Fig. 2. Block Diagram of a DCS

Transcription

1 Iformatio source Optioal Essetial From other sources Spread code ge. Format A/D Source ecode Ecrypt Auth. Chael ecode Pulse modu. Multiplex Badpass modu. Spread spectrum modu. X M m i Digital iput Digital output u i Bit stream imig ad sychroizatio g i (t) Digital basebad/ badpass waveform s i (t) h(t) chael impulse respose Waveform chael (badwidth limited) oise Format D/A mˆ i Source decode Decrypt Verify Chael decode Demultiplex û i Detect z() Demodulate & sample r(t) Spread spectrum despread R C V Iformatio sik o other destiatios Fig.. Block Diagram of a DCS

2 Iformatio source Format A/D From other sources Pulse modu. Multiplex Badpass modu. X M h(t): chael impulse respose m i g i (t) s i (t) Digital iput Digital output imig ad sychroizatio Digital basebad/ badpass waveform h(t): Waveform chael (badwidth limited) oise ISI Iformatio sik Format D/A mˆ i Demultiplex o other destiatios Detect y() Demodulate & sample r(t) R C V Block Diagram of a DCS i 411

3 Chapter Basebad rasmissio (Chapter 6 i i the text) 1. Digital PAM Sigals. Power Spectra of Discrete PAM Sigals 3. Itersymbol Iteferece 4. Nyquist Criterio for Distortioless Sigal Basebad rasmissio 5. Correlative Codig ad Equalizatio 6. Remarks o Chael Badwidth ad rasmissio Rate

4 { a }, a { 0,1} modulator h c ( t) δ ( t) x(t) Badlimited rasmissio chael + detector Y (t) Y ( ) decisio device (t) Figure 1.1 Basebad rasmissio

5 biary sequece a { 0,1} biary sequece a { 0,1} modulator pre-coder b b h ( t ) = b : the symbol duratio pulse shape b h ( t ) filter h (t) We cosider digital commuicatios by meas of PAM. he modulator does the followig tasks: 1. he iput biary data sequece is subdivided ito k-bit symbols ad each symbol is mapped to a correspodig amplitude level.. he amplitude level modulates the output of the trasmittig filter, the output of the modulator is the trasmitted sigal. hus, we ca describe the modulator as a model with a pre-code which performs the task 1 ad a pulse shape filter or the trasmittig filter which performs the task.

6 1. Digital PAM Sigals modulator = pre-coder Pulse shape filter { a } { } Pre-coder: trasformig b, desired form, which is a pre-coded sigal format.

7 biary source 1 or 0 { } a pre-coder { } b pulse shape (t) h x(t) badlimited chael ( t) δ( t) h c + (t) r(t) detector h d ( t ) y (t ) y ( ) decisio device { } â "1" or "0" x ( t) = b h ( t ) : a pulse amplitude modulatio (PAM) sigal Fig. 1.. Block diagram of digital PAM system

8 Objectives (a) a good utilizatio of trasmitted pulse eergy (b) a high badwidth efficiecy (c) a high trasmissio reliability (itersymbol iterferece (ISI) free trasmissio)

9 wo classes of digital PAM sigals: No-retur-to-zero (NRZ): a filter occupies the full duratio of a sigal. Retur-to-zero (RZ): a filter occupies a fractio (usually oehalf) of the sigal duratio. Cosider {a }, a biary sequece. Pre-coder: a a b Pulse shapig filter: b a b h ( t ) where is the bit duratio ad h (t) is a impulse respose of the filter.

10 1). Uipolar (o-off) format (review): b d = 0 a b h ( t ) b if if a a = 1 = 0 1 d 0 0 ). Polar (atipodal) format (review): or equivaletly, b = d( a 1) b = d d if if a a = 1 = 0 a 1 b d b h ( t ) 0 d

11 3). Bipolar format: b = + 0 d, d alterati a = 0 g 1 ' s i a 4). Machester code: b h ( t ) a 1 0 b d d

12 Biary data NRZ uipolar +1 NRZ polar NRZ bipolar -1 Machester +1-1 PAM x(t) for differet sigig format

13 5). Polar quaterary sigal (4-ary PAM): a a b h ( t ) Natural code Gray code Level b

14 Biary data Naturalecoded Grayecoded Polar quaterary format

15 .. Power Spectra of of Discrete PAM Sigals he trasmissio sigal is a discrete PAM: x ( t) = b h ( t ) where B = { b } is a statioary radom sequece, ad depeds o the differet data formats ad is the symbol duratio. b x(t) is a sample fuctio of a radom process X(t).

16 B = { b } pulse shape h (t) x(t) R B () R X (τ ) H ( f ) S B ( f ) SX ( f ) = H ( f ) SB( f ) (1) he power spectral of the radom sequece B = { b } is defied as 1 S B ( f ) = RB ( )exp( jπ f ) () = From (1) ad (), we obtai the psd of the PAM sigal x(t) as follows (3) 1 S X ( f ) = H ( f ) RB ( )exp( jπf ) =

17 Remark. he results i (1) - (3) illustrate the depedece of the psd S X ( f ) of the trasmitted sigal o (1) the spectral characteristics of H ( f ) of the pulse shape filter ad () the spectral characteristics of S B ( f ) of the pre-coded iformatio sequece. Coclusio. Both H ( f ) ad S B ( f ) ca be desiged to cotrol the shape ad form of the psd of the trasmitted sigal. he formula give by (3) is the formula that we frequetly use to determie the psd of x(t).

18 Example.1. Determie the psd i (3) where h (t) is a rectagular pulse show i Fig..1. h (t) A 0 t H ( f ) -3/ -/ -1/ 0 1/ / 3/ f Fig..1. A rectagular pulse h (t ad its eergy desity spectrum ) H ( f )

19 Solutio. he Fourier trasform of is as follows (t) h H ( f ) = A si c( f ) exp( jπf ) where si c( x) = si( π x) πx Hece, we have H ( f ) = A si c ( f ) Substitutig it ito (3), thus, we obtai that if the pulse shape is the rectagular pulse h (t) the the psd of the PAM x(t) is give by S X ( f ) = A si c ( f ) = R B ( )exp( jπ f ) (4)

20 Defiitio. A discrete radom sequece { X } is said to be idepedet if for ay k time istaces t 1 < t < L < t k, X t, X t, L, X 1 t k are idepedet. Defiitio. A discrete radom sequece { X } is said to be mutually ucorrelated if ay pair of X k ad X, k are ucorrelated, i.e., E X X ] = E[ X ] E[ X ]. [ k k Property. If a discrete radom sequece { X } is idepedet ad for each k, the radom variable takes each value equally likely, the { X } is a idepedet idetical distributed (i.i.d.) radom sequece. X k

21 Example.. Assume that B = { b } is a idepedet radom sequece ad each biary symbol occurs equally likely. Determie the psd of x(t) where the sigal format is NRZ polar ad the pulse shape h (t) is the rectagular pulse defied i Example.1. Solutio. We will use (4) to determie the psd of x(t). First we eed to compute the autocorrelatio R B () of the statioary radom sequece { b }. From the give coditio, { b } is a i.i.d. radom sequece. hus B E[ b k R (0) = P{ b = d} = P{ b = d} = k ] = d P k 1 { bk = d} + ( d) P{ bk = d} = d

22 For 0, R B ( ) = E bkbk +. By idepedece ad equally likely occurrece, the probability of k, b k is give by [ ] ( b + ) ( a k, a k + ) (0, 0) ( b k, b k + ) ( d, d) p( x, y) 1/4 P{ b p( x, y) = = x, b k k + = y} (0, 1) ( d, d) 1/4 (1, 0) (d, d) 1/4 (1, 1) (d, d) 1/4 hus ( ) = d (1/ 4 + 1/ 4) d (1/ 4 + 1/ 4) = 0 R B R B ( ) = d 0 = 0 0 S X ( f ) = A d si c ( f ) (by (4))

23 Summary o the psd of PAM sigal: x ( t) = b h ( t ) For a geeral pulse h (t) S X ( f ) = H ( f ) S B ( f ) = 1 H ( f ) = R B ( )exp( jπf ) If h (t) is a rectagular pulse with magitude A ad duratio, the S X ( f ) = A si c ( f ) = R B ( )exp( jπf Gog 3

24 S X (f) f b

25 S X (f) f b

26 S X (f) f b

27 NRZ polar S X (f) NRZ bipolar Machester NRZ uipolar f b Figure. Power spectra of differet biary data formats

28 Cosideratios for Selectio of Sigalig Schemes: Presece or absece of a DC level Power spectral desity, particularly its value at 0 Hz Spectral occupacy (i.e., badwidth) Bit error probability performace (i.e., relative immuity from oise ) Ease of clock sigal recovery for symbol sychroizatio Presece or absece of iheret error detectio properties

29 Example.3. Let { a } be a ucorrelated biary valued (±1) radom sequece, each havig a zero mea ad a uit variace. Let { b } whose value is determied by b = a + a 1 Determie the psd of the trasmitted sigal. Solutio. he autocorrelatio fuctio of the sequece { b } is ( ) = E[ b b ] E[ a + a )( a a )] R B k k + = ( k k 1 k + + k + 1 = 1 0 = 0 = ± 1 otherwise

30 From (), the psd of the iput sequece { b } is S B 1 ( f ) = (1 + cos πf ) = 1 cos πf By substitutig it ito (1), the psd for the modulated sigal is S X ( f ) = 4 H ( f ) cos πf Remark. his example explaied that the trasmitted sigal spectrum ca be shaped by havig a correlated sequece b as the iput to the modulator. { }

31 3. 3. Itersymbol Iterferece 1 or 0 { } a Precoder { } b Pulse Shape h (t) x ( t) = b h ( t ) r(t) badlimited chael + h c ( t) δ( t) Detector h d (t) y(t) y() 1 { } â or 0 decisio device Biary sequece AWGN: (t) Fig. 3.1 Digital PAM rasmissio through badlimited basebad chael

32 he trasmissio sigal: x ( t) = b h ( t ) he chael output: which is the received sigal at the demodulator r( t) = b h( t ) + ( t) where h(t) is the pulse of the cascade of the trasmittig filter ad the chael, i.e., h( t) = h ( t) h ( t) h c (t) is the impulse respose of the chael, ad (t) represets AWGN. he received sigal is passed through a LI with impulse respose h d (t). If h d (t) is matched to h(t), the its output SNR is a maximum at the the samplig istat t =. c

33 he output of the detect (receivig) filter where hus the sampler produces y( t) = μ b p( t ) + o ( t) p( t) = h( t) h ( t) = h ( t) h ( t) h ( t) which is ormalized such that p(0) = 1 ad o d ( t) = ( t) h ( t) d c d y( k) = μ b p( k ) + o ( k) = μb k + μ k b p( k ) + o (k) (1) desired symbol ISI compoet

34 he first term o the RHS of (1) is the desired symbol b k, scaled by the gai parameter μ whe the receivig filter is matched to H(f). he scale factor is give by he secod term o the RHS of (1) represets the effect of the other symbols at the sample istace t = k, called itersymbol iterferece (ISI). he scale factor is give by μ = h ( t) dt = = H ( f ) df H ( f ) H ( f ) df Eh which is the eergy of the pulse shape h(t). c

35 Example 3.1. Illustrate the ISI effect whe the iput data is 011. a h (t) h c 1 0 (t) b d d h ( t) h ( t) c * d -d Biary data

36 Observatio: a) he effect of passig the trasmitted pulse through a chael with memory h c ( t) δ ( t) is that the duratio of the trasmitted pulse is stretched. b) he pulse stretchig is referred to as a time dispersio ad the chael is called time dispersive chael. c) ime dispersio causes overlaps betwee adjacet symbol bits at the output of the commuicatio chael. d) he distortio that arises from the overlappig betwee adjacet symbols is called itersymbol iterferece (ISI). e) ISI is preserve oly whe the symbol is preserve. i.e. ISI is sigal depedet of oise (distortio). f) Ulike additive oise, ISI caot be suppressed by simply icreasig the sigal eergy. h (t)

37 k b p( k ) = If () the there is o ISI effect. I this case y( k) = μ b ( k) he trasmissio performace is oly degraded by AWGN. k + o 0 { d, d} h (t) ( t) δ ( t) (t) h c (t) + Questio: How to desig h (t) ad hd (t) such that the ISI term vaishes? If so, μp(t) is said to be a effect chael without ISI. h d y (t) y( ) decisio device âi

38 A trivial solutio: If p( t) = δ ( t), the the ISI ca be removed. However, it is ot practical i practice.