Semiblind Single-Carrier MIMO Channel Estimation Using Overlay Pilots

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We provide the formulation of the channel estimator and present numer- ..... ×APP. [ sdu , ˆΘ,n. ] (8) where cte denotes a constant term. To simplify our notations, ...
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[4] G. Woodward and B. S. Vucetic, “Adaptive detection for DS-CDMA,” Proc. IEEE, vol. 86, no. 7, pp. 1413–1434, Jul. 1998. [5] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous code division multiple-access communications,” IEEE Trans. Commun., vol. 38, no. 4, pp. 509–519, Apr. 1990. [6] Y. C. Yoon, R. Kohno, and H. Imai, “A spread-spectrum multi-access system with cochannel interference cancellation for multipath fading channels,” IEEE J. Sel. Areas Commun., vol. 11, no. 7, pp. 1067–1075, Sep. 1993. [7] D. Divsalar, M. K. Simon, and D. Raphaeli, “Improved parallel interference cancellation for CDMA,” IEEE Trans. Commun., vol. 46, no. 2, pp. 258–268, Feb. 1998. [8] J. H. Wen and Y. F. Huang, “Fuzzy-based adaptive partial parallel interference canceller for CDMA communication systems over fading channels,” Proc. Inst. Electr. Eng.—Commun., vol. 41, no. 2, pp. 111–116, Apr. 2002. [9] G. Xue, J. Weng, T. Le-Ngoc, and S. Tahar, “Adaptive multistage parallel interference cancellation for CDMA,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1815–1827, Oct. 1999. [10] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [11] H. L. Van Trees, Detection, Estimation and Modulation Theory. Hoboken, NJ: Wiley, 1968. [12] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [13] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications. Englewood Cliffs, NJ: Prentice-Hall, 1995. [14] Y. F. Huang and J. H. Wen, “An adaptive multistage parallel interference canceller for CDMA systems over Rayleigh fading channels,” in Proc. IEEE GLOBECOM, Taipei, Taiwan, R.O.C., Nov. 17–21, 2002, pp. 926–930. [15] Y. F. Huang and J. H. Wen, “An analysis on partial PIC multi-user detection with LMS algorithms for CDMA,” in Proc. IEEE PIMRC, Beijing, China, Sep. 7–11, 2003, pp. 17–21.

Semiblind Single-Carrier MIMO Channel Estimation Using Overlay Pilots Mohammad-Ali Khalighi, Senior Member, IEEE, and Salah Bourennane Abstract—Semiblind (SB) channel estimation based on the expectation– maximization (EM) algorithm is studied for the case of single-carrier multiple-input–multiple-output (MIMO) systems using overlay pilots. Iterative channel estimation and data detection are performed at the receiver. We provide the formulation of the channel estimator and present numerical results on the iterative receiver performance. Presented results show an interesting performance improvement and a fast convergence using the SB scheme when enough transmit power is attributed to pilots. The proposed scheme is also compared to the classical case of time-multiplexed pilots, and the tradeoff between the channel-estimation quality and the data-transmission rate is discussed. Index Terms—Channel estimation, EM algorithm, iterative detection, multiple-input–multiple-output (MIMO) systems, overlay pilots, semiblind estimation. Fig. 10. BER versus the capacity for multistage PIC multiuser detectors over a two-path frequency-selective fading channel. (a) Stage 2. (b) Stage 3. (c) Stage 4 (SNR = 18 dB).

R EFERENCES [1] K. S. Gilhousen, I. M. Jacobs, R. Padavani, A. J. Viterbi, L. A. Weaver, and C. E. Wheatley, “On the capacity of a cellular CCDMA system,” IEEE Trans. Vehicular Tech., vol. 40, no. 2, pp. 303–312, May 1991. [2] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [3] S. Moshavi, “Multi-user detection for DS-CDMA communications,” IEEE Commun. Mag., vol. 34, no. 10, pp. 124–136, Oct. 1996.

Manuscript received January 3, 2007; revised May 25, 2007, July 9, 2007, and July 17, 2007. Parts of this paper were presented at the Spring 2007 IEEE 65th Vehicular Technology Conference. The review of this paper was coordinated by Associate Prof. L. Lampe. The authors are with the École Centrale Marseille and the Institut Fresnel, 13397 Marseille Cedex 20, France (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TVT.2007.907080

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I. I NTRODUCTION Channel estimation is a real challenge in practical multipleinput–multiple-output (MIMO) systems, where the quality of data recovery is as important as attaining a high data throughput. Conventionally, time-multiplexed training sequences are employed to help channel estimation at the receiver—a scheme that is usually called pilot-symbol-assisted modulation (PSAM) [2]. The loss in spectral efficiency caused by the periodic insertion of pilot symbols becomes particularly important for relatively fast time-varying communication channels [3]. Recently, the idea of overlay pilots (OPs), which are also called superimposed or embedded pilots, has been proposed for channel estimation and has attracted great attention. By this approach, a pilot sequence is superimposed on the data sequence before transmission, providing bandwidth saving as no separate timeslot is dedicated to pilot transmission. However, we suffer from a degradation in the quality of the channel estimate due to the unknown data symbols [4]. Several recent works consider the use of an OP scheme, specifically in the case of orthogonal frequency-division multiplexing (OFDM). For instance, for a single-antenna system that employs OFDM, Ohno and Giannakis [5], Balasubramanian et al. [6], Ghogho et al. [7], and Varma et al. [8] address channel-estimation and tracking algorithms by using the OP scheme, and Cui and Tellambura [9] consider iterative channel estimation. Also, a simplified Gaussian maximum-likelihood (ML) estimator is proposed in [10] for multiuser MIMO systems using OP, and iterative turbo decoding and MIMO detection are studied in [11] for the case of OP and OFDM signaling. While, in multicarrier systems, the use of OP would be beneficial in practice [12], we can evoke a real hesitation for the case of single-carrier systems that we consider in this correspondence. Theoretical results based on the criteria of channel capacity [13] or Cramér–Rao lower bounds (CRBs) on channel-estimation errors [14] suggest that OP can have the same or better performance than the classical PSAM. However, more practical results presented in the literature show that PSAM mostly preserves its advantage over OP [15]. In fact, when estimation is based only on pilots, only for high SNR, short-channel coherence time, and a larger number of receive than transmit antennas that OP may be preferrable over PSAM [15], [16]. Recently, it was proposed in [13] to use semiblind (SB) channel estimation based on OP, and it is shown that the capacity of the optimized SB OP scheme is larger than that of pilot-only (PO)-based or SB PSAM. Our aim in this correspondence is to consider a real transmission system and to see how much the use of an SB estimation scheme is beneficial for the OP scheme in practice. We also aim to contrast the performance of such a scheme to the case of timemultiplexed pilots. The SB estimator that we propose is based on the expectation–maximization (EM) algorithm. We will discuss the compromise between the channel-estimation quality and the increase in the data-transmission rate. Compared to previous works on OP, our main contributions consist of the proposition and analysis of the EM-based estimator, as well as the receiver-performance evaluation in the presence of channel coding. This correspondence is organized as follows. In Section II, we present our system model and transmission scheme. Section III globally describes the data-detection and channel-estimation parts. Section IV formulates the SB estimator based on EM. Simulation results and the comparison of different schemes are presented in Section V. 1) Notations: Boldface uppercase letters are used for matrices, whereas boldface lowercase letters are used for vectors. Also, E{.} denotes the expected value; .∗ and .† denote the complex conjugate and conjugate transpose, respectively; .2 is the Frobenius norm; xi denotes the ith entry of vector x; and Xij stands for the (i, j)th entry of matrix X .

Fig. 1. Block diagram of the transmitter. Π represents the pseudorandom interleaver, sd denotes the data symbols, and sp,i is the pilot symbol transmitted on the antenna #i at a given time reference.

II. S YSTEM M ODEL , D ATA T RANSMISSION , AND P ILOT E MBEDDING We consider the context of single-carrier modulation and single-user communication. Ideal uncorrelated Rayleigh flat fading is considered, where channel coefficients are modeled by normalized independent identically distributed complex Gaussian random variables. Channel time variations are considered according to the simple quasi-static model, keeping in mind that the presented results can also easily be applied to the more general block-fading model. We denote the numbers of antennas at the transmitter and the receiver by MT and MR , respectively. Pilot symbols are overlaid on data symbols in the entire frame, and uniform data and pilot-power allotment is considered. This maximizes the mutual information between the transmitter and the receiver, assuming that the receiver can acquire the channel by using an efficient estimator [14]. We denote the power attributed to data and pilot symbols by σd2 and σp2 , respectively, and we define the parameter α, which is the percentage of the power dedicated to pilots, as α = (σp2 /(σp2 + σd2 )). The block diagram of the transmitter is shown in Fig. 1. Bitinterleaved coded modulation is performed prior to spatial multiplexing. Nonrecursive nonsystematic convolutional (NRNSC) channel coding is done, together with random interleaving and Gray bit-symbol mapping. Considering N encoded bits per frame and B = 2 bits per symbol, we denote the number of channel uses corresponding to a frame by Ns = N/(BMT ). Let us denote the (MT × 1) vector of the transmitted signals at time reference n by s[n]. We have s[n] = sd [n] + sp [n], where sd [n] and sp [n] denote the corresponding vectors of data and pilot symbols, respectively. Pilot sequences for MT antennas are mutually orthogonal, which  are chosen according to the Walsh–Hadamard series Ns sp,i [n]sp,j [n] = 0, where i = j. [17], i.e., we have n=1 Let H be the (MR × MT ) MIMO channel matrix. The (MR × 1) vector of the received symbols at time reference n is y[n] = H s[n] + n[n]. Here, n is the vector of additive white Gaussian noise of zero mean and variance σn2 , which is assumed to be known at the receiver. III. I TERATIVE D ATA D ETECTION AND C HANNEL E STIMATION As shown in Fig. 2, at the receiver, iterative soft MIMO detection and soft channel decoding are performed together with channel estimation. The soft MIMO detection is based either on the maximum a posteriori (MAP) criterion or on soft parallel interference cancellation (Soft-PIC). The soft-input–soft-output channel decoder, in turn, is based on the Max-Log-MAP algorithm [18]. The MAP detector is described in [19]. For the Soft-PIC, a simple minimum meansquare-error (MMSE) filtering is done at the first iteration, and in the succeeding iterations, interference cancellation is done by using the soft estimates of data symbols, together with simplified linear

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profit from these data symbols to improve the channel estimate. The estimator that we propose is based on the EM algorithm, wherein it is guaranteed to be stable and to converge to the ML estimate [23]. The formulation of EM that we provide here is analogous to the case of time-multiplexed pilots studied in [19] and [24] or of OFDM signaling in [25]. Let us stack the entries of H in a vector Θ as the vector of parameters to be estimated. Also, let us stack the entries of vectors y[n], where n = 1, . . . , Ns , in an (MR × Ns ) matrix Y = [y[1], . . . , y[Ns ]]. The (MT × Ns ) matrix S d is obtained in the same way from sd [n], where n = 1, . . . , Ns . The ML estimate of Θ, given the observations Y , is ˆ = arg max log p(Y |Θ). Θ LDet ext

Fig. 2. Block diagram of the receiver. and denote the extrinsic log-likelihood ratios (LLRs) at the detector and decoder outputs, respectively, −1 represent LDec post is the a posteriori LLR at the decoder output, and Π and Π the interleaving and deinterleaving functions, respectively.

filtering [20]. We consider a simplified implementation of the Soft-PIC described in detail in [21]. Regarding channel estimation, we start with a primary estimate at the first iteration and make use of the detector outputs to improve the channel estimate in the succeeding iterations. For the first iteration, where sd [n] is unknown, we use an approximate least squares (LS) solution that considers s[n] = sp [n] ˆ (1) = H LS

Ns 1  y[n] s†p [n]. Ns σp2

ij

MT 

(m−1)

ˆ s˜d,t [n]H it

(2)

t=1

ˆ (m) = H ij

Ns 1  ∗ (m) sp,j [n]˜ yi [n]. 2 Ns σp

(3)

n=1

IV. EM-B ASED SB C HANNEL E STIMATION SB estimators make use of data symbols, in addition to pilots, to improve the channel estimate. The interest of using SB estimation in the OP case is twofold: We can reduce the estimation errors that arise from the interfering unknown data symbols and, at the same time, 1 To







 (m−1)



(5)

.

(6)

(1)

For the sake of comparison, we will later present, in this correspondence, the results for the simple PO-based decision-directed (DD) estimator of [15], which we will briefly describe in the following. Let us consider the estimation of the entry Hij of H at iteration m > 1, ˆ (m) . To obtain H ˆ (m) , similar to (1), we calculate which we denote by H ij ij the correlation between the (finite-length) received sequence on the ith receive antenna yi [n] and the transmitted pilot sequence on the jth transmit antenna sp,j [n], where n = 1, . . . , Ns . Meanwhile, we try to reduce the estimation errors resulting from nonzero correlation of data ˜d of and pilots [15], [22]. To do this, we calculate the soft estimates s data symbols by using the a posteriori probabilities at the channel˜d is actually decoder output at the previous iteration. Notice that s calculated in the MIMO detector if Soft-PIC detection is employed ˆ (m) , we first cancel the effect of s˜d in yi , as [21]. Thus, to calculate H ij ˆ (m) from (3) shown in (2),1 before calculating H [n] = yi [n] −



ˆ (m−1) = E log p (Y , S d |Θ) |Y , Θ ˆ (m−1) Q Θ, Θ Θ

A. Simple Decision-Directed Estimator

(m)

Here, p(·|·) denotes the conditional probability density function (pdf). The EM algorithm consists of two update steps of expectation and maximization. In the expectation step, we calculate the expected likelihood function of (Y , S d ) conditioned to Y and the previous estimate of Θ. In the maximization step, we maximize this function with respect to Θ. For the mth update, the EM steps are described in the following:

ˆ ˆ (m) = arg max Q Θ, Θ Θ

n=1

y˜i

(4)

Θ

LDec ext

be more accurate in our notations, we have to specify the iteration (m−1) [n] in (2). number for ˜ sd [n], i.e., to use s˜d,t

The expected value is taken over the distribution of data symbols. Let us define S as the set of constellation points corresponding to the data-symbol sequence with the cardinality |S| = 2BMT Ns . The auxiliary function Q can be written as follows:





ˆ (m−1) = Q Θ|Θ







ˆ (m−1) . log p (Y |S d = S du, Θ) p S du|Y , Θ

S du∈S

(7) Here, S du stands for the uth possible transmitted data-symbol sequence, in contrast to S d , which is the actual transmitted data-symbol sequence. Based on the assumptions of flat channel and the independence of noise samples, y[n] samples are independent conditioned to the channel state, and we can write



Ns    

ˆ = Q Θ|Θ

cte−

n=1 s du ∈V

y[n]−H (sdu [n]+sp [n])2 σn2





ˆ n ×APP sdu , Θ,



(8)

where cte denotes a constant term. To simplify our notations, we did not specify the iteration number. Vector sdu [n] of size MT is the uth possible transmitted data-symbol vector. Whatever value that n has, sdu [n] is taken according to the constellation V of cardinality2 |V| = 2BMT , which is why we sometimes drop the time index [n] ˆ n] is the probability of to simplify the notations. Also, APP[sdu , Θ, sdu [n] = sd [n] and is calculated by using the a posteriori probabilities on its constituting bits at the channel-decoder output. For the sake of simplicity of notations, we hereafter denote it by APPu [n]. Furthermore, we simply denote the summations in (8) by their indices, i.e., n and u. For instance, the summation with index u will take into account all possible data-symbol vectors sdu . Now, differentiating

2 We used the calligraphic typefaces of S and V to denote the sets of constellation points corresponding to the Sequences and Vectors of data symbols, respectively.

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ˆ with respect to H and setting it to zero to find its local Q(Θ|Θ) ˆ = RY S R−1 , where maximum, we finally obtain H SS RY S =

 n

RSS =





y[n] s†du [n] + s†p [n] APPu [n]

u

 n

sdu [n] + sp [n]



(9)



s†du [n] + s†p [n] APPu [n]. (10)

u



is taken over 2BMT vector sdu , and the The summation u corresponding complexity is exponential in B and MT . To reduce the computational complexity of (9) and (10), we write RY S and RSS in terms of the soft estimates of the transmitted symbol vector ˜d . Using the notations in this paragraph, we can write s ˜d [n] = s  s [n]APPu [n]. Reformulating (9) and (10), we obtain u du RY S =





˜†d [n] + s†p [n] y[n] s

n

RSS = RS +





(11)



˜d [n]s†p [n] + sp [n]˜ s†d [n] + Ns σp2 I MT s

(12)

n

where I MT is the (MT × MT ) identity matrix, and RS =   s [n]s†du [n]APPu [n]. We can further simplify the caln u du culation of RS,ij , which is the (i, j)th entry of RS . Keeping in mind that APPu [n] is the product of BMT probabilities on the bits corresponding to the vector sdu [n], for each entry RS,ij , the summation over u, i.e., over V of cardinality 2BMT , reduces to the summation over Vij of cardinality 22B corresponding to the pair of symbols (sdi [n], sdj [n]). Assuming that the bits corresponding to these two symbols are independent, RS,ij can finally be written as follows:

RS,ij =

Ns σd2 ,  s˜di [n]˜ s∗dj [n],

i=j i = j.

(13)

n

To see the complexity reduction by the new formulation, let us consider the ratio of the number of real multiplications required in calculating RSS from (10) to that from (12) and (13). This ratio is rMAP = (3MT 2BMT )/(4MT + 2B ) for the case of MAP detection and rPIC = (3/4)2BMT for the case of Soft-PIC detection. Note ˜d [n] is already calculated and available. that, for the latter case, s For instance, for B = 2 and MT = 2, we have rMAP = 8 and rPIC = 12, whereas for B = 4 and MT = 4, we have rMAP = 24 576 and rPIC = 49 152 for the cases of MAP and Soft-PIC detections, respectively. We notice that a substantial complexity reduction is obtained by the proposed formulation of (11)–(13). V. N UMERICAL R ESULTS Here, we present some numerical results to study the performance of the iterative receiver using the proposed SB estimator. In particular, we make comparisons with other estimation schemes and the CRB. We consider two transmit and four receive antennas that are denoted by the 2 × 4 system. In effect, the OP scheme is more suitable for MR > MT [15], [16]. See [1] for some results for a 2 × 2 system. Unless otherwise stated, frames correspond to Ns = 100 channel uses. Both data and pilot symbols are QPSK modulated, and the channel code is the NRNSC code (133, 171)8 . Because, in the OP scheme, a part of the transmit power is dedicated to pilots, conversely to what is done in [15], we prefer to present the performance curves in terms of the actual average SNR, i.e., MR (σd2 + σp2 )/σn2 , instead of Eb /N0 , which takes into account only σd2 . In the

Fig. 3. BER for EM-based SB and DD estimations. Soft-PIC MIMO detection, fifth iteration of the receiver, 2 × 4 MIMO system, Ns = 100, and OP scheme.

results that will be presented, SNR stands for this. This way, we can directly see the compromise between the channel-estimation quality and the data-detection performance, e.g., by increasing α. 1) Comparison of EM-Based and DD Estimators: For the case of Soft-PIC MIMO detection, Fig. 3 shows the bit-error-rate (BER) curves versus SNR for EM-based and DD estimations and several values of α. Curves correspond to the fifth iteration of the receiver, where almost full convergence is attained, although three iterations are sufficient for the case of the EM-based estimation. We see the performance improvement by using the EM-based estimation. For instance, considering α = 15%, for which the overall performance is acceptable, we have a gain of about 1.5 dB in SNR at BER = 10−4 . For larger α, more power is dedicated to pilot symbols, and hence, the performances of the two estimators become closer. For both estimation methods, we notice an error floor at high SNR for α < 15%. For these cases, the channel estimate is not good enough, and the data symbols cannot be estimated effectively. As a result, with increased SNR, the interference terms in the channel estimate become increasingly important and result in an error floor. Obviously, there is a weaker constraint on α for an increased frame length (results are not shown). Note that excessively increasing α will result in an overall performance degradation because little power will be dedicated to data symbols; in other words, the detection SNR will be low. 2) Comparison of MAP and Soft-PIC Detectors: It is interesting to compare the receiver performance for the two cases of MAP and Soft-PIC detections. The corresponding BER curves are contrasted in Fig. 4 for several values of α. We see that, for high SNR values corresponding to low BER, the Soft-PIC detector, which has much less computational complexity, outperforms MAP for relatively small α. Only for α as high as 20% does the MAP detector become more efficient at high SNR. In fact, the MAP detector is more sensitive to channel-estimation errors than the Soft-PIC detector, and its performance degrades at low α. In what follows, we only consider Soft-PIC detection as it is the more appropriate solution. 3) Comparison With CRB: It is useful to compare the MSE with the CRB, which is the lower bound on MSE in the set of unbiased estimators. Here, we propose to consider the Bayesian CRB, which assumes Gaussian-distributed symbols. For symbols drawn from a given constellation set, these bounds are, in fact, approximate lower bounds [26]. Let us stack the columns of H in a vector h and denote the pdf of h by ph (h). With our assumption on the statistics of channel

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Fig. 4. Comparison of MAP and Soft-PIC MIMO detections. Fifth iteration of the receiver, 2 × 4 MIMO system, Ns = 100, OP scheme, and EM-based SB channel estimation.

Fig. 5. MSE versus CRB through receiver iterations for α = 5% and α = 15%. 2 × 4 MIMO system, Ns = 100, OP scheme, the EM-based SB channel estimation, and Soft-PIC MIMO detection. For instance, IT #i denotes the ith iteration.

coefficients, we have ph (h) = exp(−|h|2 ) [26]. The expressions of the CRB for the two cases of PO and SB estimations are [27] σn2 + Ns σp2 σn2   = 2 σn2 ρh + Ns σp2 + σd2

CRBPO = CRBSB

σn2

ρ2h

(14) (15)

where ρ2h = E{|(∂ ln ph (h)/∂h∗ )|2 } = MR MT . Notice that, in contrast to CRBSB , CRBPO depends on α for a given SNR. Note also that CRBSB in (15) can be used for the encoded data because Ns σd2 is the transmitted energy of data symbols. For the case of the EM-based estimator, Fig. 5 shows the curves of MSE of estimation errors and CRB versus SNR for α = 5% and α = 15% and three iterations of the receiver. There is almost no improvement with more iterations. Notice that what we denote by MSE is the sum of the MSE of all MT MR subchannels. We see that, for a large enough α, we closely attain

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Fig. 6. MSE versus CRB through receiver iterations for α = 15%. 2 × 4 MIMO system, Ns = 100, OP scheme, DD channel estimation, and Soft-PIC MIMO detection.

the CRB after few iterations and, practically, after only two iterations for moderate-to-high SNR. For a small α, however, we observe the problem of error floor, and the Gaussian CRB becomes useless. We have also presented in Fig. 6 the curves of the CRB and the MSE of the DD estimator for α = 15% and five iterations. For α = 5%, we again have an error floor (results not shown). We see that, unlike the SB estimator, we need to process more iterations for an increased SNR to approach the CRB. 4) Comparison of OP and PSAM Schemes: Let us now contrast the BER performance of OP and PSAM methods. In order to perform a fair comparison, we consider two approaches for PSAM: a simple PO-based estimation (denoted by PSAM-PO) that is done once at the first iteration and an iterative SB estimation based on EM (denoted by PSAM-EM) that is considered in [19]. For the OP case, we denote the estimation based on DD and EM by OP-DD and OP-EM, respectively. Parameter setting for PSAM: We use Np channel uses for the transmission of mutually orthogonal pilot sequences and Nd channel uses for the transmission of data sequences. Then, Ns = Np + Nd , where Np ≥ MT to ensure the identifiability of the MIMO channel. We attribute the same power to pilot and data symbols. We define the parameter α = Np /(Np + Nd ), which represents here the ratio of the energy dedicated to pilots to the total transmitted energy for a frame. For a given SNR, we adjust the noise power to take into account the loss in the spectral efficiency η for PSAM. To do this, we multiply the noise power by (Nd + Np )/Nd = 1/(1 − α). Fig. 7 shows the curves of BER versus α for SNR = 10 dB and different estimation schemes after five iterations. Two cases of Ns = 100 and Ns = 1000 channel uses are considered. The latter case would correspond to a ten-times larger channel coherence time. We first explain how we should interpret the results of Fig. 7. 1) OP: Given the received SNR, in order to attain a desired BER, we should look for the optimal α. There exists an optimal value for α as there is a compromise between the channel-estimation quality and the detection SNR. Here, α has no effect on η. 2) PSAM: For a given SNR, α does not affect the data symbols’ transmission power. It directly represents the loss in η as well as the increase in the equivalent receiver noise power. Obviously, here, we should look for the smallest α that gives the desired BER.

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Fig. 7. Comparison of OP and PSAM schemes for the two cases of PO and SB (EM) estimations. Soft-PIC MIMO detection, fifth iteration of the receiver, SNR = 10 dB, and 2 × 4 MIMO system.

Now, consider the case of Ns = 100, and let the desired BER be 10−3 . The required α’s for PSAM-PO and PSAM-EM are 4.5% and 2.8%, respectively, which also represent the loss in η. For the OP-EM case, α = 12.5% is sufficient to attain BER = 10−3 , whereas OP-DD can only approach this BER for α = 20%. If we impose BER = 10−4 , OPs should not be used, as even OP-EM is not able to attain this BER. However, PSAM-EM and PSAM-PO can effectively attain this BER by taking α = 6.8% and α = 12%, respectively. Consider now the case of Ns = 1000. We see that the performances of OP approach more closely those of PSAM. If we impose BER = 10−4 , α should be chosen about 1.1% and 0.63% for PSAM-PO and PSAM-EM and about 3.2% and 1.8% for OP-DD and OP-EM, respectively. We saw that, for a given received SNR, OP-DD can attain the performance of OP-EM just by choosing a larger α. However, OP-EM still conserves its interest, as it permits a faster convergence of the receiver. Moreover, in general, the performance of OP-DD does not necessarily improve by increasing α, as there is a tradeoff between the channel-estimate quality and the detection SNR. Hence, it cannot always attain the performance of OP-EM. VI. C ONCLUSION We considered MIMO channel estimation using overlay pilots and proposed the SB estimator based on the EM algorithm. We showed that an interesting performance improvement is obtained as compared to PO-based estimation. The comparison made between OP and PSAM schemes suggests to prioritize OP-EM over not-tooshort channel coherence times. Indeed, although, for a larger coherence time, there is a weaker tradeoff between BER performance and data throughput for PSAM, the OP-EM scheme still preserves its interest. R EFERENCES [1] M. A. Khalighi and S. Bourennane, “Semiblind channel estimation based on superimposed pilots for single-carrier MIMO systems,” in Proc. VTC, Dublin, Ireland, Apr. 2007, pp. 1480–1484. [2] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, no. 4, pp. 686–693, Nov. 1991.

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