ECE 6640 Digital Communications

Transcription

1 C 664 Digital Communication Dr. Bradley J. Bazuin Aitant rofeor Department of lectrical and Computer ngineering College of ngineering and Applied Science

2 Chapter 9 9. odulation and Coding Trade-Off. 1. Goal of the Communication Sytem Deigner.. rror roaility lane. 3. yquit inimum Bandwidth. 4. Shannon-Hartley Capacity Theorem. 5. Bandwidth fficiency lane. 6. odulation and Coding Trade-Off. 7. Defining, Deigning, and valuating Sytem. 8. Bandwidth-fficient odulation. 9. odulation and Coding for Bandlimited Channel. 1. Trelli-Coded odulation. C 664

3 Slar Communication Sytem ote and figure are aed on or taen from material in the coure textoo: C 664 Bernard Slar, Digital Communication, Fundamental and Application, 3 rentice Hall T, Second dition, 1.

4 Sytem Level Tradeoff The yquit theoretical minimum andwidth requirement The Shannon-Hartley capacity theorem The Shannon limit Government regulatory involvement frequency allocation, andwidth limitation Technology limitation phyically realizale component uing current technology Other ytem requirement For atellite: orit and energy limitation C 664 4

5 rror roaility lane rror proaility performance curve define acceptale B determine required /o e would prefer equivalent andwidth performance curve allow ytem level tradeoff trade-off /o for modulation type at fixed B trade off B v modulation type at fixed /o how range of expected B a /o varie C 664 5

6 B v /o Curve C 664 6

7 yquit inimum Bandwidth yquit howed that the theoretical minimum andwidth needed for aeand tranmiion of ymol per econd without ISI i / Hz. A theoretical minimum contraint on andwidth required. eferred to a ymol/ec/hz Typical ytem and filter are 1%-4% wider ore liely 1.8 to ¼ ymol//hz. in term of ymol modulation log C 664 7

8 xample 9.1: Digital Scheme Orthogonal Signaling expect improvement in B a or increae on-orthogonal ignaling expect a decreae in B a or increae a) Doe error-performance improve or degrade with increaing, for -ary ignaling? ) The choice availale almot alway involve a tread-off. If error performance improve, what price mut we pay? c) If error-performance degrade, what enefit i exhiited? C 664 8

9 xample 9.1 xpected trade-off -FSK a increae, the required tranmiion andwidth increae for minimum frequency pacing. to maintain a contant it rate, the ymol tranmiion rate decreae with increaing -SK while there i degradation a increae, the ymol tranmiion rate may e decreaed a increae -SK ytem plot equal-andwidth curve, a the it tranmiion rate increae. C 664 9

10 Shannon-Hartley Capacity Theorem The capacity relation in AG can e tated a C log 1 S where S i the ignal power, the noie power, and the andwidth the value i defined in it per econd C 664 1

11 Shannon-Hartley Capacity Theorem The normalized channel andwidth v. S may alo e plotted C log 1 S C log 1 S 1 C

12 S-H quivalent quation earranging and defining the noie power and ignal power For Letting C = C S C 1 log S 1 1 C 1 C C 1 1 C C C log 1 C C log 1

13 Shannon Capacity Theorem There i a limiting cae a C/ let C C x C C 1 log 1 C C log 1 x x 1 log 1 1 e x x x 1 log 1 log lim 1 x x 1 1 log 1 db e log 1

14 Shannon Limit db log e A C/ or /C In practice, it i not poile to reach the ound. rovide an improvement ound for encoding and decoding. For example: raw BSK require approximately 9.6 db /o to achieve a B of 1-5 which ugget that up to an 11. db improvement i poile. Turu Code can achieve ~ 1 db. C

15 ntropy To compute communication capacity, a metric for the meage content of a ytem i alo important. ntropy i defined a the average amount of information per ource output. It i expreed y: n H where p i i the proaility of the ith output and the um of all p i i 1. For a inary ytem, entropy can e expreed a: i1 p i log p i H p 1 plog p p log 1 C

16 ntropy for a Binary Sytem The entropy i aed on the proaility, p, of an event. Thi can alo e looed at a the randomne of ucceive event or how correlated individual event are. ote that maximum entropy i achieved when the proaility i 5% A ample provide no information aout a ucceeding ample. C

17 xample 9. nglih Language The nglih language i highly redundant. H 4 8 The proaility of the next letter in a word i not equally liely for all poile character. Determine the ntropy aed on the letter proailitie p=.1 for the letter a, e, o, t n p=.7 for the letter h, I, n, r, H p i log p i i1 p=. for the letter c, d, f, l, m, p, u, y p=.1 for the letter, g, j,, q, v, w, x, z log.1 5.7log.7.log. 9.1log.1 it/char nglih Language 1 1 H 6 log log it/char qual roaility C

18 quivocation A term ued y Shannon to account for the uncertainty in a received ignal. It i defined a the conditional entropy of the meage X (tranmitted ource meage), given Y (the received ignal). H X Y X, Y log X Y X, Y aed on conditional proaility X Y Y X Y log X Y H Y X C

19 quivocation xample Conider a inary equence, X, where the it are equally liely. Aume that the channel produce on error in a received equence of 1 it (=.1). H H X Y X, Y log X Y X, Y X Y 1 log 1 H log X Y.99log.99.1 log.1 H X Y. 81 Interpretation: the channel introduce.81 it/received ymol of uncertainty. C

20 ffective Tranmiion ate Uing the equivocation computation, the effective tranmiion rate of the channel can e computed a X H X Y H eff H aed on the previou example, the inary ytem would have an effective tranmiion rate (in term of it/received ymol) of H eff for a communication ytem with = 1 it/ec, the effective tranmiion rate would ecome eff H eff C 664

21 v /o Curve It appear that approache.5 a /o decreae ut the Shannon limit i /o=-1.6 db. I thi a contradiction or not? Shannon refer to received information it aed on equivocation. C 664 1

22 Deriving an ffective /o A an example, tae /o=-1 db for coherent BSK H B Q B Q X Y 1.33log log H eff from thi form an effective /o eff H eff.1.85 Thu, he effective /o i well aove the Shannon limit, -1.6dB db C 664

23 Bandwidth-fficieny lane Uing Shannon-Hartley Capacity, the normalized channel andwidth veru /o for different ymol cheme can e compared. Typically performed for a defined it-error proaility and under optimal ymol detection aumption. Let =C, then log 1 The ound and appropriate value for SK, FSK and QA ymol cheme are hown on Fig. 9.6 C 664 3

24 Figure 9.6: Bandwidth-fficiency lane Factor of note: SK and QA nominally maintain the ame andwidth will increaing the it per ymol and required /o FSK ue an increaing andwidth a the it per ymol increae while the /o i decreaing BSK and QSK have the ame /o ut different it per ymol C 664 4

25 Bit and Symol ate Conideration For SK / increae with log 1 IF IF T log log For FSK log IF T / decreae with IF log log C 664 5

26 Bandwidth veru ower For a andwidth-limited ytem pectral efficiency i important expect that ignal power may e increae to offet the limitation tudy the andwidth-efficient plane SK allow for fixed andwidth For a power-limited ytem a defined tranmiion power limit ha een etalihed expect that ignal andwidth may increae to offet the limit tudy the it-error proaility plane FSK allow for limited pectral power C 664 6

27 Digital Comm. Sytem ngineering Defining, deigning, and evaluating communication ytem. Comparing SK and FSK (tale 9.1) SK on Coherent FSK min / /o (db) min / /o (db) it/ec ym/ec (Hz) =1e 5 (Hz) =1e C 664 7

28 Sytem xample #1: Bandwidth Limited = 4 Hz, r/o=53 db-hz, =96 p, B =1e-5 quation needed for the computation (auming -SK) C r Q in B log log

29 Sytem xample #1: Bandwidth Limited = 4 Hz, r/o=53 db-hz, =96 p, B =1e-5 r/o r/o 4 Hz 53 db Hz 96 p 1. 5 B Hz r Q in log B log /o.78 /o db SK lin db qrt(*/o) in(pi/) x Q(x)=e 96 ym/ /o ym/ /o ym/ /o ym/ /o C 664 9

30 Sytem xample #: ower Limited = 45 Hz, r/o=48 db-hz, =96 p, B =1e-5 quation needed for the computation (auming -FSK) C exp B log r

31 Sytem xample #: ower Limited = 45 Hz, r/o=48 db-hz, =96 p, B =1e-5 r/o r/o 45 Hz 48 db Hz 96 p 1. 5 B Hz /o 6.57 /o 8.18 db r 1 exp 1 log B 1 1 FSK lin db exp( /o/) 1 96 ym/ 19 Hz /o ym/ 19 Hz /o ym/ 56 Hz /o ym/ 384 Hz /o ym/ 6144 Hz /o C

32 Coded Sytem xample hen the previou method do not produce a valid implementation, encoding and decoding will e required. onitor the effect of code rate on ymol/ec and andwidth C 664 3

33 Sytem xample #3: ncode-decode = 4 Hz, r/o=53 db-hz, =96 p, B =1e-9 Starting with the previou 8-SK ytem, we need additional coding gain C c c r Q in C log n c log log c n log j n c j c n t j B j n j n 1 1 1

34 Solution i Step Step 1: Compute the /o Step : Compute the codeword ymol error rate () Step 3: Compute the codeword-it-error rate Step 4: Compute the decoded it error proaility C c c r n c log log Q in C log j n c j c n t j B j n j n 1 1 1

35 xcel Computation An excel preadheet can e ued for all of the example. ee reult for xample #3 Alternate Approach the coding gain formula can e ued. db in db in db G in uncoded an encoding cheme that meet the andwidth requirement and ha.8 db or more coding gain i ufficient for olving thi prolem. C coded db G in

36 Bandwidth fficient odulation odern communication i hungry for andwidth, demanding an every increaing communication capacity within the fixed frequ3ency and availale, Additional requirement to allow for non-linear amplification put a premium on uing ignal that are minimally effected y A to converion, limiting the amplitude variation of the ignal (deiring a contant modulu). C

37 QSK and Offet QSK Conventional QSK ue conecutive it received to determine I-Q pair for tranmiion. Offet QSK alo ue the it, ut direct them to the I and Q port a they arrive in time (next lide) C

38 QSK veru Offet QSK OQSK mae 9 degree phae tranition 18 degree phae change may reult in ignificant amplitude variation C

39 inimum Shift Keying (SK) Avoiding dicontinuou phae tranition of the ignal maintain a contant amplitude ue a form of continuou-phae FSK alo a modified form of OQSK C T t T x t T d f t 1, 4 co, mod 1 1 d d x x

40 SK Quadrature epreentation xpanding the coine term co(a+) the imilarity to OQSK i aed on the amplitude weighted quadrature tructure of thi formulation C t f T t t f T t a t in in co co 1 co a x 1 co x d, mod 1 1 d d x x

41 Bandwidth Comparion: BSK, QSK & OQSK, & SK C

42 odulation and Coding for Bandlimited Channel eearch Area (a of 1 copyright): Optimum ignal contellation oundarie (chooing a cloely paced ignal uet from any regular array or lattice of candidate point) Higher denity lattice tructure (adding improvement to the ignal uet choice y tarting with the denet poile lattice for the pace) Trelli-coded modulation (comined modulation and coding technique for otaining coding gain for andlimited channel). Ungeroec artitioning C 664 4

43 volution of Telephone odem Standard (1) Telephone modem have dealt with the limited power and andwidth prolem for a coniderale time. rogre wa made at different time for oth leaed-line and dial-line ervice. C

44 volution of Telephone odem Standard () Home modem tandard otly replaced y telephony DSL or cale TV acce C

45 Signal Contellation Boundarie Variou QA contellation that have een invetigated. optimal pacing of point with maximum eparation reduce maximum amplitude optimize () C

46 Trelli-Coded odulation (TC) Developing comined modulation and coding cheme Ue a redundant noninary modulation in comination with a finite-tate machine aed encoding proce. FS could e imilar to convolutional encoding A multi-level/phae modulation cheme The concept, when performing ATLAB imulation of encoded it tream uing SK or QA ymol, i there an optimal comination? if you now the ymol eing ued, could one convolutional code leading to an appropriate trelli decoding perform etter than another? C

47 TC ncoding Ungeroec, G., "Channel coding with multilevel/phae ignal," Information Theory, I Tranaction on, vol.8, no.1, pp.55,67, Jan 198. Initial paper decriing trelli coded, oft deciion encoding and modulation technique for communication. C