Case I: 2 users In case of 2 users, the probability of error for user 1 was earlier derived to be 2 A1

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1 MUTLIUSER DETECTION (Letures 9 and 0) 6:33:546 Wireless Communiations Tehnologies Instrutor: Dr. Narayan Mandayam Summary By Shweta Shrivastava (shwetash@winlab.rutgers.edu) bstrat This artile ontinues the disussion of multiuser detetion. We first analyze the BER performane of a onventional detetor. Starting with the results for user ase, we extend it the ase of K users. We then define ertain other performane measures whih are simpler to alulate (than BER) and yet give a reasonable insight into the performane of multiuser detetors. Finally, we try to define an optimum detetor for multiuser systems. I. INTRODUCTION Multiuser Detetion deals with the demodulation of mutually interfering digital streams of information. Cellular telephony, satellite ommuniation, high-speed data transmission lines et. are some of the ommuniation systems subet to multi-aess interferene. The superposition of transmitted signals may originate from non-ideal harateristis of the transmission medium, or it may be an integral part of the multiplexing method as in the ase of CDM. Multiuser detetion exploits the onsiderable struture of the multiuser interferene in order to inrease the effiieny with whih hannel resoures are employed. Until a ouple of deades earlier, the onventional wisdom was that it was best to simply neglet the presene of multiaess interferene sine its statistial properties would be similar to additive white Gaussian noise, and therefore a single-user mathed filter should be near-optimal to ombat suh interferene. This onventional wisdom was proven wrong by the derivation and analysis of the optimal multiuser detetor by Sergio Verdu, who showed that there is, in general, a huge gap in performane between the performane of the onventional single-user mathed filter and the optimal attainable performane. In this artile, we first analyze the error probability of the onventional reeiver for synhronous and asynhronous ases. We then define ertain other maor performane measures whih are used to ompare multiuser detetors. Finally, we analyze an optimum detetor for multiuser systems. II. THE MTCHED FILTER IN THE CDM CHNNEL This setion is onerned with the analysis of the apabilities and performane of multiuser detetion using the simplest detetor: the single-user mathed filter. In the multiuser detetion literature, this detetor is frequently referred to as the onventional detetor. We onsider the basi synhronous CDM multiple aess K-user hannel K y( = b s ( + n( () =

2 where ( 0, ), b {, + } and s are the reeived amplitudes, information bit sequene and unit-energy signature waveform of the th user respetively, and n( is the additive white Gaussian noise. We first ontinue with the ase for users. Case I: users In ase of users, the probability of error for user was earlier derived to be + P = + ( ) Q Q () If interferene from user is dominant as ompared to the reeived signal from user, then the following ondition holds. > (3) Then the onventional reeiver exhibits anomalous behavior. For example, the error probability is not monotoni with. lim P ( ) = (4) lim P ( ) = (5) 0 Thus the probability of error tends to ½ in both these ases - when the noise is very high as well as when the noise is negligible. Eq (5) results beause, due to (3), as 0, the polarity of the output of the mathed filter for user is governed by b rather than b. We an say that some amount of noise ( > 0) is atually good for detetion beause it leads to P ( ) <. In fat, it an be shown that there is an optimum noise variane that minimizes the BER under ondition given by (), i.e. when interferene is dominant. This optimum variane is, = (6) tanh Further, when the ontribution from interferene is the same as the ontribution of the desired user, i.e. when = (7) Then, the probability of error is given by, P ( ) = + Q (8) 4

3 The interpretation of the above is that when ondition given by (6) holds, then with probability ½, the signal of user exatly anels the signal of user, whih means that we are left only with noise. lso with probability ½, the signal of user doubles the ontribution of the desired user. Figure plots the BER in () with = 0. as a funtion of the SNR / for several values of the relative amplitude of the interferer. Fig.. Bit-error-rate of onventional detetor with two synhronous users and = 0. We an observe that if the interferer has a stronger signal, it an ompletely suppress the signal of the desired user (shown with inreasing in the graph). This leads to the nearfar problem. In other words, the BER degrades rapidly as inreases. n alternate view of the near-far problem is given by the following graph (figure ). The graph shows power tradeoff regions for the ase of users. Power tradeoff regions tell the SNRs required from both users in order to ahieve a given BER for a given value of. The graph shows power tradeoff regions to ahieve a BER for = 0.. We have Q - (3 0-5 ) = db. When = 0, the signals from the users an t see eah other. So, the SNRs required from both need to be greater than db to ahieve the above BER. Note: The graph shows that as inreases, (a) even if amplitudes or power levels of both users are idential, the neessary energy required to ahieve a given BER inreases rapidly (shown by the points lying on the diagonal). (b) The sensitivity to imbalane in the reeived energy from the users inreases. 3

4 In mobile systems, the reeived amplitudes may vary over a wide range, whih ditates the need for strit power ontrol and also low ross-orrelation properties. Fig.. Signal-to-noise ratios neessary to ahieve bit-error-rate not higher than for both users, parameterized by. To visualize the operation of a onventional detetor in signal spae, we loo at the spae y -y, where y is the output of mathed filter and y is the output of mathed filter. Fig. 3. Deision regions in the two-dimensional spae of mathed filter outputs 4

5 In the above graph, we have assumed that = =. is the ross-orrelation between the two signals. (+ +) means that b > 0 and b > 0, (- +) means b < 0 and b > 0, and so on. We are looing at mean vetors here. Conditioned on ( b, b ) the output vetor is a Gaussian vetor with mean The ovariane matrix is b b ov( y, y + b + b ) = The reeived vetor an be viewed as the sum of the mean vetor and the zero-mean n Gaussian vetor. n Therefore, the idea is that if we eep the deision regions fixed, the transmitted vetors hange aording to the amplitudes and ross-orrelations. This may lead to degradation in performane and also anomalous behavior. Case : K Users Let s generalize the above results to a K-user system. Following the same reasoning as before, the probability of error for the th user is given by, P ( ) P b = + P y < 0 b = + + P b = P y > 0 b = (9) [ ] [ ] [ ] [ ] = If we adopt the same proedure as in the ase for users, we get P ( ) = P n > b + P n < b (0) The two probabilities (the two terms in the above equation) are symmetri. P ( ) = P n > b () Conditioning on all the interfering bits, the above probability beomes P ( ) = Q + e K e {, + } e {, + } e {, + } Observations: (i) () The BER of onventional detetor in the CDM Gaussian hannel depends on the shape of the signature waveforms only through the ross-orrelations. 5

6 (ii) BER depends on reeived amplitudes and the noise level only through the Upper Bound ratios, as the deisions are invariant to the saling of the reeived waveform. We an find an upper bound on the probability of error expression of () by replaing with, and onsidering the largest expression. We get, P ( ) Q (3) To observe the near-far behavior, note that () 0 as 0 (i.e. probability of error vanishes when baground noise diminishes) iff > (4) The above ondition is nown as the open-eye ondition. It is the ondition that the reeived signal strength of the desired user is greater than the sum of noises due to interferene from other users. Thus, if there is no baground noise in the system, error free deisions an be made if the open eye ondition is met. Under (4), the bound of (3) beomes tight as 0. lso note that the omputation of () grows exponentially as the number of users. It is very tempting to use a Gaussian approximation, i.e. replae the binomial random variable b with Gaussian with idential variane. The probability of error for th user with Gaussian approximation beomes, ~ P ( ) = Q (5) + The above Gaussian approximation is generally good at low SNRs, but may beome unreliable at high SNRs. The graphs of figures 4 and 5 draw a omparison between the exat expression of () and the Gaussian approximation of (5). The graphs show a plot of BER as a funtion of SNR for the two ases. Note from the figures that in the limit as 0, the exat expression and Gaussian approximation behave differently. For example, (4) has non-zero limit even if open-eye ondition is satisfied. This differene an be attributed to the fat that the error in replaing binomial random variable with Gaussian random variable is greatest in the tails, whih determines the BER (unless baground noise is dominant, whih is usually not the ase for M systems). 6

7 Fig. 4. Bit-error-rate of the single-user mathed filter with 0 equal-energy users and idential rossorrelations l = 0.08; (a) exat, (b) Gaussian approximation. Fig. 5. Bit-error-rate of the single-user mathed filter with 4 equal-energy users and idential rossorrelations l = 0.08; (a) exat, (b) Gaussian approximation. For the asynhronous ase, eah bit is affeted by ( -) interfering bits (as opposed to for the synhronous ase). The probability of error for th user is then given by, 7

8 P ( ) = K ( e, d) {, + } ( e, d ) {, + } ( e, d ) {, + } Q + ( e + d ) (6) Note that. and are asynhronous ross-orrelations; these are same as ontinuous-time partial ross-orrelations, and are given by, ( τ ) s ( s ( t τ ) dt (7) l = T τ l = τ l ( τ ) s ( sl ( t + T τ ) dt (8) 0 l is nown as the right ross-orrelation and l is nown as the left ross-orrelation. The open-eye ondition in ase of asynhronous system is, > + ( ) Note from the above equations that probability of error is hard to ompute even for simple mathed filter reeivers. Hene we define some other easy to ompute performane measures. (9) III. MUTLIUSER EFFICIENCY ND RELTED MESURES While BER is a main performane measure in most ommuniation systems, there are several performane measures derived from it that are useful in design, analysis and understanding of the performane of various detetors. Signal-to-Interferene Ratio (SIR) SIR is one of the performane measures of multi-user detetion. SIR gives the ratio of powers due to the desired user and due to all other omponents. In the absene of interferene, SIR =, and single-user performane is ahieved, i.e. probability of error is given by, P ( ) = Q (0) In the presene of interferene, SIR = + () Therefore, the presene of interferene inreases the BER. 8

9 It is of interest to quantify the multi-user error probability relative to the optimum singleuser BER. Hene we define the following metri. Effetive Energy Effetive energy of user, e () is the energy that user would require to ahieve a BER equal to Where P P ( ) ( ) in a single-user Gaussian hannel with the same baground noise, i.e., e ( ) P = ( ) Q () is the multi-user error probability. Sine the multi-user error probability is lower bounded by the single-user error probability, we have, P ( ) Q e ( ) (3) tells us that the effetive energy for user is upper bounded by the atual energy. If we normalize the effetive energy by e (noise variane), we obtain, ( Q ( P ( ))) (3) = (4) ( ) The power tradeoff region an then be haraterized in terms of the effetive energy as follows. The power tradeoff region for a given permissible BER P (same for all users) is the set of SNRs,,...,, suh that max P ( ) P, or equivalently, e ( ) min [ Q ( P) ] (5) Multi-user Effiieny Multi-user effiieny is defined as the ratio of effetive and atual energies. It is given by e ( ) /, and quantifies the performane loss due to the existene of other users in the hannel. Multi-user effiieny depends on the signature waveforms, the reeived amplitudes (SNRs) and on the detetor employed. It follows from (3) that multiuser effiieny belongs to the interval [0,] (or [-, 0] in db). symptoti Multi-user Effiieny symptoti multi-user effiieny haraterizes the performane loss (in effetive signalto-noise ratio (SNR)) of multi-user detetor as the baground noise vanishes. It is defined for user as, 9

10 e ( ) η = lim (6) 0 n alternate and more formal definition of multi-user effiieny is given as, P ( ) η = sup 0 r ; lim = 0 r 0 r Q = lim log 0 (. P ) Therefore, we see that when the eye is losed (i.e. BER does not vanish as 0), the asymptoti multiuser effiieny is equal to 0. If η > 0, then BER 0 as 0 and moreover, it vanishes exponentially in the SNR. η also measures the slope with whih P () 0 on a log sale in high SNR region. Typially, the asymptoti multiuser effiieny is very lose to multiuser effiieny (exept at low SNRs). (7) Near-Far Resistane The near-far resistane is the asymptoti multiuser effiieny minimized over the reeived energies of all the other users. It measures the robustness of the detetor to varying levels of interferene. It is given by, η = inf η > 0 (8) The near-far resistane depends on the signature waveforms and on the demodulator. It is sometimes easier to ompute these measures (multiuser effiieny and near-far resistane) than the probability of error. We now analyze the robustness of onventional reeiver for multi-user detetion by alulating its multi-user effiieny and near-far resistane. Conventional Reeiver We onsider the ase for users, i.e. K = with synhronous CDM model. symptoti multiuser effiieny: The probability of error for user as given by () is, P ( ) = Q + + Q 0

11 When the eye is losed, >, then as 0, P ( ) does not vanish (reall that it goes to ½ ). η = 0 If eye is open, >, then lim 0 Q P + Q + Q ( ) = lim r 0 r Q 0, =, r r Using the above equation and (7), we get, η < > = Thus the overall asymptoti multiuser effiieny (using both eye-open and eye-losed onditions) is given as, η = max 0, (30) The asymptoti multiuser effiieny is plotted as a funtion of the relative amplitude for = 0. in figure 6. Evaluating similarly for the K-user ase, we get, η = max 0, Near-far resistane: The near-far resistane is obtained by minimizing (3) with respet to,. We observe that η = 0, unless = 0 for all. We an thus onlude that the mathed filter or onventional reeiver is not near-far resistant unless the signature waveform of the th user is orthogonal to eah of the partially overlapping waveforms from all other users. (9) (3)

12 Fig. 6. symptoti multiuser effiieny of onventional detetor as a funtion of the amplitude of the interferer; = 0. (linear plo. IV. OPTIMUM DETECTOR very simple demodulator for the CDM hannel was analyzed in setion II. We turn our attention now to the derivation and analysis of optimum strategies. The analysis of optimum multiuser detetors yields the minimum ahievable probability of error (and optimum asymptoti multiuser effiieny, as well as optimum near-far resistane) in CDM hannels. This serves as a baseline of omparison for suboptimum multiuser detetors. The onventional reeiver requires no nowledge beyond the signature waveforms and timing of users it wants to demodulate. In the derivation of an optimum reeiver, we will assume that the reeiver not only nows the signature waveform and timing of every ative user, but it also nows (or an estimate) the reeived amplitudes of all users and the noise level. Consider the K-user basi synhronous CDM hannel: K y( = b s +. n(, t [ 0, T ] (3) = The optimum deision rule in this ase is the maximum a posteriori probability rule (MP). However, two optimum deision strategies using MP an be employed, and they need not result in the same deision. They are the following. Individually optimum: Jointly optimum: max b [ b y(,0 t T ] P, =,..., K [( b, b,..., b ) y(,0 t T ] (33) max P K, b,..., b (34) b The following example illustrates that the two riteria are indeed different. Consider K =. Let the noise realizations are suh that the a posteriori probabilities tae the following values:

13 [( +, + ) y( ] = 0.6 [(, + ) y( ] = 0.6 [( +, ) y( ] = 0.7 [(, ) y( ] = 0. P P P P Then the ointly optimum deisions are the one with highest a posteriori probability, i.e., (b, b ) = (+, -). The individually optimum deisions are given by evaluating [ y( ] = P[ ( b, + ) y( ] + P[ ( b, ) y( ] [ y( ] P[ ( +, b ) y( ] + P[ (, b ) y( P b P b = ] Thus the individually optimum deisions are (b, b ) = (+, +). However, we usually expet the two results to be the same with very high probability if the probability of error is low. Hene, either riteria is aeptable. Let us onsider the ointly optimum demodulation of b = [ b,..., ] For the ase of n( in (3) being WGN, the optimum reeiver is the maximum lielihood reeiver and also the minimum probability of error reeiver. bˆ ML = arg max P[ y(, t [0, T ] b] (35) It an be shown that a suffiient statisti for ML detetion is is the vetor of mathed filter outputs. nd y n K b b K T = T y s ( dt = b + 0 T y = y, y,..., ], whih [ ( b + n (36) n = n, n,..., ]. n is ointly Gaussian random vetor. [ We an write [ ] E n E = 0 [ n n l ] = l y = R b + n (37) where R = ] is the normalized ross-orrelation matrix whose diagonal elements are [ i equal to and whose (i, ) element is equal to the ross-orrelation i. is K K diagonal matrix of reeived amplitudes, = diag,..., } (38) { K y K T 3

14 ( y b) Therefore, the ML rule is, Where, p = exp T ( y Rb) ( R) ( y Rb) ( π ) K ( b) R (39) b ˆ = arg max Ω (40) ML ( b) b T T Ω = b y b H b (4) H = R (4) The above maximization is a ombinatorial optimization problem, whih implies that the omplexity grows exponentially in the number of users (need to searh over K hoies). For K =, the optimum reeiver s asymptoti multiuser effiieny is given by, opt η = min, + (43) Fig. 7. Optimum and single-user asymptoti multiuser effiienies for two synhronous users. Figure 7 shows the asymptoti effiieny of user for both optimum and onventional reeiver. It an be seen that for optimum reeiver asymptoti effiieny is not monotoni in /. tually, if > then η =. Therefore, as long as the energy of user exeeds the threshold given by above equation the asymptoti bit-error-rate of user is equivalent to the single-user ase where user is not ative The explanation of this behavior of the optimum reeiver is that if the interfering user is suffiiently powerful, then the primary soure of errors ommitted in the optimum demodulation of user is the baground Gaussian noise, rather than the randomness of the information arried by the interfering signal. This fat ould be explained using the suessive deoding tehnique. The near-far resistane is obtained by minimizing equation 0 over / 0. 4

15 The least favorable relative amplitude of user is = whih yields the near-far resistane for either user: η = Figure 8 shows the two-user power-tradeoff region so that the optimum bit-error rate of both the users is not higher than 3 0 5, for =0.8, 0.9 and If we ompare this figure with the one for onventional reeiver, we an onlude that the permissible signal-to-noise ratios are indistinguishable as long as the ross-orrelation satisfies 0.5. lso for high ross-orrelations values, equal powers for users are detrimental. The reason is that if both signature waveforms are very muh alie, then the similar amplitudes ompliate the tas of the optimum reeiver. Fig. 8. Signal-to-noise ratios neessary to ahieve optimum bit-error-rate not higher than for both users. The omplexity of optimum multi-user detetor requires one to ome-up with other suboptimum multiuser detetors that exhibit good performane and omplexity tradeoffs. 5

16 REFERENCES [] N. Mandayam, Wireless Communiation Tehnologies, Leture Notes, Spring 005, Rutgers University. [] S. Verdu, Optimum Sequene Detetion of synhronous Multiple-ess Communiations, bs. 983 Int. symp. Information Theory, St. Jovite, Canada, p. 80, sept [3] S. Verdu, Optimum Multi-User Signal Detetion, Ph.D. Thesis, Dept. of Eletrial and Computer Engineering, University of Illinois, ug [4] S. Verdu, Multi-user detetion, Cambridge University Press,