Minimum Outage Probability Transmission With Imperfect Feedback ...
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 3, MAY 2005
Minimum Outage Probability Transmission With Imperfect Feedback for MISO Fading Channels Yongzhe Xie, Costas N. Georghiades, Fellow, IEEE, and Ari Arapostathis, Senior Member, IEEE
Abstract—We study optimal transmission strategies in terms of minimum outage probability for a fading channel with multiple transmit antennas and a single receive antenna. We consider two cases of imperfect channel state information (CSI) feedback: mean feedback and covariance feedback. In both cases, the optimum strategy is shown to be transmission of multiple data streams to the same directions as when maximizing the ergodic channel capacity, but with different power allocation strategies which are closely related to the target rate. In the mean feedback case, the optimal power allocation strategy is also related to the accuracy of the feedback CSI, which affects significantly the achievable outage capacity. Index Terms—Channel capacity, channel state information, fading channel, multiple antennas.
I. INTRODUCTION
I
N WIRELESS communications, in addition to the ergodic capacity which characterizes the long-term average achievable rate limit of a fading channel, information outage capacity [1] is also used since practical codeword lengths are limited due to delay constraints. Perfect channel state information at the transmitter (CSIT) has been shown to significantly improve the channel outage capacity for both single and multiple antenna systems [2], [3]. However, if only imperfect CSIT is available, is the optimum transmit strategy changed? How is the outage capacity affected? These questions are usually hard to answer partly due to the difficulty in evaluating the distribution of the instantaneous mutual information. Some recent papers proposed some partial CSIT models in the case of multiple transmit antennas and a single receive antenna and studied optimum transmission in terms of maximizing the ergodic capacity using these models [4], [5]. Here, we are interested in the models of [5]. It is assumed that the receiver has
Manuscript received March 19, 2003; revised January 13, 2004, April 6, 2004; accepted April 10, 2004. The editor coordinating the review of this paper and approving it for publication is F.-C. Zheng. This work has been supported by a Motorola University Partnership in Research (UPR) Award. The work of A. Arapostathis was supported in part by the National Science Foundation under Grant ECS-0218207 and in part by the Office of Naval Research through the Electric Ship Research and Development Consortium. Y. Xie and is with the Prediction Company, Santa Fe, NM 87505 USA (e-mail: [email protected]). C. N. Georghiades is with the Electrical Engineering Department, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]). A. Arapostathis is with the Department of Electrical and Computer Engineering, The University of Texas, Austin, Texas 78712 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.847030
perfect channel state information, and feedback some channel information to the transmitter. Based on the feedback, the transmitter models the channel as shown in the following two cases: 1) Mean Feedback: The channel distribution is modeled at , where the mean the transmitter as could be interpreted as an estimate or prediction of the as the variance of the channel based on feedback and estimation or prediction error. This is the case of slow fading. 2) Covariance Feedback: The channel distribution is mod. This models very fast fading, eled as in which the feedback channel fails to provide an accurate estimate of the current channel value. However, , determined by the relative geometry of the propagation paths, changes slowly compared to the fading, thus can be tracked by feedback. Moreover, is practically the same for both the uplink and the downlink channels in freqency division duplex (FDD) systems and, therefore, can be estimated from uplink data, obviating the need for feedback [6]. The solution in both cases is determined by solving simple numerical optimization problems. When there is a moderate disparity between the strengths of different paths from the transmitter to the receiver, it is nearly optimal to employ the simple beamforming strategy that concentrates all the transmit power in the strongest direction indicated by the feedback. This problem is further studied by [8], [10] for some special multiple-input multiple-output (MIMO) channels in the case of covariance feedback. The condition for beamforming to be optimal is studied in [9]–[11]. In this paper, we study the optimal transmission strategies in terms of minimum outage probability for the case of mean feedback. In particular, we prove that the optimal transmission directions in this case are the same as in maximizing the ergodic capacity. We also provide some supplemental results on minimum outage probability transmission in the case of covariance feedback other than those presented in [10]. Note that for fast fading, it may not be meaningful in the covariance feedback case to consider channel outage capacity as defined in [1] and in this paper when a codeword may experience different channel states. However, our results for this case can be used to illustrate how knowledge of at the transmitter can affect the outage capacity of a block fading channel. A recent paper [12] also studied the problem dealt with in this paper. Our work was done independently and apparently concurrently with [12]. It was first disseminated to others in a report to Motorola [13]. Compared to the results of [12], our proofs and analysis follow a substantially different approach, in addition to
1536-1276/$20.00 © 2005 IEEE
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
providing some new results. For example, we show that the optimal power allocation schemes for both mean feedback and covariance feedback are a function of the target outage probability, both by using majority theory and by observing the form of the for outage probability ((10)). We present an example for the covariance feedback case in which we show that even when , the optimal power althe channel is uncorrelated, i.e., location may not be a uniform distribution among orthogonal directions. As shown in Section III, the optimal power allocation scheme in the covariance feedback case could allocate all the power to only one of the orthogonal directions (outage probability may be greater than 0.5 in this case). This is only shown by simulation in [12] and it is not clearly stated. Finally, we note that the proof of a main result regarding the optimal transmission directions in the mean feedback case presented in [12, Appendix II] is incorrect. The error occurs from (65) to (66) in , where and are which step the authors assume (dummy) double integration variables. However, the region of obviously does not guarantee , integration which makes the proof invalid. The following notation will be used throughout the paper: superscript denotes matrix/vector conjugate transpose, is the trace operator, is the th element of vector , and is the identity matrix. is the expectation operator and will also be used to denote the mean of .
. Letting , we have problem can then be expressed as
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. The (5) (6)
subject to
(7)
and
(8)
are independent circularly symmetric where is noncomplex Gaussian random variables. Therefore, central Chi-square distributed with two degrees of freedom and . To minimize , we need to a noncentrality parameter and for , subfind the optimal values of ject to their individual constraints. We first fix the power allocation to each transmit antenna by fixing the and consider the optimization with respect to . The following Lemma helps provide a solution. Lemma 1: Given two real and independent random variables , , and , then
II. MEAN FEEDBACK Consider the discrete model of a multiple-input single-output (MISO) fading channel [5], (1) where is circularly symmetric complex Gaussian noise with variance per dimension. Let denote the number of transmit antennas. is an complex and deGaussian channel vector modeled as notes the complex transmit signal vector. Define . For a given , since the optimum input distribution of that maximizes the conditional mutual information for any fixed channel realization is zero mean complex Gaussian, the problem of minimal outage transmission can be formulated as
(2) (3) subject to
subject to is attained at , for any . The proof is given in the Appendix. Based on this lemma, we have the following proposition. Proposition 1: Without loss of generality, assume for . Then, the optimal solution of the problem in , for all . (6) is achieved by Proof: Since the distribution of is not dependent on , without loss of generality, for all , where we can let and are two independent real Gaussian random variables. Therefore, the optimization over can be equivalently transferred to the optimization . over Denote the optimal solution as and the associated minimum outage probability as . As, such that . Consume there exists some , and struct . Let the cumulative distribution function (CDF) of and the probability density function (PDF) of be and , respectively. Applying Lemma 1 , we have the following inequality: to
(4)
where ; and denote the target rate and available power, respectively. Note that here we only consider short-term power control [2] so that is not a is positive semi-definite, we have function of time. Since the eigenvalue decomposition (EVD) , where is a diagonal matrix with indicating the power allocated to transmission directions indicated by the corresponding column vectors of unitary matrix
(9) where tion solution of
is the CDF associated with another solu, in which , , and for all , . Since the achieves lower outage probability than ,
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 3, MAY 2005
is not optimal, which contradicts our assumption. Thus, for any . we have When there are more than only which have equal largest value, it can be easily shown that the choice of can be arbitrary within this set, as long as their sum is equal to . Therefore, is still optimal, which indicates that allocating to a single , where is a complex the optimal choice of is scalar such that . Without loss of generality, let . Since , we have , is an arbitrary set of orwhere thonormal vectors that are orthogonal to . Therefore, the optimal transmission directions to minimize outage probability are the vector channel mean and its orthogonal directions, which are the same as in the case of maximizing the ergodic capacity [5]. is obtained, we still need to determine the optimal After in the different transmission directions. power allocation Here, we will briefly analytically study this problem and will follow up with numerical results in Section IV. Let in (6). Given the optimal as above, is noncentral -disand tributed with two degrees of freedom, mean . The , , are each a noncentrality parameter central -distributed with two degrees of freedom and mean . Therefore, the outage probability can be evaluated (after some simple change of variables) as
. Therefore, the s smaller reduces beamforming, which concentrates power in the direction of the channel mean might be optimal for larger outage probability. as a function of is relatively hard to evaluate due to its dependency on . So, the previous analysis is only approximate, but well matches the numerical results to be presented later. III. COVARIANCE FEEDBACK Given a fixed power allocation scheme, transmitting along the eigenvectors of the channel covariance matrix is proven to be necessary and sufficient to achieve minimum outage probability in the covariance feedback case [10]. Note that this is the same strategy as maximizing the ergodic capacity, as shown in [5]. Here, to make our presentation complete, we provide a slightly different, but much more concise proof using an inequality from [8]. Proposition 2: Let , the EVD of be 1. Choosing , and , is a diagonal matrix, is necessary and sufficient for where minimizing the outage probability. . Letting , Proof: We first assume we need to show that is diagonal in order to minimize outage probability. As in the last section, minimizing the outage probability at a given rate is equivalent to minimizing (11) (12) subject to
(13)
so that is white with distribution of . Let with EVD . Then, the minimum outage probability problem is rewritten as
where (10) where
(14) (15)
subject to Letting
, we can further transform (14) to
for and is the zeroth-order modified Bessel function of the as the feedback first kind. We define the ratio SNR SNR. We see that the minimum outage probability is strongly a function of SNR , which is a measure of the accuracy of CSIT. When is sufficiently small, and are approximately 1 and the outage probability is mainly affected by , which is minimized by making the equal. Therefore, for sufficiently small (associated with a very small outage probability and and noise a low target rate for given transmission power covariance ), the optimal power allocation tends to spread power over different transmission directions. On the contrary, could increase very as increases, rapidly as increases. Making larger, and, thus, the rest of
(16) It is shown in [8] that (17) Comparing (14) and (16), since and have the same distribution, the minimum outage probability achieved by the opwith at timal can always be achieved by a diagonal matrix least the same or even more stringent power constraint. Therefore, choosing as diagonal is sufficient to minimize outage probability. Equivalently, this means is diagonal since . Necessity is easily proven using the fact that the is diagonal. first equality in (17) is satisfied if and only if , because any set of orthonormal For the trivial case of 1If tr[6 6] = N , we can always normalize the channel by absorbing into the transmit power P
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
Fig. 1. Achievable minimum outage probability versus target rate of mean-feedback for N = 2. Channel received SNR SNR (= ). at 8 dB. SNR
vectors can be used as beamforming directions, the theorem is easily proved. Next, it remains to determine the optimal power allocation in different transmission directions, i.e., the di. Let agonal entries of . The optimal power allocation problem is defined as
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((P ( + )) = ) is fixed
Now, we consider two special cases for the problem in (18). . Then, Case 1:
(20)
(18)
subject to
and
(21)
(19)
where is exponentially distributed. Although tedious, in this case can be computed in a closed form expression. Therefore, we could solve the constrained optimization problem of (18) numerically. However, the optimization problem is not . convex, which makes it difficult to find the global optimal Here, we coarsely identify the conditions when the optimum strategy is to spread power over different directions and when to concentrate all the power in a single direction using majority theory [7], to which we give a brief introduction next. Majorization: Given two real positive vectors , having equal summation of all entries. is majorized by , de, if the sum of the smallest entries of is greater noted as than or equal to the same sum for for all . This is a mathematical description of the vague concept of is “less spread out” than . Schur-convexity: A real-valued function defined on a set of is Schur-convex, if . is . Schur-convexity Schur-concave, if and Schur-concavity can be viewed as extensions of the increase or decrease functions defined on .
In particular, when , we can show that is a Schur, and Schur-concave function when convex function when where the constant is a solution . Therefore, if to the equation , we need to spread power equally on two orthogonal transmission directions to minimize outage probability; otherwise, we should concentrate on a single transmission direction to minimize outage. , and . Case 2: for some fixed power allocation Consider optimizing , on the remaining directions
(22) . As a function of for all , is defined symmetrically for all . Since is a Schur-concave function for vector if and a Schur-convex function if , we is Schur-convex for can conclude that some small enough or Schur-concave for some large enough where
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 3, MAY 2005
Fig. 2. Effect of SNR on achievable minimum outage probability of mean-feedback for N = 2. Channel received SNR SNR (= ). fixed at 8 dB. SNR
((P ( + )) = ) is
Fig. 3. Optimal power allocation over different transmission directions of mean-feedback for N = 2. Channel received SNR SNR fixed at 8 dB. SNR (= ).
((P ( + )) = ) is
according to [7, Th. 3.A.5]2. Therefore, for a given channel , the optimal power allocation received SNR SNR scheme tends to spread power when target rate is low, and concentrate power when is large. However, in the latter case, the outage probability could be very large (say 0.5), and not of practical interest. So, spreading power over different trans-
mission directions is not necessarily optimum for minimizing . Through numerical results, we outage probability when , the optimal power allocation can also show that when schemes still tend to spread power when outage probability is low and concentrate power otherwise.
2Let f (x) be symmetric in each element of x, where x = [x ; x ; . . . x ]. Theorem 3.A.5 states that to prove f (x) is Schur-convex in x, it is sufficient to show f (x) is Schur-convex in a vector composed of any two elements of x by fixing the other elements.
IV. NUMERICAL RESULTS Figs. 1 and 2 plot the achievable minimum outage probability of mean-feedback for different rates and SNR , for a fixed
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
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Fig. 4. Minimum outage probability achievable by optimal and equal power allocation over the directions indicated by the two eigenvectors of covariance matrix and satisfy + = 2. Channel received SNR SNR (P= ) is fixed at 8 dB.
6. Note that the two eigenvalues
average received SNR, SNR dB. We can see outage probability decreases exponentially as the target rate decreases. SNR affects outage capacity significantly, especially at low outage probability. Note here that dB is about the capacity of an AWGN channel at SNR b/channel-use. To achieve 1% outage, the dB required SNR for 1.15 b/channel-use (40% of dB ) and for 1.72 b/channel-use (60% of ) is about 9 and 12 dB, dB respectively. Fig. 3 compares the optimal power allocation over different transmission directions for mean-feedback. When “good” channel feedback is available, the optimal solution tends to beamform to the direction indicated by the mean with all available power. On the contrary, “bad” channel feedback may require multiple beams to be transmitted. Moreover, at higher rate or larger outage probability, beamforming to the direction indicated by the channel mean with all power is optimal. Fig. 4 shows the minimum outage probability achievable by covariance feedback with the optimal power and the equal . Let the EVD of be power allocation schemes for and and be the two diagonal elements of . Even when , which indicates high spatial correlation between transmit antennas, optimal power allocation cannot significantly reduce outage probability compared to the scheme in which power is uniformly allocated to both orthogonal directions. V. CONCLUSION We have studied the problem of minimum outage probability transmission for a MISO fading channel in the cases of mean feedback and covariance feedback. In the case of mean feedback, the optimal transmission strategy is proven to be transmitting several independent data streams in the direction of the channel mean vector and its orthogonal directions.
When SNR is high, the optimal strategy tends to beamform to the direction indicated by the channel mean. The quality of the channel information, measured by SNR affects the outage probability significantly. For both mean and covariance feedback, we show that the optimum power allocation scheme which minimizes outage probability is closely related to the target rate. It is more desirable to spread the power over all transmission directions than beamforming to a single direction for sufficiently small target rates. APPENDIX Proof: Parameterize , with
and , as , and let
,
Thus
(23) We convert to polar coordinates . Then, takes the form
For
, let
,
,
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The integral in (23) transforms to
We have
(24) Note (25), located at the bottom of the page, where
(28) Decomposing the inner integral in (24) along the partition of in (25) and using the symmetry properties of the functions, we obtain
and
where (29) By (26), (28), and (29)
(26) (30)
Therefore, by (24) Therefore, by (30)
(27) The first integral in (27) is independent of
Hence, in order to establish that , it is sufficient to show that
and is equal to
for all
This completes the proof.
if if
, ,
(25)
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
ACKNOWLEDGMENT The authors would like to thank Motorola for support of this research and in particular Dr. K. Rohani of Motorola Labs for many helpful discussions on the topic. REFERENCES [1] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998. [2] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1468–1489, Jul. 1999. [3] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of blockfading channels with multiple antennas,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1273–1289, May 2001. [4] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple antenna data transmission over fading channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998. [5] E. Vistotsky and U. Madhow, “Space-time transmit precoding with imperfect feedback,” IEEE Trans. Inf. Theory, vol. 47, no. 6, pp. 2632–2639, Sep. 2001. [6] G. G. Raleigh and V. K. Jones, “Adaptive antenna transmission for frequency duplex digital wireless communications,” in Proc.IEEE Int. Communications Conf., vol. 2, 1997, pp. 641–646. [7] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. San Diego, CA: Academic, 1979. [8] S. A. Jafar, S. Vishwanath, and A. J. Goldsmith, “Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback,” in Proc. IEEE Int. Communications Conf., vol. 7, 2001, pp. 2266–2270. [9] S. A. Jafar and A. J. Goldsmith, “On optimality of beamforming for multiple antenna systems with imperfect feedback,” in Proc. IEEE Int. Symp. Information Theory, 2001, pp. 321–321. [10] S. H. Simon and A. L. Moustakas, “Optimizing MIMO antenna systems with channel covariance feedback,” Bell Labs. Tech. Memo., 2002. [11] E. Jorswieck and H. Boche, “On transmit diversity with imperfect channel state information,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, 2002, pp. 2181–2184. [12] A. L. Moustakas and S. H. Simon, “Optimizing multi-transmitter-singlereceiver (MISO) communication systems with general Gaussian channels: Nontrivial covariance and nonzero mean,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2770–2780, Oct. 2003. [13] Y. Xie and C. N. Georghiades, “Space-time minimum outage probability transmission with imperfect feedback,” Motorola Univ. Partnership Res. (UPR) Project, 2002.
Yongzhe Xie received the B.E. degree from the Department of Biomedical Engineering, Shanghai Jiaotong University, Shanghai, China, in 1996, the M.E. degree from the Department of Electrical Engineering, the National University of Singapore, in 1999, and the Ph.D. degree from the Department of Electrical Engineering, Texas A&M University, College Station, in 2004. His current research interests are in the general area of information theory, signal processing for communications, and statistical signal processing for financial applications. Mr. Xie was the recipient of the Texas Telecommunications Engineering Consortium (TxTEC) Fellowship of 1999, NOKIA Wireless Future Intern Award of 2001, and the China Instrument Scholarship of 1995.
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Costas N. Georghiades (M’85–SM’90–F’98) eceived the B.E. degree with distinction from the American University of Beirut, Beirut, Lebanon, in 1980, and the M.S. and D.Sc. degrees from Washington University, St. Louis, MO, in 1983 and 1985, respectively, all in electrical engineering. Since September 1985, he has been with the Electrical Engineering Department, Texas A&M University, College Station, where he is currently a Professor and holder of the Delbert A. Whitaker Endowed Chair. He has been involved in organizing a number of conferences, including as Technical Program Chair for the 1999 IEEE Vehicular Technology Conference and the 2001 Communication Theory Workshop and as Chair of the Communication Theory Symposium within Globecom 2001. He currently serves as General Co-Chair for the 2004 IEEE Information Theory Workshop, as Technical Program co-Chair for the 2005 Communication Theory Workshop and as Guest Editor for the EURASIP Journal on Wireless Communications and Networking. His research interests include the application of information, communication, and estimation theories to the study of communication systems. Dr. Georghiades he served in editorial positions with the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON INFORMATION THEORY, the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and the IEEE COMMUNICATIONS LETTERS. He is a Registered Professional Engineer in Texas.
Ari Arapostathis (M’82–SM’91) received the B.S. degree from the Massachusetts Instistute of Technology, Cambridge, and the Ph.D. degree from the University of California at Berkeley. He has been a Faculty Member with the University of Texas at Austin since 1982. He has served as the Vice Chairman for Invited Sessions of the Program Committee for the 1987 American Control Conference. His research interests include analysis and estimation techniques for stochastic systems, the application of differential geometric methods to the design and analysis of control systems, stability properties of large-scale interconnected power systems, and stochastic and adaptive control theory. Among his main technical contributions are in the areas of adaptive control and estimation of stochastic systems with partial observations, adaptive control of nonlinear systems, geometric nonlinear theory, and stability of large-scale interconnected power systems. Dr. Arapostathis was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. He is a member of the AMS and SIAM.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 3, MAY 2005
Minimum Outage Probability Transmission With Imperfect Feedback for MISO Fading Channels Yongzhe Xie, Costas N. Georghiades, Fellow, IEEE, and Ari Arapostathis, Senior Member, IEEE
Abstract—We study optimal transmission strategies in terms of minimum outage probability for a fading channel with multiple transmit antennas and a single receive antenna. We consider two cases of imperfect channel state information (CSI) feedback: mean feedback and covariance feedback. In both cases, the optimum strategy is shown to be transmission of multiple data streams to the same directions as when maximizing the ergodic channel capacity, but with different power allocation strategies which are closely related to the target rate. In the mean feedback case, the optimal power allocation strategy is also related to the accuracy of the feedback CSI, which affects significantly the achievable outage capacity. Index Terms—Channel capacity, channel state information, fading channel, multiple antennas.
I. INTRODUCTION
I
N WIRELESS communications, in addition to the ergodic capacity which characterizes the long-term average achievable rate limit of a fading channel, information outage capacity [1] is also used since practical codeword lengths are limited due to delay constraints. Perfect channel state information at the transmitter (CSIT) has been shown to significantly improve the channel outage capacity for both single and multiple antenna systems [2], [3]. However, if only imperfect CSIT is available, is the optimum transmit strategy changed? How is the outage capacity affected? These questions are usually hard to answer partly due to the difficulty in evaluating the distribution of the instantaneous mutual information. Some recent papers proposed some partial CSIT models in the case of multiple transmit antennas and a single receive antenna and studied optimum transmission in terms of maximizing the ergodic capacity using these models [4], [5]. Here, we are interested in the models of [5]. It is assumed that the receiver has
Manuscript received March 19, 2003; revised January 13, 2004, April 6, 2004; accepted April 10, 2004. The editor coordinating the review of this paper and approving it for publication is F.-C. Zheng. This work has been supported by a Motorola University Partnership in Research (UPR) Award. The work of A. Arapostathis was supported in part by the National Science Foundation under Grant ECS-0218207 and in part by the Office of Naval Research through the Electric Ship Research and Development Consortium. Y. Xie and is with the Prediction Company, Santa Fe, NM 87505 USA (e-mail: [email protected]). C. N. Georghiades is with the Electrical Engineering Department, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]). A. Arapostathis is with the Department of Electrical and Computer Engineering, The University of Texas, Austin, Texas 78712 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.847030
perfect channel state information, and feedback some channel information to the transmitter. Based on the feedback, the transmitter models the channel as shown in the following two cases: 1) Mean Feedback: The channel distribution is modeled at , where the mean the transmitter as could be interpreted as an estimate or prediction of the as the variance of the channel based on feedback and estimation or prediction error. This is the case of slow fading. 2) Covariance Feedback: The channel distribution is mod. This models very fast fading, eled as in which the feedback channel fails to provide an accurate estimate of the current channel value. However, , determined by the relative geometry of the propagation paths, changes slowly compared to the fading, thus can be tracked by feedback. Moreover, is practically the same for both the uplink and the downlink channels in freqency division duplex (FDD) systems and, therefore, can be estimated from uplink data, obviating the need for feedback [6]. The solution in both cases is determined by solving simple numerical optimization problems. When there is a moderate disparity between the strengths of different paths from the transmitter to the receiver, it is nearly optimal to employ the simple beamforming strategy that concentrates all the transmit power in the strongest direction indicated by the feedback. This problem is further studied by [8], [10] for some special multiple-input multiple-output (MIMO) channels in the case of covariance feedback. The condition for beamforming to be optimal is studied in [9]–[11]. In this paper, we study the optimal transmission strategies in terms of minimum outage probability for the case of mean feedback. In particular, we prove that the optimal transmission directions in this case are the same as in maximizing the ergodic capacity. We also provide some supplemental results on minimum outage probability transmission in the case of covariance feedback other than those presented in [10]. Note that for fast fading, it may not be meaningful in the covariance feedback case to consider channel outage capacity as defined in [1] and in this paper when a codeword may experience different channel states. However, our results for this case can be used to illustrate how knowledge of at the transmitter can affect the outage capacity of a block fading channel. A recent paper [12] also studied the problem dealt with in this paper. Our work was done independently and apparently concurrently with [12]. It was first disseminated to others in a report to Motorola [13]. Compared to the results of [12], our proofs and analysis follow a substantially different approach, in addition to
1536-1276/$20.00 © 2005 IEEE
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
providing some new results. For example, we show that the optimal power allocation schemes for both mean feedback and covariance feedback are a function of the target outage probability, both by using majority theory and by observing the form of the for outage probability ((10)). We present an example for the covariance feedback case in which we show that even when , the optimal power althe channel is uncorrelated, i.e., location may not be a uniform distribution among orthogonal directions. As shown in Section III, the optimal power allocation scheme in the covariance feedback case could allocate all the power to only one of the orthogonal directions (outage probability may be greater than 0.5 in this case). This is only shown by simulation in [12] and it is not clearly stated. Finally, we note that the proof of a main result regarding the optimal transmission directions in the mean feedback case presented in [12, Appendix II] is incorrect. The error occurs from (65) to (66) in , where and are which step the authors assume (dummy) double integration variables. However, the region of obviously does not guarantee , integration which makes the proof invalid. The following notation will be used throughout the paper: superscript denotes matrix/vector conjugate transpose, is the trace operator, is the th element of vector , and is the identity matrix. is the expectation operator and will also be used to denote the mean of .
. Letting , we have problem can then be expressed as
1085
. The (5) (6)
subject to
(7)
and
(8)
are independent circularly symmetric where is noncomplex Gaussian random variables. Therefore, central Chi-square distributed with two degrees of freedom and . To minimize , we need to a noncentrality parameter and for , subfind the optimal values of ject to their individual constraints. We first fix the power allocation to each transmit antenna by fixing the and consider the optimization with respect to . The following Lemma helps provide a solution. Lemma 1: Given two real and independent random variables , , and , then
II. MEAN FEEDBACK Consider the discrete model of a multiple-input single-output (MISO) fading channel [5], (1) where is circularly symmetric complex Gaussian noise with variance per dimension. Let denote the number of transmit antennas. is an complex and deGaussian channel vector modeled as notes the complex transmit signal vector. Define . For a given , since the optimum input distribution of that maximizes the conditional mutual information for any fixed channel realization is zero mean complex Gaussian, the problem of minimal outage transmission can be formulated as
(2) (3) subject to
subject to is attained at , for any . The proof is given in the Appendix. Based on this lemma, we have the following proposition. Proposition 1: Without loss of generality, assume for . Then, the optimal solution of the problem in , for all . (6) is achieved by Proof: Since the distribution of is not dependent on , without loss of generality, for all , where we can let and are two independent real Gaussian random variables. Therefore, the optimization over can be equivalently transferred to the optimization . over Denote the optimal solution as and the associated minimum outage probability as . As, such that . Consume there exists some , and struct . Let the cumulative distribution function (CDF) of and the probability density function (PDF) of be and , respectively. Applying Lemma 1 , we have the following inequality: to
(4)
where ; and denote the target rate and available power, respectively. Note that here we only consider short-term power control [2] so that is not a is positive semi-definite, we have function of time. Since the eigenvalue decomposition (EVD) , where is a diagonal matrix with indicating the power allocated to transmission directions indicated by the corresponding column vectors of unitary matrix
(9) where tion solution of
is the CDF associated with another solu, in which , , and for all , . Since the achieves lower outage probability than ,
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is not optimal, which contradicts our assumption. Thus, for any . we have When there are more than only which have equal largest value, it can be easily shown that the choice of can be arbitrary within this set, as long as their sum is equal to . Therefore, is still optimal, which indicates that allocating to a single , where is a complex the optimal choice of is scalar such that . Without loss of generality, let . Since , we have , is an arbitrary set of orwhere thonormal vectors that are orthogonal to . Therefore, the optimal transmission directions to minimize outage probability are the vector channel mean and its orthogonal directions, which are the same as in the case of maximizing the ergodic capacity [5]. is obtained, we still need to determine the optimal After in the different transmission directions. power allocation Here, we will briefly analytically study this problem and will follow up with numerical results in Section IV. Let in (6). Given the optimal as above, is noncentral -disand tributed with two degrees of freedom, mean . The , , are each a noncentrality parameter central -distributed with two degrees of freedom and mean . Therefore, the outage probability can be evaluated (after some simple change of variables) as
. Therefore, the s smaller reduces beamforming, which concentrates power in the direction of the channel mean might be optimal for larger outage probability. as a function of is relatively hard to evaluate due to its dependency on . So, the previous analysis is only approximate, but well matches the numerical results to be presented later. III. COVARIANCE FEEDBACK Given a fixed power allocation scheme, transmitting along the eigenvectors of the channel covariance matrix is proven to be necessary and sufficient to achieve minimum outage probability in the covariance feedback case [10]. Note that this is the same strategy as maximizing the ergodic capacity, as shown in [5]. Here, to make our presentation complete, we provide a slightly different, but much more concise proof using an inequality from [8]. Proposition 2: Let , the EVD of be 1. Choosing , and , is a diagonal matrix, is necessary and sufficient for where minimizing the outage probability. . Letting , Proof: We first assume we need to show that is diagonal in order to minimize outage probability. As in the last section, minimizing the outage probability at a given rate is equivalent to minimizing (11) (12) subject to
(13)
so that is white with distribution of . Let with EVD . Then, the minimum outage probability problem is rewritten as
where (10) where
(14) (15)
subject to Letting
, we can further transform (14) to
for and is the zeroth-order modified Bessel function of the as the feedback first kind. We define the ratio SNR SNR. We see that the minimum outage probability is strongly a function of SNR , which is a measure of the accuracy of CSIT. When is sufficiently small, and are approximately 1 and the outage probability is mainly affected by , which is minimized by making the equal. Therefore, for sufficiently small (associated with a very small outage probability and and noise a low target rate for given transmission power covariance ), the optimal power allocation tends to spread power over different transmission directions. On the contrary, could increase very as increases, rapidly as increases. Making larger, and, thus, the rest of
(16) It is shown in [8] that (17) Comparing (14) and (16), since and have the same distribution, the minimum outage probability achieved by the opwith at timal can always be achieved by a diagonal matrix least the same or even more stringent power constraint. Therefore, choosing as diagonal is sufficient to minimize outage probability. Equivalently, this means is diagonal since . Necessity is easily proven using the fact that the is diagonal. first equality in (17) is satisfied if and only if , because any set of orthonormal For the trivial case of 1If tr[6 6] = N , we can always normalize the channel by absorbing into the transmit power P
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
Fig. 1. Achievable minimum outage probability versus target rate of mean-feedback for N = 2. Channel received SNR SNR (= ). at 8 dB. SNR
vectors can be used as beamforming directions, the theorem is easily proved. Next, it remains to determine the optimal power allocation in different transmission directions, i.e., the di. Let agonal entries of . The optimal power allocation problem is defined as
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((P ( + )) = ) is fixed
Now, we consider two special cases for the problem in (18). . Then, Case 1:
(20)
(18)
subject to
and
(21)
(19)
where is exponentially distributed. Although tedious, in this case can be computed in a closed form expression. Therefore, we could solve the constrained optimization problem of (18) numerically. However, the optimization problem is not . convex, which makes it difficult to find the global optimal Here, we coarsely identify the conditions when the optimum strategy is to spread power over different directions and when to concentrate all the power in a single direction using majority theory [7], to which we give a brief introduction next. Majorization: Given two real positive vectors , having equal summation of all entries. is majorized by , de, if the sum of the smallest entries of is greater noted as than or equal to the same sum for for all . This is a mathematical description of the vague concept of is “less spread out” than . Schur-convexity: A real-valued function defined on a set of is Schur-convex, if . is . Schur-convexity Schur-concave, if and Schur-concavity can be viewed as extensions of the increase or decrease functions defined on .
In particular, when , we can show that is a Schur, and Schur-concave function when convex function when where the constant is a solution . Therefore, if to the equation , we need to spread power equally on two orthogonal transmission directions to minimize outage probability; otherwise, we should concentrate on a single transmission direction to minimize outage. , and . Case 2: for some fixed power allocation Consider optimizing , on the remaining directions
(22) . As a function of for all , is defined symmetrically for all . Since is a Schur-concave function for vector if and a Schur-convex function if , we is Schur-convex for can conclude that some small enough or Schur-concave for some large enough where
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Fig. 2. Effect of SNR on achievable minimum outage probability of mean-feedback for N = 2. Channel received SNR SNR (= ). fixed at 8 dB. SNR
((P ( + )) = ) is
Fig. 3. Optimal power allocation over different transmission directions of mean-feedback for N = 2. Channel received SNR SNR fixed at 8 dB. SNR (= ).
((P ( + )) = ) is
according to [7, Th. 3.A.5]2. Therefore, for a given channel , the optimal power allocation received SNR SNR scheme tends to spread power when target rate is low, and concentrate power when is large. However, in the latter case, the outage probability could be very large (say 0.5), and not of practical interest. So, spreading power over different trans-
mission directions is not necessarily optimum for minimizing . Through numerical results, we outage probability when , the optimal power allocation can also show that when schemes still tend to spread power when outage probability is low and concentrate power otherwise.
2Let f (x) be symmetric in each element of x, where x = [x ; x ; . . . x ]. Theorem 3.A.5 states that to prove f (x) is Schur-convex in x, it is sufficient to show f (x) is Schur-convex in a vector composed of any two elements of x by fixing the other elements.
IV. NUMERICAL RESULTS Figs. 1 and 2 plot the achievable minimum outage probability of mean-feedback for different rates and SNR , for a fixed
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
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Fig. 4. Minimum outage probability achievable by optimal and equal power allocation over the directions indicated by the two eigenvectors of covariance matrix and satisfy + = 2. Channel received SNR SNR (P= ) is fixed at 8 dB.
6. Note that the two eigenvalues
average received SNR, SNR dB. We can see outage probability decreases exponentially as the target rate decreases. SNR affects outage capacity significantly, especially at low outage probability. Note here that dB is about the capacity of an AWGN channel at SNR b/channel-use. To achieve 1% outage, the dB required SNR for 1.15 b/channel-use (40% of dB ) and for 1.72 b/channel-use (60% of ) is about 9 and 12 dB, dB respectively. Fig. 3 compares the optimal power allocation over different transmission directions for mean-feedback. When “good” channel feedback is available, the optimal solution tends to beamform to the direction indicated by the mean with all available power. On the contrary, “bad” channel feedback may require multiple beams to be transmitted. Moreover, at higher rate or larger outage probability, beamforming to the direction indicated by the channel mean with all power is optimal. Fig. 4 shows the minimum outage probability achievable by covariance feedback with the optimal power and the equal . Let the EVD of be power allocation schemes for and and be the two diagonal elements of . Even when , which indicates high spatial correlation between transmit antennas, optimal power allocation cannot significantly reduce outage probability compared to the scheme in which power is uniformly allocated to both orthogonal directions. V. CONCLUSION We have studied the problem of minimum outage probability transmission for a MISO fading channel in the cases of mean feedback and covariance feedback. In the case of mean feedback, the optimal transmission strategy is proven to be transmitting several independent data streams in the direction of the channel mean vector and its orthogonal directions.
When SNR is high, the optimal strategy tends to beamform to the direction indicated by the channel mean. The quality of the channel information, measured by SNR affects the outage probability significantly. For both mean and covariance feedback, we show that the optimum power allocation scheme which minimizes outage probability is closely related to the target rate. It is more desirable to spread the power over all transmission directions than beamforming to a single direction for sufficiently small target rates. APPENDIX Proof: Parameterize , with
and , as , and let
,
Thus
(23) We convert to polar coordinates . Then, takes the form
For
, let
,
,
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The integral in (23) transforms to
We have
(24) Note (25), located at the bottom of the page, where
(28) Decomposing the inner integral in (24) along the partition of in (25) and using the symmetry properties of the functions, we obtain
and
where (29) By (26), (28), and (29)
(26) (30)
Therefore, by (24) Therefore, by (30)
(27) The first integral in (27) is independent of
Hence, in order to establish that , it is sufficient to show that
and is equal to
for all
This completes the proof.
if if
, ,
(25)
XIE et al.: MINIMUM OUTAGE PROBABILITY TRANSMISSION WITH IMPERFECT FEEDBACK FOR MISO FADING CHANNELS
ACKNOWLEDGMENT The authors would like to thank Motorola for support of this research and in particular Dr. K. Rohani of Motorola Labs for many helpful discussions on the topic. REFERENCES [1] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998. [2] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1468–1489, Jul. 1999. [3] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of blockfading channels with multiple antennas,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1273–1289, May 2001. [4] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple antenna data transmission over fading channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998. [5] E. Vistotsky and U. Madhow, “Space-time transmit precoding with imperfect feedback,” IEEE Trans. Inf. Theory, vol. 47, no. 6, pp. 2632–2639, Sep. 2001. [6] G. G. Raleigh and V. K. Jones, “Adaptive antenna transmission for frequency duplex digital wireless communications,” in Proc.IEEE Int. Communications Conf., vol. 2, 1997, pp. 641–646. [7] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. San Diego, CA: Academic, 1979. [8] S. A. Jafar, S. Vishwanath, and A. J. Goldsmith, “Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback,” in Proc. IEEE Int. Communications Conf., vol. 7, 2001, pp. 2266–2270. [9] S. A. Jafar and A. J. Goldsmith, “On optimality of beamforming for multiple antenna systems with imperfect feedback,” in Proc. IEEE Int. Symp. Information Theory, 2001, pp. 321–321. [10] S. H. Simon and A. L. Moustakas, “Optimizing MIMO antenna systems with channel covariance feedback,” Bell Labs. Tech. Memo., 2002. [11] E. Jorswieck and H. Boche, “On transmit diversity with imperfect channel state information,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, 2002, pp. 2181–2184. [12] A. L. Moustakas and S. H. Simon, “Optimizing multi-transmitter-singlereceiver (MISO) communication systems with general Gaussian channels: Nontrivial covariance and nonzero mean,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2770–2780, Oct. 2003. [13] Y. Xie and C. N. Georghiades, “Space-time minimum outage probability transmission with imperfect feedback,” Motorola Univ. Partnership Res. (UPR) Project, 2002.
Yongzhe Xie received the B.E. degree from the Department of Biomedical Engineering, Shanghai Jiaotong University, Shanghai, China, in 1996, the M.E. degree from the Department of Electrical Engineering, the National University of Singapore, in 1999, and the Ph.D. degree from the Department of Electrical Engineering, Texas A&M University, College Station, in 2004. His current research interests are in the general area of information theory, signal processing for communications, and statistical signal processing for financial applications. Mr. Xie was the recipient of the Texas Telecommunications Engineering Consortium (TxTEC) Fellowship of 1999, NOKIA Wireless Future Intern Award of 2001, and the China Instrument Scholarship of 1995.
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Costas N. Georghiades (M’85–SM’90–F’98) eceived the B.E. degree with distinction from the American University of Beirut, Beirut, Lebanon, in 1980, and the M.S. and D.Sc. degrees from Washington University, St. Louis, MO, in 1983 and 1985, respectively, all in electrical engineering. Since September 1985, he has been with the Electrical Engineering Department, Texas A&M University, College Station, where he is currently a Professor and holder of the Delbert A. Whitaker Endowed Chair. He has been involved in organizing a number of conferences, including as Technical Program Chair for the 1999 IEEE Vehicular Technology Conference and the 2001 Communication Theory Workshop and as Chair of the Communication Theory Symposium within Globecom 2001. He currently serves as General Co-Chair for the 2004 IEEE Information Theory Workshop, as Technical Program co-Chair for the 2005 Communication Theory Workshop and as Guest Editor for the EURASIP Journal on Wireless Communications and Networking. His research interests include the application of information, communication, and estimation theories to the study of communication systems. Dr. Georghiades he served in editorial positions with the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON INFORMATION THEORY, the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and the IEEE COMMUNICATIONS LETTERS. He is a Registered Professional Engineer in Texas.
Ari Arapostathis (M’82–SM’91) received the B.S. degree from the Massachusetts Instistute of Technology, Cambridge, and the Ph.D. degree from the University of California at Berkeley. He has been a Faculty Member with the University of Texas at Austin since 1982. He has served as the Vice Chairman for Invited Sessions of the Program Committee for the 1987 American Control Conference. His research interests include analysis and estimation techniques for stochastic systems, the application of differential geometric methods to the design and analysis of control systems, stability properties of large-scale interconnected power systems, and stochastic and adaptive control theory. Among his main technical contributions are in the areas of adaptive control and estimation of stochastic systems with partial observations, adaptive control of nonlinear systems, geometric nonlinear theory, and stability of large-scale interconnected power systems. Dr. Arapostathis was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. He is a member of the AMS and SIAM.
