Block Diagram of a DCS in 411

Transcription

1 Informaion source Forma A/D From oher sources Pulse modu. Muliplex Bandpass modu. X M h: channel impulse response m i g i s i Digial inpu Digial oupu iming and synchronizaion Digial baseband/ bandpass waveform h: Waveform channel bandwidh limied noise ISI Informaion sink Forma D/A mˆ i Demuliplex o oher desinaions Deec y Demodulae & sample r R C V Block Diagram of a DCS in 4

2 Chaper 3 Waveform Coding echnique ex book, Chaper 5 Purpose: o sudy baseband ransmission sysem wih A/D and D/A converers.. Review of Inroducion o PCM. Binary Signal Deecion in AWGN and Error Probabiliy of PCM Sysem 3. Muliplexing of PCM Signals 4. Uniform Quanizaion and Signal-o-Noise Raio 5. Differenial Pulse-Code Modulaion DPCM 6. Dela Modulaion Gong

3 . Review of Inroducion o PCM Analog-o-Digial A/D conversion : a sample a δ { } quanize a i PCM encode.. binary codeword a : coninuous - ime, coninuous - ampliude analog signal a δ : discree - ime, coninuous - ampliude discree signal { }: discree - ime, discree - ampliude digial signal. a i where a δ a δ n i s, s is he sampling inerval.

4 Discreize: o discreize he ampliude of a δ, he mos convenien way is quanize he ampliude o L n levels. log L n log n bis hen, each signal level can be represened by an n-bi codeword a bi sream is called a Gong 4

5 For example, an n 3 bis digial signal can be ransmied o an L n 8 level PAM signal or be represened as a 3-bi codeword and ransmied using a level PAM signal. Level L bi codeword Level L bi codeword

6 Pulse Code Modulaion PCM he ransmission of an L-level signal, obained from he quanized signals, as an n-bi codeword, -level signal is referred as pulse-code modulaion PCM. Procedure of PCM a ransmier: Sep. he source informaion is sampled. Sep. Each sample is quanized o one of L levels. Sep 3. Each quanized ampliude is encoded ino an n-bi codeword. Sep 4. Each bi is represened by a baseband waveform for Gong 6

7 Code Quanizaion Example. number lever x v 4 x s s 3 s 4 s 5 s 6 s 7 s Naural sample value Quanized sample value Code number PCM sequence Pulse waveform s s 3 s 4 s 5 s 6 s 7 s

8 ransmission of PCM signals a sampler a δ f s W quanizer { } i PCM encoder PCM sequence a { } f s b i nf s Pulse generaor PCM waveform channel δ + n r X r Mached filer rae nf s Decision device { } i nf bˆ { } s PCM decoder â i f s Reconsrucion filer aˆ R X a i : an L-level digial signal wih L n ; b i : a binary sequence. Noe. One daa sample is represened by n bis. hus he bi rae nf s.

9 Operaion a x ransmier: Sampling Quanizing 3 Encoding ino binary codewords 4 Waveform represenaion of each bi of he codeword. Operaion a Rx receiver: Mached filering and equalizaion Synchronizaion for correc sampling 3 hreshold decision o produce binary codeword sequence 4 Decoding o conver PCM o PAM decoder-oupu 5 Reconsrucion he analog signal by pass PAM signal hrough a low-pass filer wih cuoff frequency equal o he message Gong 9

10 .. Binary Signal Deecion in in AWGN and Error Probabiliy of of PCM Sysem hree major sources of noise ha effecs he performance of PCM sysems: Channel noise, modeled as AWGN in his secion Quanizaion noise in he nex secion Inersymbol inerference in Chaper Gong

11 he effec of channel noise is o inroduce ransmission errors in reconsrucion of he original PCM waveform a he receiver oupu. In oher words, misaken for a symbol misaken for a Gong

12 Noe. he following reamen is general, i can be applied o any binary waveform ransmission. Objecive: o compue he probabiliy of ransmission error for he opimal receiver. In which sense is i opimal? Examples of Signal Ses for Binary Daa ransmission Waveform selecion: he possible waveforms for binary symbols and. s s s where s is a real funcion wih duraion, i.e., A s oherwise his is referred o as he on-off signaling or non-reurn-zero NRZ signaling.

13 Anipodal signal se: Orhogonal signal se: s s s s < oherwise, / /, s oherwise, /, s A A

14 binary sequence General Model modulaor s i + r si + n deecor: mached filer h Y decision variable Y sio + no AWGN: decision device N n wih SN f binary sequence Receiver srucure for deecing binary signals in AWGN Srucure of Receiver: he above figure, LI, which is he mached filer, is followed by a sampler and hreshold comparaor.

15 Descripion of he Receiver Srucure Channel noise process: he signal s only corruped by he AWGN process n wih psd N /. Here noise includes he hermal noise originaed in he receiver iself. Received signal: due o he memoryless channel, he received signal r s δ + n s i i + n, i, where n is addiive noise wih psd his is he inpu o he LI filer. S n f N he oupu of he filer i will be he mached filer in he opimal case is denoed by Y s io + n o, i, where s io is he oupu of he mached filer o he inpu s i n o is he oupu of he mached filer o he inpu n

16 s i h s io n WSS h n o WSS he filer is followed by a sampler. he oupu of he sampler is he random variable Y sio + n he hreshold device: Y is compared wih a hreshold value α. Decision rule: Y α if, decide is sen Y < α if, decide is sen

17 Noise Componen of he Filer Oupu S he psd of he noise componen n o is given by N n o f S n f H f H f n WSS h n o WSS he average power of n o is P + n E[ n S f o o ] n o df S f H f df E[ n ] + N + + N H f df h d n Gong 7

18 Inpu o he hreshold Device Noe. A random process X is called a Gaussian process if a ime, X is a Gaussian random variable, and for any,,, n, he random variables X, X,, X n are joinly Gaussian. Propery : Boh Y and n o are gaussian processes why?. Decision variable: he decision is made based on he oupu of he deecor Y a ime. We define i as a decision variable, i.e., he decision variable is given by Y sio + no Quesion: From Propery, boh Y and n o are gaussian random variables, wha are heir means and variances? If X is a gaussian random variable wih mean μ and variance σ, we denoe i as X ~ N μ, Gong 8

19 Case : suppose ha is sen. he decision variable is given by Y so + no We define a random process: Y s n o + o Case : suppose ha is sen. he decision variable is given by Y s o + no We define a random process: Y s n o + o hen Y Y, for all hen Y Y, for all he mean of he variance of Y μ E[ Y ] so Y Var Y E[ n ] R n o he mean of Y μ E[ Y ] s o he variance of Y Var Y E[ n ] R n o Observaion: he wo decision variables have he same variance which is independen of which signal is sen. Conclusion: he only difference beween wo decision variables is he mean. I is expeced ha he abiliy of he decision device o discriminae beween hese wo cases should depend on he difference of he means μ μ.

20 Propery. We se. hen σ R no Var Yi σ, for boh i and i and he decision variable Y has he following disribuions: Y Y ~ N σ Y Y ~ N o σ s o, if is ransmied s, if is Gong

21 A s A s h < oherwise, A A s o s o for all, Example. Le h s-. For he on-off or nonreurn-o-zero NRZ unipolar signaling, we have h s s o A he means and variances of he decision variables are given by ] [ s E Y o μ < < A A s E Y,, ] [ μ d h N n E Y Var i ] [ σ A N he difference of he means is given by μ μ μ

22 We denoe by he Error Probabiliies Pe he probabiliy ha decision made by he receiver is wrong when is sen Pe he probabiliy ha decision made by he receiver is wrong when is sen he receiver is wrong when is sen if and only if hus Y Y α P e P{ Y α } P{ Y α} Φ[ α s o / σ ] Q[ s o α / σ ] he receiver is wrong when is sen if and only if Y Y > α hus P e P{ Y > α } P{ Y > α} P{ Y α} Φ[ α s o / σ ] Q[ α s o / σ ] P e Q[ s o α / σ ] P e Q[ α so / σ ]

23 Alernaive way: Using condiional densiies f X x i Le X Y and be he pdf of X given ha symbol i is ransmied, i,. hen f f X X x equals he pdf of he random variable wih N x equals he pdf of he random variable wih N s, o σ s, o Gong 3

24 f X x f X x s o α s o Error occurs in he following wo cases: X α if is ransmied X < α if is Gong 4

25 s Example. Coninue he on-off signaling and h s- called he mached filer. Deermine he error probabiliies for sampling ime. Soluion. A, A < s o A oherwise o μ and μ A Q / / N E E / / N E P e α P e Q α If we se α A / E / Noe ha h d A hen he expressions becomes he energy of he signal, denoed as E. hus he variance σ R o N A N n P e Q[ α / σ ] P e Q[ A α / σ ] E P e P e Q E N Using he complemenary error funcion, P e P e erfc E N

26 he Average Probabiliy of Error he average probabiliy of error is defined as P e P{is sen} P e + P{ is sen} P e If wo signals are ransmied equally likely, i.e., P{ is sen} P{ is sen} hen he average probabiliy of error is given by P e / { Q[ s o α / ] + Q[ α so / σ ] } σ For he on-off signaling, he mached filer, he sampling ime, and he hreshold α s o / A / / E Pe Q E N erfc E Gong 6

27 3 Opimizaion of he hreshold For an AWGN channel, he opimal hreshold in he sense of minimizing he error probabiliies is given by α [ μ + μ ]/ his hreshold value is called he minimax hreshold in he lieraure. For his hreshold, he expression of error probabiliy becomes P m e P e P e Q[ μ μ /σ Gong 7

28 4 Mached Filer for he AWGN Channel Goal: Find he opimum filer such ha he error probabiliy P m e is minimized. We define a signal-o-noise raio as SNR o [ μ μ ]/σ P m e becomes P m e Q[ SNR o] Noe ha μ s o s h μ so s h SNR o s h s N h h where { } { } + / + h d H f / h df - Gong 8

29 Le us define g s s V * f H f e j πf hen maximizing SNR o is equivalen o maximize Noe ha hen s io g h + Si f H f jπf e df + + j πf * G f H f e df G f V f g * h Gong 9

30 Using he Schwarz s inequaliy Appendix G, + - G f V * f df + G f df + V * f df where equaliy holds if and only if Gf kvf where k is some consan. hus, o maximum SNR o means ha Gf kvf, i.e., H f k G * f e jπf Noe ha he value k does no change SNR o, so i can be free o choose. We conclude ha he opimum filer has impulse response h λg λ[ s s his is called he mach filer for he AWGN channel. Gong 3

31 By a proper choice of he consan, we can wrie We assume ha g SNR o g / N / g / [ s s We now express he signal-o-noise raio for he mached filer in erms of more fundamenal parameers of he signal se. Gong 3

32 We use he fac + g [ g ] d + [ s s ] d 4 + [ s u + s u ] du s u s u du 4 E + E 4 ρ where E i is he energy of s i, and ρ is he inegral in he second erm. We wrie E E + E, / r ρ / E he parameer E called he average energy for he signal se and r is he correlaion coefficien or normalized correlaion for he wo signals s and s.hus g E r / he signal-o-noise raio for a receiver wih he mached filer is herefore given by / SNR o { E r / Gong 3

33 Observaions: Signal-o-noise raion does no depend on he sampling ime if he mached filer is used. he inuiive reason for his is ha he mached filer, as defined by h λ[ s s auomaically compensaes for any changes in he sampling ime. I should also noed ha, because he noise is WSS, he variance of he oupu noise is independen of he sampling ime. his is rue for any LI filer. Signal-o-noise raio and hence he error probabiliies depend on wo signal parameers only. hese are he average energy in he wo signals s and s and heir inner produc. he deailed srucure of he signals are unimporan. Gong 33

34 Example 3. Deermine he signal-o-noise raio and he error probabiliy of he anipodal signal se for he mached filer receiver. Soluion. E E E ρ s s d [ s ] d E r he resuling signal-o-noise raio is SNR o E / N and he error probabiliies for he minimax hreshold are given by E / P e P e Q N Quesion: How abou he SNR and he error probabiliies of he orhogonal signal Gong 34