Performance Analysis and Channel Capacity for Multiple-Pulse Position Modulation on Multipath Channels

Transcription

1 Performace Aalysis ad Chael Capacity for Multiple-Pulse Positio Modulatio o Multipath Chaels Hyucheol Park ad Joh R. Barry School of Electrical ad Computer Egieerig Georgia Istitute of Techology, Atlata, GA Abstract Although multiple pulse-positio modulatio performs well o ideal chaels, its performace o multipath chaels is degraded sigificatly. I a attempt to quatify the iheret pealty due to multipath dispersio, we evaluate upper bouds for the error probability of each modulatio scheme i the presece of itersymbol iterferece, cosiderig both a uequalized receiver ad the optimal maximum-likelihood sequece detectio receiver. We also preset upper ad lower bouds of the chael capacity for multiple pulse-positio modulatio ad its variats, PPM ad overlappig PPM. Numerical results show that the PPM-based schemes are sigificatly more sesitive to multipath dispersio tha is o-off keyig. I. Itroductio [6] cosidered the chael capacity for over a photo coutig chael, without itersymbol iterferece. Hirt [7] calculated the chael capacity for a biary discrete-time Gaussia chael usig a Mote Carlo approximatio. Shamai [8] derived lower ad upper bouds of the chael capacity with i.i.d. iput symbols for a scalar discrete-time Gaussia chael. I this paper, we examie the chael capacity of various modulatio schemes o the chael () uder the costraits of (). I Sect. II, we develop the geeral system model for multiple PPM ad its variats. I Sect. III, we aalyze the performace of the uequalized receiver ad the maximum-likelihood sequece detector. I Sect. IV, we preset the expressio for chael capacity o the ideal chael ad boud the chael capacity of PPM-based schemes over ISI chaels. We preset umerical results i Sect. V. No-directed ifrared radiatio offers several advatages over radio as a medium for idoor wireless etworks, icludig a immese widow of uregulated badwidth, immuity to multipath fadig (but ot multipath distortio), ad a lack of iterferece from oe room to aother []. But the itese backgroud light of typical idoor eviromets is a severe impedimet, ad power-efficiet modulatio is eeded to achieve high data rates or log rage. The wireless ifrared chael is accurately modeled by the followig basebad AWGN model []: y(t) = x(τ)h(t τ) dτ + (t), () where x(t) represets the istataeous optical power of the trasmitter, y(t) represets the istataeous curret of the receivig photodetector, h(t) represets the multipath impulse respose, ad (t) is white Gaussia oise with two-sided power spectral desity N. The high itesity of the backgroud light makes the Gaussia oise model extremely accurate. Because x(t) represets optical power, it must satisfy: x(t) ad x(t) P, () where P is the average optical power costrait of the trasmitter. A recet paper [3] examied the performace of multiple PPM o the wireless ifrared chael assumig o multipath dispersio. Audeh ad Kah [4] examied the effects of itersymbol iterferece (ISI) o PPM. We exted these results by examiig the effects of ISI o multiple PPM ad overlappig PPM. The chael capacity is cosidered to be a fudametal limit for reliable trasmissio over a give chael [5]. Georghiades II. System Model Cosider the system model show i Fig. (a). Iformatio bits with rate R b b s eter the ecoder, which groups the bits ito blocks of legth log L ad maps each block to oe of L codewords c c L, where each codeword is a biary -tuples of weight w. The set of allowable codewords is what distiguishes the differet modulatio schemes. Whe all ( w) -tuples of weight w are valid codewords, the resultig modulatio scheme is called multiple PPM (). Whe the oly valid codewords are those biary -tuples of weight w i which the w oes are cosecutive, the result is overlappig PPM (OPPM). Fially, whe w is restricted to uity, both ad OPPM reduce to covetioal PPM modulatio. Note that the umber of codewords L for, OPPM, ad PPM is ( w), w +, ad, respectively. The output of the ecoder is a sequece of codewords {x k } with rate T = R b log L. This sequece is serialized to produce the biary chip sequece {x j } with rate T, where x k = [x k, x k +,, x k + ] T. The biary chip sequece drives a trasmitter filter with a rectagular pulse shape p(t) of duratio T ad uity height. To satisfy the power costrait of (), the filter output is multiplied by (P w) before the sigal is set across the chael. As show i Fig. (a), the receiver uses a uit-eergy filter f(t) ad samples the output at the chip rate T producig y j. The receiver groups the samples y j ito blocks of legth, producig a sequece of observatio vectors {y k }, where y k = [y k, y k +,, y k + ] T. The receiver passes each observatio vector through a decisio device to form a estimate xˆ k of x k. Supported by NSF uder grat umber NCR = T lim ( ) dt T T T To appear, Iteratioal Symposium o Persoal, Idoor ad Mobile Radio Commuicatios, Taipei, October 996.

2 The equivalet discrete-time chael betwee trasmitted ad received chips is: y j = h i x j i + j = s j + j, (3) i = where h j is the equivalet chip-rate impulse respose: P h j = p(t) g(t) f(t), (4) w t = jt ad where s j is defied by (3). We assume that f(t) * f( t) is a Nyquist pulse, which will be true whe f(t) is matched to p(t) or whe f(t) is a whiteedmatched filter. I this case, the oise samples { j } will be idepedet zero-mea Gaussia radom variables with variace N. As show i Fig. (b), the equivalet vector chael betwee trasmitted codewords x k ad observatio vectors y k is give by: µ y k = H l x k l + k = s k + k, (5) where the chael impulse respose is a Toeplitz sequece H k, with [H k ] ij = h k + i j, s k = [s k, s k +,, s k + ] T is the sigal compoet, k = [ k, k +,, k + ] T is the oise compoet ad µ is the umber of memories i vector ISI chael. Throughout this paper we costrai the iput symbols x k to be idepedet ad uiformly distributed over a or OPPM alphabet. III-A. Without Equalizatio III. Performace Aalysis By defiitio, the uequalized receiver uses the decisio device that would be optimal were there o ISI. I other words, it decides o the codeword c l that maximizes the correlatio: Λ l = c l T y k for,, L. (6) If we kew that x k = c i, ad if we kew all past ad future codewords X = {, x k, x k, x k+, x k+, }, the the probability of error xˆ k x k ca be bouded usig the uio boud: L Pr[error x k = c i, X ] Pr[Λ i Λ j x k = c i, X ] (7) L = Pr[(c i c j ) T k > (c i c j ) T s k x k = c i, X ]. But (c i c j ) T k is a zero-mea Gaussia radom variable with variace d ij N, where d ij = d H (c i, c j ) is the Hammig distace betwee codewords c i ad c j. Therefore, (7) reduces to: L ( Pr[error x k = c i, X ] Q c i c j ) T s k (8) d ij N o Averagig overall all possible codeword sequeces gives: L L ( Pr[error] , (9) L M + Q c i c j ) T s k X d i = ij N o where the first summatio is over all X C M, where C is the set of L valid codewords ad M + is the umber of ozero terms i the impulse respose {H k }. Fially, the bit-error probability is: log L Pr[bit error] = ( Pr[error]). () For o-off keyig (), the bit stream, symbol stream, ad chip stream are all oe i the same. By averagig over all possible bit streams {x }, the total bit error probability is: Pr[bit error] = , () M Q h x h j x N o where the summatio is overall all biary M-tuples {x }, where M + is the umber of ozero terms i h j. Hereafter, we preset simplified expressios for the symbol error probability for the special case of a ideal chael, without ISI. First, cosider multiple PPM: whe the chael has o ISI, the expressio (9) simplifies to: w k w Pr[error] a k Q ks , () N o k = w where a k = ( )( k ) is the umber of codewords with mutual distace k, ad s = (P w) log L R b. This expressio follows from the ISI-free results for the photo coutig chael of [6]. Whe w =, (i.e., for PPM), () simplifies further to: Pr[error] (L )Q s , (3) N o where s = P Llog L R b. Next, cosider ( w) OPPM: whe the chael has o ISI, the expressio (9) simplifies to: b x P w (t) i k x j x(t) y y j k xˆ k ENC P S p(t) g(t) f(t) S P DEC T rate rate rate (a) rate rate rate log L T T T T T T k x k y k H(z) Fig.. (a) Block diagram of multiple PPM system; (b) equivalet vector model. (b)

3 w Pr[error] --- b, (4) L k Q ks N o k = (L k) k =,,, w where b k =, (5) (L w)(l w + ) k = w ad s = (P w) log L R b. This result uses the photo-coutig results of [6]. Fially, cosider : whe the chael has o ISI, the h j from (4) reduces to (P R b )δ j, ad so the error probability from () simplifies to Q(P N R b ). III-B. ML Sequece Detectio The maximum-likelihood (ML) sequece detector for PPM is derived i [9], ad it easily geeralizes to multiple PPM ad overlappig PPM. The receiver filter is the whiteed-matched filter, so that h j is causal ad miimum phase. The probability of a symbol (block) error for the ML sequece detector is well approximated at high SNR by: Pr[error] Q(d mi N ), (6) where d mi is the miimum distace betwee received sequeces: d mi = mi H m e k m. (7) {e k } k The above miimizatio is performed over all possible error sequeces {e k } startig at time zero, usig a error alphabet of {u v: u v; u, v C}, where C is the set of valid multiple PPM codewords. IV. IV-A. Memoryless Chael m Chael Capacity If the chael is memoryless (ISI-free), so that H = I, H for all l i (5), the the chael capacity (i bits per codeword) uder the i.i.d. costrait is [5]: I iid = I(x; x + ) = e y x L l σ log L ( πσ ) e y x l σ dy. (8) --- e y x L m σ Lm = Followig [], we substitute z = (y x l ) / σ ad v m = x m / σ, so that (8) reduces to: L --- L I iid = log L e z log π ( e v l v m )z v L l v m e dz. (9) m = The above equatio cotais a dimesioal itegral ad has o simple closed form solutio. As a cosequece, we shall use the Mote Carlo method to estimate I iid i Sect. V. IV-B. Multipath Chael Followig [8], we ca also represet the chael (5) usig matrix otatio: Y = HX + N = S + N, () where Y = [y T, y T,..., y N T ] T, X = [x T, x T,..., x N T ] T, S = [s T, s T,..., s N T ] T, ad N = [ T, T,..., N T ] T are N colum vectors. The two equatios (5) ad () are equivalet as N, ad the rows of H are specified by circular shifts of {H i }:. () The chael capacity for () uder the i.i.d. costrait is [5][7]: lim lim I iid = ----I ( Y; X) = ---- ( h( Y) h( Y X) ) () = log L log L N e z π N Exact evaluatio of this expressio is ot possible, so we resort to upper ad lower bouds. I the followig discussio, we preset lower ad upper bouds for the chael capacity uder the i.i.d. costrait. Our presetatio is a straightforward geeralizatio of the scalar results of Shamai [8] to the vector chael (5). Followig [8], we represet the etropy of the output vector i () usig the chai rule: h( Y) = h( y l y l, y l, y ). (4) Sice coditioig decreases the etropy: h(y) H = L lim ---- N N H H H H µ H L ( S l S m )z S l S m e dz (3) m = h( s l + l s l,, s, l,, ) = h( s l + l s l,, s ), (5) where the last equality follows because s l, l, l,, are idepedet. Sice s l =, (5) reduces to: j H l j x j h(y) h( H x l + l ). (6) The mutual iformatio betwee the iput ad output vectors is: I( X; Y) = h( Y) h( N) { h( H x l + l ) h( l )} = I( H x l + l ; x l ). (7) This leads to the followig lower boud I L for the chael capacity uder the i.i.d. costrait:

4 lim ---- I( X; Y) We ca evaluate (8) by replacig x with H x i (8). As poited out i [9], the lower boud of the chael capacity is equivalet to the mutual iformatio betwee the iput ad the output of a error-free block zero-forcig decisio feedback equalizer. We ow preset a upper boud I U for the capacity I iid of (3). By the chai rule, we ca represet the mutual iformatio betwee the iput ad output of () as: I(Y; X) = I( Y; x l x l, x l,, x ) = Sice {x k } are i.i.d. ad coditioig decreases the etropy: = = I( x l ; H x l +, H x l +,, H µ x l + µ ) = I(x l ; Ỹ ), (3) where Ỹ = H x l + Ñ, ad where H = [H T, H T,..., H T µ ] T ad Ñ = [ T, T,..., T µ ] T. T Let V = H Ỹ = Bx l + w, where H T is a matched filter, so that B = lim ---- I H ( x l + l ; x l ) = I( H x + ; x) (8) [ h( x l x l, x l,, x ) h( x l x l, x l,, x, Y) ]. (9) I( Y; X) [ h( x l x, x,, x, l x, l + x, l +, x N ) h( x l x, x,, x, l x, l + x, l +, x Y N, ) ] I( x l ; x, x,, x, l x, l + x, l +, x Y N, ) µ m = H m T H m ad w is a zero-mea Gaussia vector with correlatio matrix E[ww T ] = σ B. Sice B is positive defiite, it ca be factored ito B = ΓΓ T for some matrix Γ. We ca white the oise by applyig Γ - to V, yieldig Z = Γ - V = Γ T x l +, where has the same distributio as l, a zero-mea Gaussia vector with correlatio matrix σ I. Sice both the matched filter ad oise whiteer are iformatio lossless, we have I(x l ; Ỹ ) = I(x l ; V) = I(x l ; Z). Therefore, (3) reduces to: I(Y; X) I(x l ; Γ T x l + l ). (3) Fially, takig the limit as N yields our upper boud I U of the chael capacity uder the i.i.d costrait: lim ---- I( X; Y) lim ---- I x ( l + l ; x l ) = I( Γ T x + ; x). (3) We ca evaluate (3) by replacig x with Γ T x i (8). Note that, for the scalar case, the matrix Γ T reduces to m h m upper boud (3) reduces to the matched filter boud [8]., ad the V. Numerical Results To geerate umerical results, we assume that the uderlyig cotiuous-time chael i Fig. has impulse respose g(t) = We Wt u(t), a first-order low-pass filter with a 3-dB badwidth of W, where u(t) is the uit step fuctio. Observe that the chael has uity d.c. gai. We also assume that the receive filter f(t) is a uit-eergy whiteed-matched filter.to reduce computatioal complexity, we trucate the vector chael of (5) to four 3 terms, so that y k = H l x k l + k. This trucatio will have o appreciable effect whe is large or whe R b W is small, although it may ot be accurate for small ad large R b W. Observe that the chael has uity d.c. gai. We calculated the optical power required to achieve a -6 bit-error rate over this ISI chael. The results are summarized i Fig., where the ormalized power requiremet is plotted versus the bit-rate-to-badwidth ratio R b W. The power requiremets are ormalized by P = N Q ( 6 R b ), the power required by i the ideal case (W = ) to achieve a -6 bit error rate. I Fig. (a) we plot power requiremet versus R b W whe equalizatio is ot used. We see that some modulatio schemes Normalized Power Requiremet (db) Normalized Power Requiremet (db) Without Equalizatio ( 4 ) 4-PPM 8-PPM ( 6 3 ) OPPM -PPM ( 8 4 ) OPPM PPM OPPM 6.. Bit Rate Badwidth (R b W) With MLSD ( 6 3 ) OPPM -PPM ( 8 4 ) OPPM ( 4 ) 4-PPM PPM 4 OPPM 8-PPM 6.. Bit Rate Badwidth (R b W) Fig.. Power requiremet vs bit rate o a ISI chael:. (a) Uequalized system (b) ML sequece detectio. (a) (b)

5 are more sesitive to ISI tha others. At large badwidth (R b W <.), the ISI pealties are small. At oe extreme is (represeted by the symbol ), with a power requiremet icreasig slowly with decreasig badwidth. At the other extreme is OPPM, for which the power requiremet grows rapidly with decreasig badwidth. It is thus highly desirable to use sigal processig at the receiver to mitigate ISI, either equalizatio or maximum-likelihood sequece detectio. I cotrast to the uequalized results of Fig. (a), the results of Fig. (b) are based o the maximum-likelihood sequece detector (MLSD). Comparig Fig. (a) ad Fig. (b), we see that MLSD offers sigificat improvemet. The power requiremets do ot grow as rapidly as i the uequalized case, ad the ormalized power requiremet is always less tha db, eve whe R b W =. We ote that MLSD is much more effective i reducig the power requiremet for tha for other modulatio schemes. The exact capacity I iid of (3) is difficult to evaluate, but we ca form a estimate Î iid by choosig N suitably large i (3) rather tha lettig N, ad by usig the Mote Carlo method with Q sample vectors to approximate the multiple itegral. I Fig. 3, I ideal (9), I L (8), ad I U (3) are show for PPM, ad compared to the approximate chael capacity Î iid (based o N = 6, Q = ). Our results show that I L ad I U are.5 db apart at moderate SNR whe R b W =.5. Whe SNR = 3.3 db, the chael capacity is.95 bits codeword whe the chael is ideal, but is oly.8 bits codeword whe R b W =.5. To achieve a capacity of.95 bits codeword usig PPM at R b W =.5, the required SNR is 9.5 db. I cotrast, we see from Fig. (b) that a ucoded -PPM system with MLSD requires a additioal 3.3 db, or.8 db SNR, to achieve -6 BER at R b W =.5. Thus, the codig gai for a code based o -PPM ca be at most 3.3 db i this case. Higher codig gais are possible for higher-order alphabets. Liks, IEEE Trasactios o Commuicatios, vol. 43, o. -4, pp , Feb.-Apr [3] H. Park ad J. R. Barry, Modulatio Aalysis for Wireless Ifrared Commuicatio, Proc. of Itl. Cof. o Comm., Seattle, WA, pp. 8-86, Jue 995. [4] M. D. Audeh ad J. M. Kah, Performace Evaluatio of L-Pulse Positio Modulatio o No-Directed Idoor Ifrared Chaels, ICC, New Orleas, LA, pp , May 994. [5] R. G. Gallager, Iformatio Theory ad Reliable Commuicatio, New York, Wiley, 968. [6] C. N. Georghiades, Modulatio ad Codig for Throughput-Efficiet Optical Systems, IEEE Trasactios o Iformatio Theory, vol. 4, o.5, pp , September 994. [7] W. Hirt, Capacity ad Iformatio Rates of Discrete- Time Chaels with Memory, Ph. D. Dissertatio, Swiss Federal Istitute of Techology, Zürich, 988. [8] S. Shamai (Shitz), L. H. Ozarow, ad A. D. Wyer, Iformatio Rates for a Discrete-Time Gaussia Chael with Itersymbol Iterferece ad Statioary Iputs, IEEE Trasactios o Iformatio Theory, vol. 37, No. 6, , November 99. [9] J. R. Barry, Sequece Detectio ad Equalizatio for Pulse-Positio Modulatio, Proc. of Itl. Cof. o Comm., New Orleas, LA, pp , May 994. [] R. E. Blahut, Priciples ad Practice of Iformatio Theory, Readig, MA, Addiso-Wesley, 987. VI. Coclusios We have examied the performace of multiple PPM ad its variats PPM ad OPPM o ISI chaels with additive white Gaussia oise. We have derived the chael capacity over ISI chaels of multiple PPM ad its variats PPM, OPPM, ad with additive white Gaussia oise, assumig the codewords are idepedet ad uiformly distributed. For umerical comparisos we cosidered a first-order low-pass filter chael with badwidth W. The error probability ad chael capacity results idicates that, although PPM modulatio schemes are extremely power efficiet across ISI-free chaels, their power efficiecy drops dramatically o ISI chaels. Bits / Codeword R b / W =.5 I ideal I U I L Î iid VII. Refereces [] F. R. Gfeller, U. Bapst, Wireless I-House Data Commuicatios via Diffuse Ifrared Radiatio, Proceedigs of the IEEE, vol. 67, o., pp , November 979. [] J. M. Kah, W. J. Krause, ad J. B. Carruthers, Experimetal Characterizatio of No-Directed Idoor Ifrared SNR (db) Fig. 3. Bouds o the chael capacity for PPM vs SNR = P / N R b for R b W =.5. (The capacity I ideal for the ideal case R b W = is icluded for referece.)