Substantiation of the Water Filling Theorem Using Lagrange Multipliers

Transcription

1 J.A. Crawford U048 Water Fillig o Multiple Gaussia Chaels.doc Substatiatio of the Water Fillig heorem Usig Lagrage Multipliers Shao s capacity theorem states that i the case of parallel statistically idepedet Gaussia chaels that the chael capacity C is give by E () C = B log + = where B is the badwidth per chael i Hz, ad E ad are the eergy per symbol ad oise variace per symbol respectively. We wat to fid the best allocatio of trasmit power i order to maximize C uder the maximum power costrait that () = E = E he Lagrage Multiplier solutio method is give by the followig rule : I order to determie the extreme values of a cotiuously differetiable fuctio f(x, x,, x ) whose variables are subjected to m cotiuously differetiable costraiig relatios give as (3) ϕ i ( x, x,... x) = 0 for i =,,..., m form the fuctio (4) m F f λϕ = + i= i i ad determie the parameters i ad the values of x k from the equatios F (5) = 0 for k =,,..., x k ad the m equatios give by (3). he objective fuctio that we wish to fid the extremum of is the give by E (6) Λ = B log + λ E + E = = Differetiatig (6) with respect to E results i I.S., Sokolikoff, R.M. Redheffer, Mathematics of Physics ad Moder Egieerig, McGraw-Hill Book, 966 May J.A. Crawford

2 J.A. Crawford U048 Water Fillig o Multiple Gaussia Chaels.doc Λ B (7) = 0 λ = E l() E + from which we obtai the requiremet that B (8) l() E + = λ for ay (positive) FKRLFH RI FRQVWDQW for all which meas that E + = µ a costat for all thereby providig us the classical water fillig criteria for maximizig the system capacity. Sice it is ot possible to have a egative value for E, the miimum allowable value for µ is the maximum value of the. Substitutig this result ito the capacity formula, we fid that (9) µ µ C = B log + B log = = = Parallel Gaussia BPSK Chaels ow assume that we have parallel Gaussia chaels each supportig ucoded BPSK, ad we wish to miimize the overall average bit error rate (BER) subject to the same total power costrait as used above. he average BER is give by E (0) BER = erfc = ad the total power costrait is give agai by () = E = E Employig the Lagrage multiplier method, the objective fuctio that we seek the extremum values for is give by E () Λ = erfc λ E E = + = Differetiatig () with respect to E ad settig the derivatives to zero results i the criteria that May J.A. Crawford

3 J.A. Crawford U048 Water Fillig o Multiple Gaussia Chaels.doc 3 (3) λ = E exp π E which is obviously a trascedetal equatio i E which must be satisfied for all give a costat SDUDPHWHU his equatio ca be rearraged to give E i terms of the other parameters as E + l E = l λ π (4) ( ) A o-trascedetal result ca be obtaied if we use the Cheroff boud expressio for BER rather tha the exact form. I this case, the objective fuctio ivolved is give by E (5) Λ = exp E + λ E = = Upo takig derivatives of (5) ad settig them to zero, the relatioship for extremum values that results is give by (6) E = l( λ) It is iterestig to compare the results for E usig the exact result foud by solvig (3) ad the o-trascedetal result provided by (6). A example case is show i the Mathcad worksheet that follows. U0333 Miimize Ave BER o Parallel Faded Chaels.mcd May J.A. Crawford

4 J.A. Crawford U0333 Miimize Ave BER o Parallel Faded Chaels.mcd Miimizig BER for Parallel Gaussia BPSK Chaels := 5 umber of parallel chaels := 0.. var0 := ( ) var := var0 var := var0 var E := 0.3 Give λ e E var var π E 05 = 0 CompE λ, var ( ) Fid E CompE( 0.5, 0.) = 0.0 := ( ) Solutio for Idividual chael eergy usig rigorous derivatio from Lagrage multiplier March 004 Copyright 004 J.A. Crawford

5 J.A. Crawford U0333 Miimize Ave BER o Parallel Faded Chaels.mcd λ := E := CompE( λ, var ) E = AvSR db 0 log E := var E BER := erfc BER = var E E = log = 9.6 var Rigorous BER Optimizatio 0 AvSR db = SR (db), E (db) λ := ( ) E := var l λ var Parallel Chael umber SR E E = E ( E ) = log = 9.68 var March 004 Copyright 004 J.A. Crawford

6 J.A. Crawford U0333 Miimize Ave BER o Parallel Faded Chaels.mcd 3 E Loadig 0.5 SR (db), E (db) Parallel Chael umber Rigorously Accuracy BER Upper Boud del := 0 log E var 0 log E var 0.04 Differece Betwee Exact & Approx Differece, db Chael March 004 Copyright 004 J.A. Crawford

7 Advaced Phase-Lock echiques James A. Crawford 008 Artech House 50 pages, 480 figures, 00 equatios CD-ROM with all MALAB scripts ISB-3: ISB-0: X Chapter Brief Descriptio Pages Phase-Locked Systems A High-Level Perspective 6 A expasive, multi-disciplied view of the PLL, its history, ad its wide applicatio. Desig otes 44 A compilatio of desig otes ad formulas that are developed i details separately i the text. Icludes a exhaustive list of closed-form results for the classic type- PLL, may of which have ot bee published before. 3 Fudametal Limits 38 A detailed discussio of the may fudametal limits that PLL desigers may have to be attetive to or else ever achieve their lofty performace objectives, e.g., Paley-Wieer Criterio, Poisso Sum, ime-badwidth Product. 4 oise i PLL-Based Systems 66 A extesive look at oise, its sources, ad its modelig i PLL systems. Icludes special attetio to /f oise, ad the creatio of custom oise sources that exhibit specific power spectral desities. 5 System Performace 48 A detailed look at phase oise ad clock-jitter, ad their effects o system performace. Attetio give to trasmitters, receivers, ad specific sigalig waveforms like OFDM, M- QAM, M-PSK. Relatioships betwee EVM ad image suppressio are preseted for the first time. he effect of phase oise o chael capacity ad chael cutoff rate are also developed. 6 Fudametal Cocepts for Cotiuous-ime Systems 7 A thorough examiatio of the classical cotiuous-time PLL up through 4 th -order. he powerful Haggai costat phase-margi architecture is preseted alog with the type-3 PLL. Pseudo-cotiuous PLL systems (the most commo PLL type i use today) are examied rigorously. rasiet respose calculatio methods, 9 i total, are discussed i detail. 7 Fudametal Cocepts for Sampled-Data Cotrol Systems 3 A thorough discussio of samplig effects i cotiuous-time systems is developed i terms of the z-trasform, ad closed-form results give through 4 th -order. 8 Fractioal- Frequecy Sythesizers 54 A historic look at the fractioal- frequecy sythesis method based o the U.S. patet record is first preseted, followed by a thorough treatmet of the cocept based o -Σ methods. 9 Oscillators 6 A exhaustive look at oscillator fudametals, cofiguratios, ad their use i PLL systems. 0 Clock ad Data Recovery Bit sychroizatio ad clock recovery are developed i rigorous terms ad compared to the theoretical performace attaiable as dictated by the Cramer-Rao boud. 5