ( ) () 1 , 1 , 1 ( ) () 0 , 1 , 1 ( ) () 0 , 1 , 1 ( ) (. otherwise , 01 , 1) ( ) 3 1 (for4321 = + + = + = + + = < = t vt vt vt vi t it u ii4-684.2.3 Orthogonal expansions of signals (usual) Orthonormal basis for digitally modulated signals = =) 2 sin(2) () 2 cos(2) (21t fTt ut fTt uccApply the basis to a bandpass signal.). ( ) (2) ( ) (2) 2 sin( ) ( ) 2 cos( ) ( ) (2 1t u t yTt u t xTt f t y t f t x t sc c = =--- Po-Ning Chen --- NCTU/CM4-694.2.3 Orthogonal expansions of signals Applications of orthogonal expansions(dis-)similarity> < + = + = + + == = = =) ( ), ( 2 || ) ( || || ) ( || 2 ) ( ... ) ( ) ( and ) ( of distance Euclean similaritybasis. same for the ) ,..., , ( ) (basis. some for ) ,..., , ( ) (2 122211 12122 21 12 1 122 1 22 1 1t s t s t s t sb a b ab a b at s t s db b b t sa a a t snii iniiniin nnn4-704.2.3 Orthogonal expansions of signals Alternative definition of similarityof two signals (measured by normalized crosscorrelation)Definition.|| ) ( ||) (|| ) ( ||) (|| ) ( || || ) ( ||) ( ), ( by defined issimilarity their ), ( and ) ( signals two Given22-112 12 1122 1dtt st st st st s t st s t st s t s= > < =+ > < = = = 2|| ||, max arg|| || , 2 || || min arg ) () ( || || min arg ) (212 2121mm M mm m M m MDML m M m MDss rs s r r r dr d s r r drv rr v r r rr r r rAlso,--- Po-Ning Chen --- NCTU/CM5-395.1.3 The optimum detectorRealization of the optimumAWGN receiver > < = 2|| ||, max arg ) (21mm M mss r r d rv r r The 1st term= Projection of received signal onto each channel symbols. The 2nd term= Compensation for channel symbols with unequal powers, such as PAM. = > < = 2) ( ) ( max arg2|| ||, max arg ) (0121mTm M mmm M mdt t s t rss r r dErv r r5-405.1.3 The optimum detector Note that the front-end part is not necessarily a correlation-type demodulation, since channel symbols may not forma basis. The metric is named the correlationmetric. Realization ofOptimal Detector--- Po-Ning Chen --- NCTU/CM5-415.1.3 The optimum detectorTerminology: Correlation metric (metric =distance function) It is named the correlation metric because it is a measure of the correlation between the received vector and the m-th signal..2|| ||, ) , (2mm mss r s r C rv r v r > =< Under AWGN, maximumdistance criterion with correlation metric = ML criterion.Terminology: PM metric). ( ) | ( ) ( ) | ( ) , ( r p r s p s p s r p s r PMm m m m r r v v v r v r= = Maximumdistance criterion with PM metric = MAP criterion5-425.1.3 The optimum detector Example. Binary PAM withProblem: Determine the optimumMAP detector under AWGN with two-sided power spectrumdensity N0/2.= = = =. ) ( 1 ) (.2 12 1p s P s Ps s E Solution.. 2 / variance and mean with d distribute Gaussian is. 2 / variance with d distribute Gaussian mean - zero is where ,00N rNn n rEE + =--- Po-Ning Chen --- NCTU/CM5-435.1.3 The optimum detector{ }{ }{ }{ } = = =====+ + otherwise spp Nr sotherwise se p pe se p pes r PMs P s r Pr P r s Pr s P r dN r N rN r N rm mm m mm mm m MAP,1ln4if ,,) 1 ( , ) 1 ( , max arg ) , ( max arg) ( | max arg ) ( | max arg | max arg ) (2012/ ) ( / ) (1/ ) ( / ) (2 12 12 12 102020202EE EE E5-44Observations. The threshold depends not only on the prior but also on the noise power. With equal prior, the noise power becomesirrelevant.5.1.3 The optimum detector =otherwise spp Nr sr dMAP,1ln4if ,) (201E--- Po-Ning Chen --- NCTU/CM5-455.1.4 The maximum-likelihood sequence detector Optimal detector for signalswith memory(not channel-with-memory or noise-with-memory, i.e, still, the noise is AWGN) It is implicitly assumed that the order of the signal memoryis known.Maximum-likelihood sequence detectorMaximuma posteriori probability based on a sequence of received signal vectors.5-465.1.4 The maximum-likelihood sequence detector Maximum-likelihood sequence detectorExample study : NRZI = =A t sA t s) () (for time. index the is and , 2 / variance with d distribute Guassian mean - zero is where, , discussion previous the From0k Nnn A rkk k + =--- Po-Ning Chen --- NCTU/CM5-475.1.4 The maximum-likelihood sequence detectorPDF of a sequence of demodulation outputs = =Kkk kK K KNs rNs s r r P1 022 /01 1) (exp) (1) ,..., | ,..., (s1, , sKhave memory, so it is advantageous to detect the original signals based on a sequence of outputs.If ML rule is employed, the resultant detector is called the maximum-likelihood sequence detector.5-485.1.4 The maximum-likelihood sequence detector ML sequence detector for NRZI signalsies. possibilit 2 of consist which ), ,..., ( ofns combinatio possible all for search to needs therefore Wedistance Euclidean , ) ( min arg) (exp) (1max arg) ,..., | ...., ( max arg ) ,..., (112} , { ) ,..., (1 022 /0} , { ) ,..., (1 1} , { ) ,..., (1111KKKkk kA A s sKkk kK A A s sK KA A s sK MLs ss rNs rNs s r r P r r dKKKKKK= = = ==--- Po-Ning Chen --- NCTU/CM5-495.1.4 The maximum-likelihood sequence detector ML sequence detector for multi-dimensional signals with memoryies. possibilit 2 of consist which ), ,..., ( ofns combinatio possible all for search to needs therefore We wheredistance, Euclidean , ) ( min arg) (exp) (1max arg) ,..., | ,..., ( max arg ) ,..., (11 12 ) ,..., (1 1 022 /0) ,..., (1 1) ,..., (1111KNKKkNjkj kjs sKkNjkj kjKN s sK Ks sK MLs s N. | |s rNs rNs s r r P r r dKKKKKKr rr r r r r rr rr rr r= = === == =SSSSThe complexity of searching the optimal solution becomes a burden.5-50 Viterbi (demodulation) AlgorithmA sequential trellis searchalgorithmthat performs ML sequence detection Transforming a searchover 2Kvector points into a sequential searchover a (vector) trellis sequential = break the vectors into components and performthe search based on each component (in sequence) of the vectors(Also, it is a decoding algorithmfor convolutional codes.) = =A t sA t s) () (5.1.4 The maximum-likelihood sequence detector--- Po-Ning Chen --- NCTU/CM5-515.1.4 The maximum-likelihood sequence detectorThe number of sequences in the trellis search may be reduced by using the Viterbi algorithm.= =Kii iA A s sK MLs r r r dKK12} , { ) ,..., (1) ( min arg ) ,..., (1memory. have not does which input, digital the is } 1 0 { memory. has which symbol, channel the is } { ) 2 1 (1 if ,0 if , : Notations111, IA,A s II sI sskk kk kk kk == == 5-515-525.1.4 The maximum-likelihood sequence detectorThe signal memory order of NRZI signals is 1(L=1). The current channel symbol only depends on the previous channel symbol. Assume the initial state is S0. Then the trellis will reach its regular formafter the reception of the first two signals. --- Po-Ning Chen --- NCTU/CM5-53Explaining the Viterbi Algorithm(fromS0at t = 0). There are two paths entering each node at t =2T.5.1.4 The maximum-likelihood sequence detector). 2 ( by denoted , 2 at node) 1 , 1 ( or ) 0 , 0 ( ) , ( path002 1T ST t SI I= =). 2 ( by denoted , 2 at node) 0 , 1 ( or ) 1 , 0 ( ) , ( path112 1T ST t SI I= =0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s) ( / 1 t s) ( / 1 t s0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s ) ( / 1 t s ) ( / 1 t s ) ( / 0 t s5-545.1.4 The maximum-likelihood sequence detector Euclidean distance for each path2221 0 02221 0 0)) ( ( ) ( ) 1 , 1 ( ) 2 ( node entering ) 1 , 1 ( path for distance Euclidean)) ( ( )) ( ( ) 0 , 0 ( ) 2 ( node entering ) 0 , 0 ( path for distance EuclideanA r A r D T SA r A r D T S + = = + = = Viterbi algorithm. Discard, among the above two paths, the one with larger Euclidean distance. The remaining path is called survivor at t =2T. Now, you should sense (at least, roughly) the key of the Viterbialgorithm.0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s ) ( / 1 t s ) ( / 1 t s ) ( / 0 t s--- Po-Ning Chen --- NCTU/CM5-55 Euclidean distance for each path2221 12221 1) ( ) ( ) 0 , 1 () ( )) ( ( ) 1 , 0 (A r A r DA r A r D + = + = Viterbi algorithm. Discard, among the above two paths, the one with larger Euclidean distance. The remaining path is called survivor at t =2T. We therefore have two survivor pathsafter observing r2.5.1.4 The maximum-likelihood sequence detector0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s) ( / 1 t s) ( / 1 t s5-565.1.4 The maximum-likelihood sequence detector Suppose the two survivor pathsare (0,0) and (0,1) Then, there are twopossible paths entering S0at t = 3T, i.e., (0,0,0) and (0,1,1).0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s ) ( / 1 t s0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s ) ( / 0 t s ) ( / 1 t s ) ( / 1 t sT t 3 =--- Po-Ning Chen --- NCTU/CM5-57 Euclidean distance for each path23 1 023 0 0)) ( ( ) 1 , 0 ( ) 1 , 1 , 0 ()) ( ( ) 0 , 0 ( ) 0 , 0 , 0 (A r D DA r D D + = + = Viterbi algorithm. Discard, among the above two paths, the one with larger Euclidean distance. The remaining path is called survivor at t =3T.5.1.4 The maximum-likelihood sequence detector0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 1 t sT t 3 =) ( / 0 t s ) ( / 1 t s ) ( / 0 t s 5-58 Euclidean distance for each path23 1 123 0 1) ( ) 1 , 0 ( ) 0 , 1 , 0 () ( ) 0 , 0 ( ) 1 , 0 , 0 (A r D DA r D D + = + = Viterbi algorithm. Discard, among the above two paths, the one with larger Euclidean distance. The remaining path is called survivor at t =3T.5.1.4 The maximum-likelihood sequence detector0 = t T t = T t 2 =0S1S) ( / 0 t s ) ( / 0 t s ) ( / 0 t s) ( / 1 t s ) ( / 1 t sT t 3 =--- Po-Ning Chen --- NCTU/CM5-595.1.4 The maximum-likelihood sequence detector Viterbi algorithmCompute two metrics for the two signal paths entering a node at each stage of the trellis searchRemove the one with larger Euclidean distanceThe survivor path for each node is then extended to the next state. The elimination of one of the two paths is done without compromising the optimalityof the trellis search, because any extension of the path with the larger distance will always have a larger metric than the survivor that is extended along the same path. 5-605.1.4 The maximum-likelihood sequence detector The number of paths searched reduced by a factor of two at each stage (cf. the next slide). The Viterbi algorithmdoes not reduce the computational complexity (still, 2K metric computations are required). What the Viterbi algorithmminimizes is the number of trellis paths searched in performing ML sequence detection.= =Kii iA A s sK MLs r r r dKK12} , { ) ,..., (1) ( min arg ) ,..., (1--- Po-Ning Chen --- NCTU/CM5-615.1.4 The maximum-likelihood sequence detector) ( / 0 t s ) ( / 1 t s) ( / 0 t s ) ( / 1 t s) ( / 0 t s ) ( / 1 t s) ( / 0 t s ) ( / 1 t s) ( / 0 t s) ( / 1 t s ) ( / 0 t s) ( / 1 t s ) ( / 0 t s) ( / 1 t s 10SS) ( / 0 t s ) ( / 1 t s) ( / 0 t s ) ( / 1 t s) ( / 0 t s ) ( / 1 t s) ( / 0 t s ) ( / 1 t s) ( / 0 t s) ( / 1 t s ) ( / 0 t s) ( / 1 t s ) ( / 0 t s) ( / 1 t s survivor paths=(0,0) and(0,1)These dotted Paths are removed.5-625.1.4 The maximum-likelihood sequence detector Apply the Viterbi algorithmto Delay modulation2entering paths for each nodeL = memory order = 14survivor paths at each stage--- Po-Ning Chen --- NCTU/CM5-63 In the previous discussion, we only discussed how to remove the paths?, but did not touch the issue of how to make decision?.Definition of decision delay for Viterbi decoding delay= k means that the transmitted bits corresponding to channel symbol at time instant i should be estimated after the reception of rk+i. = = = 12} , { ) ,..., (1 1) ( min arg ) ,..., ( ), ,(is decision optimal The1ii iA A s sML MLs r r r d I I LViterbipath remover(after k delays)1,...,r ri k+ i is s of estimate ML =5.1.4 The maximum-likelihood sequence detector5-64Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 5.1.4 The maximum-likelihood sequence detectorUnder the premise that the Viterbi algorithmyields the ML decision, what is the maximumdecision delay possibly encountered?= = = 12} , { ) ,..., (1 1) ( min arg ) ,..., ( ), ,(is decision optimal The1ii iA A s sML MLs r r r d I I L--- Po-Ning Chen --- NCTU/CM5-655.1.4 The maximum-likelihood sequence detector Lets borrow an example fromExample 14.3-1 of Digital and Analog Communications by J . D. Gibson.A code with L =2Assume the received codeword is(10,10,00,00,00,)5-66At time instant 2, one still does not know what the first two transmitted bits are. (There are two possibilities for time period 1; hence, decision delay > 1.)(If decision were made now, the decision delay = T.)Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2,r r1 estimator s ML = k =1 ?minimum (Hamming)distance associatedwith each path1r2r--- Po-Ning Chen --- NCTU/CM5-67Hence, we get r3and compute the accumulated metrics for each path.minimum (Hamming)distance associatedwith each path1r2r3r5-68At time instant 3, one still does not know what the first two transmitted bits are. (Still, there are two possibilities for time period 1; hence, decision delay > 2.)(If decision were made now, the decision delay = 2T.)Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2 3, , r r r1 estimator s ML = k =2 ?1r2r3r--- Po-Ning Chen --- NCTU/CM5-69Hence, we get r4and compute the accumulated metrics for each path.1r2r3r4r5-70At time instant 4, one still does not know what the first two transmitted bits are. (There are two possibilities for time period 1; hence, decision delay > 3.)(If decision were made now, the decision delay = 3T.)Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2 3 4, , , r r r r1 estimator s ML = k =3 ?1r2r3r4r--- Po-Ning Chen --- NCTU/CM5-71Hence, we get r5and compute the accumulated metrics for each path.1r2r3r4r5r5-72At time instant 5, one still does not know what the first two transmitted bits are. (There are two possibilities for time period 1; hence, decision delay > 4.)(If decision were made now, the decision delay = 4T.)Time instant :State012300011011Codeword Received : 10 10 00 00 000 1 2 3 4 52333Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2 3 4 5, , , , r r r r r1 estimator s ML = k =4 ?1r2r3r4r5r--- Po-Ning Chen --- NCTU/CM5-73Hence, we get r6and compute the accumulated metrics for each path.Time instant :State012300011011Codeword Received : 10 10 00 00 000 1 2 3 4 5 600233323441r2r3r4r5r6r5-74At time instant 6, one still does not know what the first two transmitted bits are. (There are two possibilities for time period 1; hence, decision delay > 5.)(If decision were made now, the decision delay = 5T.)Time instant :State012300011011Codeword Received : 10 10 00 00 000 1 2 3 4 5 6002344Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2 3 4 5 6, , , , , r r r r r r1 estimator s ML = k =5 ?1r2r3r4r5r6r--- Po-Ning Chen --- NCTU/CM5-75Hence, we get r7and compute the accumulated metrics for each path.Time instant :State012300011011Codeword Received : 10 10 00 00 000 1 2 3 4 5 600234400724441r2r3r4r5r6r7r5-76At time instant 7, one still does not know what the first two transmitted bits are. (There are two possibilities for time period 1; hence, decision delay > 6.)(If decision were made now, the decision delay = 6T.)10 10 00 00 000 1 2 3 4 5 600 0072444Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2 3 4 5 6 7, , , , , , r r r r r r r1 estimator s ML = k =6 ?1r2r3r4r5r6r7r--- Po-Ning Chen --- NCTU/CM5-77Hence, we get r8and compute the accumulated metrics for each path.10 10 00 00 000 1 2 3 4 5 600 007244400824551r2r3r4r5r6r7r8r5-78At time instant 8, one finally knows what the first two transmitted bits are, which is 00. Hence, the decision delay = 7T.10 10 00 00 000 1 2 3 4 5 600 007240082455Viterbipath remover(after k delays)1,...,r ri k+ is ML estimator = 1 2 3 4 5 6 7 8, , , , , , , r r r r r r r r1 estimator s ML = k =7 !--- Po-Ning Chen --- NCTU/CM5-795.1.4 The maximum-likelihood sequence detectorOptimal Viterbi ML-decision maker : To wait until there is only one possibility for the transmitted symbol. (The decision delay could be as large as the transmitted codeword length.)Suboptimal Viterbi ML-decision maker : Set a limit to the decision delay. 1identical. are paths survivor all of symbols ed transmitt 5 previous ThePr L iViterbipath remover(after 5L delays)1,...,r riL i L is s5 5 of estimate ML =5-80 Suboptimal Viterbi algorithm= If there are more than one survivor paths (which results in more than one possible decoding results)remain for time period i5L, just select the one with smaller metric, and forcefully remove the others. For example, NRZI codes with L=1.5.1.4 The maximum-likelihood sequence detector. state for ) , , , , , ( and state for ) , , , , , ( on base willdecision next the I.e., results. decision following the changing it without drop can we paths, both for same the is 1 bit before metric d accumulate to on contributi the Since. state for ) , , , , , ( and state for ) , , , , , ( become paths survivor new two the Then,former. say the smaller, is )) , , , , , ( and ) , , , , , ( (between metric whose ), , ( from one select , If. state for ) , , , , , ( and state for ) , , , , , ( are paths survivor that two Suppose1 7 6 5 4 3 2 0 7 6 5 4 3 21 6 5 4 3 2 10 6 5 4 3 2 16 5 4 3 2 1 16 5 4 3 2 1 0 1 1 1 11 6 5 4 3 2 10 6 5 4 3 2 1S b b b b b b S b b b b b bS b b b b b bS b b b b b bb b b b b b Db b b b b b D b b b bS b b b b b bS b b b b b b--- Po-Ning Chen --- NCTU/CM5-815.1.4 The maximum-likelihood sequence detector The suboptimal Viterbi algorithmmay still yield a bad decision (but with a very small probability); As far as the decision delay is concerned (with practical constraint), how about to minimize the probability of decision error based on a fixed delay D. Apparently, this will result in a smaller error probability than the suboptimal Viterbi algorithm.The example in the previous slide is actually a symbol-by-symbol detector withsymbol length =6information bits.5-825.1.5 A symbol-by-symbol detector for signals with memory Abend and Fritchman (1970) algorithmOptimal in the sense of minimizing the symbol error for a given delay D. delay with decision the is where), ,..., , | ( max arg 1 1 )} 1 ),...,( 3 ( ), 1 ( {L D Dr r r s P sD k D k k M M M s kk= + + . ) | ,..., , ( ) ( max arg,) ,..., , () | ,..., , ( ) (max arg) ,..., , | ( max arg 1 1 )} 1 ),...,( 3 ( ), 1 ( {1 11 1)} 1 ),...,( 3 ( ), 1 ( {1 1 )} 1 ),...,( 3 ( ), 1 ( {k D k D k k M M M sD k D kk D k D k kM M M sD k D k k M M M s ks r r r P s Pr r r Ps r r r P s Pr r r s P skkk + + + + + + + + === ) , , ,..., ( of functionmemory have not do which input, digital block the is } 1 ,..., 1 0 {memory has which symbol, channel the is ), , ,..., , ( of function where12 1= + = = k k L k k k k kkk L k k k kn I s s n s rM , II s s s s--- Po-Ning Chen --- NCTU/CM5-835.1.5 A symbol-by-symbol detector for signals with memoryEstimation formula + +++ + + ++===)} 1 ),...,( 1 ( )} 1 ),...,( 1 (1 1 1)} 1 ),...,( 1 ()} 1 ),...,( 1 ( )} 1 ),...,( 1 (1 111 1 1)} 1 ),...,( 1 (1 1 1 1 1 1)} 1 ),...,( 1 (12 112 111) ,..., ( max arg) ,..., | ( ) ,..., ( max arg) | ,..., , ( ) ( max arg M M s M M sDM M sM M s M M sDDDM M sD DM M sDDs s qs s r P s s Ps r r r P s P s{ {{{ {{{LL). ,..., , ,..., , () ,..., | ,..., , ( ) ,..., ( ) ,..., ( where1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1s s r r r Ps s r r r P s s P s s qD D DD D D D D+ + ++ + + + +==5-845.1.5 A symbol-by-symbol detector for signals with memory + +++ + + ++===)} 1 ),...,( 1 ( { )} 1 ),...,( 1 ( {2 2 2)} 1 ),...,( 1 ( {)} 1 ),...,( 1 ( { )} 1 ),...,( 1 ( {2 221 2 2)} 1 ),...,( 1 ( {2 1 1 1 2 2)} 1 ),...,( 1 ( {23 223 222) ,..., ( max arg) ,..., | ( ) ,..., ( max arg) | ,..., , ( ) ( max arg M M s M M sDM M sM M s M M sDDDM M sD DM M sDDs s qs s r P s s Ps r r r P s P sLL--- Po-Ning Chen --- NCTU/CM5-85[ ][ ] + + + ++ + + + + + + + + +++ + + ++ + + + + ++ + + + + + ++ + + + + ++ + + + + ++ + + + = = = =====)} 1 ),...,( 1 ( {11 112 222 2)} 1 ),...,( 1 ( {1 1 1 2 1 2 2 2 2)} 1 ),...,( 1 ( {1 1 12 12 22 2 22 1)} 1 ),...,( 1 ( {1 1 12 2 2 2 22 1)} 1 ),...,( 1 ( {1 1 1 12 2 2 2 22 1 1 1 2 2 2 2 22 2 2 1 1 2 2 2 2 22 2 1 2 2 2 2 2 211111) ( ) | ( ) | () ,..., ( ) ,..., | ( ) ,..., | () ,..., () ,..., () ,..., () ,..., | () ,..., () ,..., () ,..., | ( ) ,..., () ,..., () ,..., , ,..., () ,..., | ( ) ,..., () ,..., | ,..., ( ) ,..., | ( ) ,..., () , ,..., | ,..., ( ) ,..., | ( ) ,..., () ,..., | ,..., ( ) ,..., ( ) ,..., (M M sD DL D DDL D DM M sD D D L D D DM M sDDDL D D DDM M sDL D D D DDM M sD DL D D D DD D L D D D DD D D D D DD D D Ds q s s P s r Ps s q s s s P s s r Ps s qs s Ps s Ps s r Ps s Ps s qs s r P s s Ps s Ps s r r Ps s r P s s Ps s r r P s s r P s s Pr s s r r P s s r P s s Ps s r r P s s P s s q5.1.5 A symbol-by-symbol detector for signals with memory5-86 + +=)} 1 ( ),..., 1 ( { )} 1 ( ),..., 1 ( {3 3 3)} 1 ( ),..., 1 ( {34 33) ,..., ( max arg M M s M M sDM M sDs s q s L + + + ++ + + + =)} 1 ),...,( 1 ( {22 223 333 3 3 3 32) ( ) | ( ) | ( ) ,..., (M M sD DL D DDL D D Ds q s s P s r P s s q + + +=)} 1 ( ),..., 1 ( { )} 1 ( ),..., 1 ( {)} 1 ( ),..., 1 ( {1) ,..., ( max arg M M s M M sk D k kM M skk D kks s q s L + + + ++ + + + =)} 1 ),...,( 1 ( {11 111) ( ) | ( ) | ( ) ,..., (M M sD kk kD kL D k D kD kL D k D k k D k kks q s s P s r P s s q5.1.5 A symbol-by-symbol detector for signals with memory--- Po-Ning Chen --- NCTU/CM5-875.1.5 A symbol-by-symbol detector for signals with memoryAdvantage of the above approach Optimality (in MAP sense) under the constraint of delay = D. Hence, it can be applied to the case of non-equal prior.Disadvantage of the above approach Computational complexity, especially when M or L is large.5-885.2 Performance of the optimum receiver for memoryless modulation Performance = Probability of error In this section, our goal is to derive the probability of error for memoryless modulation/demodulation.--- Po-Ning Chen --- NCTU/CM5-895.2.1 Probability of error for binary modulation Antipodal signalOne-dimensional signal (Refer to slide 5-42) + + = + = = + = = < = + = 2 /1 ) 1 (2 / ) 2 / , ( ) 1 ( ) 2 / , ( ) | ( ) ( ) | ( ) (,1ln4if ,) (. 2 / variance and mean with d distribute Gaussian is2variance with d distribute Gaussian mean - zero is where ,0 00 0,000NpNpN Normal p N Normal ps error P P s error P P Potherwisepp Nrr dN rNn n rBPAM eMAPE EE EE E E EEEEEE 5-905.2.1 Probability of error for binary modulationObservation 1: POE is completely characterized by SNR.( ) ) 2 ( . 2 ). / ( as defined usually is .22 /1212 / 21. 0 prior, equal For 00 , 00 0 0,NSNR SNR P N SNRN N NPBPAM eBPAM eEEE E E= = = + = = The larger the SNR, the smaller the probability of error.Observation 2: POE can also be expressed in terms of the distance between the two signals. The larger the distance among signals, the smaller the probability of error.. 2 ) ( where ,212012, E E E = = = dNdPBPAM e--- Po-Ning Chen --- NCTU/CM5-915.2.1 Probability of error for binary modulation1e-061e-050.00010.0010.010.10 2 4 6 8 10 12 14BPAMdBb( )( )( )( )20 / dB20 / dB,10 erfc 5 . 02 1022bbQQPbb Antipodal e === =b=SNR per information bit5-92On white noise The power of white noise processes (an erroneous view).[ ]2 / ) ( , 00 if , 2 / ) ( ) ( ) ( . function ation autocorrel its establishfirst to need we ), ( of power the derive To00 *N fotherwiseNt n t n Et nnn= == + = = = = =- -22. ) ( ) 0 ( ) ( ) ( use to is it compute way to e alternativ An. ) 0 ( ] [ process WSS a of power average that the Recalldf f df e fX Ef jt ! ! ! ) ( )] ( [ ) 0 ( Hence,2 = = = df f t n En nNon-consistent--- Po-Ning Chen --- NCTU/CM5-93On white noise Definition of white noiseDiscrete white noise Definition. 2 / 1 1/2 for constant ) ( if white be to said is } { process random discrete A< = f fXxn Autocorrelation function of discrete white noise = == = = ,... 2 1, if , 00 if ,) ( ) (22 12 / 12 2 22 12 / 1 x/f jxf j/x xdf e df e f Average power of discrete white noise. ) ( (0)21/22 / 1x x xdf f = =5-94On white noise A white process is often implicitly assumed zero-mean.Advantage of adopting discrete white noise concept It is well-defined and free of analytical difficulties.--- Po-Ning Chen --- NCTU/CM5-95On white noise = . all for constant ) ( if white be to said is } { process random continuous Af fXxt Autocorrelation function of discrete white noise). ( ) ( ) (2 2 2 2 xf jxf jx xdf e df e f = = = 0. for 0 ) ( (3) . (0) (2) .) 1 ) ( (Hence, ). 0 ( ) ( ) ( ) 1 ( : function delta Dirac ), ( of Properties = = = = d h d h{ } = = ===0 , 00 ,) ( cases) continuous for efined function(d delta Dirac,... 2 , 1 , 00 , 1) ( cases) discrete for (defined function delta ker Kronec: functions delta Two Continuous white noise Definition5-96On white noise Solution to the previous non-consistence.[ ]2 / ) ( , 00 if ,) (2) ( ) ( ) ( . function ation autocorrel its establishfirst to need we ), ( of power the derive To00 *N fotherwiseNt n t n Et nnn= = = = + = = = = =- -22. ) ( ) 0 ( ) ( ) ( use to is it compute way to e alternativ An. ) 0 ( ] [ process WSS a of power average that the Recalldf f df e fX Ef jt ! ! ! ) ( )] ( [ ) 0 ( Hence,2 = = = df f t n En n. ) 0 ( noise, white continuous For = Consistent--- Po-Ning Chen --- NCTU/CM5-97On white noise Therefore, we say that the average power of a continuous white noise is infinite. It should not be a problemfor its analysis because the post-sampling noise power is finite!5-98[ ] = = = =+ = = ==TTT TTT TTnsTnTsnsd T h Nd T h sdt d t T h T h t Nd T h sdt d t T h T h n t n Ed T h sT y ET ySNRn s rd T h n T y d T h s T yT y ET ySNR020200 00200 0202200 0220) ( ) 2 / () ( ) ( ) ( ) ( ) ( ) 2 / () ( ) ( ) ( ) ( ) ( ) () ( ) ()] ( [) ( (ii)) ). ( ) ( ) ( that (Note . ) ( ) ( ) ( and ) ( ) ( ) ( where,)] ( [) ( Define (i) That explains why we adopt a quantity such as SNR0.! ! ! integrable is ) ( On white noise--- Po-Ning Chen --- NCTU/CM5-99On white noise So impractical, why the continuous white noise is so popular? There do exist noise sources that have a flat power spectral densityover a range of frequencies that is much larger than the bandwidths of subsequent filters or measurement devices.. experiment by closely verified been has fact that a bandwidth, large a over white like looks process noise the Hence,. when , 2) 2 (2) ( is density spectrum power Itsmedium. physical the of parameters the are and re, temperatu absolute the is constant, s Boltzman' the is where, ) ( form the has noise the of funciton tion autocorrea the that shown have some t, measuremen phsically on Based2 22| |

+ A process satisfying spectral factorization theoremis physically realizable and possibly seen in practice. This is the end of the discussion on white noises.--- Po-Ning Chen --- NCTU/CM5-1035.2.1 Probability of error for binary modulation Orthogonal two-dimensional signalsEEE2 || ||002 121= =s sssr rrrd. transmite is ) ( if ], , [1 2 1t s n n r + = Er5-1045.2.1 Probability of error for binary modulation Determine the optimumMAP detector under AWGN with power spectrumdensity N0/2and M= 2.. 2 / variance marginal and mean zero with Gaussianst independen are and where , ] , [ or ] , [02 1 2 1 2 1Nn n n n n n r + + = E Er{ }= = == ++ otherwise. ,1ln2, ) 1 ( , max arg ) ( | max arg ) (201 2 1) ( ) ( 2 10222102221spp Nr r se p pes P s r P r dNr rNr rm m m MAPEE Er r. 1 ) ( and ) (2 1p s P p s P = =--- Po-Ning Chen --- NCTU/CM5-1055.2.1 Probability of error for binary modulation + + = + =+ = 0 00 02 2 1 1 ,) 1 ( 1) , ( ) 1 ( ) , ( ) | ( ) ( ) | ( ) (NpNpN Normal p N Normal ps error P s P s error P s P PBFSK eE EE E = = ed. transmitt is if ), , (ed. transmitt is if ), , (2 0 1 21 0 1 2s N Normal r rs N Normal r rEE5-1065.2.1 Probability of error for binary modulationObservation 1( ) SNR PNSNRN N NPBFSK eBFSK e = = = + + = =,00 0 0, , Since.21121. 0 prior, equal For EE E E The larger the SNR, the smaller the probability of error.Observation 2 The larger the distance among signals, the smaller the probability of error (Cf. slide 5-90).. 2 where ,212012, E = = dNdPBFSK e--- Po-Ning Chen --- NCTU/CM5-1075.2.1 Probability of error for binary modulation Comparison between binary antipodal signals (BPAM) and binary orthogonal signals (BFSK)Binary orthogonal signal requires twicethe transmitted power than antipodal signal to achieve the same probability of error.Binary orthogonal signal is 3dB poorer than antipodal signal since 10log102= 3dB.( )( ) SNR PSNR Pl BOrthogona eAntipodal e = =,,25-1081e-061e-050.00010.0010.010.10 2 4 6 8 10 12 14BPAMBFSK5.2.1 Probability of error for binary modulation( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 10 erfc 5 . 0 1010 erfc 5 . 0 2 10 2 220 / dB 20 / dB,20 / dB 20 / dB,b bb bQ Q PQ Q Pb b l BOrthogona eb b Antipodal e = = = = = = = =) 2 (bQ ) (bQ dB 3 dBb--- Po-Ning Chen --- NCTU/CM5-1095.2.2 Probability of error for M-ary orthogonal signals We now turn to M-ary signals.Probability of error under ML detection and equal prior for M-ary equal-power orthogonal signalsML detector ( )) , ( ' max arg, max arg2 / || || , max arg ) (1121m M mm M mm m M m MLs r Cs rs s r r dv rv rr v r r = > < = > < ==EEE0 0 0 00 0 0 00 0 0 021LM M O M M MLLrMrrMsssM-ary equal-power orthogonal signals5-1105.2.2 Probability of error for M-ary orthogonal signals Outputs of M correlatorsM M M MMMMr n n n n s r Cr n n n n s r Cr n n n n s r Ct s n n n rE E E EE E E EE E E E EE= = + == = + = = + = + =+ =] ,..., 0 , 0 [ ] ,..., , [ ) , ( ' ] 0 ,..., , 0 [ ] ,..., , [ ) , ( ') ( ] 0 ,..., 0 , [ ] ,..., , [ ) , ( 'd. transmite is ) ( if ], ,..., , [2 12 2 2 1 21 1 2 1 11 2 1r rMr rr rr. max ly, equivalent or , max only when correct is decision The) , ( ' max arg ) (2 1 2 11m M m m M mm M m MLr r r rs r C r d =E Ev r r--- Po-Ning Chen --- NCTU/CM5-1115.2.2 Probability of error for M-ary orthogonal signals It is sometimes convenient to first derive the probability that the detector makes a correct decision.( )( )( )( ) [ ]. variance and mean with pdf Gaussian the is ) ( where ,12 ) ( ) ( ) | Pr( , Pr ) | Pr( , ] [ ] [ Pr ] [ ] [ Pr max Pr ) | Pr(2) , (1/ ) (0 01 11 1 ) 2 / , (12 2 ) 2 / , 0 (1 1 1 111 1 1 1 21 1 1 1 1 1 1 1 2 11 1 2 11 ,..., 2 1 12021010 === = < == = > > => > = > = =NN NEEdr eN Nrdr r dr rdr s r R r R s r Rdr s r R r R s R r R rs R R R Rs R R s correctN r MNMrNMMMm M mr rr rLrLr r) | Pr( ) | Pr(1) Pr(11s correct s correctMcorrectMmm r r= ==5-1125.2.2 Probability of error for M-ary orthogonal signals( ) [ ]( ) [ ] == = = =dx e xN r x dr e N rNdr eN Nrcorrect P/N x MN r MN r MMFSK e2 / ) 2 ( 10 1 1/ ) (0 1101/ ) (0 01 1,20021021121. / 2 Let . / 2 11121) Pr( 1EEE A numerically evaluated expression is established.--- Po-Ning Chen --- NCTU/CM5-1135.2.2 Probability of error for M-ary orthogonal signals Probability of error (POE) versus SNR(-per-bit)In previous section, we establish the relation between POE and SNR for binaryantipodal signals (PAM) and binaryorthogonal signals:A fair comparison must be made based on the POE-per-bit and SNR-per-bit.( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 10 erfc 5 . 0 1010 erfc 5 . 0 2 10 2 220 / dB 20 / dB,20 / dB 20 / dB,b bb bQ Q PQ Q Pb b l BOrthogona eb b Antipodal e = = = = = = = =5-1145.2.2 Probability of error for M-ary orthogonal signalsSNR-per-bit for M-ary orthogonal signals. still is power Noise. log if , ) ( symbols, channel For 02 symbolNM k k k Mb bit = = = = E E E EPOE-per-bit for M-ary orthogonal signals ( ) = =.) signals orthogonal of symmetry the on based y probabilit error the to on contributi equal (Assume , ) 1 2 ( y probabilit with symbols ) 1 2 (1 y probabilit with symbol 1symbols. ) 1 2 ( other the decision if Error,symbol; ed transmitt the decision if Correct,symbols, 2 of out one ng transmiti When,,kMFSK ekMFSK ekkPP( ) [ ] ( ) [ ] = = dx e x dx e x Pb bk x M /N k x MMFSK e2 / ) 2 ( 1 2 / ) 2 ( 1,2 20121121 E--- Po-Ning Chen --- NCTU/CM5-1155.2.2 Probability of error for M-ary orthogonal signals.211 22. error_bit] number_of_ [ that assumes usually it Also, ,1,,MFSK e MFSK e kkMFSK ebMFSK ebP P PP k E= =( ) [ ] = dx e x Pbk x MkkMFSK eb2 / ) 2 ( 11,211 2221 error symbol one error bit two ), 1 2 ( y probabilit with 11 00error symbol one error bit one ), 1 2 ( y probabilit with 10 00error symbol one error bit one ), 1 2 ( y probabilit with 01 00error symbol no error bit no ), 1 ( y probabilit with 00 00. 2 Example.,,,,= = = = =kMFSK ekMFSK ekMFSK eMFSK ePPPPk.1 221 2] error_bit number_of_ [ Hence,,11,MFSK e kk kikMFSK eP kikiPE== =5-1165.2.2 Probability of error for M-ary orthogonal signals Concluding remarksThe larger the M is, the better the systemperformance.For example, to achieve bit-POE=105, one needs b= 12dB for M= 2; but only requires b= 6dB for M= 64, a 6dB save in transmission power !!!--- Po-Ning Chen --- NCTU/CM5-1175.2.2 Probability of error for M-ary orthogonal signals Since bit-POE is decreasing with respect to M, one may question that whether ? 0 lim, = MFSK ebMPAnswer: Not necessary. Union Boundon the probability of errorThe previous formula is too complicated to evaluate its ultimate behavior with respect to M. We therefore develop the (famous) Union bound.5-1185.2.2 Probability of error for M-ary orthogonal signals( ). max Pr ) | Pr( that Recall1 ,..., 2 1 1s R R s correctm M m r r=> =( )( )( )( )( )( )BFSK eMMm M mMFSK eP Ms R R Mdr s r R r R s r R Mdr s r R r R s r R Mdr s r R r R s R r R rs R R R Rs R Rs error P,1 2 11 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 21 1 1 1 1 1 1 1 2 11 1 2 11 ,..., 2 11 ,) 1 (Pr ) 1 () | Pr( , Pr ) 1 () | Pr( , Pr ) 1 () | Pr( , ] [ ] [ Pr ] [ ] [ Prmax Pr) | Pr( = = = = == = = = = = == =rr rr rr rLrLrr. prior, equal For 0, =NPBFSK e E--- Po-Ning Chen --- NCTU/CM5-1195.2.2 Probability of error for M-ary orthogonal signals( ) + + = L6 4 22 /5 3 1 3 1 1121) ( 02u u ueuu uu A famous approximation formula for Gaussian cdfwhere the approximation error is less than the last termused.( ) .21) ( 02 /2ueuu u ( ) .1121) ( 02 /22ueu uu u 5-120( )[ ][ ][ ][ ] [ ]( ) [ ] 2 ) 2 log( 2 exp212 / exp ) 2 log( exp212 / exp212) 2 ( exp21) 1 2 () 2 ( exp21) 1 () 1 () 1 (00000l , , = = = = bbbbbbkbbkBFSK e MFSK ekkk kkkkN kN kNNMN MP M P EEEEE5.2.2 Probability of error for M-ary orthogonal signals(Union bound)--- Po-Ning Chen --- NCTU/CM5-121Therefore, if then=== 0211 22lim12 /lim lim021lim lim)) 2 log( 2 (1, ,)) 2 log( 2 (,bbkbkkkMFSK eMMFSK ebMkbkMFSK eMekPMMPekP 5.2.2 Probability of error for M-ary orthogonal signals, 42 . 1 ) 2 ln( 2 dBb = > 5-1225.2.2 Probability of error for M-ary orthogonal signals1.42dBError approaches zero as M grows.--- Po-Ning Chen --- NCTU/CM5-1235.2.2 Probability of error for M-ary orthogonal signalsWhat if Answer: In other words, 1.42 dB is not a tight bound!!In fact, it can be shown that( ).2) 2 log(, bkMFSK ee P Therefore if , 6 . 1 ) 2 log( dBb = > then bit-POE approaches zero as M tends to infinity.91.6dB is called the Shannon limit for M-ary orthogonal signals under AWGN channel. This is the minimumbit-SNR to achieve arbitrarily small bit-POE for M-ary orthogonal signals.? 42 . 1 ) 2 log( 2 dBb = ? 0 lim guarantee not does 42 . 1, > MFSK eMbP dB 5-1247.1.2 Channel capacity (with orthogonal signals for AWGN channels)T 21T 21T 21T 21M.2ly equivalent or 2WTMTMW = = Y1=X1+N1Y2=X2+N2:YM=XM+NM+ =+ = =02021 log2 /1 log21) ; ( l bits/symbo ) ; (WNPWTNpMY X I M Y X IavxM MShannon(1948) proved the upper limit of bitsper symbol for arbitrary small bit-POE is:xxTavWpMpTdt t x ETP21)] ( [102=== --- Po-Ning Chen --- NCTU/CM5-1257.1.2 Channel capacity (with orthogonal signals for AWGN channels). d bits/secon 1 logbol second/sym l bits/symbo ) ; (02 + = =WNPWTY X ICavM MShannon proved that:if R > C, then the bit-POE of any transmission scheme is bounded away from zero;if R < C,then for any given >0, there exists one transmission scheme (such as by letting Mlarge) whose bit-POE + =+ =+ = >EJ oule/bit d bits/secon nd J oule/seco b avR P E = Assume that the transmission rate is R bits/second.--- Po-Ning Chen --- NCTU/CM5-1277.1.2 Channel capacity (with orthogonal signals for AWGN channels) .2 / 21 2/1 2 Then .22 2 1.2that recall Also, . 2 and l bits/symbo mbol seconds/sy bits/sec that Observe fixed. is mbol seconds/sy Suppose2 / 2 /kkbW RbkkkrW RrkRMTRW WRTMWM k T RTk< > < ( )( ) . for POE - bit 0 then , ) 2 log( If0 0k k kb > < > > Now, we need to show that if b> log(2), then we can make bit-POE arbitrarily small by taking k large enough, i.e.,5-130The Union Bound is loose. This can be seen fromthat7.1.3 Achieving channel capacity with orthogonal signalsdoes not provide a meaningful bound when( )[ ] ) 2 ( exp21) 1 () 1 () 1 (000l , ,NNMN MP M PBFSK e MFSK eEEE = [ ] small. is ) ( ly, equivalent or , 1 ) 2 ( exp21) 1 (0 00N NNM E EE> --- Po-Ning Chen --- NCTU/CM5-131Recall that we have the close formfor Pe,MFSK. 7.1.3 Achieving channel capacity with orthogonal signals( ) [ ]( ) [ ] ( ) [ ]. 0 for , )] ( ) 1 [(2121121121121) Pr( 102 / ) 2 ( 2 / ) 2 (2 / ) 2 ( 1 2 / ) 2 ( 12 / ) 2 ( 1,02002002002020 + + = = = y dx e x M dx edx e x dx e xdx e xcorrect Py/N xy/N xy/N x My/N x M/N x MMFSK eE EE EE ( ) ( ) . ) ( ) 1 ( ) ( 1 ) 0 ( and 1 1 ) ( Since1 1x M x x x xM M .21for 2 21212 2102 / ) 2 ( 2 / 2 / ) 2 (2 / ) 2 ( 2 / 2 / ) 2 (0202 0200202 020 + + ydx e eMdx edx e exMdx ey/N x xy/N xy/N x xy/N xE EE E5-1327.1.3 Achieving channel capacity with orthogonal signals + 02 2 0202 / ) 2 ( 2 / 2 / ) 2 (21,2221minyk x xkyk xyMFSK edx e e dx e Pb b . 114806 . 0) 2 log( 41 if ,21) 2 log( 22022212221*02 /2 / ) 2 ( 2 / 2 / ) 2 (2 / ) 2 ( 2 / 2 / ) 2 (02020202002 2 02= = = = = + k k yee e edx e e dx eyk yk y ykk yyk x xkyk xb bb b--- Po-Ning Chen --- NCTU/CM5-133( )( ) ( ) ) 2 log( 4 2 2 ) 2 log( 2. 121 since ,5 . 02 /1 2 2) 5 . 0 ( 21222122212 /*0 2 / *0) 5 . 0 ( 2) 2 / (2 /2 / ) 2 (2 / ) 2 ( 2 / 2 / ) 2 (,*02*02*02 2*02k k e k kk ye k ydx eedx edx e e dx e Pbk kbb k kbyk xk kyk xyk x xkyk xMFSK ebbbbbb b + =< + => =dx eN Nxdx x dr rdx s x P s x r x r x Pr s rdx s x P s x r r x r x r r Ps r r r r r P s correct PN xMNMxxNs rMM m ms r M Mm M m MEEN Nr rrr rr rrr5.2.3 Probability of error for M-ary biorthogonal signals( ) [ ] = = = = = 22 /1 2 /0/ ) (01 2 /01, ,202212 2 1 1122 1 1) | Pr(11 1dx e x dx eN Nxs correctMP PxMN xMMmm nal MBiOrthogo c nal MBiOrthogo eErwhere =E/N0.--- Po-Ning Chen --- NCTU/CM5-1395.2.3 Probability of error for M-ary biorthogonal signals Re-formulating the formula using bit-SNR instead of the symbol-SNR, we obtain:Larger M, better performanceexcept M =2and M = 4. Note that symbol-POE comparisondoes not really tell the winner in performance. Symbol-POE BPSK (M = 2)< symbol-POE QPSK (M = 4), but bit-POE BPSK (M = 2) = bit-POE QPSK (M = 4), The Shannon-limit remains the same.5-1405.2.4 Probability of error for simplex signals Simplex signals fromorthogonal signalssignal. simplex the called is } ,..., , { Then . as symbols channel new Define.1,...,1,1 is center its , ,..., 1 for ] ,..., , [symbols channel power) - equal and l (orthogona of tions representa vector the Given' '2'1'1 12112 1Mm mMmmkMmmMmmmk m m ms s ss s saMaMaMsM m a a a s === = = = =--- Po-Ning Chen --- NCTU/CM5-1415.2.4 Probability of error for simplex signalsTransmitted energy of simplex signals is reduced.2) for 3dB ( .1log 10 save ) log 10 (1log 10 ) log 10 (1110 10= += =M dBMMdB dBMMdBME EE E''Since the inter-symbol distances (as well as their relative positions) remain the same, the POE of the simplex signal is the same as M-ary orthogonal signals.5-1425.2.5 Probability of error for M-ary binary-coded signals Example. Multi-dimensional BPSK.. / where. 1 ], ,..., , [ : symbols Channel02 1N sM m s s s smjmK m m mE = =Possibly M> = + + >> +>> E EE[ ] ) ( ) () )( (1) ( ) () ( ) ( Therefore,2 , 1 ,*2 , 1 1 , 1) (*2 , 1 1 , 1 2 , 1 , 2 , 1 ,*2 , 1 1 , 1) (*2 , 1 1 , 1 2 , 1 ,11k k k kjk k k k k k k kjk k k kn j n n j n ejn n jn n n j n n j n ejr r jr rk kk k+ + + + + + + + + + + =+ + EEEE5-1705.2.8 Differential PSK (DPSK) and its performanceNormalized pdfwith small varianceNormalized pdfwith large variance[ ]XYNNNormal Y Xn n n n Y Xk kk kk kk k k k k k k k1100112 2 , 1 1 1 1 , 1 1tan ) ( : estimate to is goal Our .00,) sin() cos() , ( Then ) sin( , ) cos( ) , ( Let = + + + = EEE E--- Po-Ning Chen --- NCTU/CM5-171[ ] [ ][ ].00,0) , ( and tan 0 when : DPSK.2 / 00 2 /,0, r and tan , , 0 when : PSK00 1d002 112 12 1 2 1 = = = + = = =NNNormal Y XXYNNNormal rrrn n r r rdEEE r Comparison of POE between PSK and DPSK5.2.8 Differential PSK (DPSK) and its performanceObserve that the noise variance of DPSK is twice as large as that of PSK. Hence, the POE of DPSK is larger than that of PSK.The above formula of DPSK performance is only an approximation. (Cf. Appendix B of the textbook.)5-1725.2.8 Differential PSK (DPSK) and its performanceThe POE degradation of DPSK to PSK is better than double or 3 dB. When symbol-POE is less than 105, the difference in bit-SNR for binarycasesis only 1 dB.--- Po-Ning Chen --- NCTU/CM5-1735.2.8 Differential PSK (DPSK) and its performance The good performance of DBPSK is due to that (k k1)is either 0or . Hence, we only need to consider the real part of { } + + + + =+ + + + + + + + + =+ + ) (1 ) ( ) ( Re., .) )( (1) ( ) () ( ) (2 , 1 2 , 1 , 1 1 , 1 , 1 , 1*2 , 1 1 , 1 2 , 1 ,*2 , 1 1 , 1 2 , 1 , 2 , 1 ,*2 , 1 1 , 1) (*2 , 1 1 , 1 2 , 1 ,1k k k k k kk k k kk k k k k k k kjk k k kn n n n n njr r jr re Ijn n jn n n j n n j n ejr r jr rk kEEEEEE Then, the optimal decision maker is:{ }*2 , 1 1 , 1 2 , 1 ,) ( ) ( Re + +k k k kjr r jr r> + + + === --- Po-Ning Chen --- NCTU/CM5-175Appendix B Answer:= + +> + + + + + + + = = = =1 if , ) (21exp ) (/ 1/) , (1 if ,1 2) () / 1 (] 2 / ) ( exp[ 1 2) / 1 (] 2 / ) ( exp[ ) () (21exp ) ( ) , (2 201 21 2111101 212121 21 22 210 121 21 22 20 2 20 1L b a ab Iv vv vb a QLvvbavvabkLab Iv vb avvkLv vb a ab Ib a ab I b a QPLnn Lkk L n k nn LLkkLb+ + => > =+ + + = + = + + +=+ + = + == =+ +=+ = =+=+ = =kind first the of function Bessel modified order th , 0 ,) 1 ( !) 2 / () (function - Q Marcum , 0 , ) ( ) , (| | | |), | | | )(| | (| 2,) | )(| | | ( 4) | )(| | | ( 41,) | )(| | | ( 41, , ,) () ( 2,) () ( 2 where0202 / ) (1*, ,*,*,2,2, 2* *, , ,*,2,2,212 2* *2 222 2 22112 211 12 / 122 12 1 122 12 / 122 12 1 1 2212 2j xk j kxx Ia b ab Ibae b a Qm m C m Cm m B m Am m m m m m AB CAB CC C B AwwAB Cw v wAB Cw vv vv v vbv vv v vakk jjkkkb ay k x k y k x k y k x k kxy y k x k xy y k x k xx y k yy x k kxy yy xxxy xy yy xxxy yy xx xy yy xxLkkLkk 5-176Appendix B Apply the Appendix B to DBPSK for (k k1) = . = 0exp21NEPb= = = + = == == = 1 ) ( lim , 0 ) / 2 , 0 (2 ) ( , 2,1,2, 0 0 0 0 1) ( ) (2 0 102 12 / 101 1x I N E QE e e E ENNv vNEb axj jk k k k = = = = == + = = = = = + + = = + + = = 0 ,2, ,} Re{ 2 , 1 , 0 , 1) () (0 ) ( ) (*1 1 1*1*1 12 , 1 1 , 1) (1 12 1) (111xy yy xxjyjxk kjkk kjkNe m e mY X Y X Y X D C B A Ljn n e r Yjn n e r Xk kkk E EEE.) 0 ) ( when occurs error (Because1 = k k --- Po-Ning Chen --- NCTU/CM5-1775.2.8 Differential PSK (DPSK) and its performanceDBPSK is in general 1 dB inferior in bit-POE to BPSK/QPSK.DQPSK (see the formula in the next slide) is in general 2.3dB inferior in bit-POE to DBPSK.5-178 The performance of DQPSK can be derived as (k k1)is either 0or /2 or or 3/2. Hence, we only need to consider: + + + + + + += + + + + + + + = *2 , 1 1 , 1 2 , 1 , 2 , 1 ,*2 , 1 1 , 1*2 , 1 1 , 1 2 , 1 , 2 , 1 ,*2 , 1 1 , 1) ( *1) )( (1) ( ) (10 for ,11 for ,01 for ,00 for , ) )( (1) ( ) (1k k k k k k k kk k k k k k k kjk kjn n jn n n j n n j njjjn n jn n n j n n j n e r rk kEEEEEEE E 5.2.8 Differential PSK (DPSK) and its performanceThen, the optimal decision maker is:{ } { }{ } { }*1*1*1*1Im Re bit, second for theIm Re bit, first for the +k k k kk k k kr r r rr r r r> 4 that 3 dB is a more accurate estimate.Hence, considering the implementation complexity (DPSK do not require an elaborate method for estimating the carrier phase) and its smaller-than-expectation performance degradation, DPSK is often used in practice especially for M = 2 and M = 4.--- Po-Ning Chen --- NCTU/CM5-1835.2.9 Probability of error for QAM( ) ( ) t f t g A t f t g A t sc ms c mc m 2 sin ) ( 2 cos ) ( ) ( : symbol channel QAM = Our goal is to find the relation of SNR versus POE. POE is (partially) decided by the minimum Euclidean distanceamong signal points. Hence, we can fix the minimum Euclidean distance among signal points (i.e., to some extent, fix the POE), and find the required signal power instead (i.e., to improve the SNR).5-1845.2.9 Probability of error for QAM Fix the minimum distance among signal points to be 2.--- Po-Ning Chen --- NCTU/CM5-1855.2.9 Probability of error for QAM Average transmitted power73 . 4 ]) 3 2 4 [ 4 2 4 (81) (1 ) (6 ) 8 4 4 4 (81) (1 ) (83 . 6 ]) 2 4 6 [ 4 2 4 (81) (1 ) (. 6 ) 10 4 2 4 (81) (1 ) (12 212 212 212 2= + + = + == + = + == + + = + == + = + =====Mmms mc avMmms mc avMmms mc avMmms mc avA AMP dA AMP cA AMP bA AMP aHence, the best 8QAM constellation is (d). In fact, (d) is the 8QAM constellation that minimizes average transmitted power subject to a minimum distance of 2.However, since rectangular QAM is more easily implemented,it is used more often in practice.5-1865.2.9 Probability of error for QAM POE of rectangular QAMEach dimension often contains 2k/2signal points, if M = 2k.It therefore can be treated as two 2k/2PAM.( )) ( 21 1) (1) (.) equivalent the of half - one is of power ed transmitt average that (Note.121212 22,,PAM M P AMAMA AMMQAM PMQAM PAM MP PavMmmsMmmcMmms mc avPAM M cMQAM c= + = + == = = =148.) - 5 slide (See ,) 1 (6 ) 1 ( 202 , ||.|

\| =N M MMPavMPAM e E||.|

\| =0,) 1 (6 ) 1 ( 2N M MMPavPAM M e E--- Po-Ning Chen --- NCTU/CM5-1872, ,22, ,220,20,,) 1 () ( log 32 1 1) 1 () ( log 3 ) 1 ( 21 1) 1 (3 ) 1 ( 21 1) 1 (6 ) 1 ( 21 1||.|

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\| =MQAM av bMQAM av bMQAM avPAM M avMQAM eMMMMMMN M MMN M MMPEE5.2.9 Probability of error for QAM5-1885.2.9 Probability of error for QAM--- Po-Ning Chen --- NCTU/CM5-189( ) 158.) - 5 slide (See ) / sin( 2 2,M PMPSK e ||.|

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\| 134134134132 1 122,MM MMPMQAM e Comparison in POE between QAM and PSK5.2.9 Probability of error for QAM5-1905.2.9 Probability of error for QAM) / ( sin ) 1 ( 23) / sin( 2) 1 /( 322M M MMRM =(((

= ObservationQPSK and 4QAM yield comparable (i.e., almost the same) performance, while MPSK performs worse than MQAM when M > 4.. n better tha performs somehow 1, , MPSK e MQAM e MP P R >M 10 log10RM4816326401. 654.207.029.95--- Po-Ning Chen --- NCTU/CM5-191 1e-06 1e-05 0.0001 0.001 0.01 0.1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 304QAM16QAM32QAM64QAM4PSK16PSK32PSK64PSK5.2.9 Probability of error for QAM4PSK16PSK 32PSK 64PSK5-1925.2.9 Probability of error for QAM From the performance comparison in the previous slide4PSK 4QAM in performance as predicted by the table at right)16PSK performs 4dB poorer than 16QAM (as predicted by the table at right)32PSK performs 7dB poorer than 32QAM (as predicted by the table at right)64PSK performs 10dB poorer than 64QAM (as predicted by the table at right)M 10 log10RM4816326401. 654.207.029.95--- Po-Ning Chen --- NCTU/CM5-1935.2.9 Probability of error for QAM Performance bound for non-rectangular QAMIt can be shown by a similar procedure as establishing (cf. slide 5-118) thatpoints. signal between distance Euclidean minimum the is d whereNdM PMQAM e 2) 1 ( min02min, ||.|

\| BFSK e MFSK eP M P, ,) 1 ( 5-1945.2.10 Comparison between digital modulation methods Is POE-versus-SNR a sufficient basis when comparing different digital modulation methods?Bit-POE versus bit-SNR : Comparison should be done under a fixed data rate of transmission Higher data rate surely has poorer performance.Bandwidth required Some researchers view the transmission data rateas a measure of the required bandwidth. In fact, such equivalencedepends on the modulation method and pulse shape used.--- Po-Ning Chen --- NCTU/CM5-1955.2.10 Comparison between digital modulation methods DSB-PAMPulse shape g(t).log is bandwidth consumed and rate data between relation the Hence, ) 3 (. / log / rate data the So bits. log transmits system the , During ) 2 (.1 bandwidth bandpass duration ) ( ) 1 (222MRWT M T k RM k TTW T t g== = = Bandwidth efficiency bit-rate to bandwidth ratioMWR2log ) Hz bit/sec ( efficiency Bandwidth = =5-1965.2.10 Comparison between digital modulation methods SSB-PAMPulse shape g(t).log 2 is bandwidth consumed and rate data between relation the Hence, ) 3 (. / log / rate data the So bits. log transmits system the , During ) 2 (.21 bandwidth bandpass duration ) ( ) 1 (222MRWT M T k RM k TTW T t g== = = Bandwidth efficiencyPAM DSB MWR > = =2log 2 efficiency Bandwidth--- Po-Ning Chen --- NCTU/CM5-197 DSB-QAM (no SSB-QAM)It doubles the data rate of DSB-PAM; and hence, double the bandwidth efficiency of DSB-PAM, which is equivalent to SSB-PAM. (FSK) Orthogonal signals (and its simplex signals)M frequency bands, separate in frequency by 1/(2T). Bi-orthogonal signals5.2.10 Comparison between digital modulation methodsMMWRRMMR MMTM W22 2log 2log 2 ) / (log 2 21= = = =MMWRRMMR MMTMW22 2log 4log 4 ) / (log 4 212 = = = =5-1985.2.10 Comparison between digital modulation methodsNormalized data rate R/W versus SNR per bit required to achieve a given error probabilityPower-limitedregion: R/W< 1Band-limitedregion: R/W> 1) ( log2M) ( log 22M) ( log 22MM M / ) ( log 22) ) / ( 1 ( log /2 bW C W C + =--- Po-Ning Chen --- NCTU/CM5-199 (Bits per second per hertz) versus (bit-SNR), under a fixed bit- or symbol-POE.M-ary orthogonal signals Increase M, Decrease bit-per-second-per-hertz. Usually, R/W < 1. Bad for channels with bandwidth constraints. But, decrease bit-SNR required for fixed symbol-POE. Good for channels with power constraints Note that as M approaches infinity, POE achieves zero (by increasing M) if bit-SNR > 1.6dB. In such case, W = infinity and R/W = 0. (cf. Slide 5-134)5.2.10 Comparison between digital modulation methods5-2005.2.10 Comparison between digital modulation methods (Bits per second per hertz) versus (bit-SNR), under a fixed bit- or symbol-POE.PAM, QAM, PSK Increase M. Increase bit-per-second-per-hertz. Good for channels with bandwidth constraints (since each hertz can be used more efficiently.) Usually, R/W > 1. But, increase bit-SNR required for fixed POE. Bad for channels with power constraints. Example of channel with bandwidth constraint : (digital)telephone channels and (digital) radio channels.--- Po-Ning Chen --- NCTU/CM5-2015.2.10 Comparison between digital modulation methods Normalized capacity under AWGN = Highest bit-per-second-per-hertz achievable (C/W)Shannon upper bound (1948) : A general upper bound for any modulation methods (For details, see chapter 7.)This is the bound for arbitrary small error probability, not just the bound corresponding to POE = 105. /1 21 log 1 log 1 log/20202W CrWCWWNCWWNPW CW Cb bb av = |.|

\| + =||.|

\|+ =||.|

\|+ = E--- Po-Ning Chen --- NCTU/CM6-1Digital CommunicationsChapter 6 : Carrier and Symbol Synchronization6-2Why synchronization?nl(t)Channelsl,m(t)rl(t)=sl,m(t-)+nl(t) +TransmittedsignalReceivedsignal Channel will introduce unknownpropagation delay .m = 1, 2, , M--- Po-Ning Chen --- NCTU/CM6-3 Example. Correlation demodulatorNeed to sample at the right time.( )Tdt0( )Tdt0( )Tdt01r2rNrdetector To{ } l orthonorma 1Nn nf =Why synchronization?Received signal rl(t)f1(t)f2(t)fN(t)Sample at t = T6-46.1 Signal parameter estimation Why signal parameter estimation?Output of the demodulator must be sampled at the correct symbol timing.--- Po-Ning Chen --- NCTU/CM6-56.1 Signal parameter estimation Received signal model due to AWGN{ }{ }t f jlt f j jlcce t z t se t z e t st n t s t r 22)] ( ) (~[ Re)] ( ) ( [ Re) ( ) ( ) (+ = + = + =where| | | |= = == = = . AWGN ) (2 delay; n propagatiosignal lowpass equivalent ) (; ) ( Re ) ( ) ( Re ) () ( 2 2t nf t se t s t s e t s t sclt f jlt f jlc c 6-66.1 Signal parameter estimation. and by determinedbe can where estimate to needs one ), ( recover to ) ) ( (or ) ( ) (~ From cl lft s t r t z t s +This is only theoretically correct; but not necessarily true in practice!--- Po-Ning Chen --- NCTU/CM6-76.1 Signal parameter estimation Estimation of phase shift and the propagation delay .Dependence on the accuracy of fc It seems that there is only one signal parameter to be estimated, because = 2fc. But since the receiver carrier frequency may drift away from fc at the transmitter site. Hence, the phase turns out not only depending on . Estimate accuracy consideration Estimate of must be within a relative small fraction, say 1%, of T for the sake of sampling. However, for =2fc, within 1% of accuracy in the estimate of may still lead to a large deviation of because fcis usually large.Thus, separate estimate of and is technically necessary. 6-86.1 Signal parameter estimation Estimation of phase shift and the propagation delay .Estimation criterions ML (Maximum likelihood) estimator MAP (Maximum a posteriori probability) estimatorEstimation approaches One-shot estimator, based on one observation Tracking loop estimator, continuously update the estimate.--- Po-Ning Chen --- NCTU/CM6-96.1.1 The likelihood function Assumptions and notations.The transmitted signal s(t) and received signal r(t) can be spanned by N orthonormal functions {fn} (over interval T0)Let ) ( ) , ; ( ) ( ) ( ) ( t n t s t n t s t r + = + = Then||.|

\|= = = Nnn nN s rNNeNr r p102/ )] , ( [011) , | ) ,..., ( ( rwhere. ) ( ) , ; ( ) , ( and ) ( ) (0 0 = =Tn nTn ndt t f t s s dt t f t r r 6-106.1.1 The likelihood function Although it is possible to derive the parameter estimates based on [r1, r2, , rN], it is convenient to deal directly with the signal waveforms. Assume that and are deterministic (or slowly time-varying).Hence, ML is adopted as the criterion in subsequent discussion. T0, which is referred to as observation interval, is usually (much) larger than T.--- Po-Ning Chen --- NCTU/CM6-116.1.1 The likelihood function == + = == =((

+ =((

=Nnn nTNnNn mm n m m n nNnn n nTNnn n nTs rNdt t f t f s r s r t f s rNdt t f s rNdt t s t rN12001 1 12 200210020)] , ( [1) ( ) ( )] , ( )][ , ( [ 2 ) ( )] , ( [1) ( )] , ( [1)] , ; ( ) ( [1 Observe that||.|

\|= 020)] , ; ( ) ( [101) , | ) ( (Tdt t s t rNNeNt r p This becomes a function of the signal waveform6-126.1.1 The likelihood functionThe ML estimator is therefore: =02) , ()] , ; ( ) ( [ min arg ) ,(TML MLdt t s t r --- Po-Ning Chen --- NCTU/CM6-136.1.2 Carrier recovery and symbol synchronization in signal demodulation Exemplified block diagram : Binary PSK| | ( ) ( )2 , 1 and , ) 3 2 ( re whe 2 cos ) ( Re ) ( : symbol channel) 2 (= = + = = +m d m At f t g A e t g A t smc mt f jm mc This is a bandpass-based correlation-type demodulator! Here, the integrator acts like a lowpass filter. Later, we will actually replace the integrator by a low-pass filter in PLL analysis. 6-136-146.1.2 Carrier recovery and symbol synchronization in signal demodulation Exemplified block diagram : M-ary PAM| | ( ) ( )M m d M m At f t g A e t g A t smc mt f jm mc,..., 2 , 1 and , ) 1 2 ( re whe 2 cos ) ( Re ) ( : symbol channel) 2 (= = + = = + Received Receivedsignal signalAGC at the front end is to eliminate channel gain variations, which is important to amplitude-sensitive demodulator such as M-ary PAM and QAM.--- Po-Ning Chen --- NCTU/CM6-15Effect of phase error: An example on BPSK and M-ary PAMThen,). ( ) ( where ), 2 cos( )] ( / ) ( [ ) ( Let = + = t g A t A t f t g t A t sm c m)cos( ) (21)cos( ) (21)cos( ) (21)2 cos( ) 2 cos( ) ( )]2 cos( ) ( )[ (Low Pass + + 4 + =+ + = + t At f t A t At f t f t A t f t g t scc c c m Hence, signal power is reduced by a factorA phase error of 10oleads to 0.13 dB of signal power loss.A phase error of 30oleads to 1.25dB of signal power loss.)( cos2 6-166.1.2 Carrier recovery and symbol synchronization in signal demodulation Exemplified block diagram : M-ary PSK| |( ) ( )( ) ( ) ( ) ( ) ( ) ( )M m M mt f t g t f t gt f t ge e t g t smc m c mm ct f j M m jmc,..., 2 , 1 and / ) 1 ( 2 re whe 2 sin sin 2 cos cos 2 cos ) ( Re ) ( : symbol channel) 2 ( / ) 1 ( 2= = + + = + + = = + --- Po-Ning Chen --- NCTU/CM6-176.1.2 Carrier recovery and symbol synchronization in signal demodulation Exemplified block diagram example : M-ary PSKReceived Receivedsignal signal6-186.1.2 Carrier recovery and symbol synchronization in signal demodulation Exemplified block diagram( ) ( ) + + = t f t g A t f t g A t sc ms c mc m2 sin ) ( 2 cos ) ( ) ( : symbol channel PAM - Q--- Po-Ning Chen --- NCTU/CM6-19Effect of phase error: An example on QAM and M-ary PSK) 2 sin( ) ( ) 2 cos( ) ( ) ( + + = t f t B t f t A t sc c m + = =)sin( ) (21)cos( ) (21) ()sin( ) (21)cos( ) (21) ( t A t B t yt B t A t yQI Hence, not only the desired signal (first term) power is reduced by a factor of but a cross-talk interference is induced between in-phase and quadrature components. Therefore, the phase error requirement for QAM and M-ary PSK is much higher that that of M-ary PAM.)( cos2 6-206.2 Carrier phase estimation In this topic, we assume that is known. In other words, only needs to be estimated. Common approaches Multiplex a pilot signal, often in frequency, that allows the receiver to extract and to sync its local oscillator to the carrier frequency of the received signal. Derive the carrier phase directly from the (suppressed carrier) modulated signal More prevalent in practice Advantage All the transmission power is allocated to the information-bearing signals.--- Po-Ning Chen --- NCTU/CM6-216.2.1 Maximum-likelihood carrier phase estimation Assume the propagation delay is known. Then the ML estimator of phase becomes| |=+ = =000) ; ( ) ( max arg) ; ( ) ; ( ) ( 2 ) ( min arg)] ; ( ) ( [ min arg2 22TTTMLdt t s t rdt t s t s t r t rdt t s t r 6-226.2.1 Maximum-likelihood carrier phase estimation Example. ML phase estimator.) ( ) 2 cos() ( ] ) ( Re[ ) ( Then) 2 (t n t f At n e t s t rct f jlc+ + = + = + . ) ( Assume A t sl =, ) ; ( ) ( max argobtain To0=TMLdt t s t r . 0) ; ( ) ( have must we0==MLTdt t s t r (cf. slide 6-5)6-22--- Po-Ning Chen --- NCTU/CM6-236.2.1 Maximum-likelihood carrier phase estimation(((

= = += + = = 0 00 00 0 0) 2 cos( ) ( ) 2 sin( ) ( arctan. 0 ) 2 cos( ) ( ) sin( ) 2 sin( ) ( ) cos( ly, Equivalent. 0 ) 2 sin( ) ( ) ; ( ' ) ( ) ; ( ) (TcTc MLTcTcTcT Tdt t f t r dt t f t rdt t f t r dt t f t rdt t f t r A dt t s t r dt t s t r ). ( ) ; ( ) ( and ) 2 cos( ) ; ( t n t s t r t f A t sc + = + = 6-236-246.2.1 Maximum-likelihood carrier phase estimation A one-shot ML phase estimator for a time-independent sl(t)Oscillator21 |.|

\|=XYMLarctanXYr(t)cos(2fct)sin(2fct)0) (Tdt 0) (Tdt (((

= 0 0) 2 cos( ) ( )] 2 sin( )[ ( arctanTcTc MLdt t f t r dt t f t r --- Po-Ning Chen --- NCTU/CM6-25 A PLL (Phase-Locked Loop) ML phase estimator for a time-independent sl(t)0 )2 sin( ) (0= +TML cdt t f t r VCOr(t))2 sin(ML ct f + 0) (Tdt Tuning VCO to make it zero..6.2.1 Maximum-likelihood carrier phase estimation6-26 An (equivalent) PLL ML phase estimator for a time-independent sl(t)| | 0 )2 sin( ) ( )2 sin( ) (0 0 = + = +TML cTML cdt t f t r dt t f t r VCOr(t))2 sin(ML ct f + 0) (Tdt Tuning VCO to make it zero..6.2.2 The phase-locked loop--- Po-Ning Chen --- NCTU/CM6-276.2.2 The phase-locked loop Basic diagram of a PLL6-286.2.2 The phase-locked loop Analysis of the tracking loop estimatorAssume noise-free environment, i.e., n(t)=0.VCO) 2 cos( ) ( + = t f A t rc0) (Tdt Tuning VCO to make it zero.. )2 sin(ML ct f + e(t))4 sin(2)sin(2)2 sin( ) 2 cos( ) (ML c MLML c ct fA At f t f A t e + + =+ + =--- Po-Ning Chen --- NCTU/CM6-296.2.2 The phase-locked loop). 2 / 1 ( of multiple if , 0 )sin(2)4 sin(2)sin(2) (000 0 0c MLTML cTMLTf TATdt t fAdtAdt t e= =+ + = VCO) 2 cos( ) ( + = t f A t rc0) (Tdt )2 sin(ML ct f + 0 )sin(2) (00 = =MLTATdt t e )4 sin( ) 2 / ()sin( ) 2 / ( ) (ML cMLt f AA t e + + = The effect of Integration is similar to a low-pass filter.6-296-306.2.2 The phase-locked loopVCO) 2 cos( ) ( + = t f A t rc0) (Tdt )2 sin(ML ct f + 0 )sin(2) (00 = =MLTATdt t e )4 sin( ) 2 / ()sin( ) 2 / ( ) (ML cMLt f AA t e + + =VCO) 2 cos( ) ( + = t f A t rc)2 sin(ML ct f + 0 )sin(2 = MLA )4 sin( ) 2 / ()sin( ) 2 / ( ) (ML cMLt f AA t e + + =LowpassFilter--- Po-Ning Chen --- NCTU/CM6-316.2.2 The phase-locked loopVCO) 2 cos( ) ( + = t f A t rc)2 sin(ML ct f + )sin(2MLAv =)4 sin( ) 2 / ()sin( ) 2 / ( ) (ML cMLt f AA t e + + =LowpassFilter VCO can be modeled as a sinusoidal signal generator with an integration-based phase adjustment. + = +tc cd v K t f t t f ) ( 2 ) (2 Sinusoidal term of VCO output = =td v K t ) ( ) (Or phase of VCO output = Starting from this view, we can reduce the above block diagram to an equivalent closed-loop PLL system model.6-32Equivalent close-loop system model of PLLVCO) 2 cos( ) ( + = t f A t rc)2 sin(ML ct f + )sin(2MLAv =)4 sin( ) 2 / ()sin( ) 2 / ( ) (ML cMLt f AA t e + + =LowpassFilter G(s)sin( )) (t ) () ( t t VCO=K/s) (t vLoopFilter G(s) (t) +)] () ( sin[ t t =td v K VCO ) (: The existence of sin( ) function yields a non-linear effect in the system, which causes difficulty in its analysis.--- Po-Ning Chen --- NCTU/CM6-33Equivalent close-loop system model of PLLVCO=K/s) (t ) (t v ) () ( t t LowpassFilter G(s) (t) +Linearized PLL modelsmall. is )( when ,)sin( function. transfer system loop - closed (linear) a us gives Thiss s KGs s KGs s s s KG s ss s s s KGs s sssss H/ ) ( 1/ ) ()] () ( )[ / ) ( ( )] () ( [)] () ( )[ / ) ( () ()] () ( [) () () () (+= + =+ = = 6-34Analysis of equivalent close-loop system model of PLL Analysis based on G(s) for second-order loop transfer function. (Linearized PLL model)filter.) lowpass a for (Usually, 11) (2 112 >>++=sss G21 22) / ( ) / 1 ( 11) ( function transfer loop closeds K s Kss H + + + += where. 2 / ) / 1 ( factor damping loop and / loop the of frequency natural21KKnn+ = = 1 ) / ( 2 ) / (1 ) / )( / 2 (2) / 2 () , , ; (2 2 22 2+ + + =+ + + = n nn nn nn n nns ss Ks ss KK s H --- Po-Ning Chen --- NCTU/CM6-35-20-18-16-14-12-10-8-6-4-2024680.1 0.2 0.3 0.4 0.5 0.7 1.0 2 3 4 5 7 10=0.3=0.5=0.707=1.0=2.0=5.0n /| ) ( | log 2010 j HCritical dampingOverdampingUnderdamping=0.3=0.5=0.707=1.0=2.0=5.0Analysis of equivalent close-loop system model of PLL Damping factor = response speed (for changes)|.|

\| >> >> 0 as 22Kn. 2111 221 2 2 Assume2 ||.|

\|+ =||.|

\| = K K Kn n6-36Analysis of equivalent close-loop system model of PLL (one-sided) Noise-equivalent bandwidth of H(f)(cf. Problem 2-24)nnnfeqdf f Hf HB 84 1/ 8) ( 1| ) ( || ) ( | max12 22022 ++= = Tradeoff of parameter selection in PLL It is desirable to have larger PLL bandwidth in order to track any time variations in the phase of the received carrier. However, with larger PLL bandwidth, more noise will be passed into the loop; and hence, the phase estimate is more inaccurate. The above analysis is based on a noise-free assumption. The effect of noise will be considered in the next subsection.|H(f)|2f Beq 2 22 =Knn--- Po-Ning Chen --- NCTU/CM6-3784 12+Analysis of equivalent close-loop system model of PLL 0 0.5 1 1.5 2 2.5 30.3 0.50.707 1 2 56-386.2.3 Effect of additive noise on the phase estimate Assume n(t) is the wideband white noise, and represent it as: =W/Hz. ) 2 / 1 ( density spectral power sided - two with ) ( ) () 2 sin( ) ( ) 2 cos( ) ( ) (0N t y t xt f t y t f t x t nc c Analysis + + = + + =)] ( 2 sin[ ) ( )] ( 2 cos[ ) ( ) () ( )] ( 2 cos[ ) (t t f t n t t f t n t nt n t t f A t rc s c cc c. 5) - 2 Problem (cf. ) ( and ) ( as statistics same the have ) ( and ) ( Hence,.)] ( cos[ )] ( sin[)] ( sin[ )] ( cos[) () () () (t y t xt n t nt tt tt yt xt nt ns csc||.|

\| ||.|

\|=||.|

\| where--- Po-Ning Chen --- NCTU/CM6-396.2.3 Effect of additive noise on the phase estimateVCO)] ( 2 sin[ ) ()] ( 2 cos[ )] ( [ ) (t t f t nt t f t n A t rc sc c c + + + =)] (2 sin[ t t fc + VCO)] ( 2 sin[ ] / ) ( [)] ( 2 cos[ ] / ) ( 1 [ ) (t t f A t nt t f A t n t rc c sc c c + + + =)] (2 sin[ t t fc + 0 ) (21)] () ( sin[212 = + t n t t )] (2 sin[ ) ( ) ( t t f t r t ec + =)]). () ( cos[ ) ( )] () ( sin[ ) ( (1) ( Let 2t t t n t t t nAt ns cc + =0) (10TdtT0 )] () ( cos[2) ( )] () ( sin[2) ( )] () ( sin[2) (0= + + t tt nt tt nt tAdt t esccT )4 cos(2) ()cos(2) ()4 sin(2)] ( [)sin(2)] ( [)2 sin( ) 2 sin( ) ( )2 sin( ) 2 cos( )] ( [)] (2 sin[ ) ( ) (ML csMLsML cc cMLc cML c c s ML c c c cct ft n t nt ft n A t n At f t f t n t f t f t n At t f t r t e + + ++ ++ +=+ + + + + + = + =0) (10TdtT6-406.2.3 Effect of additive noise on the phase estimateVCO)] ( 2 sin[ ] / ) ( [)] ( 2 cos[ ] / ) ( 1 [ ) (t t f A t nt t f A t n t rc c sc c c + + + =)] (2 sin[ t t fc + 0 ) (21)] () ( sin[212 = + t n t t 0) (10TdtTVCO)] ( 2 sin[ ] / ) ( [)] ( 2 cos[ ] / ) ( 1 [ ) (t t f A t nt t f A t n t rc c sc c c + + + =)] (2 sin[ t t fc + 0 ) (21)] () ( sin[212 = + t n t t )] (2 sin[ ) ( ) ( t t f t r t ec + =)4 cos(2) ()cos(2) ()4 sin(2] / ) ( 1 [)sin(2] / ) ( 1 [)] (2 sin[ ) ( ) (ML ccsMLcsML cc cMLc cct fAt nAt nt fA t n A t nt t f t r t e + + ++ ++ +=+ =Lowpass Filter--- Po-Ning Chen --- NCTU/CM6-416.2.3 Effect of additive noise on the phase estimateVCO)] ( 2 sin[ ] / ) ( [)] ( 2 cos[ ] / ) ( 1 [ ) (t t f A t nt t f A t n t rc c sc c c + + + =)] (2 sin[ t t fc + ) (21)] () ( sin[21) (212t n t tt v+ = )] (2 sin[ ) ( ) ( t t f t r t ec + =Lowpass Filter G(s)=VCO=K/s) (t ) (t v) () ()) () ( sin(t tt t LoopFilter G(s) (t) +) (2t nLinearizedPLL model.2) ( with ) (2021cnANf t n = white6-426.2.3 Effect of additive noise on the phase estimate( )22002202202222222| ) ( | max | ) ( | | ) ( |2 | ) ( | ) ( ) (tic. determinis assumed is ) ( where ), ( * )] ( ) ( [ ) () ( )] ( ) ( [ ) (/ ) ( 1/ ) () () ( ) () () () ( ) () ( ) ( f H BANdf f HANdf f HANdf f H fndf ft t H t n t ts H s n s ss s KGs s KGs Hs n sssK s Gs n s s sfeqccc === = = + = + = += =+ + = --- Po-Ning Chen --- NCTU/CM6-436.2.3 Effect of additive noise on the phase estimatenoise. to due error estimation the of variance the is 22| ) ( | max f HfL cfeqAf H B N1power signalpower noise) | ) ( | max 2 )( 2 / (2202 = = = Thus, the (approximate-by-linearized model) signal-to-noise ratio is:2022| ) ( | max1f H B NASNRf eqcL = = =eqB eqB2| ) ( | f Heq6-44sin( )) (t ) () ( t t VCO=K/s) (t vLoopFilter G(s) (t)+Exact PLL model) (2t nVCO=K/s) (t ) (t v) () ( t t LoopFilter G(s) (t)+Linearized PLL model) (2t n6.2.3 Effect of additive noise on the phase estimate Exact PLL model versus Linearized PLL model--- Po-Ning Chen --- NCTU/CM6-456.2.3 Effect of additive noise on the phase estimate It turns out that the exact PLL model is tractableunder G(s) = 1. (Viterbi, 1966) Specifically, the pdf for phase error (under Gaussian distributed n2) is established as{ } ) cos( exp) ( 21) (0 = LLIpwhere zero. order of function Bessel modified the is ) ( and loop, order - first the of bandwidth equivalent noise the being with| ) ( | max002202= =IBB NAf H B NAeqeqcf eqcL6-466.2.3 Effect of additive noise on the phase estimateThe variance for the linear model is close to the exact variance for L > 3 dB.Hence, the linear model is adequate for practical purposes.00.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1 1.220/ / 1c eq LA B N = Variance of VCOphase estimateExactLinear model1/32(First-order PLL)--- Po-Ning Chen --- NCTU/CM6-476.2.3 Effect of additive noise on the phase estimate DiscussionsTransient behavior at initial acquisition is important for PLL. Notably, the analysis based on linearized model assumes that is small so that sin() can be well approximated by . Behavior of PLL at low SNR is also important. There is a rapid deterioration in the performance of the PLL at low SNR. In such case, the loop will lose lock.6-486.2.4 Decision-directed loops In the previous analysis, the input to the phase estimator does not carry information. Such input is often referred to as unmodulated carrier signal. In this subsection, modulated carrier signal will be considered (in terms of the form of equivalent lowpass signal.)) ( ) 2 cos( ) ( t n t f A t rc + + = ) ( ) () ( ) ( ) (t z nT t g I et z e t s t rnnjjl l+ =+ = =in imaginarycomplex domain.in real domain.) ( )] ( 2 cos[ ) ( t n t t f A t rc c + + = input--- Po-Ning Chen --- NCTU/CM6-496.2.4 Decision-directed loops Analysis = =nn lnT t g I t s ) ( ) ({ } ( ) { }) 2 ( 2) ( Re ) ( Re ) ( + = =t f j jlt f jlc ce e t s e t s t s{ }) 2 () ( Re ) ( +=t f jlce t r t r) ( ) ( ) ( t z e t s t rjl l + = 6-50 AssumptionsBoth the Re-part and Im-part of the transmitted signal sl(t)ejand received signal rl(t) can be spanned by N orthonormal functions {fn}.Let Then ( )||.|

\|= = = + Nii i i is r s r NNN NeNr r r r p12 20)] ( [ )] (~ ~[ ) / 1 (201 11) | ) ,..., ,~,...,~( ( rwhere= == = 0 00 0. ) ( ) ; ( ) ( and ) ( ) (; ) ( ) ; (~) (~ and ) ( ) (~ ~Ti l iTi l iTi l iTi l idt t f t s s dt t f t r rdt t f t s s dt t f t r r 6.2.4 Decision-directed loops)] ( ) (~[ )] ; ( ) ; (~[ ) ( ) (~) ( t z j t z t s j t s t