Optimal Discrete-Level Power Control for Adaptive ... - Semantic Scholar

Optimal Discrete-Level Power Control for Adaptive Coded Modulation Schemes with Capacity-Approaching Component Codes Anders Gjendemsjø

Geir E. Øien

P˚al Orten

NTNU, Dept. of Electronics and Telecom. NTNU, Dept. of Electronics and Telecom. UniK / Nera Research N-7491 Trondheim, Norway N-7491 Trondheim, Norway N-1375 Billingstad, Norway Email: [email protected] Email: [email protected] Email: [email protected]

Abstract— In wireless communications, bandwidth is a scarce resource. By employing link adaptation we achieve bandwidthefficient wireless transmission schemes. Using a fixed number of codes we propose a variable-power transmission scheme for slowly flat-fading channels. Assuming that capacity-achieving codes for AWGN channels are available, we develop new combined discrete-rate discrete-power adaptation algorithms with limited feedback for wireless systems. The adaptation schemes are optimized in order to maximize the average spectral efficiency (ASE) for any finite number of available rates. We show that the new transmission schemes can achieve significantly higher ASE when compared to constant power schemes, almost reaching the upper bound of continuous power adaptation. Specifically, using just four rates and four power levels per rate results in a spectral efficiency that is within 1 dB of the continuous-rate continuouspower Shannon capacity. Further, the novel discrete transmission schemes reduce the probability of outage and are more robust against imperfect channel estimation and prediction.

I. I NTRODUCTION Link adaptation, in particular adaptive coded modulation (ACM), is a promising technique to increase throughput in wireless communication systems affected by fading. Today, adaptive schemes are already proposed for implementation in wireless systems such as Digital Video Broadcasting - Satellite Version 2 (DVB-S2) [1]. The underlying premise of ACM is the ability to adapt to a time-varying channel through variation of channel codes, modulation constellations, and transmitted power [2]–[8]. Consider a wireless channel with additive white gaussian noise (AWGN) and fading. Under the assumption of slow, frequency-flat fading, we may use a block fading model to approximate the wireless channel by an AWGN channel within the length of a codeword [9], [10]. Hence, an adaptive system, such as shown in Fig. 1 can be designed to use codes that guarantee a certain bit error rate (BER) within a range of channel-signal-to-noise-ratios (CSNRs) on an AWGN channel. Based on a prediction of the channel, the highest spectral efficiency (SE) code satisfying the BER constraint is chosen. Compared to a non-adaptive system, adapting to the channel in such a way makes it possible to achieve a significant gain in average spectral efficiency (ASE), measured in information bits/s/Hz.

Information bits

Adaptive encoding and modulation

Power control

Zero−error return channel

Fig. 1.

Frequency−flat fading channel

Adaptive decoding and demodulation

Channel predictor

Channel estimator

Decoded information bits

System model.

In [3], [4], [11] ACM systems are designed subject to N given codes. The switching levels between the N codes are obtained by analyzing the codes in order to find at which CSNR level they will fulfill a certain bit error rate requirement, BER ≤ BER0 , on an AWGN channel. Under the assumption of perfect channel knowledge at the transmitter, this approach will always provide an instantaneous BER ≤ BER0 . The average BER will then typically be strictly smaller than BER0 . Since the lower average BER can be increased until it reaches BER0 , bit error rate performance can be traded for a larger ASE, and thus the approach is not optimal from an ASE point of view. In [12] a different approach is proposed. Given the number of codes N , and assuming that capacity-approaching codes for AWGN channels are available for any rate, the switching levels and corresponding rates are chosen such that they are optimal with respect to maximal ASE. An upper bound on the ASE of a practical ACM scheme with a finite set of discrete rates to choose between, is then denoted by the maximum ASE for ACM (MASA) [12]. Without power adaptation, the adaptive coded modulation scheme derived in [12] is restricted to N degrees of freedom corresponding to the number of permissible rates. From previous work by Chung and Goldsmith [11] we know that the spectral efficiency of such a restricted adaptive system increases if more degrees of freedom are allowed. In this paper we optimize discrete-rate discrete-power ACM systems to maximize the average spectral efficiency, while satisfying an average power constraint. We show that introducing power adaptation provides substantial average spectral efficiency and outage probability gains when the number of rates is finite. In [13] continuous power adaptation for discrete-rate link

adaptation is investigated, showing the optimal power control to be piecewise channel inversion. The resulting transmission scheme enables very high transmission rates using a limited number of codes. However, continuous power control is not realistic as it would require an infinite capacity feedback channel. In addition, it is not likely that the transmitter is able to transmit at a very large (infinite) number of power levels. Discrete-rate discrete-power schemes solve these problems by using only a finite set of rates and power levels, requiring the receiver to feed back only an indexed rate and power pair for each fading block. Further, completely discrete schemes are more resilient towards errors in channel estimation and prediction [14]. The remainder of our paper is organized as follows. We introduce the wireless system model under investigation in Section II. In Section III we show previous results for constant and continuous-power MASA schemes and derive the optimal discrete-power adaptation schemes for discrete-rate MASA systems. Section IV shows numerical results and plots of the maximum average spectral efficiencies for different MASA schemes, comparing them to the theoretical upper bounds given by the corresponding Shannon capacities. Finally, conclusions and discussions are given in Section V.

Outage

γ1,1 γ1,2

γ1,K γ2,1

γn,k

γN,K

Fig. 2. The range of γ is partitioned into pre-adaptation SNR regions where γn,k are the switching thresholds.

fading is so slow that capacity-achieving codes for AWGN channels can be employed, giving relatively tight upper bounds on the MASA [15], [16]. Now, recall that the pre-adaptation SNR range is divided into regions lower bounded by γn,1 . ¢ ¡ S(γ ) Thus, we let Rn = Cn , where Cn = log2 1 + Sn,1 γn,1 is shown below to be the highest spectral efficiency that can be supported within the range [γn,1 , γn+1,1 ) for 1 ≤ n ≤ N , after transmit power adaptation. Note that the fading is nonergodic within each codeword, and the results of [17, Section IV] do not apply. Following [13], the basic principle is to create an expression for the MASA, using a parameterized power adaptation scheme. The optimal switching thresholds and power adaptation parameters are then found by optimization techniques. In general the MASA, under the restriction of utilizing capacityachieving codes designed for AWGN channels, is given by MASA =

II. S YSTEM M ODEL

N X

Cn Pn ,

(1)

n=1

We consider the single-link wireless system depicted in Fig. 1. The discrete-time channel is a wide-sense stationary (WSS) fading channel with time-varying gain. The fading is assumed to be slowly varying and frequency-flat. We denote the instantaneous pre-adaptation received signal-to-noise ratio (SNR) by γ[i] and the average pre-adaptation received SNR by γ. These are the SNRs that would be experienced using signal constellations of average power S without power control [4]. Assuming that the transmitter receives perfect channel predictions, sent over a zero-error feedback channel, we can adapt the transmit power instantaneously at time i according to a power adaptation scheme S(γ[i]) The received post-adaptation SNR at time i is then given by γ[i]S(γ[i])/S. By virtue of the WSS assumption, the distribution of γ[i] is independent of i, and is denoted by fγ (γ). To simplify the notation we omit the time reference i from now on. The transmission scheme is to be based on a set of N codes, associated with K power levels per code. Following [3], [12], we partition the range of γ into N K + 1 pre-adaptation SNR regions, which are defined by the switching thresholds {γn,k }N,K n,k=1,1 as illustrated in Fig. 2. Code n, with spectral efficiency Rn , is selected whenever γ is in the interval [γn,1 , γn+1,1 ). Within this interval the transmission rate is constant, but the system can adapt the transmitted power to the channel conditions in order to maximize the average spectral efficiency. For convenience, we let γ0,1 = 0 and γN +1,1 = ∞.

R γn+1,1 where Pn = γn,1 fγ (γ) dγ is the probability that code n be employed at any given time. For a fading channel with AWGN, the MASA is thus given by (in bits/s/Hz) N ³ ´ Z γn+1,1 X S(γn,1 ) MASA = log2 1 + γn,1 fγ (γ) dγ, (2) S γn,1 n=1

III. MASA A NALYSIS

where Fγ (·) denotes the cumulative density distribution function (CDF) of γ. From (4) we see that the received postadaptation SNR monotonically increases within [γn,1 , γn+1,1 ) ¡ ¢ S(γ ) for 1 ≤ n ≤ N . Hence, log2 1 + Sn,1 γn,1 is the highest

Using N distinct codes we review the MASA for constant and continuous transmit power schemes and derive optimal discrete-power adaptation schemes. We shall assume that the

subject to the average power constraint, N Z X n=0

γn+1,1

S(γ)fγ (γ) dγ ≤ S,

(3)

γn,1

where S denotes the average transmit power. (2) is seen to be a discrete-sum approximation to the Shannon capacity given in [2, Eq. (4)]. We now identify and analyze several interesting power adaptation schemes. A. Constant-Power Transmission Scheme When a single transmission power is used for all codes, we adopt the term constant-power transmission scheme [18]. The optimal constant power policy is seen to be [13]  1 , if γn,1 ≤ γ < γn+1,1 ,  S(γ)  1−Fγ (γ1,1 ) = (4) 1 ≤ n ≤ N,  S  0, if γ < γ1,1 ,

possible spectral efficiency that can be supported over the whole of region n. Introducing (4) in (2) we obtain a new expression for the MASA, denoted by MASAN : N ³ ´ Z γn+1,1 X γn,1 MASAN = log2 1 + fγ (γ) dγ. 1 − Fγ (γ1,1 ) γn,1 n=1 (5) The optimal switching thresholds {γn,1 }N n=1 are found by numerical optimization.

We now extend the MASA analysis by introducing the MASAN ×K scheme, where we allow for K ≥ 1 power regions within each of the N rate regions. For each rate region we again use a capacity-achieving code which ensures an arbitrarily low probability of error for any AWGN channel S(γn,1 ) with a received SNR greater than or equal to γn,1 , S imposing the following restriction on the power adaptation function, for γ ∈ [γn,1 , γn+1,1 ) S(γ) S(γn,1 ) γ≥ γn,1 . (6) S S Since the rate is restricted to be constant in each region, it intuitively makes sense from a capacity perspective to reduce the transmitted power, constrained by (6), when the channel S(γn,1 ) conditions are more favorable. Now, define γn,1 = βn . S The optimal power adaptation scheme is then of the form

S(γ) S

=

{βn }N n=1

if γn,k ≤ γ < γn,k+1 , K ≥ 2 1 ≤ n ≤ N, 1 ≤ k ≤ K − 1, if γn,K ≤ γ < γn+1,1 ,

(7)

1 ≤ n ≤ N, if γ < γ1,1 ,

{γn,k }N,K n,k=1,1

and are parameters to be optiwhere mized. We can interpret (7) as follows, for 1 ≤ n ≤ N : Inside each region [γn,1 , γn+1,1 ) we reduce the power in a stepwise manner, and at each step obtaining equality in (6), thus using the least possible power, while still ensuring transmission with an arbitrarily low error rate. The value of βn corresponds to the minimum required post-adaptation SNR within region n. Using (7) in (2) and (3) we arrive at the following optimization problem: Maximize Z γn+1,1 N X MASAN ×K = log2 (1 + βn ) fγ (γ) dγ, (8) n=1

γn,1

such that ³K−1 X 1 Z γn,k+1 fγ (γ) dγ βn γn,k γn,k n=1 k=1 Z γn+1,1 ´ 1 + fγ (γ) dγ ≤ 1. (9) γn,K γn,K N X

MASAN ×K =

N X

³ γn,1 γn+1,1 ´ log2 (1 + βn ) e− γ − e− γ , (10)

n=1

such that N X

βn

n=1

B. Discrete-Power Transmission Scheme

 β n   γn,k ,      βn γn,K ,        0,

If we make a Rayleigh fading channel model assumption the maximization problem can be written as follows. Maximize

³K−1 X 1 ¡ γn,k γn,k+1 ¢ e− γ − e− γ + γn,k k=1 γn+1,1 ¢´ 1 ¡ − γn,K ≤ 1. (11) e γ − e− γ γn,K

With a small increase of feedback, relative to the constant transmit power scheme, the MASAN ×K scheme will give a significantly increased average spectral efficiency as we show in the Section IV. C. Continuous-Power Transmission Scheme Letting S(γ) vary continuously we obtain a discrete-rate continuous-power adaptation scheme, which we denote by MASAN ×∞ . In [13] the optimal power adaptation scheme is derived to be 1 ( βn , if γn,1 ≤ γ < γn+1,1 , 1 ≤ n ≤ N, S(γ) (12) = γ S 0, if γ < γ1,1 , i.e., piecewise channel inversion. Substituting (12) in (2) and (3) the optimization problem is to find {γn,1 }N n=1 and in order to maximize {βn }N n=1 MASAN ×∞ =

N X

Z log2 (1 + βn )

n=1

such that

N X n=1

Z

γn+1,1

βn γn,1

γn+1,1

fγ (γ) dγ,

(13)

γn,1

1 fγ (γ) dγ ≤ 1. γ

(14)

As shown in [13] this scheme is highly spectrally efficient, almost reaching the optimal simultaneous continuous power and rate adaptation [2, Eq. 4], using only 4 codes. However, the requirements on the transmitter and the feedback channel to be able to continuously adapt the power, are not physically tractable. IV. N UMERICAL R ESULTS For the following numerical results, a Rayleigh fading channel model has been assumed. A. Switching Thresholds and Power Adaptation Schemes Fig. 3 shows the set of optimal switching levels {γn,1 } for some selected MASA schemes as a function of average pre-adaptation SNR. (For the MASA2×2 and MASA4×4 schemes the internal switching thresholds {γn,k }N,K n=1,k=2 , are not shown in Fig. 3 due to clarity reasons).

35

35

β

MASA4x4

1

Min. req. SNR, βn for n=1,2,3,4 (dB)

MASA2x2

30

MASA

MASA1

20

15

10

β3

25

β4

20 15 10 5 0 −5

0

5

10 15 20 Average pre-adaptation SNR, γ¯ (dB)

25

30

5

Fig. 4. Minimum received SNR values {βn }4n=1 for MASA4×4 , plotted as a function of average pre-adaptation SNR.

0

2

0

5 10 15 20 25 Average pre-adaptation SNR, γ¯ (dB)

30

Fig. 3. Switching thresholds {γn,1 }N n=1 as a function of average preadaptation SNR. For each data series, the lowermost curve shows γ1,1 , while the uppermost shows γN,1 .

{βn }4n=1

for the The minimum required SNR values MASA4×4 transmission scheme are depicted in Fig. 4. We can interpret Figs. 3 and 4 as follows: Given a number of codes N , and power levels K, we find the switching thresholds for a given γ. Then find the corresponding spectral efficiencies, given by SEn = log2 (1 + βn ). (15) For each γ of interest we then design optimal codes for these SEs, and use the power adaptation scheme as given in (7). Examples of optimized power adaptation schemes for MASA2×2 and MASA4×4 are shown in Figs. 5(a) and 5(b), respectively, for γ = 5 dB. In the analysis of Section III no stringent peak power constraint has been imposed, and as such it is interesting to note the limited range of S(γ) for both schemes. B. Comparison of MASA schemes Under the average power constraint of (3) the average spectral efficiencies corresponding to MASAN , MASAN ×K and MASAN ×∞ are plotted in Figs. 6 and 7. From Fig. 6(a) we see that the ASE increases with the number of codes, while Fig. 7 shows that the spectral efficiency also increases with flexibility of power adaptation. Although, note that the MASAN ×1 and MASAN schemes perform identical. This is an expected result since the MASAN ×1 scheme is unable to reduce the power within the transmission regions when the channel conditions improve. 1 Slightly reformulated here to fit the current notation, but equivalent to the formulation in [13].

2 MASA2x2

1.8 1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

5 10 15 Pre-adaptation SNR, γ (dB)

(a)

S(γ) S

MASA4x4

1.8

¯ S(γ)/S

−5

¯ S(γ)/S

Switching thresholds, {γ n,1} N n=1 (dB)

2

25

β2

30

20

0

0

for MASA2×2 .

5 10 15 Pre-adaptation SNR, γ (dB)

(b)

S(γ) S

20

for MASA4×4 .

Fig. 5. Power adaptation schemes for γ = 5 dB versus pre-adaptation SNR.

Fig. 6(b) compares four MASA schemes with the product N × K = 8, showing that number of codes has a (somewhat) larger impact on the spectral efficiency than the number of power levels. However, we see that the three schemes with N ≥ 2 performs almost similar, indicating that the number of rates and power levels can be traded against each other, while still achieving approximately the same ASE. From an implementation point of view this is highly valuable as it gives more freedom to design the system. C. MASA versus Shannon Capacities Assume that the channel state information γ is known to the transmitter and the receiver. Then, given an average transmit power constraint, the channel capacity of a Rayleigh fading channel with optimal continuous rate adaptation and constant transmit power, CORA , is given in [2], [19] as 1

CORA = log2 (e)e γ E1

³1´ γ

.

(16)

Furthermore, if we include continuous power adaptation, the channel capacity, COPRA , becomes [2], [19]

for each SNR there exists an optimal code and power level. Alternatively, if the fading is ergodic within each codeword, MASA MASA as opposed to the assumptions in this paper, COPRA can be 7 7 MASA obtained by a fixed rate transmission system using a single 6 6 gaussian code [17], [20]. N=4 5 5 As the number of codes (switching thresholds) goes to infin4 4 N=3 ity, MASAN will reach the CORA capacity, while MASAN ×K 3 3 will reach the COPRA capacity, when also K → ∞. Of N=2 2 2 course, this is not a practically feasible approach; however, as N=1 1 1 illustrated in Figs. 6(a) and 7, a small number of optimally 0 0 designed codes, and possibly power adaptation levels, will 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Average pre-adaptation SNR, γ¯ (dB) Average pre-adaptation SNR, γ¯ (dB) indeed yield a performance that is close to the theoretical upper (a) MASAN ×1 and MASAN for N = (b) MASA schemes with N × K = 8. bounds, CORA and COPRA , for any given γ. 1, 2, 4, 8. CORA for reference. From Fig. 7 we see that the power adapted MASA schemes Fig. 6. Average spectral efficiencies versus average pre-adaptation SNR for perform close to the theoretical upper bound (COPRA ) using selected MASA schemes. only four codes. Specifically, restricting our adaptive policy to just four rates and four power levels per rate results in 9 a spectral efficiency that is within 1 dB of the efficiency COPRA obtained with continuous-rate and continuous power (17), MASA4x∞ demonstrating the remarkable impact of power adaptation. 8 MASA4x4 This is in contrast to the case of continuous rate adaptation, MASA4 where introducing power adaptation gives negligible gain [2]. 7 9

9

MASA8x1

CORA

8

8

MASANx1

MASA

4x2

N

2x4

ASE (bits/s/Hz)

ASE (bits/s/Hz)

1x8

D. Probability of no transmission ASE (bits/s/Hz)

6

The probability that the SNR is so low that the lowest code cannot guarantee a BER ≤ BER0 is equivalent to the probability that the pre-adaptation SNR falls below γ1,1 , thus the outage probability can be calculated as Z γ1,1 Pout = fγ (γ) dγ. (19)

5

4

0

3

2

1

0

0

5

10 15 20 25 Average pre-adaptation SNR, γ¯ (dB)

30

Fig. 7. Average spectral efficiency versus average pre-adaptation SNR for MASA schemes employing N = 4 codes. COPRA as reference.

COPRA

³ e −γγ c ´ = log2 (e) γc − γ ,

(17)

γ

where the “cutoff” value γc can be found by solving Z ∞³ 1 1´ − fγ (γ) dγ = 1. γc γ γc

(18)

Thus, MASAN is compared to CORA , while MASAN ×K and MASAN ×∞ are measured against COPRA . The capacity in (17) can be achieved in the case that a continuum of capacity-achieving codes for AWGN channels, and corresponding optimal power levels, are available. That is,

Fig. 8 illustrates the outage probability of selected MASA schemes as a function of the average pre-adaptation SNR. When the number of codes is increased, the SNR range will be partitioned into a larger number of regions. As shown in Fig. 3, the lowest switching level γ1,1 will then become smaller. Pout will therefore decrease, as illustrated in Fig. 8. Similarly, as seen from Fig. 3, γ1,1 also decreases with an increasing number of power levels, when N is constant. Thus, both rate and power adaptation flexibility reduces the probability of outage. It is not necessarily a disadvantage that the outage probability is high, unless the service under consideration has strict real-time or low-delay requirements. For data-centric services, such as file or email transfer, the most important qualityof-service factor is the total time of data transmission. For large data sets, this time will be minimized independently of the value Pout , as long as the average spectral efficiency is maximized. V. C ONCLUSIONS AND D ISCUSSIONS We have investigated a capacity-optimal variable-rate and variable-power link adaptation scheme adapting to a slowly varying flat-fading wireless communication channel, extending the previous work in [12], [13] to include discretelevel power adaptation. Specifically, we have introduced

R EFERENCES

0

Outage probability, Po ut

10

−1

10

MASA4x∞ MASA4 MASA2x2

−2

10

MASA2 MASA1 0

Fig. 8.

5 10 15 20 25 Average pre-adaptation SNR, γ¯ (dB)

30

Probability of outage as a function of average pre-adaptation SNR.

transmission schemes that are spectrally efficient and have low feedback load. As long as capacity-achieving codes for AWGN channels are available, the proposed discrete-power adaptation schemes maximize the average spectral efficiency given average power and adaptation constraints. The adaptive discrete-power schemes presented in this paper are shown to be comparable to the continuous-power adapted MASA schemes, and even approaching the theoretical continuousrate and continuous-power bound on spectral efficiency. In particular, using just four codes, each associated with four power levels, we achieve a spectral efficiency within 1 dB of both the optimal four-code continuous-power transmission scheme and the COPRA Shannon capacity. The discrete nature of the proposed adaptation algorithms renders the system not only more robust against imperfect channel estimation and prediction, but also resolves implementation issues. Further, the discrete-power adaptation schemes have significant ASE and outage probability gains over the constant power scheme with a relatively low increase in feedback. As a result, we see that if the possible number of rates in a system is limited there is a significant achievable gain by introducing power adaptation. We have also seen that the number of rates, N , to some extent, can be traded against power levels, K. This flexibility is of practical importance since it may be easier to implement the proposed power adaptation schemes than to design capacity-achieving codes for a large number of a rates. Finally, we note that the adaptive power algorithms presented in this paper require that the RF power amplifier is operated in the linear region, implying a higher power consumption. For devices with limited battery capacity it is apparent that there will be a tradeoff between efficiency and linearity. This can be a topic for further research. VI. ACKNOWLEDGEMENT The authors wish to thank Doctoral Fellows S´ebastien de la Kethulle de Ryhove and Vegard Hassel (both NTNU) and M.Sc. Marius R. Hanssen for valuable discussions.

[1] Digital Video Broadcasting - Satellite Version 2. [Online]. Available: http://www.dvb.org. [2] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Transactions on Information Theory, vol. 43, no. 6, pp. 1986–1992, Nov. 1997. [3] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power M-QAM for fading channels,” IEEE Transactions on Communications, vol. 45, no. 10, pp. 1218–1230, Oct. 1997. [4] K. J. Hole, H. Holm, and G. E. Øien, “Adaptive multidimensional coded modulation over flat fading channels,” IEEE Journal on Selected Areas in Communications, vol. 18, no. 7, pp. 1153–1158, July 2000. [5] ——, “Analysis of adaptive coded modulation with antenna diversity and feedback delay,” in IST Mobile Communications Summit, Barcelona, Spain, Sept. 2001. [6] H. Holm, “Adaptive coded modulation performance and channel estimation tools for flat fading channels,” Ph.D. dissertation, Norwegian University of Science and Technology, 2002. [Online]. Available: http: //www.tele.ntnu.no/projects/beats/Documents/HenrikHolmThesis.pdf [7] B. Holter, “Adaptive coded modulation in spatial and multiuser diversity systems,” Ph.D. dissertation, Norwegian University of Science and Technology, 2005. [Online]. Available: http://www.tele.ntnu.no/projects/ beats/Documents/holter-thesis.pdf [8] L. Hanzo, C. H. Wong, and M.-S. Yee, Adaptive Wireless Transceivers: Turbo-Coded, Turbo-Equalized and Space-Time Coded TDMA, CDMA and OFDM Systems. John Wiley & Sons, Inc., 2002. [9] R. J. McEliece and W. E. Stark, “Channels with block interference,” IEEE Transactions on Information Theory, vol. IT-30, no. 1, pp. 44–53, Jan. 1984. [10] L. H. Ozarow, S. Shamai (Shitz), and A. D. Wyner, “Information theoretic considerations for cellular mobile radio,” IEEE Transactions on Vehicular Technology, vol. 43, no. 2, pp. 359–378, May 1994. [11] S. T. Chung and A. J. Goldsmith, “Degrees of freedom in adaptive modulation: A unified view,” IEEE Transactions on Communications, vol. 49, no. 9, pp. 1561–1571, Sep. 2001. [12] H. Holm, G. E. Øien, M.-S. Alouini, D. Gesbert, and K. J. Hole, “Optimal design of adaptive coded modulation schemes for maximum average spectral efficiency,” in Proc. IEEE Workshop on Signal Processing Advances in Wireless Communications, Rome, Italy, June 2003. [13] A. Gjendemsjø, G. E. Øien, and H. Holm, “Optimal power control for discrete-rate link adaptation schemes with capacity-approaching coding,” to appear in Proc. IEEE Globecom 2005, St. Louis, MO, Nov.-Dec. 2005. [Online]. Available: http://www.iet.ntnu.no/projects/ cuban/archive.htm [14] F. F. Digham and M.-S. Alouini, “Diversity combining with discrete power loading over fading channels,” in Proc. IEEE Wireless Communications and Networking Conference, Atlanta, GA, March 2004. [15] S. Dolinar, D. Divsalar, and F. Pollara, “Code performance as a function of code block size,” in JPL TDA Progress Report 42-133, 1998. [Online]. Available: http://tmo.jpl.nasa.gov/tmo/progress report/42-133/133K.pdf [16] ——, “Turbo code performance as a function of code block size,” in Proc. IEEE International Symposium on Information Theory, Cambridge, MA, Aug. 1998, p. 32. [17] G. Caire and S. Shamai (Shitz), “On the capacity of some channels with channel state information,” IEEE Transactions on Information Theory, vol. 45, no. 6, pp. 2007–2019, Sept. 1999. [18] C. K¨ose and D. L. Goeckel, “On power adaptation in adaptive signalling systems,” IEEE Transactions on Communications, vol. 48, no. 11, pp. 1769–1773, Nov. 2000. [19] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Transactions on Vehicular Technology, vol. 48, no. 4, pp. 1165–1181, July 1999. [20] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.