Nearest Neighbor Decoding and Pilot-Aided Channel Estimation for Fading Channels

We study the information rates of noncoherent, stationary, Gaussian, and multiple-input multiple-output (MIMO) flat-fading channels that are achievable with nearest neighbor decoding and pilot-aided channel estimation. In particular, we investigate the behavior of these achievable rates in the limit as the signal-to-noise ratio (SNR) tends to infinity by analyzing the capacity pre-log, which is defined as the limiting ratio of the capacity to the logarithm of the SNR as the SNR tends to infinity. We demonstrate that a scheme estimating the channel using pilot symbols and detecting the message using nearest neighbor decoding (while assuming that the channel estimation is perfect) essentially achieves the capacity pre-log of noncoherent multiple-input single-output flat-fading channels, and it essentially achieves the best so far known lower bound on the capacity pre-log of noncoherent MIMO flat-fading channels. Extending the analysis to fading multiple-access channels reveals interesting relationships between the number of antennas and Doppler bandwidth in the comparative performance of joint transmission and time division multiple-access.


I. INTRODUCTION
The capacity of coherent multiple-input multiple-output (MIMO) channels increases with the signal-to-noise ratio (SNR) as min(n t , n r ) log SNR, where n t and n r are the number of transmit and receive antennas, respectively, and SNR denotes the SNR per receive antenna [1], [2].
The growth factor min(n t , n r ) is sometimes referred to as the capacity pre-log [3] or spatial multiplexing gain [4].This capacity growth can be achieved using a nearest neighbor decoder which selects the codeword that is closest (in a Euclidean distance sense) to the channel output.
In fact, for coherent fading channels with additive Gaussian noise, this decoder is the maximumlikelihood decoder and is therefore optimal in the sense that it minimizes the error probability (see [5] and references therein).The coherent channel model assumes that there is a genie that provides the fading coefficients to the decoder; this assumption is difficult to achieve in practice.
In this paper, we replace the role of the genie by a scheme that estimates the fading via pilot symbols.This can be viewed as a particular coding strategy over a non-coherent fading channel, i.e., a channel where both communication ends do not have access to fading coefficients but may be aware of the fading statistics.Note that with imperfect fading estimation, the nearest neighbor decoder that treats the fading estimate as if it were perfect is not necessarily optimal.
Nevertheless, we show that, in some cases, nearest neighbor decoding with pilot-aided channel estimation achieves the capacity pre-log of non-coherent fading channels.(The capacity pre-log is defined as the limiting ratio of the capacity to the logarithm of the SNR as the SNR tends to infinity.) The capacity of non-coherent fading channels has been studied in a number of works.Building upon [6], Hassibi and Hochwald [7] studied the capacity of the block-fading channel and used fading channels was derived by Etkin and Tse in [10].
Lapidoth and Shamai [11] and Weingarten et.al. [12] studied non-coherent stationary fading channels from a mismatched-decoding perspective.In particular, they studied achievable rates with Gaussian codebooks and nearest neighbor decoding.In both works, it is assumed that there is a genie that provides imperfect estimates of the fading coefficients.
In this work, we add the estimation of the fading coefficients to the analysis.In particular, we study a communication system where the transmitter emits pilot symbols at regular intervals, and where the receiver separately performs channel estimation and data detection.Specifically, based on the channel outputs corresponding to pilot transmissions, the channel estimator produces estimates of the fading for the remaining time instants using a linear minimum mean-square error (LMMSE) interpolator.Using these estimates, the data detector employs a nearest neighbor decoder that detects the transmitted message.We study the achievable rates of this communication scheme at high SNR.In particular, we study the pre-log for fading processes with bandlimited power spectral densities.(The pre-log is defined as the limiting ratio of the achievable rate to the logarithm of the SNR as the SNR tends to infinity.)For SISO fading channels, using some simplifying arguments, Lozano [13] and Jindal and Lozano [8] showed that this scheme achieves the capacity pre-log.In this paper, we prove this result without any simplifying assumptions and extend it to MIMO fading channels.We show that the maximum rate pre-log with nearest neighbor decoding and pilot-aided channel estimation is given by the capacity pre-log of the coherent fading channel min(n t , n r ) times the fraction of time used for the transmission of data.Hence, the loss with respect to the coherent case is solely due to the transmission of pilots used to obtain accurate fading estimates.If the inverse of twice the bandwidth of the fading process is an integer, then for MISO channels, the above scheme achieves the capacity pre-log derived by Koch and Lapidoth [9].For MIMO channels, the above scheme achieves the best so far known lower bound on the capacity pre-log obtained in [10].
The rest of the paper is organized as follows.Section II describes the channel model and introduces our transmission scheme along with nearest neighbor decoding and pilots for channel estimation.Section III defines the pre-log and presents the main result.Section IV extends the use of our scheme to a fading multiple-access channel (MAC).Sections V and VI provide the proofs of our main results.Section VII summarizes the results and concludes the paper.December 11, 2013 DRAFT

II. SYSTEM MODEL AND TRANSMISSION SCHEME
We consider a discrete-time MIMO flat-fading channel with n t transmit antennas and n r receive antennas.Thus, the channel output at time instant k ∈ (where denotes the set of integers) is the complex-valued n r -dimensional random vector given by Here x k ∈ ¼ nt denotes the time-k channel input vector (with ¼ denoting the set of complex numbers), H k denotes the (n r ×n t )-dimensional random fading matrix at time k, and Z k denotes the n r -variate random additive noise vector at time k.
The noise process {Z k , k ∈ } is a sequence of independent and identically distributed (i.i.d.) complex-Gaussian random vectors with zero mean and covariance matrix I nr , where I nr is the n r × n r identity matrix.SNR denotes the average SNR for each received antenna.
The fading process {H k , k ∈ } is stationary, ergodic and complex-Gaussian.We assume that the n r • n t processes {H k (r, t), k ∈ }, r = 1, . . ., n r , t = 1, . . ., n t are independent and have the same law, with each process having zero mean, unit variance, and power spectral density f H (λ), − 1 2 ≤ λ ≤ 1 2 .Thus, f H (•) is a non-negative (measurable) function satisfying where (•) * denotes complex conjugation, and where i √ −1.We further assume that the power spectral density f H (•) has bandwidth λ D < 1/2, i.e., f H (λ) = 0 for |λ| > λ D and f H (λ) > 0 otherwise.We finally assume that the fading process {H k , k ∈ } and the noise process {Z k , k ∈ } are independent and that their joint law does not depend on {x k , k ∈ }.
The transmission involves both codewords and pilots.The former conveys the message to be transmitted, and the latter are used to facilitate the estimation of the fading coefficients at the receiver.We denote a codeword conveying a message m, m ∈ M (where M = 1, . . ., ⌊e nR ⌋ is the set of possible messages, and where ⌊b⌋ denotes the largest integer smaller than or equal to b) at rate R by the length-n sequence of input vectors x1 (m), . . ., xn (m).The codeword is selected from the codebook C, which is drawn i.i.d.from an n t -variate complex-Gaussian Pilot Data No transmission distribution with zero mean and identity covariance matrix such that where • denotes the Euclidean norm.
To estimate the fading matrix, we transmit orthogonal pilot vectors.The pilot vector p t ∈ ¼ nt used to estimate the fading coefficients corresponding to the t-th transmit antenna is given by p t (t) = 1 and p t (t ′ ) = 0 for t ′ = t.For example, the first pilot vector is where (•) T denotes the transpose.To estimate the whole fading matrix, we thus need to send the n t pilot vectors p 1 , . . ., p nt .
The transmission scheme is as follows.Every L time instants (for some L ∈ , where is the set of all positive integers), we transmit the n t pilot vectors p 1 , . . ., p nt .Each codeword is then split up into blocks of L − n t data vectors, which will be transmitted after the n t pilot vectors.The process of transmitting L − n t data vectors and n t pilot vectors continues until all n data vectors are completed.Herein we assume that n is an integer multiple of L − n t . 1 Prior to transmitting the first data block, and after transmitting the last data block, we introduce a guard period of L(T − 1) time instants (for some T ∈ ), where we transmit every L time instants the n t pilot vectors p 1 , . . ., p nt , but we do not transmit data vectors in between.The guard period ensures that, at every time instant, we can employ a channel estimator that bases its estimation on the channel outputs corresponding to the T past and the T future pilot transmissions.This facilitates the analysis and does not incur any loss in terms of achievable rates.The above transmission scheme is illustrated in Fig. 1.The channel estimator is described in the following.
Note that the total block-length of the above transmission scheme (comprising data vectors, pilot vectors and guard period) is given by where n p denotes the number of channel uses reserved for pilot vectors, and where n g denotes the number of channel uses during the silent guard period, i.e., We now turn to the decoder.Let D denote the set of time indices where data vectors of a codeword are transmitted, and let P denote the set of time indices where pilots are transmitted.
The decoder consists of two parts: a channel estimator and a data detector.The channel estimator considers the channel output vectors Y k , k ∈ P corresponding to the past and future T pilot transmissions and estimates H k (r, t) using a linear interpolator, so the estimate where the coefficients a k ′ (r, t) are chosen in order to minimize the mean-squared error. 2ote that, since the pilot vectors transmit only from one antenna, the fading coefficients corresponding to all transmit and receive antennas (r, t) can be observed.Further note that, since the fading processes {H k (r, t), k ∈ }, r = 1, . . ., n r , t = 1, . . ., n t are independent, estimating H k (r, t) only based on {Y k (r), k ∈ } rather than on {Y k , k ∈ } incurs no loss in optimality.
Since the time-lags between H k , k ∈ D and the observations Y k ′ , k ′ ∈ P depend on k, it follows that the interpolation error is not stationary but cyclo-stationary with period L. It can be shown that, irrespective of r, the variance of the interpolation error tends to the following expression as T tends to infinity [14] ǫ 2 ℓ (t) lim where ℓ k mod L denotes the remainder of k/L.Here f L,ℓ (•) is given by and fH (•) is the periodic continuation of f H (•), i.e., it is the periodic function of period In this case, irrespective of ℓ and t, the variance of the interpolation error is given by which vanishes as the SNR tends to infinity.Recall that λ D denotes the bandwidth of f H (•).
Thus, (13) implies that no aliasing occurs as we undersample the fading process L times.Note that in contrast to (11), the variance in ( 15) is independent of the transmit antenna index t.See Section V-A for a more detailed discussion. where On the RHS of (17), assuming that the first pilot symbol is transmitted at time k = 0, we have defined as a set of time indices for a single codeword transmission.

III. THE PRE-LOG
We say that a rate is achievable if there exists a code with ⌊e nR ⌋ codewords such that the error probability tends to zero as the codeword length n tends to infinity.In this work, we study the set of rates that are achievable with nearest neighbor decoding and pilot-aided channel estimation.We focus on the achievable rates at high SNR.In particular, we are interested in the maximum achievable pre-log, defined as where R * (SNR) is the maximum achievable rate, maximized over all possible encoders.
The capacity pre-log-which is given by (20) but with R * (SNR) replaced by the capacity3 C(SNR)-of SISO fading channels was computed by Lapidoth [3] as where µ(•) denotes the Lebesgue measure on the interval [−1/2, 1/2].Koch and Lapidoth [9] extended this result to MISO fading channels and showed that if the fading processes {H k (t), k ∈ }, t = 1, . . ., n t are independent and have the same law, then the capacity pre-log of MISO fading channels is equal to the capacity pre-log of the SISO fading channel with fading process {H k (1), k ∈ }.Using (21), the capacity pre-log of MISO fading channels with bandlimited power spectral densities of bandwidth λ D can be evaluated as To the best of our knowledge, the capacity pre-log of MIMO fading channels is unknown.
For independent fading processes {H k (r, t), k ∈ }, t = 1, . . ., n t , r = 1, . . ., n r that have the same law, the best so far known lower bound on the MIMO pre-log is due to Etkin and Tse [10], and is given by For power spectral densities that are bandlimited to λ D , this becomes Observe that (24) specializes to (22) for n r = 1.It should be noted that the capacity pre-log for MISO and SISO fading channels was derived under a peak-power constraint on the channel inputs, whereas the lower bound on the capacity pre-log for MIMO fading channels was derived under an average-power constraint.Clearly, the capacity pre-log corresponding to a peak-power constraint can never be larger than the capacity pre-log corresponding to an average-power constraint.It is believed that the two pre-logs are in fact identical (see the conclusion in [3]).
In this paper, we show that a communication scheme that employs nearest neighbor decoding and pilot-aided channel estimation achieves the following pre-log.
Theorem 1.Consider the Gaussian MIMO flat-fading channel with n t transmit antennas and n r receive antennas (1).Then, the transmission and decoding scheme described in Section II achieves where L * = 1 2λ D .
Proof: See Section V.
Remark 1.We derive Theorem 1 for i.i.d.Gaussian codebooks, which satisfy the averagepower constraint (3).Nevertheless, it can be shown that Theorem 1 continues to hold when the channel inputs satisfy a peak-power constraint.More specifically, we show in Section V-C that a sufficient condition on the input distribution with power constraint E X 2 ≤ n t for achieving the pre-log is that its probability density function (p.d.f.) p X ( x) satisfies for some K satisfying The condition (26) is satisfied, for example, by truncated Gaussian inputs, for which the n t elements in X are independent and identically distributed and If 1/(2λ D ) is an integer, then (25) becomes Thus, in this case nearest neighbor decoding together with pilot-aided channel estimation achieves the capacity pre-log of MISO fading channels (22) as well as the lower bound on the capacity pre-log of MIMO fading channels (24).
Suppose that both the transmitter and the receiver use the same number of antennas, namely Then, as the codeword length tends to infinity, we have from ( 4)- (6) that the fraction of time consumed for the transmission of pilots is given by Consequently, from the achievable pre-log (25), namely we observe that the loss compared to the capacity pre-log of the coherent fading channel n t ′ = min(n t , n r ) is given by the fraction of time used for the transmission of pilots.From this we infer that the nearest neighbor decoder in combination with the channel estimator described in Section II is optimal at high SNR in the sense that it achieves the capacity pre-log of the coherent fading channel.This further implies that the achievable pre-log in Theorem 1 is the best pre-log that can be achieved by any scheme employing n t ′ pilot vectors.
To achieve the pre-log in Theorem 1, we assume that the training period L satisfies L ≤ 1 2λ D , in which case the variance of the interpolation error (15), namely vanishes as the inverse of the SNR.The achievable pre-log is then maximized by maximizing Note that as a criterion of "perfect side information" for nearest neighbor decoding in fading channels, Lapidoth and Shamai [11] suggested that the variance of the fading estimation error should be negligible compared to the reciprocal of the SNR.Using the linear interpolator (7), we obtain an estimation error with variance decaying as the reciprocal of the SNR provided that L ≤ 1 2λ D .Thus, the condition L ≤ 1 2λ D can be viewed as a sufficient condition for obtaining "nearly perfect side information" in the sense that the variance of the interpolation error is of the same order as the reciprocal of the SNR.
Of course, one could increase the training period L beyond 1 2λ D .Indeed, by increasing L, we could reduce the rate loss due to the transmission of pilots as indicated in (32) at the cost of obtaining a larger fading estimation error, which in turn may reduce the reliability of the nearest neighbor decoder.To understand this trade-off better, we shall analyze the achievable pre-log when L > 1 2λ D .Note that for L > 1 2λ D , the variance of the interpolation error follows from (11) The former integral vanishes as the SNR tends to infinity.However, we prove in Appendix B that as the SNR tends to infinity, the latter integral is bounded away from zero.This implies that the interpolation error (35) does not vanish as the SNR tends to infinity, and the pre-log achievable with the scheme described in Section II is zero.It thus follows that the condition L ≤ 1 2λ D is necessary in order to achieve a positive pre-log.
Comparing ( 25) and ( 24) with the capacity pre-log min(n t , n r ) for coherent fading channels [1], [2], we observe that, for a fading process of bandwidth λ D , the penalty for not knowing the fading coefficients is roughly (min(n t , n r )) 2 • 2λ D .Consequently, the lower bound (25) does not grow linearly with min(n t , n r ), but it is a quadratic function of min(n t , n r ) that achieves its maximum at This gives rise to the lower bound which cannot be larger than 1/(8λ D ).The same holds for the lower bound (23).

IV. FADING MULTIPLE-ACCESS CHANNELS
In this section, we extend the use of nearest neighbor decoding with pilot-aided channel estimation to the fading MAC.We are interested in the achievable pre-log region that can be achieved with this scheme.
We consider a two-user MIMO fading MAC, where two terminals wish to communicate with a third one, and where the channels between the terminals are MIMO fading channels.Extension to more than two users is straightforward.The first user has n t,1 antennas, the second user has n t,2 antennas and the receiver has n r antennas.The channel model is depicted in Fig. 2. The channel output at time instant k ∈ is a complex-valued n r -dimensional random vector given by Here x s,k ∈ ¼ nt,s denotes the time-k channel input vector corresponding to User s, s = 1, 2; H s,k denotes the (n r × n t,s )-dimensional fading matrix at time k corresponding to User s, s = 1, 2; SNR denotes the average SNR for each transmit antenna; and Z k denotes the n r -variate additive noise vector at time k.The fading processes {H s,k , k ∈ }, s = 1, 2 are independent of each other and of the noise process {Z k , k ∈ }, and follow the same setup as the one used in the point-to-point channel (Section II).
Both users transmit codewords and pilot symbols over the channel (40).To transmit the message m s ∈ {1, . . ., ⌊e nRs ⌋}, s = 1, 2, (where m 1 and m 2 are drawn independently) each user's encoder selects a codeword of length n from a codebook C s , where C s , s = 1, 2 are drawn i.i.d.from an n t,s -variate, zero-mean, complex-Gaussian distribution of covariance matrix I nt,s .
Similar to the single-user case, orthogonal pilot vectors are used.The pilot vector p s,t ∈ ¼ nt,s , s = 1, 2, t = 1, . . ., n t,s used to estimate the fading coefficients from transmit antenna t of User s is given by p s,t (t) = 1 and p s,t (t ′ ) = 0 for t ′ = t.For example, the first pilot vector of User s is given by (1, 0, . . ., 0) T .To estimate the fading matrices H 1,k and H 2,k , each training period requires transmission of (n t,1 + n t,2 ) pilot vectors p 1,1 , . . ., p 1,n t,1 , p 2,1 , . . ., p 2,n t,2 .
Assuming transmission from both users is synchronized, the transmission scheme extends the December 11, 2013 DRAFT point-to-point setup in Section II to the two-user MAC setup as illustrated in Fig. 3. Every L time instants (for some L ≥ n t,1 + n t,2 , L ∈ ), User 1 first transmits the n t,1 pilot vectors p 1,1 , . . ., p 1,n t,1 .Once the transmission of the n t,1 pilot vectors ends, User 2 transmits its n t,2 pilot vectors p 2,1 , . . ., p 2,n t,2 .The codewords for both users are then split up into blocks of (L − n t,1 − n t,2 ) data vectors, which are transmitted simultaneously after the (n t,1 + n t,2 ) pilot vectors.The process of transmitting (L − n t,1 − n t,2 ) data vectors and (n t,1 + n t,2 ) pilot vectors continues until all n data symbols are completed.Herein we assume that n is an integer multiple of (L − n t,1 − n t,2 ). 4 Prior to transmitting the first data block, and after transmitting the last data block, a guard period of L(T − 1) time instants (for some T ∈ ) is introduced for the purpose of channel estimation, where we transmit every L time instants the (n t,1 + n t,2 ) pilot vectors but we do not transmit data vectors in between.Note that codewords from both users are jointly transmitted at the same time instants whereas pilots from both users do not interfere and are separately transmitted at different time instants.The total block-length of this transmission scheme (comprising data vectors, pilot vectors and guard period) is given by where n p and n g are Similarly to the single-user case, the receiver guesses which messages have been transmitted using a two-part decoder that consists a channel estimator and a data detector.The channel estimator first obtains matrix-valued fading estimates { Ĥ(T) s,k , k ∈ D}, s = 1, 2 from the received pilots Y k ′ , k ′ ∈ P using the same linear interpolator as (7).From the received codeword {y k , k ∈ D} and the channel-estimate matrices { Ĥ(T) s,k , k ∈ D}, s = 1, 2 (which are the realizations of { Ĥ(T) s,k , k ∈ D}, s = 1, 2), the decoder chooses the pair of messages ( m1 , m2 ) that minimizes the distance metric Fig. 3: Structure of joint-transmission scheme, n t,1 = 2, n t,2 = 1, L = 7 and T = 2.

Pilot Data
No transmission where and where D (n ′ ) is defined in the same way as (18).In the following, we will refer to the above communication scheme as the joint-transmission scheme.
We shall compare the joint-transmission scheme with a time-division multiple-access (TDMA) scheme, where each user transmits its message using the transmission scheme illustrated in Fig. 4. Specifically, during the first βn ′ channel uses (for some 0 ≤ β ≤ 1), User 1 transmits its codeword according to the transmission scheme given in Section II (see also Fig. 4), while User 2 is silent.(Here n ′ is given in (41).)Then, during the next (1 − β)n ′ channel uses, User

A. The MAC Pre-Log
Let R * 1 (SNR), R * 2 (SNR) and R * 1+2 (SNR) be the maximum achievable rate for User 1, the maximum achievable rate for User 2 and the maximum achievable sum-rate, respectively.The achievable-rate region is given by the closure of the convex hull of the set [15] We are interested in the pre-logs of R 1 (SNR) and R 2 (SNR), defined as the limiting ratios of R 1 (SNR) and R 2 (SNR) to the logarithm of the SNR as the SNR tends to infinity.Thus, the pre-log region is given by the closure of the convex hull of the set where The capacity pre-logs Π C We next present our result on the pre-log region of the two-user MIMO fading MAC achievable with the joint-transmission scheme.

Theorem 2. Consider the MIMO fading MAC model (40). Then, the pre-log region achievable with the joint-transmission scheme is the closure of the convex hull of the set
where Proof: See Section VI.
The pre-log region given in Theorem 2 is the largest region achievable with any transmission scheme that uses (n t,1 + n t,2 )/L * of the time for transmitting pilot symbols.Indeed, even if the channel estimator would be able to estimate the fading coefficients perfectly, and even if we could decode the data symbols using a maximum-likelihood decoder, the capacity pre-log region (without pilot transmission) would be given by the closure of the convex hull of the set [1], [2], [15] which, after multiplying by 1−(n t,1 +n t,2 )/L * in order to account for the pilot symbols, becomes (51).Thus, in order to improve upon (51), one would need to design a transmission scheme that employs less than (n t,1 + n t,2 )/L * pilot symbols per channel use.
where L * = 1 2λ D .This follows directly from the pre-log of the point-to-point MIMO fading channel (Theorem 1) where the number of transmit antennas from Users 1 and 2 is given by n t,1 and n t,2 , respectively.
Note that the sum of the pre-logs Π R 1 + Π R 2 is upper-bounded by the capacity pre-log of the point-to-point MIMO fading channel with (n t,1 +n t,2 ) transmit antennas and n r receive antennas, since the point-to-point MIMO channel allows for cooperation between the transmitting terminals.
While the capacity pre-log of point-to-point MIMO fading channels remains an open problem, the capacity pre-log of point-to-point MISO fading channels is known, cf.(22).It thus follows from (22) that, for n r = n t,1 = n t,2 = 1, we have which together with the single-user constraints implies that TDMA achieves the capacity pre-log region of the SISO fading MAC.The next section provides a more detailed comparison between the joint-transmission scheme and TDMA.

B. Joint Transmission versus TDMA
In this section, we discuss how the joint-transmission scheme performs compared to TDMA.To this end, we compare the sum-rate pre-log Π R * 1+2 of the joint-transmission scheme (Theorem 2) with the sum-rate pre-log of the TDMA scheme employing nearest neighbor decoding and pilot-aided channel estimation (Remark 2) as well as with the sum-rate pre-log of the coherent TDMA scheme, where the receiver has knowledge of the realizations of the fading processes {H s,k , k ∈ }, s = 1, 2. In the latter case, the sum-rate pre-log is given by The following corollary presents a sufficient condition on L * under which the sum-rate pre-log of the joint-transmission scheme is strictly larger than that of the coherent TDMA scheme (57), as well as a sufficient condition on L * under which the sum-rate pre-log of the joint-transmission scheme is strictly smaller than the sum-rate pre-log of the TDMA scheme given in Remark 2.
Since ( 57) is an upper bound on the sum-rate pre-log of any TDMA scheme over the MIMO fading MAC (40), and since the sum-rate pre-log given in Remark 2 is a lower bound on the sum-rate pre-log of the best TDMA scheme, it follows that the sufficient conditions presented in Corollary 1 hold also for the best TDMA scheme.

Corollary 1. Consider the MIMO fading MAC model (40). The joint-transmission scheme
achieves a larger sum-rate pre-log than any TDMA scheme if where we define a/0 ∞ for every a > 0. Conversely, the best TDMA scheme achieves a larger sum-rate pre-log than the joint-transmission scheme if Recall that L * is inversely proportional to the bandwidth of the power spectral density f H (•), which in turn is inversely proportional to the coherence time of the fading channel.Corollary 1 thus demonstrates that the joint-transmission scheme tends to be superior to TDMA when the coherence time of the channel is large.In contrast, TDMA is superior to the joint-transmission scheme when the coherence time of the channel is small.Intuitively, this can be explained by observing that, compared to TDMA, the joint-transmission scheme uses the multiple antennas at the transmitters and at the receiver more efficiently, but requires more pilot symbols to estimate the fading coefficients.Thus, when the coherence time is large, the number of pilot symbols required to estimate the fading is small, so the gain in achievable rate by using the antennas more efficiently dominates the loss incurred by requiring more pilot symbols.On the other hand, when the coherence time is small, the number of pilot symbols required to estimate the fading is large and the loss in achievable rate incurred by requiring more pilot symbols dominates the gain by using the antennas more efficiently.
We next evaluate (58) and (59) for some particular values of n r , n t,1 , and n t,2 .
1) Receiver employs less antennas than transmitters: Suppose that n r ≤ min(n t,1 , n t,2 ).Then, the right-hand sides (RHSs) of ( 58) and (59) become ∞, so every finite L * satisfies (59).Thus, if the number of receive antennas is smaller than the number of transmit antennas, then, irrespective of L * , TDMA is superior to the joint-transmission scheme.
2) Receiver employs more antennas than transmitters: Suppose that n r ≥ n t,1 + n t,2 , and suppose that n t,1 = n t,2 = n t .Then, (58) and (59) become and Thus, if L * is greater than 4n t , then the joint-transmission scheme is superior to TDMA.In contrast, if L * is smaller than 3n t , then TDMA is superior.This is illustrated in Fig. 5 for the case where n r = 2 and n t,1 = n t,2 = 1.Note that if L * is between 3n t and 4n t , then the joint-transmission scheme is superior to the TDMA scheme presented in Remark 2, but it may be inferior to the best TDMA scheme.
3) A case in between: Suppose that n r ≤ n t,1 + n t,2 and n t,2 < n r ≤ n t,1 .Then, (58) becomes and (59) becomes Thus, in this case the joint-transmission scheme is always inferior to the coherent TDMA scheme (57), but it can be superior to the TDMA scheme in Remark 2.
Noncoherent Joint-transmission Noncoherent TDMA Coherent TDMA Fig. 5: Pre-log regions for a fading MAC with n r = 2 and n t,1 = n t,2 = 1 for different values of L * .Depicted are the pre-log region for the joint-transmission scheme as given in Theorem 2 (dashed line), the pre-log region of the TDMA scheme as given in Remark 2 (solid line), and the pre-log region of the coherent TDMA scheme (57) (dotted line).

C. Typical Values of L *
We briefly discuss the range of values of L * that may occur in practical scenarios.To this end, we first recall that L * ≤ ⌊1/(2λ D )⌋, and that λ D is the bandwidth of the fading power spectral density f H (•), which can be associated with the Doppler spread of the channel as [10] Here f m is the maximum Doppler shift given by where v is the speed of the mobile device, c = 3 • 10 8 m/s is the speed of light and f c is the carrier frequency; and W c is the coherence bandwidth of the channel approximated as [10], [16] where σ τ is the delay spread.Following the order of magnitude computations of Etkin and Tse [10], we determine typical values of λ D for indoor, urban, and hilly area environments and for carrier frequencies ranging from 800 MHz to 5 GHz and tabulate the results in Table I.
For indoor environments and mobile speeds of 5 km/h, we have that L * is typically larger  GHz.The values of the delay spread are taken from [10], [16] for indoor and urban environments and from [17] for hilly area environments.
than 5 • 10 4 .For urban environments, L * is typically larger than 2.5 • 10 3 for mobile speeds of 5 km/h and larger than 125 for mobile speeds of 75 km/h.For hilly area environments and mobile speeds of 200 km/h, L * ranges typically from 10 to 250.Thus, for most practical scenarios, L * is typically large.It therefore follows that, if n r ≥ n t,1 + n t,2 , the condition (58) is satisfied unless n t,1 +n t,2 is very large.For example, if the receiver employs more antennas than the transmitters, and if n t,1 = n t,2 = n t , then L * > 4n t is satisfied even for urban environments and mobile speeds of 75 km/h, as long as n t < 30.Only for hilly area environments and mobile speeds of 200 km/h, this condition may not be satisfied for a practical number of transmit antennas.Thus, if the number of antennas at the receiver is sufficiently large, then the joint-transmission scheme is superior to TDMA in most practical scenarios.On the other hand, if n r ≤ min(n t,1 , n t,2 ), then TDMA is always superior to the joint-transmission scheme, irrespective of how large L * is.This suggests that one should use more antennas at the receiver than at the transmitters.

V. PROOF OF THEOREM 1
Theorem 1 is proven as follows.We first characterize the estimation error from the linear interpolator (7).We then compute the rates achievable with the communication scheme described in Section II.Finally, we analyze the pre-log corresponding to these rates.

A. Linear Interpolator
We first note that the estimate of H k (r, t) is given by ( 7), namely, We denote the interpolation error by k (r, t).For future reference, and for any k ∈ , we express k = jL + ℓ, so ℓ = k mod L. Assuming that the first pilot symbol is transmitted at k = 0, it follows that ℓ = 0, . . ., n t − 1 for k ∈ P and ℓ = n t , . . ., L − 1 for k ∈ D. The statistical properties of the channel estimator for a given window size T are summarized in the following lemma.
Lemma 1.For a given T , the linear interpolator (67) has the following properties.
2) a) For a given transmit antenna t and ℓ ∈ {n t , . . ., L − 1}, the n r processes jL+ℓ (n r , t)), j ∈ } are independent and have the same law.
b) For a given receive antenna r and ℓ ∈ {n t , . . ., L − 1}, the n t processes jL+ℓ (r, n t )), j ∈ } are independent but have different laws.
Proof: See Appendix A.

B. Achievable Rates and Pre-Logs
In the following proof, we only consider the case where n t = n r .The more general case of n t = n r follows then by employing only n r transmit antennas or by ignoring n r − n t antennas at the receiver.This yields a lower bound on the maximum achievable rate and does not incur a loss with respect to the pre-log.Indeed, it can be shown that the nearest neighbor decoder described in Section II achieves the pre-log min(n r , n t ).Thus, increasing n t beyond n r or n r beyond n t does not improve the pre-log achievable by such a decoder.In fact, increasing n t beyond n r requires the transmission of more pilot symbols and does therefore even reduce the pre-log achievable with the communication system described in Section II.
To prove Theorem 1, we analyze the generalized mutual information (GMI) [18] for the channel and communication scheme in Section II.The GMI, denoted by I gmi T (SNR), specifies the highest information rate for which the average probability of error, averaged over the ensemble of i.i.d.Gaussian codebooks, tends to zero as the codeword length n tends to infinity (see [5], [11], [12] and references therein).The GMI for stationary Gaussian fading channels employing nearest neighbor decoding has been evaluated in [11], [12] for the case where a genie provides the receiver with an estimate of the fading process.However, the estimate considered in [11], [12] is assumed to be jointly stationary ergodic with { We therefore need to adapt the work in [11], [12] to our channel model.For completeness, we present all the main steps here, even though they are very similar to the ones in [11], [12].
We prove Theorem 1 as follows: 1) We compute a lower bound on I gmi T (SNR) for a fixed window size T .2) We analyze the behavior of this lower bound as T tends to infinity.
3) We evaluate the limiting ratio of this lower bound to log SNR as SNR tends to infinity.

1) I gmi
T (SNR) for a fixed T : We analyze the GMI for a fixed T using a random coding upper bound on the average error probability.Note that due to the symmetry of the codebook construction, it suffices to consider the error behavior, conditioned on the event that message 1 was transmitted.Let E(m ′ ) denote the event that D(m ′ ) ≤ D(1).The ensemble-average error probability, where the average is over the ensemble of i.i.d.Gaussian codes, corresponding to message m = 1 is thus given by To evaluate the GMI from the RHS of (69), we define some useful quantities in the following.
Recall the channel and the transmission model in Section II.Without loss of generality, assume December 11, 2013 DRAFT that the first pilot vector is transmitted at time k = 0. Define F (SNR) as where E (T ) ℓ is a random matrix with element at row r and column t given by E (T ) ℓ (r, t), and where • F denotes the Frobenius norm.Further define a typical set with D (n ′ ) = {0, . . ., n ′ − 1} ∩ D as provided in (18), and some δ > 0, where we have recalled n ′ in (4), namely Then, we have the following convergence as n tends to infinity.
Lemma 2. For the communication scheme described in Section II, we have that where we have used the notation U n ′ to denote the sequence U 0 , . . ., U n ′ −1 .
Proof: We have converges to F (SNR) almost surely, which in turn implies that it also converges in probability, thus proving (73).
Considering the typical set (71) and following the derivation in [11], [12], Pe ( 1) in ( 69) can be upper-bounded as where T c δ denotes the complement of T δ .It follows from Lemma 2 that the second term on the RHS of (80) can be made arbitrarily small by letting n tend to infinity.

The GMI characterizes the rate of exponential decay of the expression
as n → ∞ [11], [12].The computation of the GMI requires the conditional log momentgenerating function of the metric D(m ′ ) associated with the wrong message output m ′ = 1conditioned on the channel outputs and on the fading estimates-which we shall denote by Here we define Proceeding along the lines of [11], [12], we can express the conditional log moment-generating function in (82) as the sum of conditional log moment-generating functions for the individual We then have that for all θ < 0 lim n→∞ where the last step should be regarded as the definition of κ(θ, SNR).The convergence in ( 88) is due to the ergodicity of {(Y jL+ℓ , Ĥ(T) jL+ℓ ), j ∈ }, ℓ = n t , . . ., L − 1 (see Part 3) of Lemma 1) and the ergodic theorem.
Following the same steps as in [11], [12], we can then show that for all δ ′ > 0, the ensembleaverage error probability can be bounded as for some ε(δ ′ , n) satisfying On the RHS of (89), I gmi T (SNR) denotes the GMI as a function of SNR for a fixed T , which is given by Herein the pre-factor (L − n t )/L equals the fraction of time used for data transmission.The bound (89) implies that for rates below I gmi T (SNR), the communication scheme described in Section II has vanishing error probability as n tends to infinity.Combining (70) and ( 88) with (91) yields Following the steps used in [20, App.D], it can be shown that for θ < 0 As observed in [20, App.D], a good lower bound on I gmi T (SNR) for high SNR follows by choosing where ǫ 2 * ,T = max r=1,...,nr, t=1,...,nt, ℓ=nt,...,L−1 Hence, substituting the choice of θ in (94) and applying (93) to the RHS of (92) yields 2) Achievable Rates as T → ∞: We next analyze the RHS of (96) in the limit as T tends to infinity.To this end, we note that, for L ≤ 1 2λ D , the variance of the interpolation error tends to (15), namely irrespective of ℓ and t.We shall therefore denote the variance of the interpolation error ǫ 2 ℓ (t) by ǫ 2 .Note that for a fixed T , the entries of are independent but not i.i.d., which follows from Part 2) of Lemma 1.However, as T tends to infinity, their distribution becomes identical due to (97) and hence they converge in distribution to 1 where the entries of H are i.i.d.complex-Gaussian random variables with zero mean and variance Note that is a continuous function with respect to the entries of the matrix It therefore follows from Portmanteau's lemma [21] that, as T → ∞, the RHS of (96) can be lower-bounded by where the last inequality follows from the lower bound log det (I + A) ≥ log det A. Combining December 11, 2013 DRAFT (103) with (96) yields where in the last inequality we have used the assumption n t = n r .
3) The Pre-Log: We next compute a lower bound on the pre-log by computing the limiting ratio of the RHS of (105) to log SNR as SNR tends to infinity.To this end, we first consider Since the integrand is bounded by it follows that 0 ≤ SNR ǫ 2 ≤ Ln t , which implies that We next consider the term E log det H H † − 1.Note that by [22,Lemma A.2] and by the assumption n t = n r , we have where ψ(•) is Euler's digamma function [23].Furthermore, since we have by the Dominated Convergence Theorem [19] that so log(1 − ǫ 2 ) vanishes as the SNR tends to infinity.Combining (112) with (110) yields It thus follows from (105), ( 109) and (113) that where we have used that n t = n r = min(n t , n r ).Note that the condition L ≤ 1 2λ D is necessary since otherwise (97) would not hold.This proves Theorem 1.

C. A Note on Input Distribution
The pre-log in Theorem 1 is derived using codebooks whose entries are drawn i.i.d.from an n t -variate Gaussian distribution with zero mean and identity covariance matrix.However, Gaussian inputs are not necessary to achieve the pre-log (25).In fact, (25) can be achieved by any i.i.d.inputs having density satisfying E[ X 2 ] ≤ n t and ( 26) and ( 27), namely, Note that the fact that the inputs have a density implies that E[ X 2 ] > 0. To show that the conditions ( 26) and ( 27) suffice to achieve (25), we follow the steps in Section V-B but with We then upper-bound F (SNR) and κ(θ, SNR) as follows.Using that for any two matrices A and 24,Sec. 5.6] and that E (T ) ℓ and Xℓ are independent, we can upper-bound F (SNR) by As for κ(θ, SNR), we have Here (121) follows from (116), and (122) follows by evaluating the integral as in [12,App. A].
By following the steps used in Section V-B, and by choosing where ǫ 2 * ,T is given in (95), we obtain from ( 119) and ( 122) Taking the limit as T tends to infinity, and repeating the steps used in Section V-B yield where we have again used the assumption n t = n r .We conclude by evaluating the limiting ratio of the RHS of ( 127 This concludes the proof.

VI. PROOF OF THEOREM 2
In contrast to the proof of Theorem 1, for the fading MAC, it is not sufficient to restrict ourselves to the case of n t,1 = n t,2 = n r .For example, increasing n r beyond n t,1 and n t,2 does not increase the single-rate pre-logs Π R * 1 and Π R * 2 , but it does increase the pre-log of the achievable sum-rate Π R * 1+2 .For the proof of Theorem 2, we therefore consider a general setup of n t,1 , n t,2 and n r .
We derive the achievable pre-logs for the MAC case using a similar approach to the pointto-point case.We first consider the average error probability, averaged over the ensemble of i.i.d.Gaussian codebooks.Let Pe and Pe (m 1 , m 2 ) be the ensemble-average error probability and the ensemble-average error probability corresponding to message m 1 and m 2 being transmitted, respectively.Due to the symmetry of the codebook construction, Pe is equal to Pe (1, 1) and it therefore suffices to consider Pe (1, 1) to derive the achievable rates.Let . Using the union bound, the error probability Pe (1, 1) can be upper-bounded as We next analyze these probabilities corresponding to the error events (m s,k (r, t) be the estimation-error matrix in estimating H s,k , i.e., To facilitate the analysis, we first generalize F (SNR) and T δ in the point-to-point case (cf.( 70) and ( 71)) to the MAC case, i.e., for some δ > 0, with n ′ given in (41) and D (n ′ ) = {0, . . ., n ′ − 1} ∩ D. Using F (SNR) and the typical set T δ , we continue by evaluating the GMI for each of the three probabilities on the RHS of (132) corresponding to the error events (m : Following the steps as used in Section V-B to derive (80), we can upper-bound the ensemble-average error probability for the error event E(m ′ 1 , 1), m ′ 1 = 1 using T δ and its complement T c δ as Note that the second probability on the RHS of (136) vanishes as n tends to infinity, which can be shown along the lines of the proof of Lemma 2.
The GMI for User 1 gives the rate of exponential decay of the term as n → ∞.The evaluation of the GMI for User 1 requires the expression of the log momentgenerating function of the metric D(m ′ 1 , 1) associated with an incorrect message m ′ 1 = 1conditioned on the channel outputs, on m ′ 2 = 1, and on the fading estimates-which we shall denote by κ 1,n (θ, where we have defined Following the steps used in Section V-B to obtain (84) and ( 85), it can be shown that − log det Then, following the steps used in Section V-B to derive (86)-(88), we have that for all θ < 0 lim n→∞ almost surely, where (141) should be regarded as the definition of κ 1 (θ, SNR).Here we define Following the derivation in [11], [12], we can then upper-bound the ensemble-average error for some ε On the RHS of (143), I gmi 1,T (SNR) denotes the GMI for User 1 as a function of SNR for a fixed T and is given by The pre-factor (L − n t,1 − n t,2 )/L equals the fraction of time used for data transmission.The bound (143) implies that for all rates below I gmi 1,T (SNR), decoding the message from User 1 using the scheme described in Section IV has vanishing error probability as n tends to infinity.
Combining (134) and ( 141) with (145) yields As the supremum (146) is difficult to evaluate, we next consider a lower bound on I gmi 1,T (SNR).By noting g 1,ℓ (θ, SNR) ≤ 0 for θ ≤ 0 (which can be shown using the technique developed in [20,  we obtain a lower bound on I gmi 1,T (SNR) We continue by analyzing the RHS of (149) in the limit as the observation window T of the channel estimator tends to infinity.To this end, we note that, for L ≤ irrespective of s, ℓ, r and t.Hence, irrespective of ℓ, the estimate Ĥ(T) 1,ℓ tends to H1 in distribution as T tends to infinity, so where the n r × n t,1 entries of H1 are i.i.d., circularly-symmetric, complex-Gaussian random variables with zero mean and variance (1 − ǫ 2 ).Using Portmanteau's lemma (as used in (102)), we obtain that where Here the last inequality follows by lower-bounding log det (I + A) ≥ log detA.
By evaluating the RHS of (154) in the same way as evaluating the RHS of (105) in Section V-B, we obtain a lower bound for the maximum achievable pre-log for User 1 as Here instead of assuming n t = n t,1 + n t,2 = n r , we have used a general setup of n t,1 , n t,2 and n r .Note that the condition L ≤ 1/(2λ D ) is necessary since otherwise (15) would not hold.This yields one boundary of the pre-log region presented in Theorem 2.
2) Error Event (m ′ 1 = 1, m ′ 2 = 1): This follows from the proof for the error event (m ′ 1 = 1, m ′ 2 = 1) by swapping User 1 with User 2. We thus have yielding the second boundary of the pre-log region presented in Theorem 2.
3) Error Event (m ′ 1 = 1, m ′ 2 = 1): The analysis on the achievable sum rate corresponding to the joint error event ) in the MAC case follows the same analysis as in the point-to-point case (Section V-B).More specifically, the sum of the GMI I gmi 1+2,T (SNR) can be viewed as the GMI of an n r × (n t,1 + n t,2 )-dimensional point-to-point MIMO channel with fading matrix at time k, [H 1,k , H 2,k ], and fading estimate matrix at time k, Ĥ(T) 1,k , Ĥ(T) 2,k .The maximum achievable sum-rate pre-log can therefore be obtained using the same approaches as in Section V-B, but with arbitrary n r and n t = n t,1 + n t,2 .It can be shown that the maximum achievable sum-rate pre-log Π R * 1+2 is lower-bounded by On the RHS of (158), the term min (n r , n t,1 + n t,2 ) corresponds to the MIMO gain, which is given by the minimum number of receive and transmit antennas, and the term 1 − n t,1 +n t,2 L corresponds to the fraction of time for data transmission, which changes for arbitrary number of transmit antennas in comparison to the proof of the point-to-point channel.This yields the third boundary of the pre-log region presented in Theorem 2.

VII. CONCLUSION
In this paper we studied a communication scheme for MIMO fading channels that estimates the fading via transmission of pilot symbols at regular intervals, and feeds the fading estimates to the nearest neighbor decoder.Restricting ourselves to fading processes with a bandlimited power spectral density, we studied the information rates achievable with this scheme at high SNR.Specifically, we analyzed the achievable rate pre-log, defined as the limiting ratio of the achievable rate to the logarithm of the SNR in the limit as the SNR tends to infinity.
We showed that, in order to obtain fading estimates whose variance vanishes as the SNR tends to infinity, the portion of time required for pilot transmission must be greater or equal to the number of transmit antennas times twice the bandwidth of the fading power spectral density.We demonstrated that, in this case, the nearest neighbor decoder achieves the capacity pre-log of the coherent fading channel times the fraction of time used for the transmission of data.Hence, the loss with respect to the coherent case is solely due to the transmission of pilots used to obtain accurate fading estimates.Our achievability bounds are tight in the sense that any scheme using as many pilots as our proposed scheme cannot achieve a higher pre-log using a nearest neighbor decoder.Furthermore, if the inverse of twice the bandwidth of the fading process is an integer, then, for MISO channels, our scheme achieves the capacity pre-log of the non-coherent fading channel derived by Koch and Lapidoth [9].For non-coherent MIMO channels, our scheme achieves the best so far known lower bound on the capacity pre-log obtained by Etkin and Tse [10].Since the last result only yields a lower bound on the capacity pre-log of MIMO channels, there may exist other schemes achieving a better pre-log than our scheme.
APPENDIX A PROOF OF LEMMA 1 1) By the orthogonality principle [25], we have that Ĥ(T) k (r, t) and E (T ) k (r, t) are uncorrelated.Noting that the pilot symbols are unity, we can write (67) as Since the processes {H k (r, t), k ∈ } and {Z k (r), k ∈ } are zero-mean complex-Gaussian processes, we have from (159) and the orthogonality principle that Ĥ(T) k (r, t) and E (T ) k (r, t) are independent zero-mean complex-Gaussian random variables.
2) Recall from Section V-A that the time index k can be written as k = jL + ℓ.Then, for k ∈ D, we have ℓ = n t , . . ., L − 1, and for k ∈ P we have ℓ = 0, . . ., n t − 1.Since the pilot vectors are transmitted sequentially from p 1 to p nt , we have for k ∈ P that x jL+ℓ = p ℓ+1 , ℓ = 0, . . ., n t − 1 (160) namely the (ℓ+1)-th pilot vector, ℓ = 0, . . ., n t −1 is used to estimate the fading coefficients from transmit antenna t.We next note that, in order to estimate H k (r, t), there is no loss in optimality by considering only the outputs Y k ′ (r) for k ′ ∈ P ∩ {k − T L, . . ., k + T L} satisfying Indeed, the channel outputs Y k ′ (r), k ′ mod L = t − 1 correspond to H k ′ (r, t ′ ), t ′ = t, which are independent from H k (r, t) since we have assumed that the fading processes corresponding to different transmit and receive antennas are independent.It follows that for the estimation at k = jL + ℓ, the coefficients a k ′ (r, t) that minimize the mean-squared Recall that, as T tends to infinity, we have that, irrespective of j and r, the variance of the interpolation error (11), namely where In order to analyze the behavior of ǫ 2 ℓ (t) for L > 1 2λ D , we first express L as for some ε > 0. The variance of the interpolation error (167) can be lower-bounded as where the inequality is because the first integral in (170) is non-negative.Let ℓ ′ ℓ − t + 1.We have that Since the summands are non-negative, it follows that The RHS of (171) can thus be lower-bounded as SNRf L,0 (λ) + n t dλ (177) Since L is of positive Lebesgue measure, and since the integrand on the RHS of (178) is strictly positive, it follows from [31] that Recall that ℓ ′ = ℓ − t + 1.Thus, for ℓ = n t , . . ., L − 1, we have Then, combining (180) and ( 179) with (178), ( 175) and (171) yields

Remark 2 (
TDMA Pre-Log).Consider the MIMO fading MAC model (40).Then, the pre-log region achievable with the TDMA scheme employing nearest neighbor decoding and pilot-aided December 11, 2013 DRAFT channel estimation is the closure of the convex hull of the set

Herein ( 76 )
follows from Part 3) of Lemma 1 and the ergodic theorem [19, Chap.7]; (77) follows from Part 4) of Lemma 1; and (78) follows since Xℓ has zero mean and covariance matrix I nt , and is independent from

TABLE I :
Typical values of L * for various environments with f c ranging from 800 MHz to 5 ) to log SNR as SNR tends to infinity.Using (108) and that E[ X 2 ] ≤ n t App. D]) and by choosing5 θ = −1 n r + n r (n t,1 + n t,2 ) SNR ǫ 2 * ,T (15)t)| 2 ] tends to(15)(with SNR in (15) replaced by n t SNR), 6 so lim