Dgtal Modems Lecture
Revew We have shown that both Bayes and eyman/pearson crtera are based on the Lkelhood Rato Test (LRT) Λ ( r ) < > η Λ r s called observaton transformaton or suffcent statstc The crtera dffer on the choce of threshold η Bayes crteron s used for mnmum probablty of error detectors (e.g., communcaton systems) eyman crteron s used for mnmum false alarm probablty (e.g., radar) Categorzaton of problems: Smple hypotheses (the noseless sgnal s perfectly known) Composte hypotheses (there are unknown parameters n the sgnal) The parameters have a statstcal descrpton The parameters do not have a statstcal descrpton
Dscrete representaton of contnuous-tme sgnals Up tll now, we have consdered optmal detectors for problems nvolvng dscrete-tme sgnals (vectors) In realty, a contnuous-tme sgnal (waveform) s transmtted What s the optmal detector n ths case? It turns out that the optmal detector s stll based on dscretetme processng, by transformng the problem from contnuous to dscrete tme
Dscrete representaton of contnuous-tme sgnals s() t φ s s s 3 s φ nt () φ 3 φ n n 3 n φ r s n r s n = + r s n φ 3
Determnstc sgnals () Let s t be a (complex) determnstc sgnal and be an orthonormal bass,.e., * (), () () () φ t φ t = φ t φ t dt = δ j j j It can be shown that x t can be wrtten as A bass s called complete when t holds { φ( t), φ( t),, φ ( t) } The most common orthonormal bass s t e j π nt T φ n =, t, T () = φ (), = (), φ () s t s t s s t t = T lm s() t s () φ t dt = = [ ]
Determnstc sgnals Orthonormal expanson Equvalent realzatons., φ t s s( t)., φ ( t) s s( t)., φ t., φ ( t ) s s s t Correlator s( t) Matched flter * φ t ( ) * φ T t T dt t = T s s
Comments { } s Snce the coeffcents can be used to reconstruct the orgnal sgnal s( t), we say that ths dscretzaton procedure s nformaton lossless. Recall that the samplng theorem states the same thng. Exercse: State the samplng theorem as an orthonormal sgnal expanson. What s the orthonormal bass? In general, a contnuous tme sgnal s of nfnte dmensonalty, and therefore requres an nfnte number of coeffcents to descrbe t, whch s clearly not practcal!! In communcatons, we employ contnuous tme sgnals of fnte dmensons, so that ther dscrete tme representaton s also of fnte dmenson
Examples BPSK: =± cos( π ), [, ], = { () } s t A f t t T f k T Usng φ t = cos π ft, φ t = φ3 t =. = the BPSK sgnal can be wrtten as st = ± Aφ t BPSK s one-dmensonal wth M-PSK: s t = Acos π f t+ θ, m=,,, M, t, T Rewrtng the sgnal as we can choose () M-PSK s two-dmensonal wth s = ± A () [ ] { s, s } = Acos ( θ ), Asn( θ ) m st = Acos( θm) cos( π ft ) Asn ( θm) sn( π ft ) ( t) = cos( f t), ( t) = sn( f t) φ t { φ π φ π } { m m } A A φ ( t ) φ ( t )
Examples BFSK: Frequences, are chosen such that Choose () = cos( π ), =,, [, ] s t ft t T BFSK s two-dmensonal wth T f f ( ft) ( ) ( t) = cos( ft), ( t) = cos( f t) { s, s } = {, } or { s, s } = {,} { φ π φ π } φ ( t ) cos π cos π f t dt = MFSK: M-dmensonal sgnal (show t) φ () t FSK s an example of an orthogonal transmsson scheme
Examples φ ( t ) M-QAM: A, A I Q () = cos( π ) + sn( π ), [, ] s t A f t A f t t T : n-phase/quadrature symbols M-QAM s two-dmensonal I Q φ () t OFDM: M: number of orthogonal sub-carrers OFDM s an M-dmensonal sgnal DS-CDMA: M () M: number of users M As ( t) s t = (,, ) [ ] st = A cos π ft+ A sn π ft, t, T, f = πkt = = I k Q k k M { ()} = s t : P sequences ( s ) t, sj t = δj DS-CDMA s an M-dmensonal sgnal wth φ ( t) = s( t), =,. M
Stochastc sgnals Just lke determnstc sgnal, stochastc sgnals can also be represented by an nfnte sum of weghted orthogonal bass functons ( ; ) = φ, = (, ), φ ntu n u t n u ntu t = However, coeffcents n = n u are random Snce the sgnal s random we assume convergence of the seres n the Mean Square Error (MSE) sense,.e, lm E ntu ( ; ) n( u) φ( t) = =
Karhunen-Loeve theorem Let nt be a stochastc process wth autocorrelaton Rnn t, t. nt Sdscan be expanded on the orthonormal bass φ t that s a { } soluton of the followng system of ntegral equatons: T * (, ) φ = λφ R t t t dt t nn Each soluton φ t s called egenfuncton of R t, t and λ s the correspondng egenvalue { φ } { n } For that choce of t, coeffcents are uncorrelated nn
Comments T E n t dt λ cφ t are also vald egenfunctons Dfferent egenvalues λ λ j correspond to dfferent egenfunctons φ t φ t Each egenvenlue wth multplcty greater than one s assocated wth a set of egenfunctons of the same multplcty that can also be chosen to be orthonormal (Gram-Schmdt procedure) { } E n () = = = { } = λ j * Mercer s theorem: Rn t, t = = λφ t φ t For nt Gaussan, the K-L coeffcents { n } are also Gaussan and therefore ndependent (snce they are uncorrelated)
Example : Band-lmted process Gaussan process nt wth strctly band-lmted power spectrum observed n the tme perod [, T] { } The solutons φ t to the K-L ntegral equaton are called prolate spherodal functons The number of essentally non-zero egenvalues s equal to BT S ( f ) Rnn ( τ ) nn B B
Example : Whte nose nt R ( τ ) = σδτ Gaussan process wth autocorellaton nn It can be shown that the K-L ntegral equatons are satsfed by any orthonormal set of functons { } E E { j} = = n λ σ nn * =, j { φ () t }
Applcaton to the contnuoustme detecton problem The receved sgnal s r t = s t + n t; u, t, T st () s an -dmensonal sgnal ntu ( ; ) s whte nose (nfnte dmensonal sgnal) r t s nfnte-dmensonal (due to the presence of nose) () { ()} = () Let φ t be the set of bass functon for { } φ t can be (arbtrarly) extended to = so that t represents a bass for Therefore, we end up wth [ ] ntu ( ; ) s( t) { { φ ()}, { ()} } t φ t = = + = φ + φ r t s t n t = = = rφ = r = s + n ( s n ) φ () t nφ () t = + + = = + () t
Applcaton to the contnuoustme detecton problem ote that snce are ndependent of, only coeffcents r = s + n provde nformaton about the transmtted sgnal { r } = { n } { } = n = + { } { } = = The set s called suffcent statstc Thus, we have establshed the followng equvalence () = () + ( ; ), [, ] { } = { + } r t s t n t u t T r s n = = The problem s now formulated as a dscrete-tme problem, so that we can use the optmal detecton crtera (Bayes, eyman)