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Capacity of Rayleigh Fading Channels Under Different Adaptive Transmission and Diversity-Combining Techniques
Mohamed-Slim Alouini, Member IEEE, and Andrea J. Goldsmith, Member, IEEE
Abstract We study the Shannon capacity of adaptive transmission techniques in conjunction with diversity combining. This capacity provides an upper bound on spectral efciency using these techniques. We obtain closed-form solutions for the Rayleigh fading channel capacity under three adaptive policies: optimal power and rate adaptation, constant power with optimal rate adaptation, and channel inversion with xed rate. Optimal power and rate adaptation yields a small increase in capacity over just rate adaptation, and this increase diminishes as the average received carrier-to-noise ratio (CNR) or the number of diversity branches increases. Channel inversion suffers the largest capacity penalty relative to the optimal technique, however, the penalty diminishes with increased diversity. Although diversity yields large capacity gains for all the techniques, the gain is most pronounced with channel inversion. For example, the capacity using channel inversion with two-branch diversity exceeds that of a single-branch system using optimal rate and power adaptation. Since channel inversion is the least complex scheme to implement, there is a tradeoff between complexity and capacity for the various adaptation methods and diversity-combining techniques. Index Terms Adaptive transmission techniques, diversity combining, Shannon capacity.

at the communication link level through a skillful combination of bandwidth efcient coding and modulation techniques. In this paper, we focus on the link spectral efciency, dened as the average transmitted data rate per unit bandwidth for a specied average transmit power and bit error rate (BER). Over the last three decades, researchers have looked at various ways to improve the link spectral efciency of wireless systems. In what follows, we rst briey summarize the major steps and progress achieved in that arena. We then present the objectives and outline of our paper. A. Spectral Efciency over Fading Channels Multilevel modulation schemes, such as MQAM, increase link spectral efciency by sending multiple bits per symbol [3]. Unfortunately, mobile radio links are subject to severe multipath fading due to the combination of randomly delayed reected, scattered, and diffracted signal components [4]. Fading leads to serious degradation in the link carrier-to-noise ratio (CNR), resulting in either a higher BER or a higher required transmit power for a given multilevel modulation technique. Thus, fading compensation is typically required to improve link performance. One compensation technique, proposed by Sampei and Sunaga [5], uses pilot symbolassisted modulation (PSAM). This technique inserts a training sequence into the stream of MQAM data symbols to extract the channel-induced attenuation and phase shift, which are then used for symbol detection. Space diversity, which combines signals received over several antenna branches, is another powerful technique to combat fading [6]. Diversity can often be combined with other fading compensation methods to mitigate most of the fading degradation. For example, joint use of PSAM and maximal ratio combining (MRC) antenna diversity was proposed in [7], and eld trials with this technique demonstrated considerable performance improvement over MQAM without compensation [8], [9]. Other fading compensation techniques include an increased link budget margin or interleaving with channel coding [2]. However, these techniques are designed relative to the worst case channel conditions, resulting in poor utilization of the full channel capacity a good percentage of the time (under negligible or shallow fading conditions). Adapting certain parameters of the transmitted signal to the channel fading leads to better utilization of the channel capacity. The concept

I. INTRODUCTION HE RADIO spectrum available for wireless services is extremely scarce, while demand for these service is growing at a rapid pace [1]. Spectral efciency is therefore of primary concern in the design of future wireless data communications systems. High overall spectral efciency of a wireless cellular system may be achieved at several levels of the system design [2]: at the radio coverage planning level by minimizing cell area and the cochannel reuse distance; at the network/system level by using sophisticated channel allocation schemes that maximize the overall carried trafc;
Manuscript received January 9, 1997; revised March 19, 1998. This work was supported in part by the NSF CAREER Development Award NCR9501452 and by the Ofce of Naval Research under Grant N149510861. M.-S. Alouini was with the Communication Group, Department of Electrical Engineering, California Institute of Technology (Caltech), Pasadena, CA 91125 USA. He is now with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: alouini@ece.umn.edu). A. J. Goldsmith was with the Communications Group, Department of Electrical Engineering, California Institute of Technology (Caltech), Pasadena, CA 91125 USA. She is now with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: andrea@ee.stanford.edu). Publisher Item Identier S 0018-9545(99)05741-2.

00189545/99$10.00 1999 IEEE

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Fig. 1. Transmission system block diagram.

of adaptive transmission, which requires accurate channel estimation at the receiver and a reliable feedback path between that estimator and the transmitter, was rst proposed around the late 1960s [10][12]. Interest in these techniques was short lived, perhaps due to hardware constraints, lack of good channel estimation techniques, and/or systems focusing on point-to-point radio links without transmitter feedback. The fact that these issues are less constraining in current land mobile radio systems, coupled with the need for spectrally efcient communication, has revived interest in adaptive modulation methods. The main idea behind these schemes is real-time balancing of the link budget through adaptive variation of the transmitted power level [10], symbol rate [11], constellation size [13][15], coding rate/scheme [16], or any combination of these parameters [12], [17][22]. Thus, without sacricing BER these schemes provide a much higher average spectral efciency by taking advantage of the timevarying nature of the wireless channel: transmitting at high speeds under favorable channel conditions and responding to channel degradation through a smooth reduction of their data throughput. The performance of these schemes is further improved by combining them with space diversity [23]. The disadvantage of these adaptive techniques is that they require an accurate channel estimate at the transmitter, additional hardware complexity to implement adaptive transmission, and buffering/delay of the input data since the transmission rate varies with channel conditions. B. Objective and Outline The aim of this paper is to investigate the theoretical spectral efciency limits of adaptive modulation in Rayleigh fading channels. This fading channel model applies to land mobile radio channels without a line-of-sight path between the transmitter and receiver antennas, as well as to ionospheric [24] and tropospheric scatter [25] channels. Our analyses can also be generalized to Nakagami fading channels [26]. The Shannon capacity of a channel denes its theoretical upper bound for the maximum rate of data transmission at an arbitrarily small BER, without any delay or complex-

ity constraints. Therefore, the Shannon capacity represents an optimistic bound for practical communication schemes, and also serves as a benchmark against which to compare the spectral efciency of all practical adaptive transmission schemes [19]. In [27], the capacity of a single-user atfading channel with channel measurement information at the transmitter and receiver was derived for various adaptive transmission policies. In this paper, we apply the general theory developed in [27] to obtain closed-form expressions for the capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques. In particular, we consider three adaptation policies: optimal simultaneous power and rate adaptation, constant power with optimal rate adaptation, and channel inversion with xed rate. We investigate the relative impact of MRC and selective combining (SC) diversity schemes in conjunction with each of these adaptive transmission schemes. Note that an analytical evaluation of the capacity in a Rayleigh fading environment with the constant power policy was carried out in [28][30]. We extend this analysis to derive closed-from expressions for capacity with MRC and SC and compare it to the capacity of the other adaptive methods. The remainder of this paper is organized as follows. In Section II, we outline the channel and communication system model. We derive the capacity of a Rayleigh fading channel (with and without diversity) for the optimal adaptation policy, constant power policy, and channel inversion policy in Sections IIIV, respectively. In Section VI, we present some numerical examples comparing: 1) the Rayleigh channel capacity with the capacity of an additive white Gaussian noise (AWGN) channel and 2) the Rayleigh channel capacity for the various adaptation policies and diversity-combining techniques under consideration. We review our main results and offer some concluding remarks in Section VII. II. CHANNEL
AND

SYSTEM MODEL

In this section, we describe the channel and communication system model. A block diagram of the transmission system is shown in Fig. 1.

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We assume that the channel changes at a rate much slower than the data rate, so the channel remains constant over hundreds of symbols. We call this a slowly varying channel. We assume a Rayleigh fading channel so that the probability distribution function (PDF) of CNR ( ) is given by an exponential distribution [4, eq. (5.2-13), p. 314] (1) where is the average received CNR. We consider both MRC and SC diversity combining of the received signal. MRC diversity combining requires that the individual signals from each branch be weighted by their signal voltage to noise power ratios then summed coherently [4, p. 316]. In our MRC analyses, we assume perfect knowledge of the branch amplitudes and phases, which we call perfect combining. MRC with perfect combining is the optimal diversity scheme [4, p. 316] and therefore provides the maximum capacity improvement relative to all combining techniques. The disadvantage of MRC is that it requires knowledge of the branch parameters and independent processing of each denote the average CNR on the th branch. For branch. Let independent branch signals and equal average branch CNR for all the PDF of the received CNR at the output of a perfect branch MRC combiner is a chi-square distribution with degrees of freedom [4, eq. (5.2-14), p. 319] given by (2) -

We also assume that the variation in is sent back to the transmitter via an error-free feedback path. The time delay in this feedback path is assumed to be negligible compared to the rate of the channel variation. All these assumptions, which are reasonable for high-speed data transmission over a slowly fading channel, allow the transmitter to adapt its power and/or rate relative to the actual channel state. III. OPTIMAL SIMULTANEOUS POWER AND RATE ADAPTATION Given an average transmit power constraint, the channel capacity of a fading channel with received CNR distribution and optimal power and rate adaptation ( [b/s]) is given in [27] as (7) is the optimal where [Hz] is the channel bandwidth and cutoff CNR level below which data transmission is suspended. This optimal cutoff must satisfy (8) To achieve the capacity (7), the channel fade level must be tracked at both the receiver and transmitter, and the transmitter has to adapt its power and rate accordingly, allocating highpower levels and rates for good channel conditions ( large), and lower power levels and rates for unfavorable channel , the conditions ( small). Since no data is sent when equal to optimal policy suffers a probability of outage the probability of no transmission, given by

(3) SC diversity only processes one of the diversity branches. Specically, the combiner chooses the branch with the highest CNR [4, p. 313]. Thus, this diversity-combining technique is simpler than MRC, but also yields suboptimal performance. Since the output of the SC combiner is equal to the signal on one of the branches, the coherent sum of the individual branch signals is not required.1 Assuming independent branch signals and equal average branch CNR (2), the PDF of the received CNR at the output of an -branch SC combiner is given by [4, eq. (5.2-7), p. 316] (4) Note that using the binomial expansion, we can rewrite (4) as (5) where denotes the binomial coefcient given by (6) We assume throughout our analyses that the variation in the combiner output CNR is tracked perfectly by the receiver.
1 The SC scheme can therefore be used in conjunction with differential modulation techniques, in contrast to MRC, which is restricted by design to coherent modulations.

(9) We now obtain closed-form expressions for the optimal cutoff capacity and outage probability of CNR this optimal adaptation technique with and without diversity combining. No numerical integrations are required, though numerical root nding is needed to nd A. No Diversity Substituting (1) in (8) we nd that must satisfy (10) where is the exponential integral of order dened by (11) In particular, so (10) reduces to (12)

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Let

and dene (13)

Substituting (3) in (7) we obtain the channel capacity with MRC diversity, [b/s], in terms of the integral as (21)

for all MoreNote that and over, from (13), Thus, we conclude that there is a unique for which or, equivalently, there is a unique which satises (12). An asymptotic expansion of (12) shows that as Our numerical results show that increases as increases, so always lies in the interval [0, 1]. as Substituting (1) in (7), and dening the integral

is derived in the Appendix A and The evaluation of given in (70). Using that result, we obtain the capacity per [b/s/Hz] as unit bandwidth (22) denotes the Poisson distribution dened in (67). where is obtained by The corresponding probability of outage substituting (3) into (9), and using [31, eq. (8.381.3), p. 364] and (66) (23)

(14) we can rewrite the channel capacity as (15) is derived in the Appendix A and The evaluation of given in (71). Using that result we obtain the capacity per unit [b/s/Hz] as bandwidth (16) Using (12) in (16), the optimal capacity per unit bandwidth reduces to the simple expression (17) Using (1) in the probability of outage (9) yields (18) B. Maximal Ratio Combining Inserting the CNR distribution (3) in (8) we see that with must satisfy MRC combining

C. Selection Combining Substituting the CNR distribution (5) into (8) we nd that must satisfy

(24) Let and dene

(25) Note that where Since , , Moreover, from (25), ,

we have

(19)

is the complementary incomplete gamma funcwhere and dene tion dened in (65). Let (20) Note that and for all Since and , we conclude that there such that or, equivalently, there is a unique which satises (19). An asymptotic expansion is a unique as so on (19) shows that Comparing this with the results in Section III-A, we see that as the average received CNR grows to innity, the optimal cutoff value is unaffected by diversity.

and, for all for all and Thus, we conclude that there is a unique for which or, equivalently, there is a unique which satises (24). An asymptotic expansion of (24) shows that Therefore, as in the no diversity and as always lies in the interval [0, 1]. MRC diversity cases, Inserting (5) into (7), we can express the channel capacity [b/s], in terms of the integral with SC diversity, as

(26) Using (71) from Appendix A, we obtain the capacity per unit [b/s/Hz], as bandwidth,

(27)

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The corresponding probability of outage substituting (5) into (9)

is obtained by

Using the rst series expansion for substituting it into (34) yields

[28, eq. (6)], and

(28) tends We see from (28) that as tends to innity (and tends to zero, as expected. Similarly, we can to one) tends to zero (i.e., total channel also see from (28) that as tends to zero. inversion) IV. OPTIMAL RATE ADAPTATION CONSTANT TRANSMIT POWER where for

(35) is the Euler constant Therefore, we get the following approximation for (34): (36) Using the second series expansion for eq. (7)] and substituting it in (34) yields given by [28,

WITH

With optimal rate adaptation to channel fading with a [b/s] constant transmit power, the channel capacity becomes [27], [32] (29) was previously introduced by Lee [28] as the average channel capacity of a at-fading channel, since it is obtained by averaging the capacity of an AWGN channel (30) over the distribution of the received CNR In fact, (29) represents the capacity of the fading channel without transmitter feedback (i.e., with the channel fade level known at the receiver only) [29], [33], [34]. In the following analysis, we rst obtain the channel capacity without diversity (correcting some minor errors in [28]) and then derive analytical expressions as well as simple accurate asymptotic approximations of the capacity improvement with both MRC and SC diversity. A. No Diversity Substituting (1) into (29), the channel capacity Rayleigh fading channel is obtained as of a

(37) is a remainder term. Taking the limit as the channel where bandwidth approaches innity yields (38) [W] is the average carrier power and where is the noise density power per unit bandwidth.2 B. Maximal Ratio Combining Substituting (3) into (29), we obtain the channel capacity [b/s] with MRC in terms of the integral as (39) Using (78) from Appendix B, we can rewrite [b/s/Hz] as (40) is the complementary incomplete gamma funcwhere tion dened in (65). Note that by using [31, eq. (8.359.1), p. 951], the capacity with the MRC diversity scheme (40) reduces to (34), as expected. with a single branch Note also that by using [31, eq. (8.352.3), p. 950] and [31, eq. (8.359.1), p. 951], one may express (40) in terms of the Poisson distribution as [30, eq. (7)] [W/Hz]

(31) Dening the integral as

(32) we can rewrite as (33) Using the result of (80) from the Appendix B, we can write [b/s/Hz] as (34) , Note that the exponential-integral function of rst order, , used in is related to the exponential-integral function, [28] by

(41)
2 In [28], the average CNR is denoted by 0: Note the typographical error in [28, eq. (4)] ( instead of 0 in the denominator) and the resulting sign difference in the argument of the exponential term e1= between our result (34) and [28, eq. (5)] and between (36) and [28, eq. (9)]. There is also a sign difference between (37) and [28, eq. (10)]. However, the limit (38) matches the limiting expression [28, eq. (11)], so the sign error in [28, eq. (10)] was corrected in the limit.

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Moreover, using the rst series expansion for [28, eq. (6)] in (41) we obtain an asymptotic approximation for as

is given in [27] as

(46)

(42) Fig. 5 compares plots of (40) with its asymptotic approximation (42) and the results are discussed in Section VI-A. C. Selection Combining Substituting (5) into (29), we obtain the channel capacity [b/s] with SC in terms of the integral as

Channel inversion with xed rate suffers a large capacity penalty relative to the other techniques, since a large amount of the transmitted power is required to compensate for the deep channel fades. A better approach is to use a modied inversion policy which inverts the channel fading only above a xed The capacity with this truncated channel cutoff fade depth [b/s]) was derived in inversion and xed rate policy [27] to be

(47) (43) Then, using (80) from Appendix B, we can write [b/s/Hz] as is given by (9). The cutoff level can be selected where to achieve a specied outage probability or, alternatively, to is examined in more detail maximize (47). The choice of in the following sections. We now derive closed-form expressions for the capacity under channel inversion with the different diversity combining techniques. A. No Diversity By substituting the CNR distribution (1) in (46) we nd that the capacity of a Rayleigh fading channel with total channel is zero. However, with truncated channel inversion [b/s/Hz] inversion the capacity per unit bandwidth as can be expressed in terms of and (48) on for different Fig. 2 shows the dependence of values. All these curves show that capacity is maximized which increases as a function for an optimal cutoff CNR of Recall that we proved the existence of a unique optimal cutoff CNR for the optimal adaptation policy in Section III-A. However, for optimal adaptation the optimal cutoff CNR was always bounded between [0, 1] (i.e., smaller than 0 dB), is bigger than 0 dB when dB. whereas for this policy This means that for a xed , truncated channel inversion has both a smaller capacity (see Fig. 9) and a higher probability of outage (see Fig. 10) than the optimal policy of Section III. B. Maximal Ratio Combining We obtain the capacity per unit bandwidth for total channel , by substiinversion with MRC diversity combining, tuting (3) into (46) and using [31, eq. (3.381.4), p. 364] (49)

(44) , (44) reduces We again note that for a single branch to (34) as expected. [28, eq. (6)] and Using the rst series expansion for substituting it into (44) yields a simple asymptotic approximafor given by tion

(45) Fig. 6 compares plots of (44) with its asymptotic approximation (45) and the results are discussed in Section VI-A. V. CHANNEL INVERSION
WITH

FIXED RATE

The channel capacity when the transmitter adapts its power to maintain a constant CNR at the receiver (i.e., inverts the channel fading) was also investigated in [27]. This technique uses xed-rate modulation and a xed code design, since the channel after channel inversion appears as a time-invariant AWGN channel. As a result, channel inversion with xed rate is the least complex technique to implement, assuming good channel estimates are available at the transmitter and receiver. The channel capacity with this technique [b/s]) is derived from the capacity of an AWGN channel and

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Fig. 2. Channel capacity per unit bandwidth for a Raleigh fading channel versus cutoff CNR o with truncated channel inversion and = (a) 5, (b) 10, (c) 15, and (d) 20 dB.

Fig. 3. Channel capacity per unit bandwidth for a Raleigh fading channel versus cutoff CNR o with truncated channel inversion, MRC diversity (M = 2); and = (a) 5, (b) 10, (c) 15, and (d) 20 dB.

Note that the capacity of this policy for a Rayleigh fading channel with an -branch perfect MRC combiner (49) is the parallel independent same as the capacity of a set of AWGN channels [35, eq. (15)]. Truncated channel inversion improves the capacity (49) The capacity of at the expense of outage probability is truncated channel inversion with MRC combining, obtained by inserting (3) in (47) and using [31, eq. (3.381.3), p. 364] (50) Using property (66) of the complementary incomplete gamma function, we can rewrite (50) as the simple expression

Fig. 3 shows the dependence of on at different values for two-branch MRC diversity. Comparing Figs. 2 and 3 we see that, for a xed , the maximizing cutoff CNR increases when MRC diversity is used. In addition, the indicates relative atness of the curves in Fig. 3 for that the capacity improvement provided by truncated channel compared to total channel inversion inversion is relatively small, and this little improvement comes at the expense of a higher probability of outage (see Fig. 12). This suggests that as long as diversity is used, total channel inversion is a better alternative than truncated channel inversion. C. Selection Combining We obtain the capacity per unit bandwidth of a Rayleigh fading channel with total channel inversion and SC diversity, , by substituting (5) in (46), as given in (52) at the

(51)

(52)

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Fig. 4. Average channel capacity per unit bandwidth for a Raleigh fading and an AWGN channel versus average carrier-to-noise ratio (with no diversity).

bottom of the previous page. When truncated channel inversion is used in combination with SC the capacity per unit bandwidth becomes (53), given at the bottom of the page. Recall that tends to zero as tends to zero (i.e., total channel inversion). Hence, as expected, we see that (53) reduces to (52) when tends to zero. VI. NUMERICAL RESULTS
AND

unit bandwidth of an AWGN channel (30). With these results we nd, for example, the following. dB, whereas For For dB, whereas

COMPARISONS

In this section, we start by comparing the capacity of an with the capacity of a Rayleigh AWGN channel , constant transchannel with optimal rate adaptation, mit power, and various diversity-combining techniques. We then compare the Rayleigh channel capacities for the various adaptation policies and diversity-combining techniques. A. Comparison with AWGN Channel Capacity , given In Fig. 4, channel capacity without diversity, by (34), as well as its asymptotic approximation (36), are This gure also displays the capacity per plotted against

Therefore, the channel capacity of a Rayleigh fading channel dB and by 9.75% for is reduced by 15.9% for dB.3 Note in Fig. 4 that the asymptotic approximation (36) closely matches the exact average capacity (34) when dB. (40) as well as its asympFig. 5 shows plots of totic approximation (42) as functions of the average CNR and We use the CNR per per branch for branch for this comparison so that we can later compare the improvements provided by MRC and SC diversity schemes on a fair basis, as we will explain in more detail below. Note the
3 These results correct the 32% and 11% values reported in [28]. Moreover, from Fig. 4, note that a smaller value of results in a smaller difference between Cawgn and hC iora ; contrary to what was concluded in [28].

(53)

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Fig. 5. Average channel capacity per unit bandwidth for a Raleigh fading channel with MRC diversity versus average CNR per branch [(a) (b) = 2, and (c) = 4].

M = 1,

large diversity gain obtained by two-branch combining: the capacity with two branches in fading exceeds that of a singlebranch AWGN channel. Fig. 5 also displays the capacity per unit bandwidth of an array of -independent AWGN channels with optimal combining (MRC) [35, eq. (15)]

of

-independent AWGN channels with selection combining (55)

(54)

Note that the capacity of an array of -independent Rayleigh channels with MRC combining approaches the capacity of tends to an array of -independent AWGN channels as innity. Note also the diminishing capacity returns that are obtained as the number of branch increases. This diminishing returns characteristic is also exhibited when the performance evaluation is based on outage probabilities [4]. Finally, note again that the asymptotic approximation (42) closely matches dB. the exact average capacity (40) when (44) as well as its asympFig. 6 shows plots of totic approximation (45) as function of the average CNR per and Comparing Figs. 5 and 6 we branch for see that, as expected, the SC scheme provides less diversity gain and a lower rate of improvement than the MRC scheme. However, the greatest improvement is still obtained in going from single- to two-branch combining, which again yields a higher capacity than that of a single-branch AWGN channel. Fig. 6 also displays the capacity per unit bandwidth of an array

We again note that the capacity of an array of -independent Rayleigh channels approaches the capacity of an array of tends to innity. Finally, independent AWGN channels as the asymptotic approximation (45) closely matches for all dB. the exact average capacity (44) when -branch MRC diversity scheme, the average For the is related to the average CNR per combined CNR We can branch by [4, eq. (5.2-16), p. 319] in terms of as therefore also express

(56) versus the average combined Fig. 7 shows plots of for and These plots show the same CNR and [30, Fig. 1] numerical results as [28, Fig. 2] for These results show that, when expressed in terms for of the average combined CNR, the capacity of a Rayleigh channel can never beat the capacity of a single-branch AWGN channel, but it comes close to the AWGN channel approaches capacity as the number of diversity branches innity.

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Fig. 6. Average channel capacity per unit bandwidth for Raleigh fading channels with selection-combining diversity versus the average CNR per branch [(a) = 1, (b) = 2, and (c) = 4].

Fig. 7. Average channel capacity per unit bandwidth for a Raleigh fading channel with MRC diversity versus average combined CNR h imrc [(a) M = 1, (b) M = 2, and (c) M = 4].

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Fig. 8. Average channel capacity per unit bandwidth for Raleigh fading channels with selection combining diversity versus the average combined CNR h isc [(a) = 1, (b) = 2, and (c) = 4].

For the -branch SC diversity scheme, the average comis related to the average CNR of a single bined CNR branch by [4, eq. (5.2-8), p. 316] We can therefore also express in terms of as

branch (Figs. 5 and 6). However, when looked at individually, Figs. 7 and 8 are also of interest since they show that the capacity of a single-branch AWGN channel with CNR is always bigger than the capacity of a fading channel with the same average received CNR at the output of the diversity combiner, regardless of the adaptation and diversity-combining strategy in the latter case.

B. Comparison of the Different Policies Fig. 9 shows the calculated channel capacity per unit bandfor the different adaptation poliwidth as a function of cies without diversity combining. These curves conrm the previous numerical results reported in [27] using the closedform expressions (17), (34), and (48) instead of numerical integration. From this gure, we see that the optimal power and rate adaptation (17) yields a small increase in capacity over just rate adaptation (34), and this small increase in capacity diminishes as increases. The corresponding outage probability (18) for the optimal adaptation and truncated ) are shown in channel inversion (with optimal cutoff Fig. 10. Fig. 11 shows the channel capacity per unit bandwidth for the different policies with MRC as a function of and As the number of combining diversity for branches increases the capacity difference between optimal power and rate adaptation versus optimal rate adaptation alone For any , xed rate becomes negligible for all values of

(57) versus the average combined Fig. 8 shows plots of for and In fact, Figs. 7 and 8 CNR approaches the capacity of the are very similar, since as the number of diversity single-branch AWGN channel increases. However, a close look at the numerical branches is always slightly results shows that for a xed for an equal average CNR at the output bigger than Note that this slim of the combiner (i.e., for differences is due to the fact that the average CNR per branch is bigger than the average for SC for an equal CNR per branch for MRC average combined CNR. That is why, as mentioned above, we believe that a fair comparison between the MRC and SC diversity schemes should be based on the average CNR per

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Fig. 9. Channel capacity per unit bandwidth for a Raleigh fading channel versus average carrier-to-noise ratio for different adaptation policies with no diversity.

Fig. 10.

Outage probability of the optimal adaptation and truncated channel inversion.

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Fig. 11. Channel capacity per unit bandwidth for a Raleigh fading channel versus average carrier-to-noise ratio for different adaptation policies with MRC diversity [(a) = 1, (b) = 2, and (c) = 4].

transmission with total channel inversion suffers the largest capacity penalty relative to the other policies. However, as increases, the fading is progressively reduced, and this penalty increases, all capacities diminishes remarkably. Thus, as of the various policies converge to the capacity of an array of -independent AWGN channels (54). However, it is not possible in practice to completely eliminate the effects of fading through space diversity since the number of diversity branches is limited. This is especially true for the downlink (base station to mobile), since mobile receivers are generally constrained in size and power. Since channel inversion is the least complex technique, there is a tradeoff of complexity and capacity for the various adaptation methods and diversity-combining techniques. The diversity gain for all policies is quite important, especially for total channel inversion. For example, in Fig. 11 we see that the capacity with total channel inversion and two-branch MRC exceeds that of a single-branch system with optimal adaptation. Note that this gure also illustrates the typical diminishing returns obtained as the number of branches intotal channel inversion suffers creases. In addition, for a large capacity penalty relative to truncated channel inversion. increases, However, as the number of combining branches the effect of fading is progressively reduced, and this penalty increases, we see diminishes remarkably. In particular, as that truncated channel inversion yields a small increase in capacity over total channel inversion, and this small increase in capacity diminishes as the average CNR and/or the number

of combined branches increase. The corresponding outage probability (23) for the optimal adaptation and truncated channel inversion (with optimal cutoff ) policies with MRC are shown in Fig. 12. Fig. 13 shows the channel capacity per unit bandwidth as a function of for the different adaptation policies with SC and As expected, SC provides diversity for less diversity gain than MRC, with rapidly diminishing returns increases. However, the diversity impact on channel as inversion is still important for SC diversity, since the capacity with this policy and with two-branch combining exceeds that of a single-branch system with optimal adaptation for all dB. Fig. 14 compares the channel capacity per unit bandwidth as function of for the different adaptation policies with: 1) four-branch MRC diversity and 2) four-branch SC diversity. These curves illustrate the extra diversity gain provided by MRC over SC. We see from this gure that MRC provides about 1-b/s/Hz improvement over SC at any and for any of the considered adaptive transmission policies.

VII. CONCLUSION We have examined the Shannon capacity or, equivalently, the upper-bound on spectral efciency of three adaptive transmission techniques over Rayleigh fading channels. In particular, we obtained closed-form expressions for the capacity when

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Fig. 12.

Outage probability of the optimal adaptation and truncated channel inversion with MRC diversity.

these adaptive schemes are used in conjunction with diversity combining. We rst compared the capacity of an AWGN channel with the capacity of a Rayleigh channel with optimal rate adaptation and constant transmit power. When expressed in terms of the average CNR per branch, the capacity of an array of independent Rayleigh channels is bigger than the capacity of a single-branch AWGN channel, but is always smaller than the capacity of an array of -independent AWGN channels tends to innity. On the other hand, and converges to it as when expressed in terms of the average combined CNR, the capacity of a Rayleigh channel with diversity cannot beat the capacity of a single-branch AWGN channel, but comes close to it as the number of diversity branches approaches innity. We also compared the channel capacities of the various adaptation policies both with and without diversity combining. Optimal power and rate adaptation yields a small increase in capacity over just optimal rate adaptation, and this small increase in capacity diminishes as the average received CNR and/or the number of diversity branches increases. In addition, channel inversion suffers the largest capacity penalty relative to the two other policies. However, this capacity penalty diminishes and all capacities approach the capacity of the AWGN channel with increasing diversity. Diversity yields large capacity gains for all the techniques with diminishing returns on the number of branches. The diversity gain is most pronounced for channel inversion. In addition, selection combining provides less diversity gain than MRC for all the adaptive policies, as expected.

Although the results derived herein are Shannon bounds, similar analysis has been applied to adaptive MQAM modulation without diversity [21]. Thus, we expect that the same general trends will be observed on any adaptive modulation method, although the spectral efciency will be smaller. APPENDIX A EVALUATION OF We evaluate the integral integration, namely

(14)

dened in (14) using partial

(58) First, let (59) Thus (60) Then, let (61) successive integration by parts yields [31, Performing eq. (2.321.2), p. 112] (62)

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Fig. 13. Channel capacity per unit bandwidth for a Raleigh fading channel versus average carrier-to-noise ratio for different adaptation policies with SC diversity [(a) = 1, (b) = 2, and (c) = 4].

Substituting (59), (62), and (60) in (58), we see that the rst two terms go to zero. Hence (63) The integral in (63) can be written in a closed form with the help of [31, eq. (3.381.3), p. 364], giving (64) is the complementary incomplete gamma funcwhere tion (or Pryms function as it is sometimes called) dened by [31, eq. (8.350.2), p. 949] (65) For positive integers (66) where denotes the Poisson distribution dened as

where dened as

is the exponential integral of rst-order function

(69) Thus, for positive integers, (64) can be written as (70) In particular, when (70) reduces to (71) APPENDIX B EVALUATION OF We evaluate the integral integration, namely

(32)

dened in (32) using partial

(72) First, let

(67) Thus For [31, eq. (8.359.1) p. 951] (68)

(73)

(74)

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Fig. 14. Channel capacity per unit bandwidth for a Raleigh fading channel versus average carrier-to-noise ratio for different adaptation policies with (a) four-branch MRC diversity combining and (b) four-branch SC diversity combining.

Then, let (75) successive integration by parts yields [31, Performing eq. (2.321.2), p. 112] (76) Substituting (73), (76), and (74) in (72), we see that the rst two terms go to zero. Hence (77) The integral in (77) can be written in a closed form with the help of [31, eq. (3.383.10), p. 366], giving (78) is the complementary incomplete gamma funcwhere , (78) reduces tion dened in (65). Note that when to (79) which can be written as (80)

where by (69).

is the exponential integral of rst order dened

ACKNOWLEDGMENT The authors wish to thank Prof. J. R. Cruz of the University of Oklahoma for bringing [30] to their attention. They would also like to thank the anonymous reviewers for their valuable comments and suggestions. REFERENCES
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Mohamed-Slim Alouini (S94M99), for a photograph and biography, see this issue, p. 1066.

Andrea J. Goldsmith (S94M95), for a photograph and biography, see this issue, p. 1066.