What is the Value of Joint Processing of Pilots Data in Block-Fading Channels? Nihar Jindal University of Minnesota Minneapolis, MN 55455 Email: nihar@umn.edu Angel Lozano Universitat Pompeu Fabra Barcelona 08003, Spain Email: angel.lozano@upf.edu homas L. Marzetta Bell Labs Alcatel-Lucent Murray Hill, NJ 07974, USA Email: tlm@research.bell-labs.com Abstract he spectral efficiency achievable with joint processing of pilot data symbol observations is compared with that achievable through the conventional separate approach of first estimating the channel on the basis of the pilot symbols alone, subsequently detecting the data symbols. Studied on the basis of a mutual information lower bound, joint processing is found to provide a non-negligible advantage relative to separate processing, particularly for fast fading. It is shown that, regardless of the fading rate, only a very small number of pilot symbols at most one per transmit antenna per channel coherence interval should be transmitted if joint processing is allowed. I. INRODUCION Pilot symbols a.k.a. training or reference symbols are an inherent part of virtually every wireless system. Motivated by this prevalence, the spectral efficiency achievable when coherently detecting data with the assistance of pilots has been the object of much analysis e.g., [] [5]. A large fraction of such work has focused on the spectral efficiency achievable with Gaussian inputs under the assumption that the fading channel is estimated on the basis of the pilot observations then, using such estimate as it were the true channel, the data is detected. Although suboptimal, such separate processing reflects the operating conditions of existing systems. In this paper, we move beyond this approach quantify the advantage of jointly processing pilot data observations when Gaussian codebooks are utilized. Since the general mutual information expression is intractable, we rely on lower bounds to the achievable spectral efficiency. hese bounds allow assessing the optimum number of pilot symbols under such joint processing, also quantify the minimum improvement in spectral efficiency that joint processing brings about relative to separate processing. Although there has been prior work on receiver design for joint processing e.g., [6]-[8], to the best of our knowledge there is not yet a general understing of the conditions in terms of signal-to-noise ratio, fading rate, antenna configurations in which joint processing provides a substantial improvement. Given that joint processing is more complex than separate processing, such a quantification appears very useful. As a starting point, a simple block-fading ergodic channel model is considered. Section II restricts itself to scalar channels, from which many of the insights can already be derived. he generalization to MIMO multiple-input multiple-output follows in Section III. A. Channel Model II. SISO Let H represent a discrete-time scalar fading channel. Under block Rayleigh-fading, the channel is drawn from a zero-mean complex Gaussian distribution at the beginning of each block it then remains constant for the symbols composing the block, where corresponds to the coherence time/bwidth. his process is repeated for every block in an IID independent identically distributed fashion. A total of τ pilot symbols are inserted within each block leaving τ symbols available for data. During the transmission of pilot symbols, y p = H + n p where the received signal, y p, the noise, n p, are τ- dimensional vectors. he entries of n p are IID zero-mean unit-variance complex Gaussian. he channel satisfies E[ H ] = thus indicates the average signalto-noise ratio. During the transmission of data symbols y d = Hx + n d where y d, n d, the transmitted data x, are all τ- dimensional. he noise n d is independent of n p but it abides by the same distribution. As argued in the Introduction, the entries of x are IID zero-mean unitvariance complex Gaussian. Each transmitted codeword spans a large number of fading blocks, which endows ergodic quantities with operational meaning. B. Perfect CSI If the receiver is provided with perfect CSI channelstate information, Gaussian codebooks are capacityachieving the ergodic capacity, in bits/s/hz, equals C = E [ log + H ] 3 = e / E log e 4
where E k is the exponential integral of order k. For compactness, C is often abbreviated as C. C. Separated Processing of Pilots Data If the receiver uses the pilot observations, y p, to first produce an MMSE estimate of the channel, Ĥ, then performs nearest-neighbor decoding while treating Ĥ as if it were H, the maximum spectral efficiency is [5] with I S = max τ: τ< eff = { τ C eff MMSE + MMSE MMSE = E[ H Ĥ ] = / + τ. he maximization in 5 must be computed numerically as no closed form exists. D. Spectral Efficiency Lower Bounds for Joint Processing In the general case, the receiver decodes the data based upon y p y d without any constraints on how these observations are used. he per-symbol mutual information Ix; y p, y d / is the maximum achievable spectral efficiency is achieved by a maximum-likelihood decoder based on the true channel description py p, y d x. Since the expression for this mutual information is intractable, we instead utilize the following lower bound. heorem he ergodic spectral efficiency in bits/s/hz when τ pilot symbols τ complex Gaussian data symbols are transmitted on every fading block jointly processed at the receiver satisfies where I J = } 5 6 Ix; y p, y d I J I J 7 τ C log e I J = Proof: See Appendix A. τ e τ+/ k= E k τ + 8 τ C + log. 9 + τ he bound I J or, more precisely, its MIMO form given in Section III was first derived in [4]. However, it was not given as in 8 but rather left as an expectation over the distribution of x. As shown in the Appendix, where we provide an alternative derivation, this expectation can be expressed in closed form using the results of [9]. When no pilots are transmitted τ = 0, I J reduces to the bound given for data-only transmission in [0]. E. Optimization of Number of Pilot Symbols An initial assessment of the optimum number of pilot symbols can be made on the basis of I J, whose maximization w.r.t. τ reduces to maximizing the concave function log + τ τ C. By relaxing τ to a continuous value, the optimum number of pilots is τ = log e C 0 which satisfies 0 τ. his points to τ being, when restricted to integers, either 0 or. Furthermore, C < log + by Jensen s implying τ =. In order to sharpen the above assessment, we turn to the tighter I J consider the low- high-power regimes separately. In the low-power regime, using C = log e + O 3 e τ+/ E k τ + = k + τ + O 3 it is found that maximizing I J to second order entails maximizing the concave function τ + τ. hus, the optimum is again either τ = 0 or τ =. While both values yield the same I J to second order, an exact computation of 8 reveals that τ = for 0. In the high-power regime, using e / E / = log γ log e + O e / E k / = k + O 3, k >, 4 where γ = 0.577... is the Euler-Mascheroni constant, it is found that I J τ=0 = I J τ= = C log e C log e k= k 5 e E k. 6 k= Since e E k < /k strictly, τ = is preferrable over τ = 0 for. For τ, I J falls rapidly. Altogether, the optimum number of pilots is τ = in both the low- high-power regimes. Setting τ = 0 results in a slight loss quantified in Section II-G, whereas τ is decidedly suboptimal at moderate/high. Extrapolating this result to more realistic continuousfading channels i.e., the channel varies from symbolto-symbol according to a rom process, we can infer that, with joint processing, it is desirable to have at most roughly one pilot symbol per coherence interval.
Spectral Efficiency b/s/hz 3 Perfect CSI Separate Joint = 0 db = 0 db Power Advantage db 3 = 0 db Asymptotic = 0 db 0 0 00 00 300 400 500 0 0 00 00 300 400 500 Fig.. Spectral Efficiency vs. for a SISO channel at = 0 db = 0 db. he curves correspond to C, I S I J with τ =. Fig.. Power advantage of joint relative to separate processing asymptotically at = 0 db = 0 db. F. Comparison with Separate Processing of Pilots Data he value of joint processing is illustrated by examining how the spectral efficiency converges to the perfect- CSI capacity as the blocklength increases. From 9, the difference between C I J is C I J = τ C + log log = O + + τ 7 8 for any fixed value of τ. On the other h, the difference between C the spectral efficiency achievable with separate processing, I S, vanishes only as O/ []. his contrast is evidenced in Fig.. With joint processing, as grows the spectral efficiency converges to C even though τ is fixed because the possibly implicit channel estimation process can take advantage of the data symbols. On the other h, if τ were kept fixed the spectral efficiency of the separate approach would not converge to C; I S converges to C only because τ is properly increased, as per 5, with. G. High-Power Behavior Further insight is obtained by studying the highpower behavior of the various bounds. At high, for τ =, the lower bounds converge absolutely to I J I J C e log e k= E k C log while, with separate processing [3], 9, 0 I S C. All the above quantities have the same pre-log factor, /, thus the difference between the terms inside the brackets directly gives the power penalty relative to the perfect-csi capacity, i.e., the horizontal shift in a plot of spectral efficiency vs. db. When the information units are bits, this horizontal shift is in 3-dB units []. he asymptotic difference between I J I J is log e log e E k, k= in 3-dB units. his quantity decreases with is minute even for small values of e.g., 0.0 db for = 0 thus, at high, we can consider the simpler I J with only a negligible loss in accuracy. Based on I J then, the asymptotic power advantage of joint processing relative to separate is log 3 in 3-dB units. In Fig., this quantity is plotted versus, along with the numerically computed advantage at = 0 db = 0 db. he difference between the respective curves indicates that the convergence of I S to its asymptote occurs ever more slowly as grows. Using I J 3, it is also straightforward to compute the high-power advantage of transmitting one pilot symbol τ = rather than none τ = 0 as γ log e 4 in 3-dB units. For short blocks the single pilot is useful, but for larger blocklengths it makes little difference. Finally, we can also quantify the distance to the true capacity of the block-fading channel. In [3], such capacity indicated by C to distinguish it from C, the capacity with perfect CSI is shown to converge, for, to C C e log!. 5
Using Stirling s approximation, C C log 6 for large, coinciding with the high- expansion of I J save for the factor /. his indicates that the spectral efficiency with joint processing scales with the blocklength in the same manner as the true capacity in the high-power regime. Furthermore, the power offset between I J the true capacity is only approximately log 7 in 3-dB units. his evaluates, for instance, to 0.55 db 0. db for = 0 = 00, respectively. A. Channel Model III. GENERALIZAION O MIMO With n transmit n R receive antennas, the SISO input-output relationships in become Y p = HP + N p 8 n Y d = HX + N d 9 n where H, P, X, N p N d are, respectively, n R n, n τ, n τ, n R τ n R τ. Matrices H, X, N p N d have IID zero-mean unit-variance complex Gaussian entries while P must satisfy power constraint r{pp } n τ. B. Perfect CSI For notational convenience, define C t,r as the function [ C t,r ρ = E log det I + ρ t ZZ ] 30 where Z is an r t matrix with IID zero-mean unitvariance complex Gaussian entries. he MIMO perfect- CSI capacity with n transmit n R receive antennas at equals C n,n R. C. Separated Processing of Pilots Data he SISO expressions for I S in Section II-C apply verbatim with, τ, C replaced, respectively, by /n, τ = τ/n, C n,n R. D. Spectral Efficiency Lower Bounds for Joint Processing In the MIMO case, we allow for the possibility of either no pilot symbols τ = 0 or of at least one pilot symbol per antenna τ n. heorem Let τ = 0 or τ n. he ergodic spectral efficiency in bits/s/hz when τ pilot symbols τ complex Gaussian data symbols are transmitted on every fading block jointly processed at the receiver satisfies IX; Y p, Y d I J I J 3 where I J = τ C n,n R n R C n, τ + n τ 3 I J = τ C n,n R n n + R log n + τ n 33 Proof: See Appendix B. As a by-product of the proof, we show that I J is maximized when the pilot matrix P satisfies P P = τi 34 which coincides with the optimality condition derived in [3] for the case of separate processing. Henceforth, we shall focus on the case n = n R. Corollary If n = n R = n, then I J n = τ/n Cn,n + /n n /n log /n + τ/n 35 which coincides with its SISO counterpart in 9 only with an effective fading blocklength of /n, an effective number of pilot symbols of τ/n, C replaced by C n,n /n. E. Optimization of Number of Pilot Symbols In the low-power regime, the number of pilot symbols can be optimized on the basis of I J. Using C t,r ρ = r log e ρ t + r ρ + Oρ 3 36 t it is found that maximizing I J to second order requires maximizing the concave function τ + τ n R. his implies that either τ = 0 or τ = n is optimal, the two are indistinguishable to second order. Drawing parallels with its SISO counterpart, the maximization of I J w.r.t. to τ is equivalent to the maximization of log + τ τ C n,n /n w.r.t τ = τ/n. Hence, τ = log e C n,n /n 37 if τ is relaxed to continuous values. his quantity is below unity whenever C n,n /n log e, which implies that the optimum number of pilots is either 0 or n. Since C n,n /n log +, τ = n is preferred over τ = 0. F. High-Power Behavior Because I J I S mirror their SISO counterparts, the asymptotic power advantage in 3-dB units of joint relative to separate processing for MIMO is the SISO advantage for an effective blocklength of /n, i.e., log /n /n 38
APPENDIX A By the chain rule, the mutual information with perfect receiver knowledge of H exps as Ix; y p, y d, H = Ix; y p, y d + Ix; H y p, y d. hus, Ix; y p, y d = Ix; y p, y d, H Ix; H y p, y d 39 = Ix; y p, y d, H hh y p, y d +hh y p, y d, x 40 Ix; y p, y d, H hh y p +hh y p, y d, x 4 where h denotes differential entropy 4 holds because conditioning reduces entropy. he signal-to-noise ratio when estimating H on the basis of y p is τ. hus, H y p is conditionally Gaussian with variance / + τ therefore hh y p = log πe log + τ. 4 In turn, the signal-to-noise ratio when estimating H on the basis of y p, y d, conditioned on x d, is τ + τ k= x k thus ] τ hh y p, y d, x = E [log + τ + x k k= + log πe. 43 Using Ix; y p, y d, H = τ C, plugging 4 43 into 4, scaling all the terms by /, [ I J = τ C ] τ E log + k= x k. + τ 44 A closed form for the expectation in 44 is given in [9], leading directly to 8. he subsequent lower bound, I J, follows from application of Jensen s inequality to 44. Since E[ x k ] =, ] τ E [log + k= x k τ log + τ + + τ 45 APPENDIX B Starting at 4, we need only compute hh y p hh y p, y d, x. Because the n R antennas are decoupled when conditioned on either y p or y p, y d, x, these terms can be evaluated separately for each receive antenna. From [3], the covariances of one row of H conditioned on y p on y p, y d, x, respectively, are K H y p = K H y p,y d,x = I + PP 46 n. 47 I + n PP + XX Defining = hh y p hh y p, y d, x, we have [ ] = n R E log detk H y p,y d,x n R log detk H y p 48 [ ] = n R E log det I + I + PP XX 49 n n o obtain I J we must find the pilot sequence P that minimizes 49. his amounts to choosing the worstcase noise covariance when the input the channel are both spatially white. Since the distribution of X is rotationally invariant, we need only consider diagonal forms for PP. o show that the best choice is PP = τi, we apply the argument in [4, Sec. 4.] to the function in 49, which is convex w.r.t. PP. With PP = τi, [ ] = n R E = n R C n, τ n log det I + + τ XX n + τ n 50. 5 I J is reached by applying Jensen s inequality to 50. REFERENCES [] M. Medard, he effect upon channel capacity in wireless communications of perfect imperfect knowledge of the channel, IEEE rans. Inform. heory, vol. 46, no. 3, pp. 933 946, May 000. [] L. Zheng D. N. C. se, Communication on the Grassman manifold: A geometric approach to the non-coherent multipleantenna channel, IEEE rans. Inform. heory, vol. 48, no., pp. 359 383, Feb. 00. [3] B. Hassibi B. M. Hochwald, How much training is needed in multiple-antenna wireless links? IEEE rans. Inform. heory, vol. 49, no. 4, pp. 95 963, Apr. 003. [4] S. Furrer D. Dahlhaus, Multiple-antenna signaling over fading channels with estimated channel state information: Capacity analysis, IEEE rans. Inform. heory, vol. 53, no. 6, pp. 08 043, Jun. 007. [5] A. Lapidoth S. Shamai, Fading channels: How perfect need perfect side information be? IEEE rans. Inform. heory, vol. 48, no. 5, pp. 8 34, May 00. [6] L. ong, B. M. Sadler, M. Dong, Pilot-assisted wireless transmissions: general model, design criteria, signal processing, IEEE Signal Proc. Magazine, vol., no. 6, pp. 5, Nov. 004. [7] W. Zhang J. N. Laneman, How good is phase-shift keying for peak-limited Rayleigh fading channels in the low- regime? IEEE rans. Inform. heory, vol. 53, no., pp. 36 5, Jan. 007. [8]. Li O. Collins, A successive decoding strategy for channels with memory, Proc. of ISI, Sep. 005. [9] H. Shin J. H. Lee, Capacity of multiple-antenna fading channels: Spatial fading correlation, double scattering keyhole, IEEE rans. Inform. heory, vol. 49, pp. 636 647, Oct. 003. [0] M. Godavarti,. L. Marzetta, S. Shamai, Capacity of a mobile multiple-antenna wireless link with isotropically rom Rician fading, IEEE rans. Inform. heory, vol. 49, no., pp. 3330 3334, Dec. 003. [] N. Jindal A. Lozano, Optimum pilot overhead in wireless communication: A unified treatment of continuous blockfading channels, Submitted to IEEE rans. Wireless Comm., 009. [] A. Lozano, A. M. ulino, S. Verdu, High- power offset in multiantenna communication, IEEE rans. Inform. heory, vol. 5, no., pp. 434 45, Dec. 005. [3] B. H. Hochwald. L. Marzetta, Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading, IEEE rans. Inform. heory, vol. 46, Mar. 000. [4] I. E. elatar, Capacity of multi-antenna Gaussian channels, Eur. rans. elecom, vol. 0, pp. 585 595, Nov. 999.