Multiple-Accessing over Frequency-Selective Fading Channels

Transcription

1 Multiple-ccessing over Frequency-elective Fading Channels bstract-this work considers the transmission o inormation rom many independent sources to a common receiver over a channel impaired by multipath propagation. In cellular radio communications this is the case o the uplink. e start by examining the achievable rate region o the multiuser requency-selective ading channel without knowledge o the channel on the transmission end. It has been shown that M (pread pectrum Multiple ccess) is theoretically capable o higher data rates than FDM (Frequency Division Multiple ccess) or slow requency-hopping[1]. hen the average received power or all the users is equal, which corresponds to a perect slow power control, we show that the maximum spectral eiciency o M exceeds that o FDM or slow requency-hopping by.577 nats/s/hz or many users and ayleigh ading. lso, we ormulate the optimal multiple-access scheme when all the channels are known to the transmitters. In turns out that only one user should transmit at any given requency. Moreover, the input power spectra or the transmitters are water-illing ormulae both in requency and time. It is shown that the spectral eiciency or the optimal scheme is signiicantly higher than both those o M and FDM. IINTODUCTION By ar the greatest challenge or mobile radio communication engineers is the design o robust and eicient systems which are transparent to the hostile eects o multipath propagation. Because o the enormous growth o cellular communications in recent years, this has become even more important. The ocus o this work will be on undamental perormance limits o dierent multipleaccess methods in multipath ading channels. ecent papers dealing with the inormation capacity o time-varying channels which are unknown to the transmitters include [], in which the capacity vs. outage characteristics o two-path ayleigh ading channels are addressed. This work handled the TDM (Time Division Multiple ccess) case and is essentially a single-user result. Comparisons are made or varying degrees o path delays and dierent diversity techniques. The multiuser case is addressed in [1] where the achievable rate regions o M (pread-pectrum Multiple-ccess) and FDM (Frequency Division Multiple-ccess) are compared or arbitrary requency-selective ading channels. It is shown that the achievable rate region o FDM is contained within that o M and consequently higher data rates are theoretically possible with M. The achievable rate region o FDM is also that o a slow requency-hopping system such as the Global ystem or Mobile Communications (GM) (see [3, Chap 8]). In [4] the single-user requency-lat ading channel is examined rom the point o view o power control, which is simply a orm o channel eedback. It turns out that a. Knopp, P.. Humblet Mobile Communications Group Institut Eurécom ophia ntipolis FNCE EMIL: {knopp,humblet}@eurecom.r power control law which is water-illing in time can increase spectral eiciency or low average signal-to-noise ratio. In [5] the optimal power control laws or a multiuser system with channel eedback in requency-lat ading are considered. They are optimum in the sense that they maximize the total sum-o-rates, which is a good undamental measure o the perormance o a multiuser system when the average received power o the users are equal. This requires perect control o the slowly-varying or average statistics o the channels. In the optimal system, only one user is permitted to transmit at any given time. The total sum-o-rates or this scheme is signiicantly greater than the non-ading Gaussian channel when the number o users is large. In this work we irst discuss the model o the requencyselective ading channel ollowed by the achievable rate region o the multiuser channel. hen the channels are known to the receivers, but not to the transmitters, we show that the total sum o rates or M, with perect slow power control, exceeds that o FDM or slow requency-hopping by.577 nats/s/hz in ayleigh ading. e then consider the case o channels known to the transmitters. Using the total sum o rates as a igure o merit we derive the optimal multiple-access scheme when the basestation can allocate the users transmits powers arbitrarily in the spectrum. e show that as the number o users grows, the optimal scheme has a signiicantly higher spectral eiciency than M and FDM. II MULTIUE FDING CHNNEL MODEL In the uplink o a single-cell multiuser communication system we have K users sharing a ixed bandwidth K. Each user signal, u i () t, is transmitted over a dierent channel with time-varying impulse response h i ( τ, t). This is the response at time t to an impulse at time t τ. e have the ollowing composite complex baseband signal at the basestation, K 1 rt () = u, (1) ( t τ)h i i ( τ, t)dτ + zt () i = where zt () is complex gaussian noise with power spectral density N. The impulse response is the result o multipath propagation, and can be expressed as h i ( τ, t) = a ij ()δτ t ( τ ij () t ), () j where a ij () t and τ ij () t are the zero-mean complex signal strength and time delay or the j th path o channel i respectively. e ignore any slow variations in received signal level due to path loss and shadowing [6]. This is done since we will be concerned later with the average signal-

2 to-noise ratio (N) which can incorporate any slowlyvarying or constant terms, provided they are perectly controlled. From this point onward, we reer to this as perect slow power control. Furthermore, the multipath components are scaled such that Σ j a ij () t = 1 and are assumed to be uncorrelated. In the ollowing section, we will assume that the channels are time-invariant or blocks o length T b. In addition, we place a guard-time T G between such blocks upon transmission which is at least as long as the multipath delay spread. This assures that the received signal in any block is independent o the inormation rom previous blocks. The channel responses or blocks [ nt ( b + T G ), ( n + 1) ( T b + T G )] are denoted by h i ( τ,. These restrictions will allow or requency-domain representations or the average mutual inormation unctionals. III CHIEVBLE TE EGION (NO CHNNEL KNOLEDGE) In this section we consider the achievable rate region when the users have no complete knowledge o the channels. e assume, however, that all the channels are known to the receiver. The multiaccess coding theorem [7][8] states that the achievable inormation rates over an observation interval [, T ], i ( T), i =,, K 1, are bounded by i ( T) < 1 -- I( t ();{ U T i () t, i } { U i () t, i }, { h j ( t, τ), j =,, K 1} ) { 1,,, K 1} where I( Xt ()Yt ; ) is the average mutual inormation unctional or the processes xt () and yt (). Using similar arguments to those or the single-user case in [] and the stationary block assumption rom ection II, the H o the inequalities in (3) can be expressed as i ( T) < N( T b + T G ) N 1 I( t ()U ; i () t, i { U i () t, i }, n = { h j ( t,, j =,, K 1}, t [ n( T b + T G ), ( n + 1) ( T b + T G )]) { 1,,, K 1}, which can be inner-bounded by 1 N 1 i --- i H i < K N ln + KN n = 1 1 { 1,,, K 1}. (3) (4) d, (5) The inormation processes are chosen to be Gaussian in order to maximize the mutual inormation unctional, and i and i are respectively the power spectrum o the i th users signal and the requency response o the i th channel i the n th block were ininite in duration. The channel responses are taken to be ergodic random processes, which is valid given our assumption o perect slow power control, so that reliable communication is possible i 1 K 1 i H i i < E ln KN 1 { 1,,, K 1}. e also assume that the average transmit power (not including slow power control) is constrained to N K lim (7) N i d = K E[ i ] d n = 1 1 ithout channel knowledge on the transmission end, the optimal input spectrum is lat over the entire bandwidth (i.e. i = i ( K) ). This ollows rom the act that the statistics o the channel responses are independent o requency. In order to achieve these rates, long observation times are required, or equivalently long interleaving depths, in order to average over many possible channel realizations. For the case o equal average received N, the inequality in (6) which interests us the most will be the one corresponding to = { 1,,, K 1}, which is the total sum o rates. This is because, in the symmetric case, the equalrate line = 1 = = M 1 intersects the portion o the achievable rate region deined by this inequality. This intersection point determines the maximum rate at which all the users can transmit reliably. This is shown in Fig. 1 or a two-user system, where denotes the maximum achievable rate or a single-user (which is the same or both users in the symmetric case) and max denotes the sum o rates max max, Figure 1: The chievable um o ates The capacity region in (6) is essentially that o an M system, since all the users transmit over the entire available bandwidth. For the sake o comparison, consider an FDM system where each user occupies one o K equal size sub-bands. In this case, the achievable rate region is given by 1 i i H i < E ln d N i 1 (8) i H i = E ln N { 1,,, K 1}, It is shown in [1] that this rate region is strictly enclosed by the region o an M system s. Moreover, a slow re- d, (6)

3 quency-hopping system is also bounded by these inequalities. This implies that higher data rates are theoretically possible with M than with systems which use slow requency-hopping. Consider the case when i =, which corresponds to the case when the average powers are equalized. The achievable sum o rates or an M system is given by K 1 C M = E K ln (9) KN H i (nats/s). i = imilarly, or FDM or slow requency-hopping we have H i C FDM = K E ln (nats/s). (1) N It is clear that C FDM C M by the convexity o the logarithm [1]. Furthermore, or large K and, it can be shown that C M C FDM K E[ ln( H i )](nats/s). (11) I we assume a ayleigh ading model, H i is exponentially distributed with unit mean so that C M C FDM + γ Euler K (nats/s), (1) where γ Euler =.577 is Euler s constant. IV OPTIML POE LLOCTION UING CHNNEL KNOLEDGE In the previous section we considered the case when the channels were not known to the users, aside rom the average received N. Now we assume the channels are known completely to the users and they can choose to transmit their signals in arbitrary parts o the available bandwidth. The DECT system employs a technique along these lines. In this system, the available bandwidth is divided into several channels, which we assume to be requency lat. The users measure the strength o each channel, and choose the best available channel on which to transmit. e show that under certain conditions the optimal scheme is a generalization o such techniques. In many situations, the tap weights change slowly enough (with respect to the data rate) to be estimated accurately. ny KE receiver (see [9]), or instance, must perorm this type o estimation. ssuming this is possible, consider a situation where the basestation estimates the channel response or each channel and instructs the users (via the downlink) to transmit with power spectrum, ( H, H 1,, H M 1. Channel state inormation o this sort is a generalization o conventional power control which is normally requency independent. Moreover, the instantaneous power spectrum or a given user depends on the channel responses o all the users in the system, which is not the case in conventional power control. e would like to choose the power controllers i (we have dropped explicit mention o the channel responses rom the power controllers to simpliy notatio to maximize a useul perormance measure under the power constraint in (7). ssuming perect slow power control and equal average received powers, the most logical choice or the perormance measure is the total sum o rates. ince we have complete channel knowledge at the transmission end and equal average received powers, the maximization will yield the equal rate point on the capacity region. For a more detailed discussion see [1]. Using the Kuhn-Tucker theorem to solve this maximization problem, we obtain the ollowing amiliar water-illing ormulae or the input power spectra (power control laws) 1 KN i λ H i H i > H j = (13) otherwise where λ is a Lagrange multiplier chosen to satisy (7). It is a water-illing solution since i + KN H i is constant when user i is transmitting. This eect is similar to the single-user case [4,11] with a two added eatures. Firstly, the act that there are multiple users and that the basestation can allocate dierent parts o the bandwidth arbitrarily, only one user is permitted to transmit at any given requency. This user has the strongest instantaneous response at that requency. econdly, it is also water-illing in time; this is because more power is allocated when the channel is good and less when it is bad, which changes dynamically in time. The requencylat case [5] is a special case where we have a water-illing eect in time only, and only one user transmits at any given time over the entire bandwidth. The total sum o rates when the channels are known, C KC, can be shown to be K C KC K ( 1) i 1 K =, (14) i Ei iλk γ s (nats/s) i = 1 where Ei(. ) is the irst order exponential integral and γ s = N. Both the Lagrange multiplier λ and the capacity are independent o the requency-selectivity o the channel. e have, thereore, that the sum o rates is the same as or the requency-lat channel [5]. The important dierence between the requency-lat and requency-selective cases is that in the latter several users can share the entire band. This means that in spite o the act that requency-selectivity has no advantage in terms o average capacity, users may have to wait less time to access the channel. For the same total bandwidth, however, the instantaneous data rate or a given user will be lower than in the requency-lat case. e should also mention that in ayleigh ading with high N, the optimal power adjustment (i.e. water-illing) does not yield a signiicant improvement in terms o spectral eiciency over the case when the transmit power is kept constant, but dynamic time/requency allocation is still perormed. Dynamic allocation is the key actor and can be interpreted as exploiting an inherent diversity in multiuser channels with ading. e now compare the spectral eiciencies o FDM, M and dynamic allocation or ayleigh ading. e irst note that (1) can be expressed as [] N N C FDM Ke = Ei , (15) and (9) can be computed numerically. The three per user spectral eiciencies (expressed now in bits/s/hz) are

4 7 6 1 C KC (K = 16) 1 C KC (K = 4) 5 1 C KC (K = ) 1 C G 1 C (bits/s/hz) C FDM 1 C M (K = 16) 1 C M (K = 4) 1 C M (K = ) db N Figure : Multiuser pectral Eiciency in ayleigh Fading shown in Fig., along with the spectral eiciency o a non-ading Gaussian channel with the same transmit power, C G = ln (16) N s the number o users increases, the spectral eiciency o M approaches the Gaussian channel. The gap between the curves or M and FDM is on the order o.577 ln (bits/s/hz) or K = 16 and high N as in (1). In terms o spectral eiciency (with an ininite observation interval), M is not much better than FDM, however we believe that the capacity vs. outage characteristics o M as deined in [] are better than those o FDM because o added diversity. e see that eedback o the channel responses can yield a signiicant improvement in capacity, especially with a large number o users. Moreover, we believe the same is true or the capacity vs. outage, since a user transmits only where and when his channel is good. V PCTICL CONIDETION Partitioning o the available bandwidth in the optimal ashion may be diicult to achieve practically. more practical alternative would be to divide the entire bandwidth into N equal size sub-bands and allocate a single user to each these sub-bands based on their instantaneous requency response over the entire bandwidth. In general, a user may occupy more than one sub-band at any given time, or may not occupy any sub-band at all. e can look at this as an OFDM system with one user per carrier and dynamic allocation o the users on the carriers based on the instantaneous requency responses o the users in each subband. e illustrate this in Fig. 3 or K = 4 and N = 8, where we see that at a particular time it is possible that a particular user occupies more o the available bandwidth than the others. This would be to take ull advantage o the strength o the channels at a particular time. uch a scheme may be very appropriate or high speed wireless data networks. The bandwidth o each o the smaller bands is s = N. I N is large enough, the sub-bands [ + m s, + ( m + 1) s ], n = 1,,, N 1, can be considered as being requency-lat (i.e. s «1 T m ). This reduces the problem to one with N statistically identical, but not independent, parallel chan-

5 H i H 1 H H H 4 user user 4 user 3 user 3 user Figure 3: ubband llocation Example K = 4, N = nels. The sum o rates or this simpliied model must also be given by (14). This will be a good approximation to an optimal system provided that the requency-lat subband assumption holds. VI CONCLUION In this work we considered multiple-access methods or requency-selective ading channels rom an inormation theoretic point o view. Using the achievable sum o rates as a igure o merit or systems with perect slow power control and unknown channels, we have shown that the spectral eiciency o M exceeds that o FDM or slow requency-hopping by.577 nats/s/hz or a large number o users and high N in ayleigh ading. e then derived the optimal time-varying input power distributions when all the channel impulse responses are known to the transmitters. The orm o the input power distributions are multiuser generalizations o the amiliar water-illing ormula or GN channels with a non-lat requency response, and permit only one user to transmit at any given requency, at any given time. Expressions or the spectral eiciency in ayleigh ading were given and compared with those o M and FDM. e show that or a large number o users there is a signiicant improvement when perect channel state inormation is employed at the transmission end. It would also be interesting to determine the capacity vs. outage characteristics o the various schemes, since this may be a more practical measure o the perormance o a multiple-access method. [] Ozarow L.H., hamai., yner.d., Inormation Theoretic Considerations or Cellular Mobile adio, IEEE Trans. on Vehic. Tech., vol. 43, no., May 1994, pp [3] teele., Mobile adio Communications, Pentech Press 199, chap 8. [4] Goldsmith., Varaiya P., Increasing pectral Eiciency Through Power Control, International Conerence on Communications, ICC 93, Geneva, witzerland. [5] Knopp., Humblet P.., Inormation Capacity and Power Control in ingle-cell Multiuser Communications, International Conerence on Communications (ICC 95), eattle, ash, June [6] Lee.C.Y., Mobile Communications Engineering, McGraw-Hill, 198 [7] hlswede,., Multi-ay Communication Channels, Proc. nd Int. ymp. Inormation Theory, Tsahkadsor, rmenien, [8] Liao, H., Coding Theorem or Multiple ccess Communications, Proc. Int. ymp. IT, silomar, C., 197. [9] Proakis, John G., Digital Communications, McGraw- Hill, 1989 [1]Knopp., Humblet P.., Multiuser Diversity in Fading Communication Channels, in preparation [11]Gallager,.G., Inormation Theory and eliable Communication, iley & ons, VII EFEENCE [1] Gallager.G., n Inequality on the Capacity egion o Multiaccess Multipath Channels, in Communications and Cryptography - Two ides o One Tapestry -, Kluwer 1994, pp