Throughput maximization in wireless communication systems Number:7,522,657 from the United States Patent and Trademark Office (PTO) owispatent A communication method comprising characterizing a communications channel, determining a data rate and optionally a power allocation strategy that maximizes channel throughput, and configuring a transmitter to send a transmit signal with said data rate and said optional power allocation strategy.<H communication, maximization, Throughput, maxi, 7522657.html, Patent, pto, 7522657

Title: Throughput maximization in wireless communication systems

Abstract: A communication method comprising characterizing a communications channel, determining a data rate and optionally a power allocation strategy that maximizes channel throughput, and configuring a transmitter to send a transmit signal with said data rate and said optional power allocation strategy.

Patent Number: 7,522,657 Issued on 04/21/2009 to Ahmed, et al.

Inventors: Ahmed; S. Nadeem (Allen, TX), Baraniuk; Richard G. (Houston, TX)
Assignee: William Marsh Rice University (Houston, TX)

Appl. No.: 10/969,637

Filed: October 20, 2004

Related U.S. Patent Documents


Application Number	Filing Date	Patent Number	Issue Date
60512712	Oct., 2003

Current U.S. Class: 375/219

Current International Class: H04B 1/38 (20060101); H04L 5/16 (20060101)

Field of Search: 375/219

References Cited [Referenced By]

U.S. Patent Documents


6947407	September 2005	Ayyagari et al.
2001/0040881	November 2001	Ayyagari et al.
2003/0206559	November 2003	Trachewsky et al.

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Primary Examiner: Payne; David C
Assistant Examiner: Bolourchi; Nader
Attorney, Agent or Firm: Conley Rose, P.C. Carroll; Rodney B.

Parent Case Text

PRIORITY CLAIM

This application claims priority to U.S. Provisional Patent Application Ser. No. 60/152,712, filed on Oct. 20, 2003, entitled, "Throughput Maximization In A Wireless Communication System," incorporated herein by reference.

Claims

What is claimed is:

1. A communication method comprising: characterizing a commutations channel using a transmitter; determining a data rate that maximizes channel throughput using said transmitter; and configuring said transmitter to send a transmit signal with said data rate; wherein characterizing the communications channel comprises modeling a channel gain probabilty density function; wherein the power allocation strategy sets .gamma..sub.k, a transmit gain for a kth interval, to .gamma..function..alpha..function..function..lamda..function..alpha..alph- a. ##EQU00153## wherein .alpha. is a vector of the channel attenuation .alpha..sub.k for the last K intervals, P.sub.P is a peak power constraint, and .lamda.(.alpha.) is the solution to .times..times..times..function..function..lamda..function..alpha..alpha. ##EQU00154##

2. The method of claim 1, wherein said determining further includes determining a power allocation strategy that jointly maximizes the channel throughput with said data rate.

3. The method of claim 2, wherein said power allocation strategy provides for adjustment of the transmit power to compensate for channel gain variation.

4. The method of claim 3, wherein said power allocation strategy minimizes outage probability subject to peak power and average power constraints.

5. The method of claim 1, wherein the power allocation strategy stochastically sets .gamma..sub.k, a transmit gain for a kth interval, to .gamma..function..alpha..times..times..times..times..gamma..function..alp- ha..function..function..lamda..function..alpha..alpha. ##EQU00155## wherein .alpha. is a vector of the channel attenuation .alpha..sub.k for the last K intervals, P.sub.Pis a peak power constraint, and .lamda.(.alpha.) is the solution to .times..times..function..alpha..times..function..function..lamda..functio- n..alpha..alpha. ##EQU00156## wherein R is the data rate, and wherein the stochastic probability is based on a current channel gain and a history of transmit gains.

6. The method of claim 1, wherein the communications channel is a wireless channel.

7. The method of claim 6, wherein said characterizing includes: identifying a coherence time for the channel; and identifying a noise power.

8. The method of claim 1, wherein said determining includes: maximizing a throughput function that accounts an expected service time for transmitting a codeword to a receiver and an average amount of data carried by each codeword.

9. The method of claim 8, wherein the expected service time is expressible as a function of data rate.

10. The method of claim 8, wherein the expected service time is expressible as a function of coding delay.

11. The method of claim 8, wherein the expected service time accounts for a power allocation strategy.

12. The method of claim 8, wherein the expected service time accounts for delay constraints.

13. The method of claim 8, wherein the expected service time accounts for retransmission constraints.

14. The method of claim 8, wherein the expected service time accounts for outage probability.

15. The method of claim 8, wherein the expected service time accounts for receiver decoding strategy.

16. The method of claim 15, wherein the receiver decoding strategy includes: discarding incorrectly received codewords; and requesting re-transmission of the incorrectly received codewords.

17. The method of claim 15, wherein the receiver decoding strategy includes: requesting re-transmission of incorrectly decoded codewords; and combining re-transmitted codewords with incorrectly decoded codewords to decode the re-transmitted codewords.

18. A communication method comprising: characterizing a communications channel; determining a data rate that maximizes channel throughput; and configuring a transmitter to send a transmit signal with said data rate; wherein characterizing the communications channel comprises modeling a channel gain probability density function, wherein the throughput function is expressible as: .function..gamma..function..function..gamma. ##EQU00157## wherein R is the data rate, .gamma. is the transmit power, K is the coding delay, and E[S(R,.gamma.,K)] is the expected service time.

19. The method of claim 18, wherein the expected service time is expressible as: .function..function..function. ##EQU00158## wherein P.sub.av is the average transmit power, and P.sub.out(R,P.sub.av,K) is the probability of a channel outage.

20. The method of claim 18, wherein the expected service time is expressible as: .function..function. ##EQU00159## wherein P.sub.av is the average transmit power.

21. The method of claim 18, wherein the expected service time is expressible as: .function..function..function..function. ##EQU00160## wherein P.sub.av is the average transmit power, P.sub.out(R,P.sub.av,K) is the probability of a channel outage, and L is the maximum number of transmission attempts per codeword.

22. A transceiver that comprises: a receiver configured to receive information characterizing a communications channel; and a transmitter configured to process said information to determine a data rate that maximizes a throughput for the communications channel, and further configured to provide a transmit signal to the communications channel using said data rate wherein, as part of determining a data rate, the transceiver maximizes a channel throughput function that accounts for an expected service time for transmitting a codeword to a remote receiver; wherein said throughput function is expressible as a function of the data rate and an expected service time, said expected service time being a function of the data rate, transmit power and coding delay.

23. The transceiver of claim 22, wherein as part of determining a data rate that maximizes a throughput for the communications channel, the transmitter is configured to jointly determine a power allocation strategy that maximizes the throughput subject to a power constraint.

24. The transceiver of claim 23, wherein the power allocation strategy minimizes a channel outage probability.

25. The transceiver of claim 22, wherein the communications channel is a fading channel.

26. The transceiver of claim 25, wherein the information characterizing the channel includes a coherence time for the channel, a noise power, and model for a channel gain probability density function.

27. The transceiver of claim 22, wherein the expected service time accounts for data rate and coding delay.

28. The transceiver of claim 27, wherein the expected service time further accounts for constraints on power and retransmission attempts.

29. The transceiver of claim 27, wherein the expected service time further accounts for outage probability and receiver decoding strategy.

30. A transceiver that comprises: a receiver configured to receive information characterizing a communications channel; and a transmitter configured to process said information to determine a data rate that maximizes a throughput for the communications channel, and further configured to provide a transmit signal to the communications channel using said data rate; wherein, as part of determining a data rate, the transceiver maximizes a channel throughput function that accounts for an expected service time for transmitting a codeword to a remote receiver; wherein the throughput function is expressible as: .function..gamma..function..function..gamma. ##EQU00161## wherein R is the data rate, .gamma. is the transmit power, K is the coding delay, and E[S(R,.gamma.,K)] is the expected service time.

31. The transceiver of claim 30, wherein the expected service time is expressible as: .function..function..function..function. ##EQU00162## wherein P.sub.av is the average transmit power, P.sub.out(R,P.sub.av,K) is the probability of a channel outage, and L is the maximum number of transmission attempts per codeword.

Description

BACKGROUND

The channels encountered by many wireless communication systems often scatter the transmitted signal along its transmission path. Time variation of the channel results in random fluctuations of the received power level, or fading, making reliable communications difficult.

Transmitters typically employ channel coding techniques that map sequences of input data to codewords that add redundancy to combat the effects of fading and noise prior to transmission. Codewords consist of a number of symbols carrying data at the transmission rate, the number of information bits communicated with each symbol. The channel coherence time is the amount of time the time-varying channel is assumed constant; signals transmitted within the coherence time are affected by a single fading state. During transmission, each codeword is affected by one or more fading states with the specific number affecting the communications performance The coding delay is proportional to the codeword length and is often quantified in terms of the number of fading states affecting each codeword; it significantly affects a system's reliable communications performance. A system is considered delay unconstrained if it uses infinite-length codewords resulting in infinite coding delays Practical communication systems are delay-limited; they use finite-length codewords and therefore have a finite coding delay.

Conventional analysis of fading channels has been performed from the single-attempt paradigm. That is, the amount of information that can be reliably communicated with a single codeword transmission attempt has been quantified. This approach works well for idealized, delay-unconstrained systems that transmit a single, infinite-length codeword. However, practical systems are delay-limited since they use finite-length codewords. Therefore, the conventional performance metrics based on the single-attempt paradigm have drawbacks for delay-limited systems: .epsilon.-capacity--the highest transmission rate that can be supported with a probability of data loss no greater than .epsilon.--does not provide a measure of error-free performance, while single-attempt delay-limited capacity--.epsilon.-capacity when data loss cannot be tolerated; that is, when .epsilon.=0--underestimates achievable performance.

SUMMARY

The problems noted above are solved in large part by a technique for through-put analysis and maximization in wireless communication systems One illustrative embodiment may be a a communication method comprising characterizing a communications channel, determining a data rate that maximizes channel throughput, and configuring a transmitter to send a transmit signal with said data rate.

Another embodiment may comprise a transceiver that comprises a receiver configured to receive information characterizing a communications channel, and a transmitter configured to process said information to determine a data rate that maximizes a throughput for the communications channel, and further configured to provide a transmit signal to the communications channel using said data rate.

Yet another embodiment may be a wireless communications system that comprises a remote transceiver configured to send information characterizing a communications channel, and a local transceiver configured to receive said information and to process said information to determine a data rate that maximizes a throughput for the communications channel, and further configured to transmit data to the remote transceiver using said data rate.

BRIEF DESCRIPTION OF THE DRAWINGS

For a detailed description of illustrative embodiments of the invention, reference will now be made to the accompanying drawings in which:

FIG. 1A shows a wireless communication system with a feedback channel;

FIG. 1B shows at least some of the internal components of a transceiver that may be used as a transmission or reception device in the system of FIG. 1A;

FIG. 1C shows a graph of ergodic capacity, with and without power control, as a function of signal-to-noise ratio (SNR);

FIG. 1D shows a graph of minimum outage probability as a function of transmitted power for a transmission rate of 1 nat/sec/Hz with constant power and under long-term power constraint;

FIG. 1E shows a queueing model of a wireless communication system;

FIG. 2A shows a graph of codeword error probability vs. rate for 1) K=.infin. and 2) K=1;

FIG. 2B shows a graph of throughput vs. transmission rate when average power is approximately 10 dB for coding delays of one and infinity;

FIG. 2C shows a graph of throughput vs. transmission rate for various values of K where average power is approximately 10 dB;

FIG. 2D shows a graph of outage probability vs. transmission rate for various values of K where average power is approximately 10 dB;

FIG. 2E shows a graph of maximum zero-outage throughput (MZT) as a function of average power and coding delay vs. average power for various coding delays;

FIG. 2F shows a graph of MZT (re-transmission scheme) as a function of average power and K, where average power is 0, 5, and/or 10 dB,;

FIG. 2G shows a graph of transmission rate as a function of K for average power values of 0, 5 and/or 10;

FIG. 2H shows a graph of throughput vs. K achieved with an SNR of 10 dB for various transmission rates;

FIG. 2I shows a graph of throughput vs. outage probability for various values of K, where average power is 10 dB;

FIG. 2J shows a graph of optimal outage probability as a function of K for various average power values;

FIG. 2K shows a graph of fading throughput vs. transmission rate for an SNR. of 10 dB using simple re-transmission and incremental diversity;

FIG. 2L shows a graph of MZT vs. SNR for various values of K;

FIG. 2M shows a graph of MZT vs. coding delay for various values of average power;

FIG. 2N shows a graph of transmission rate vs. coding delay for various values of average power;

FIG. 2O shows a graph of throughput vs. rate for various quantities of transmission attempts;

FIG. 3A shows a graph of fading state/power allocation vs. block index;

FIG. 3B shows histograms of transmitted power for a rate R of 2 nats/sec/HZ, K=5, a long-term average power constraint Pav of 10 dB and the transmission of 10000 codewords;

FIG. 3C shows histograms of transmitted power for R=2 nats/sec/Hz, K=5, Pav=10 dB, and the transmission of 10000 codewords;

FIG. 3D shows a graph of minimum outage probability vs. average power for R=1 nats/sec/Hz;

FIG. 3E shows a graph of minimum outage probability vs. R for a long-term power constraint of Pav=10 dB;

FIG. 3F shows a graph of minimum outage probability vs. R for a short-term power constraint of Pav=10 dB;

FIG. 4A shows a graph of delay-limited capacity and throughput for K=2 vs. SNR;

FIG. 4B shows a graph of spectral efficiency for K=1 as a function of SNR;

FIG. 4C shows a graph of MZT vs. K for constant, short-term and long-term power allocation strategies;

FIG. 4D shows a graph of throughput vs. transmission rate for various K and Pav=10 dB;

FIG. 4E shows a graph of optimal transmission rate vs. K for constant, short-term and long-term power allocation strategies;

FIG. 4F shows a graph of optimal outage probability vs. K for Pav=10 dB;

FIG. 4G shows a graph of MZT vs. Pav with K=5 for various values of peak power;

FIG. 4H shows a graph of MZT vs. Pav for K=5 for various values of peak power;

FIG. 4I shows a graph of MZT with a delayed transmission scheme vs. K for a long-term average power constraint Pav=10 dB;

FIG. 4J shows a graph of MZT with a delayed transmission scheme vs. K under a short-term average power constraint Pav=10 dB;

FIG. 4K shows a graph of throughput vs. transmission rate under long-term and peak power constraints for K=5, Pav=10 dB, and various peak power constraint values;

FIG. 4L shows a graph of throughput vs. transmission rate under short-term and peak power constraints for K=5, Pav=10 dB, and various values of peak power constraints;

FIG. 4M shows a graph of optimal transmission rate vs. K under a long-term average power constraint of Pav=10 dB and various peak power constraints;

FIG. 4N shows a graph of optimal transmission rate vs. K under a short-term average power constraint of Pav=10 dB and various values of peak power constraint;

FIG. 4O shows a graph of optimal outage probability vs. coding delay K under the long-term average and peak power constraints with Pav=10 dB;

FIG. 4P shows a graph of optimal outage probability vs. coding delay K under the short-term average and peak power constraints with Pav=10 dB;

FIG. 5A shows a graph of MZT and near-optimal throughput as a function of average waiting time for K=1 and Pav=10 dB;

FIG. 5B shows a graph of optimal transmission rate and near-optimal transmission rate as a function of the average waiting time for K=1 and Pav=10 dB;

FIG. 5C shows a graph of optimal arrival rate as a function of average waiting time for K=1 and Pav=10 dB;

FIG. 5D shows a graph of queue utilization for both optimal and suboptimal strategies as a function of average waiting time for K=1 and Pav=10 dB;

FIG. 5E shows a graph of MZT as a function of K for a waiting time of D=20 and for Pav=10 dB; and

FIG. 5F shows a flow diagram of a technique used to optimize throughput during data transmission over a wireless channel.

Notation and Nomenclature

Let Z, Z, and represent a scalar, vector, and matrix, respectively Then diag(Z)=Z is a diagonal matrix with diagonal elements Z, and I.sup.L=diag(1, 1, . . . , 1) is the L.times.L identity matrix. Let [g(z)] represent the expected value of g(z). Let f(.alpha.) and F(.alpha.) represent the probability density function (PDF) and cumulative distribution function (CDF) of the random vector .alpha., respectively. Let and .sub.+ represent the real line and the positive real line. Then .sup.L and .sup.L.times.M are the set of length-L vectors and L.times.M matrices with elements in , respectively. Similarly, .sub.+.sup.L and .sub.+.sup.L.times.M are the set of length-L vectors and L.times.M matrices with elements in .sub.+, respectively. For {a,b}.epsilon., let I.sub.F(a,b) be the indicator function, which is 1 if a>b and 0 if a<b. Let w.about.N(m, V) represent a jointly Gaussian random vector with mean m and covariance matrix V. Similarly let x.about..chi..sub.a.sup.2 with a=1, 2, 3, . . . represent a chi-squared random variable with a degrees of freedom. Finally let (b) be Lambert's W function, the solution to xe.sup.z=b.

DETAILED DESCRIPTION

Described below is an analysis framework for delay-limited systems based on the multi-attempt paradigm. Average communications throughput is maximized by optimizing system parameters and using the maximum throughput as a measure of delay-limited communication performance. Discussed below awe two common scenarios, the first being only when the receive has channel state information (CSI-R), while in the second both transmitter and receiver have information pertaining to the channel (CSI-RT). With CSI-R, the average transmit power is held constant and throughput is maximized by performing optimal transmission rate selection. With CSI-RT, the transmitter knows the condition of the channel at the time of transmission and can vary the power accordingly. The analysis described below is performed for an average power constraint on the transmitted signal. Also considered is the scenario if an additional peak power constraint on the transmitted signal is added. Therefore, throughput is maximized by performing optimal rate selection and power control. As a prerequisite for throughput maximization, the outage minimization problem is solved for signals with both peak and average power constraints.

Maximum .epsilon.-throughput (M.epsilon.T) and maximum zero-outage throughput are shown to be measures of best-case communications performance when there is, and is not, a restriction on the maximum number of transmission attempts per codeword, respectively. A greater throughput is achieved with the multi-attempt approach than the single-attempt approach. The increased throughput comes at the cost of queueing delays that are not present when transmitters are limited to a single transmission attempt. Therefore, also discussed is the situation in which throughput is maximized with a constraint on the queueing delay.

Historically, communication systems have been examined and designed using a layered approach. The Open System Interconnection (OSI) model separates communications systems into seven layers, including the physical, data-link, network, and upper layers. The physical layer deals with the transmission of unstructured data across the physical medium, while the data-link layer is responsible for creating a reliable data pipe between transmitter and receiver. This separation works well for analyzing idealized communication systems; however, in practical systems there can be significant coupling between layers This suggests that cross-layer optimization, rather than optimizing each layer independently, should be performed to maximize the performance of practical communication systems.

The field of information theory has concerned itself primarily with understanding the performance of the physical layer. Information theoretic measures traditionally characterize the amount of information that can be transmitted reliably with a single transmission attempt for any codeword Single-attempt measures, for delay-limited and delay-unconstrained systems, are motivated by the fact that the upper layers will ensure reliable delivery of the data if there are errors in the physical link. For delay-unconstrained systems the communications performance is quantified by the ergodic capacity, the ultimate reliable data rate over a fading channel. The concept of outage has been introduced for delay-limited systems If the transmission rate exceeds what the channel condition will reliably allow then an outage occurs, resulting in a decoding error at the receiver. The outage concept leads to .epsilon.-capacity (or outage capacity) and delay-limited capacity as measures of delay-limited communication performance. .epsilon.-capacity is the highest transmission rate that can be supported with outage probability no greater than .epsilon., while delay-limited capacity is simply .epsilon.-capacity when outages cannot be tolerated; that is, when .epsilon.=0.

Multi-Attempt Communication Paradigm

The single-attempt paradigm works well, theoretically, for delay-unconstrained systems. Such systems buffer an infinite amount of data and then transmit a single infinite-length codeword. Here, error-free communications is possible as long as the transmission rate is less than the ergodic capacity of the channel. Since error-free communications is possible, data retransmission is unnecessary, making the purely physical-layer, single-attempt approach perfectly suited for delay-unconstrained systems. For delay-limited systems, the single-attempt approach makes error-free communications very difficult. Traditional communication measures for delay-limited systems reflect this: .epsilon.-capacity does not provide a measure of error-free communications performance, while delay-limited capacity tends to underestimate communication performance.

The multi-attempt paradigm is more suitable for delay-limited systems than the single-attempt paradigm Delay-limited systems need not restrict themselves to a single transmission attempt for each codeword--multiple transmission attempts can be performed since codewords are finite length. In practical systems upper layers will often retransmit data to ensure reliable communication. For example, variants of the link-layer ARQ or transport layer TCP protocols are often used in real-world systems. There is a disconnect between how delay-limited systems are designed and used (practical, multi-attempt) and the measures (idealized, single-attempt) used to quantify their performance. Characterizing the maximum communications through-put, when multiple transmission attempts per codeword is permitted, may lead to a more accurate reflection of communications performance of delay-limited systems than the single-attempt measures used today.

For delay-limited systems, transmitters need not restrict themselves to a single transmission attempt per codeword. In fact, practical communication protocols, such as TCP or ARQ, retransmit data when errors occur. There is a disconnect in the design of delay-limited systems (multi-attempt) and the conventional measures used to quantify their performance (single-attempt) in an effort to achieve optimal throughput. The following discussion lays out a foundation for the new analysis framework disclosed herein.

In many applications, the condition of the fading channel changes on a time scale that is much slower than the communications signalling. This motivates modeling the channel as a discrete-time, block-fading, additive white Gaussian noise (BF-AWGN) channel. In this model, each "block" of N symbols corresponds to the amount of time the channel remains constant, the channel coherence time. The system in the k.sup.th block can be written y.sub.k=x.sub.kh.sub.k+w.sub.k, (1) with x.sub.k, y.sub.k.epsilon..sup.N representing the system input and output. A Gaussian noise process w.sub.k.about.N(0,I.sup.N) is assumed. Scattering by the environment results in reflections of the transmitted signal that add constructively or destructively with the original signal. The multipath interference due to scattering is represented by a random multiplicative gain h.sub.k.epsilon. on the transmitted signal. Below, x, y, w and h will be used to refer to the channel input, output, noise and gain when the relative position in the codeword is not important.

FIG. 1A provides a block diagram of a wireless communication system model. The model contains a transmission channel 96 used to transmit data from a transmitter 102 to a receiver 100. The model also contains a delay-less, error-free feedback link 98 used to relay acknowledgements of codewords (whether they were successfully decoded or not) back to the transmitter 102. The receiver 100 and the transmitter 102 may each be a transceiver, as shown in FIG. 1B. Specifically, a transceiver 110 may comprise an antenna 122 coupled to a hybrid 120. The hybrid 120 may convert between the bi-directional data stream 132 and the unidirectional data streams 134, 136.

The hybrid 120 may be coupled to a receive chain comprising a gain and filter 118, an analog-to-digital converter 116, a demodulator 114, and a processor 112. The processor 112 may communicate with a user or some other entity that uses the transceiver 110 to transmit or receive information. The processor 112 may be coupled to a memory 130 that may be used to store data and embedded software. The processor 112 also may be coupled to a transmit chain comprising a modulator 128, a digital-to-analog converter 126 and a driver 124 that couples to the hybrid 120. A data signal received by the antenna 122 may be directed to the receive chain by the hybrid 120. After the signal is filtered and the gain is adjusted by the gain and filter 118, the signal may be converted from analog to digital form by the converter 116 and demodulated by the demodulator 114. The processor 112 then may process the demodulated signal to extract receive signal information. Conversely, processor 112 may convert user data into a transmit data stream, which is modulated by the modulator 128 and converted to analog form by the converter 126. The signal may have its gain adjusted by the driver 124, which also drives the antenna 122. After passing through the hybrid 120, the signal may be transmitted by the antenna 122.

Codewords span K blocks of the BF-AWGN channel, contain KN symbols, and correspond to a K block coding delay. Each of the KN symbols contain information encoded at the transmission rate

.times..times..times..times..times..times..function..function. ##EQU00001## More specifically, R denotes spectral efficiency, but also can be used to denote transmission rate and/or encoding rate. The time-variations of the channel are assumed to be independent and identically distributed (i.i.d.) from block to block Blocks can physically correspond to slots in time, frequency, or both. The K i.i.d. channel fades affecting each codeword are .alpha.:=[(.alpha..sub.0,.alpha..sub.1, . . . , .alpha..sub.K-1], (2) with .alpha..sub.k=|h.sub.k|.sup.2 (or .alpha.=|h|.sup.2 when the relative position in the codeword generally is not of substantial importance). This model applies, for example, to wireless multicarrier modulated systems with K parallel subchannels.

It is assumed that the fading states follow a .chi..sub.2.sup.2 (chi-squared with 2 degrees of freedom) distribution with f(.alpha.)=e.sup.-.alpha., (3) and F(.alpha.)=1-e.sup.-.alpha. (4) the PDF and CDF, respectively. Such a distribution results when the |h| are Rayleigh distributed. This model is commonly used for wireless communication systems without line-of-sight between transmitter and receiver. Constructive interference results in a large .alpha. and thus a large received signal power that is conducive to communication; this situation is a "good" fade. Destructive interference results in a small .alpha..apprxeq.0 and thus a small received signal power that is not conducive to communication; this situation is a "bad" fade.

A system's capacity is normally measured with an average power constraint on the input, denoted P.sub.av. Without such a restriction the capacity of the channel may be infinite since the cardinality of the input distribution is infinite; that is, x.epsilon..sup.N. The transmitted power in the k.sup.th block of codeword is

.gamma..times..times..times..function. ##EQU00002## Random fading results in a received power of .alpha..sub.k.gamma..sub.k. Since a unit variance noise process [w]=1 is assumed, .alpha..sub.k.gamma..sub.k also equals the received signal-to-noise ratio (SNR) in the block. Additionally, since [.alpha.]=1, the average received SNR is also .gamma.. The results described herein can easily be generalized to cover non-unity variance noise processes.

Two channel state information (CSI) scenarios are considered. The first is when only the receiver has perfect, delay-less and error-free, CSI (CSI-R). In this case the transmitter cannot vary the average power based on the condition of the channel since it is unknown. Therefore, performance is maximized by transmitting at the average power. That is .gamma..sub.k=P.sub.av, .A-inverted.k{0, 1, . . . , K-1}.

For the second scenario when both transmitter and receiver have perfect CSI (CSI-RT), the average transmit power need not be constant; it can be varied in different blocks of the codeword based on the condition of the channel. Let .gamma. represent a power allocation policy, a strategy that assigns the power allocation vector .gamma.(.alpha.):=[.gamma..sub.0(.alpha.),.gamma..sub.2(.alpha.), . . . , .gamma..sub.k-1(.alpha.)] (6) given the channel .alpha.. When performing power control, the transmitter must be careful not to violate the specified power constraint. A common example is the short-term average power constraint

.gamma..function..alpha..times..times..times..gamma..ltoreq. ##EQU00003## Here the average power in any block of the codeword can exceed P.sub.av, while the average within the entire codeword cannot. Another widely used example is the long-term average power constraint .sub..alpha.[.gamma.(.alpha.)].ltoreq.P.sub.av. (8) This is a more relaxed condition since it allows the average power for any particular codeword to exceed P.sub.av as long as the average long-term power across all codewords does not.

In practical communication systems there is often a peak power constraint on the channel input in addition to the average power constraint. Non-linearities in power amplifiers force transmitters to limit the peak power to avoid distortion of the transmitted signal. Similarly, peak power may be limited to comply with communication standards that limit the interference to other communication systems. The peak power constraint is defined as .gamma..sub.k.ltoreq.P.sub.p, .A-inverted.k.epsilon.{0, 1, . . . , K-1} (9) which limits the maximum average power that can be allocated in any block of a codeword. While not a constraint on the absolute peak, such an approach allows the constraint of the peak power of the transmitted signal while remaining in the class of capacity achieving Gaussian channel inputs. The peak-to-average power ratio (PAR)is defined as

##EQU00004## Note that P.sub.p=.infin. corresponds to no peak power constraint on the channel input.

The constraints defined above are denoted as: O.sub.K.sup.st(P.sub.av)={.gamma.:.gamma.(.alpha.).ltoreq.P.sub.av} (11) O.sub.k.sup.lt(P.sub.av)={.gamma.:.sub..alpha.[.gamma.(.alpha.)].ltoreq.P- .sub.av} (12) O.sub.K.sup.st(P.sub.av, P.sub.p)={.gamma.:.gamma.(.alpha.).ltoreq.P.sub.av,.gamma..sub.k.ltoreq.P- .sub.p .A-inverted.k=0, 1, . . . , K-1} (13) O.sub.K.sup.lt(P.sub.av, P.sub.p)={.gamma.:.sub..alpha.[.gamma.(.alpha.)].ltoreq.P.sub.av, .gamma..sub.k.ltoreq.P.sub.p .A-inverted.k=0, 1, . . . , K-1} (14) or in words, as the set of all K-block power allocation policies that satisfy the short-term average (11), long-term average (12), short-term average and peak (13), and long-term average and peak (14) power constraints.

The instantaneous capacity (spectral efficiency), the highest reliable data rate for a codeword, is found by maximizing the mutual information over a frame of K i.i.d. channel fades, .alpha.. Assuming a Gaussian noise process and with CSI-R, and a constant average transmit power of P.sub.av, the instantaneous capacity is given by

.function..alpha..times..times..times..function..alpha..times. ##EQU00005## With CSI-RT and for power allocation vector .gamma.(.alpha.), is it given by

.alpha..gamma..function..alpha..times..times..times..function..alpha..time- s..gamma..function..alpha. ##EQU00006## In both cases it is achieved using random coding at the transmitter, with the elements of x.sub.k drawn from a Gaussian codebook.about.N(0, 1). Prior to transmission, each of the K blocks in the codeword is scaled by {square root over (P.sub.av)} (CSI-R) or {square root over (.gamma..sub.k(.alpha.))} (CSI-RT), respectively. Maximum a posteriori (MAP) detection is used at the receiver. Instantaneous capacity is an asymptotic quantity that is achieved as N.fwdarw..infin..

Communication performance measures based on the instantaneous capacity depend on the coding delay K. The delay unconstrained (K=.infin.) and delay-limited (K<.infin.) cases are discussed below.

Delay Unconstrained Systems (K=.infin.)

If the sequence of fading states .alpha..sub.k for k.epsilon.{0, 1, . . . , K-1} is asymptotically ergodic as K.fwdarw..infin., then the channels indexed by the block length N form a family that have the same capacity. This quantity is known as ergodic capacity and with CSI-R is given by

.times..times..tau..kappa..fwdarw..infin..times..kappa..function..alpha..t- imes..times. .alpha..function..function..alpha..times..times. ##EQU00007## for an average power constraint on the channel input. The expectation is performed with respect to the distribution of the channel fading process f(.alpha.). It is found by talking K.fwdarw..infin. in (15).

With CSI-RT ergodic capacity is given by

.times..times..tau..times..times..function..gamma..di-elect cons..infin..times. .alpha..function..function..alpha..gamma. ##EQU00008## for an average power constraint P.sub.av on the channel input. Again the expectation is performed with respect to f(.alpha.). It is found by taking K.fwdarw..infin. in (16) and selecting the optimal power allocation strategy that satisfies the average power constraint. The capacity achieving power allocation strategy

.gamma..function..alpha..lamda..alpha. ##EQU00009## assigns power .gamma..sup.C(.alpha.) to any block affected by fading state .alpha.. Here, .lamda..sup.C chosen such that the power constraint is satisfied,

.intg..lamda..infin..times..lamda..alpha..times..times.d.function..alpha. ##EQU00010## For the model in which .alpha..about..chi..sub.2.sup.2, ergodic capacitor can be written

e.times..times..intg..infin..times.e.times.d ##EQU00011## with CSI-R and as

.intg..lamda..infin..times.e.times.d ##EQU00012## with CSI-RT, with .lamda..sup.C as the solution to

e.lamda..lamda..intg..lamda..infin..times.e.times.d ##EQU00013##

For both CSI scenarios codewords are drawn from an infinite-length codebook with i.i.d. symbols.about.N(0, 1). Prior to transmission, the N symbols in each block ale scaled by either {square root over (P.sub.av)} (CSI-R) or {square root over (.gamma..sup.C(.alpha.))} (CSI-RT). Since codewords are affected by infinitely many fading states, the effect of the fading channel can be "averaged out" and reliable transmission at ergodic capacity is possible Contrary to what (17) and (18) seem to suggest, ergodic capacity is not actually an average capacity, but rather the highest rate that can be sustained on all channel states with arbitrarily small probability of error.

FIG. 1C compares ergodic capacity, C.sub.erg-pc, with power control (18) to ergodic capacity for constant power allocation, C.sub.erg, as a function of the average power P.sub.av for .chi..sub.2.sup.2 fading. For small transmit powers, the capacity with power control is larger than capacity with constant power For larger transmit powers, the difference between variable and constant power transmission shrinks, leading to the accepted wisdom that power control yields negligible capacity gains over constant power transmission. This shrinking difference occurs because the power allocated for each fading state (19) differs very little when P.sub.av is large.

Delay Constrained Systems (K<.infin.)

For finite K<.infin., the sequence of fading states .alpha..sub.k for k.epsilon.{0, 1, . . . , K-1} cannot be considered asymptotically ergodic. As such the instantaneous capacity becomes a random quantity When the channel condition is good, a number of the K channel fades affecting a codeword are good and a large amount of information can be transmitted per codeword- Conversely, when the channel condition is bad only a small amount of information can be reliably transmitted. An outage is declared if the transmission rate is larger than the instantaneous capacity, R>C.sub.K(.alpha., P.sub.av) (CSI-R) or R>C.sub.K(.alpha.,.gamma.(.alpha.)) (CSI-RT). For large N the outage probability closely approximates the codeword error probability.

Since the instantaneous capacity is a random quantity, outages can occur no matter how small or how large the transmission rate. The outage probability, the likelihood of outage events, is given by

.function. .times..function.>.function..alpha. .times. .alpha..function..function..function..alpha. ##EQU00014## with CSI-R and by

.function..gamma..times..function.>.function..alpha..gamma..function..a- lpha..times. .alpha..function..function..function..alpha..gamma..function..alpha. ##EQU00015## with CSI-RT. That is, for any transmission rate R and power allocation policy (including constant power transmission) there is an associated outage probability P.sub.out(R, P.sub.av, K) (CSI-R) or P.sub.out(R, .gamma., K) (CSI-RT). Using this, .epsilon.-capacity is defined as

.function..times..function..ltoreq..di-elect cons. ##EQU00016## with CSI-R and by

.function..times..function..gamma..ltoreq..di-elect cons..gamma..di-elect cons. ##EQU00017## with CSI-RT, where O.sub.K is the set of all valid power allocation strategies over which the optimization is performed and can represent either O.sub.K.sup.st(P.sub.av),O.sub.K.sup.st(P.sub.av, P.sub.p), O.sub.K.sup.lt(P.sub.av), or O.sub.P.sup.lt(P.sub.av, P.sub.p). .epsilon.-capacity represents the highest rate that can be supported with outage probability less than .epsilon. and may be used to quantify the communications performance of delay-limited communications systems in fading channels.

The need for a measure of error-free performance leads to the notion of delay-limited capacity

.function. .times..times..function..alpha..times. ##EQU00018## with CSI-R, and by C.sub.dl.sup.pc(P.sub.av, K):=C.sub.c.sup.pc|.sub.c=0 (30) with CSI-RT. When the minimum channel gain .alpha..sub.min:=min{.alpha.}=0 (31) which is the case for many common fading distributions including .chi..sub.2.sup.2 delay-limited capacity is 0 for all K<.infin. with CSI-R. While for CSI-RT delay-limited capacity is 0 for K=1, however, it is possible to have non-zero delay-limited capacity for K>1.

Since the transmit power can be varied based on the condition of the channel with CSI-RT, the power allocation policy used affects performance One policy of particular importance is the one that minimizes the outage probability. This policy can also be used to maximize the transmission rate for a target outage probability; that is, it can be used to achieve C.sub.c.sup.pc. The outage minimization problem can be stated as

.gamma..times..function..gamma..gamma..di-elect cons. ##EQU00019## The solution to (32) is known as the outage minimizing power allocation strategy and has been found for O.sub.K=O.sub.K.sup.st(P.sub.av) and O.sub.K=O.sub.K.sup.lt(P.sub.av), the short-term and long-term average power constraints. Overviewed below are the solutions for these cases.

Under the short-term average power constraint O.sub.K=O.sub.K.sup.st(P.sub.av) in (32) and the outage minimizing power allocation policy is

.gamma..function..alpha..lamda..function..alpha..alpha. ##EQU00020## with

.lamda..function..alpha..mu..times..times..mu..function..alpha..times..alp- ha..mu..function..alpha..times. ##EQU00021## for .mu.(.alpha.).epsilon.{1, 2, . . . , K} and .alpha..sub.(0).gtoreq..alpha..sub.(1).gtoreq. . . . .gtoreq..alpha..sub.(K-1) an ordered permutation of the fading states affecting the codeword.

Under the long-term average power constraint O.sub.K=O.sub.K.sup.lt(P.sub.av) in (32), and the outage minimizing power allocation policy takes the form

.gamma..function..alpha..gamma..function..alpha..times..times..times..time- s..times..times..times..alpha..di-elect cons..function..gamma..function..alpha..times..times..times..times..times- ..times..times..alpha..di-elect cons..function..function..times..times..times..times..times..times..times- ..alpha..di-elect cons..function..function..times..times..times..times..times..times..times- ..alpha..function..function. ##EQU00022## where R.sub.1(s)={.alpha.:{circumflex over (.gamma.)}(.alpha.)<s} (36) R.sub.1(s)={.alpha.:{circumflex over (.gamma.)}(.alpha.).ltoreq.s} (37) R.sub.1(s)-R.sub.1(s)={.alpha.:{circumflex over (.gamma.)}(.alpha.)=s} (38) represent sets of fading states differentiated by the amount power allocated for each fading state. Then P.sub.1(s)=.intg..sub.R.sub.1.sub.(s){tilde over (.gamma.)}(.alpha.)dF(.alpha.) (39) P.sub.1(s)=.intg..sub. R.sub.1.sub.(s){tilde over (.gamma.)}(.alpha.)dF(.alpha.) (40) is the average power allocated over these sets. Then s.sub.1*=sup{s:P.sub.1(s)<P.sub.av} (41) is maximum aver age power allocated for any fading state and

.function. .function. .function. ##EQU00023## is the probability that the codeword is transmitted when this maximum is achieved. Both s.sub.1* and w* ensure the average transmitted power across all fading states is P.sub.av as desired. Finally,

.gamma..function..alpha..lamda..function..alpha..alpha. ##EQU00024## is the form of the power allocated for fading state .alpha., with

.lamda..function..alpha.e.mu..function..alpha..times..alpha. ##EQU00025## .mu.(.alpha.).epsilon.{1, 2, . . . , K}.

FIG. 1D plots the minimum outage probability for K=1 as a function of P.sub.av under constant power allocation and under the long-term average power constraint. The gain of power control is seen for a target outage probability, the average power required when performing power control is less than when using constant power transmission.

Throughput and Fading Channels

Within the single attempt paradigm, zero-outage (error-free) communications is often viewed as an all-or-nothing phenomenon. For delay-unconstrained systems it is possible to transmit reliably at rates approaching ergodic capacity, while for delay-limited systems delay-limited capacity is zero for many fading distributions of interest.

A new analysis framework for delay-limited systems in fading channels is described below. FIG. 1E shows a queue 302 receiving data at a rate .lamda. and transferring data to a server 300 using a first-in, first-out (FIFO) methodology. By modeling the communications systems as a queue, it is possible to relate the throughput of the system with the amount of information passing through the queue 302. The server 300 in the queueing model encompasses the details of both the physical and data-link layers, shown in FIGS. 1A-1B. The server 300 takes codewords that arrive in the queue 302 and attempts transmission repeatedly until the channel condition allows successful transmission. The service time for a codeword is based on the number of transmission attempts required and can vary from system to system based on the particular retransmission scheme. Using this approach the throughput is simply the transmission rate divided by the service time--the amount of data in each codeword divided by the number of transmission attempts required for successful decoding. Maximizing the throughput through the queue is equivalent to maximizing the throughput of the delay-limited communication system

As discussed above, two main CSI scenarios are considered: when only the receiver has CSI (CSI-R) and when both transmitter and receiver have CSI (CSI-RT). With CSI-R, throughput is maximized using optimal rate selection, while with CSI-RT it is maximized by optimal rate selection and power control. For both scenarios, the maximum throughput under the multi-attempt paradigm exceeds that under under the single-attempt paradigm, That is, for the same coding delay a higher throughput is possible my allowing multiple transmission attempts per codeword, rather than a single attempt.

1. Multi-Attempt Throughput Maximization

1.1 Cross-Layer Queueing Model

By maximizing the communications throughput within the multi-attempt framework, the communications throughput of the physical layer (which is responsible for selecting the transmission rate R) and data-link- layer (which is responsible for data retransmission in the face of errors) are jointly maximized. This joint optimization can be used to predict the best case performance for any retransmission scheme in fading channels.

The physical and data-link layers can be modeled jointly as a queue. In this model, codewords arrive into the queue encoded at rate R, and-therefore contain RKN nats. The server takes a codeword from the queue and attempts transmission. When an outage occurs, the codeword is retransmitted until successful transmission or until a maximum number of transmission attempts is reached. The number of transmission attempts for each codeword, the service time, is a random quantity due to the random nature of the fading channel. The number of transmission attempts is used to quantify the service time, since each transmission attempt corresponds to K blocks and therefore corresponds to the channel coherence time scaled by a factor of K. The service time distribution, the probability that s attempts are required for successful transmission, depends on the nature of the retransmission scheme, the transmission rate and power, and the statistics of the fading channel In general, the probability that a codeword's service time, S, will be s attempts for successful transmission is

.function..times..function..times..times..function..times..times..times..t- imes. ##EQU00026## which is the probability of outage events on the first s-1 attempts multiplied by the probability of successful transmission on the s.sup.th attempt given that it was previously in error.

The service time distribution can be used to determine the expected service time [S] and the expected service rate

.function. ##EQU00027## of a codeword. The average amount of data passing through the queue with each transmission attempt is

.function..times..times..times..times..times. ##EQU00028## the encoding rate divided by the average number of attempts for successful transmission. For example if the data transmission rate is R=10 nats/sec/Hz and takes on average [S]=2 transmission attempts per codeword, then the average throughput is 5 nats/sec/Hz. Using this idea, the maximum throughput of the system is defined as

.function..times..times..gamma..times. .function..function..gamma..gamma..di-elect cons. ##EQU00029## where the supremum is taken over all transmission rates and power allocation strategies in O.sub.K. Either constant power transmission, .gamma.=P.sub.av for a system with CSI-R, or the transmitter performing power control with O.sub.K .epsilon.{O.sub.K.sup.st(P.sub.av), O.sub.K.sup.lt(P.sub.av), O.sub.K.sup.st(P.sub.av, P.sub.p), O.sub.k.sup.lt(P.sub.av, P.sub.p)} for a system with CSI-RT, may be considered. It is noted that T.sub.max(P.sub.av, K, P.sub.p) predicts the best case performance for a particular multi-attempt scheme, coding delay K, average power constraint P.sub.av and peak power constraint P.sub.p. By matching the multi-attempt scheme used in the analysis to one that is used in practice, this analysis can be used to predict the best case communication performance of practical retransmission algorithms, (i.e., ARQ), in fading channels.

If the transmission rate is R, then the amount of data successfully decoded with any transmission attempt is either 0 or R, depending on whether an outage does or does not occur, respectively. The maximum average throughput is a representative measure of communications performance.

2. Throughput Maximization with Optimal Rate Selection

When only the receiver has CSI, the transmitter does not vary the transmit power level based on the condition of the channel. As such, for this scenario, the transmitter uses the average power, .gamma..sub.k=P.sub.av, .A-inverted.k{0, 1, . . . , K-1}. In this case the optimization in (1.2) is over the encoding rate and

.function..times..times. .times..function. ##EQU00030##

T.sub.max (P.sub.av,K) represents the optimal balance between the amount of information in each codeword and the frequency at which codewords pass through the queueing system. As R.fwdarw.0, the amount of information carried per codeword shrinks and the throughput approaches 0. Similarly, as R.fwdarw..infin., outages become frequent and [S].fwdarw..infin., resulting in a throughput that approaches 0.

The optimal transmission rate depends significantly on the coding delay K. FIG. 2A(1)(2) illustrates the codeword error probability when K=.infin. and K=1 (for scheme RT), respectively. The optimal operating point when K=.infin. is obvious, the transmit rate is set at a rate as close to ergodic capacity as possible with codeword error probability close to zero. However, for K=1 the optimal transmission rate is not immediately obvious. Examining the system from a throughput perspective in FIG. 2B, both systems are shown to have a transmission rate that maximizes throughput. For K=.infin., R=C.sub.erg is the unique throughput maximizing transmission rate For K=1, for scheme RT, there is also a unique throughput maximizing transmission rate For delay-limited systems, the optimal transmission rate depends on the particular retransmission scheme being used and its expected service time. In general, it is possible to specify conditions on the expected service time, for a particular retransmission scheme, that guarantee the existence of a unique throughput maximizing transmission rate

Theorem 2.0.1. If

.function..function. ##EQU00031## is a log-concave function of R, then (1.2) has a unique global maximum. Proof. Let

.function..function..function..times..times..function..times..times..funct- ion..times..times..times..function..function..times..times..times..times..- function..function. ##EQU00032## is a concave function then f(R) is also concave, since log R is concave and the sum of two concave functions is also concave. Then from convex optimization theory, f(R) has a unique maximizer on the convex set .sub.+. Let R* be the argument that maximizes f(R). If f(R) is composed with the monotonically increasing function e.sup.z, then e.sup.f(R)=T(R) has the same maximizer R*. Hence (1.2) has a unique maximum. .quadrature.

This is a sufficient, but not necessary, condition for the existence of a unique solution. It is possible for

.function..function. ##EQU00033## to be log-concave without

.function. ##EQU00034## being log-concave. This scenario would also have a unique maximizer for the throughput. The uniqueness of the optimal transmission rate is of practical importance. Often (1.2) cannot be solved explicitly and numerical techniques must be used. Fortunately, if

.function. ##EQU00035## is log-concave, any numerical solution to (1.2) is globally optimal.

Since only the receiver has CSI, the tra