Table of Contents
ISRN Signal Processing
Volume 2011 (2011), Article ID 758918, 13 pages
http://dx.doi.org/10.5402/2011/758918
Research Article

Fisher-Information-Matrix Based Analysis of Semiblind MIMO Frequency Selective Channel Estimation

1Department of Electrical Engineering, I.I.T Kanpur, Kanpur 208016, India
2Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0407, USA

Received 8 May 2011; Accepted 21 June 2011

Academic Editors: Y. H. Ha, C.-C- Hu, A. Ito, A. Maier, A. Wong, and B. Yuan

Copyright © 2011 Aditya K. Jagannatham and Bhaskar D. Rao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a study of semiblind (SB) estimation for a frequency-selective (FS) multiple-input multiple-output (MIMO) wireless channel using a novel Fisher-information matrix (FIM) based approach. The frequency selective MIMO system is modeled as a matrix finite impulse response (FIR) channel, and the transmitted data symbols comprise of a sequence of pilot symbols followed by the unknown blind symbols. It is demonstrated that the FIM for this system can be expressed as the sum of the blind FIM Jb and pilot FIM Jp. We present a key result relating the rank of the FIM to the number of blindly identifiable parameters. We then present a novel maximum-likelihood (ML) scheme for the semiblind estimation of the MIMO FIR channel. We derive the Cramer-Rao Bound (CRB) for the semiblind scheme. It is observed that the semi-blind MSE of estimation of the MIMO FIR channel is potentially much lower compared to an exclusively pilot-based scheme. Finally, we derive a lower bound for the minimum number of pilot symbols necessary for the estimation of an FIR MIMO channel for any general semi-blind scheme. Simulation results are presented to augment the above analysis.

1. Introduction

Multiple-input multiple-output (MIMO) communication systems [1] have gained widespread popularity as technology solutions for current and future wireless systems such as 3G/4G, WBA (wireless broadband access), and 802.11n for high-speed WLAN (wireless local area network) applications amongst others. The detection performance of the designed MIMO receivers depends on the accuracy of the available channel estimate. Accurate estimates are also valuable for feedback-based schemes such as water-filling power allocation to enhance capacity [2] or design optimum precoders for MIMO transmission [3]. Channel estimation thus plays a key role in the performance gains achievable through deployment of MIMO systems.

In current signal processing research, there are two dominant and widely prevalent approaches for channel estimation. The first one is termed as pilot-based estimation [4, 5] and employs exclusively pilot symbols. These schemes have a low complexity of implementation and are robust in nature which makes them amenable for implementation in wireless systems. A downside to the employment of pilot-based schemes is that they cause a significant bandwidth overhead since pilot symbols convey no information. This overhead is higher in MIMO systems where the number of channel parameters to be estimated grows as the product of the number of receive and transmit antennas, requiring the transmission of an increasing number of pilot symbols, thus reducing the effective spectral efficiency (defined as the ratio of information bearing bits to total bits per unit bandwidth). The alternative to pilot-based estimation, is blind estimation [6, 7] which employs exclusively the statistical information available about the input symbols and channel. Since, in principle, it employs no training symbols, blind schemes have maximum effective spectral efficiency (ratio of information to total bits equals unity). However, blind schemes are computationally complex and typically identify a channel only up to a scaling, phase or permutation indeterminacy. To avoid the shortcomings of the above schemes, semi-blind (SB) techniques [810], which utilize both pilot and blind symbols (Figure 1), have gained popularity. These schemes utilize the statistical information available in the data symbols, thus enhancing the accuracy of the channel estimate, while employing a few pilot symbols to greatly reduce overhead and improve robustness. For instance, in the context of a SIMO wireless channel, it has been shown that with just one pilot symbol placed anywhere in the transmitted data block, the channel can be estimated without any residual indeterminacy [11]. Other interesting semi-blind schemes can be found in works such as [12, 13].

758918.fig.001
Figure 1: Schematic representation of an SB system.

The focus of this work is to present a new semi-blind MIMO estimation scheme and quantify the improvements in bandwidth efficiency and robustness of SB estimation schemes in general. For this purpose, it is necessary to address multiple aspects of SB estimation. The problem of identifiability from blind symbols forms a cornerstone of SB estimation, since SB schemes rely on estimating a significant component of the wireless channel from the blind symbols. Thus, by leveraging the blind statistical information to the greatest extent, they reduce the dependence on pilots or, in other words, achieve a greater accuracy of estimation from the limited pilot data. Therefore, we present a novel approach based on the Fisher information matrix (FIM) to characterize the blind identifiability of the MIMO channel. In a key result presented in this work, we demonstrate that the number of blindly identifiable parameters of the MIMO FIR system is equal to the rank of the FIM. This observation has been employed intuitively in literature without formal justification, such as in [14] where the authors make this observation in the context of a SIMO channel. Identifiability results in the context of SIMO and MIMO channels have been presented in works such as [11, 15], respectively. Such results in the context of systems employing space-time block codes (STBC) can be found in works such as [16, 17]. Our general FIM result can be employed as a unifying framework in such a wide variety of scenarios to characterize the identifiability of the underlying system.

Further, most works on identifiability such as [18] and others consider the transmitted blind symbols to be deterministically unknown in nature, due to the relative ease of analysis of such models. However, in our analysis, we assume that the blind symbols are stochastic in nature for the following reason. In the context of direction of arrival (DOA) estimation, it has been demonstrated in [19] that if 𝐶𝑠,𝐶𝑑 denote the Cramer-Rao bounds (CRB) on the covariance matrices of the stochastic and deterministic estimation schemes, then 𝐶𝑑𝐶𝑠. In other words, the stochastic model is statistically more efficient and has a lower mean square error (MSE) of estimation than its deterministic counterpart since the stochastic signal model has the advantage that the number of unknown parameters in the system no longer grows with the number of transmitted data symbols. Hence, our estimation schemes which are based on stochastic rather than deterministic blind symbols achieve a lower MSE of estimation by exploiting the statistical information available in the blind symbols.

For the context of MIMO frequency selective channel estimation, consider an 𝑟×𝑡 MIMO system (𝑟 = no. of receive antennas, 𝑡 = no. of transmit antennas) with 𝐿 channel taps. This system has 2𝑟𝑡𝐿 real parameters (i.e., 𝑟𝑡𝐿 complex parameters) to be estimated. The FIM-based approach can be used to demonstrate the result that the above MIMO FIR system has 𝑡2 parameters that cannot be identified from exclusively employing blind data. This is a central result for the estimation of MIMO FIR matrices and is derived from the MIMO identifiability results in [15, 20], wherein it has been shown that these 𝑡2 blindly unidentifiable parameters correspond to a 𝑡×𝑡 unitary matrix indeterminacy. Motivated by this observation, we present an SB estimation scheme for a FIR MIMO channel, which achieves the dual objective of robustness and spectral efficiency. This scheme employs a whitening-rotation-like decomposition of the FIR MIMO channel, where the whitening matrix is frequency selective. We then utilize the work in [7] where Tugnait and Huang have elaborated a blind algorithm based on linear prediction for the estimation of the frequency-selective whitening matrix of the MIMO frequency selective channel. Utilizing this scheme, in conjunction with an orthogonal pilot sequence-based maximum-likelihood (ML) scheme for the estimation of the unitary matrix indeterminacy [21], we describe a scheme for the semi-blind estimation of a MIMO FIR channel. Thus, we present a novel procedure for SB estimation of the MIMO FIR channel.

The final concern in this endeavor of SB channel estimation is to quantify the estimation accuracy of such schemes for a given number of pilot symbols. Once again, using the FIM-based approach above, a concise result can be found for the MSE of estimation of the SB scheme. We demonstrate that asymptotically, as the block length of data symbols becomes very large, the MSE of SB estimation is directly proportional to the number of blindly unidentifiable parameters or 𝑡2. Hence, while for exclusively pilot-based estimation the MSE grows proportional to the total number of parameters 2𝑟𝑡𝐿, the MSE bound for SB estimation is only related to a much smaller set of 𝑡2 blindly unidentifiable parameters. Thus, in summary, by exploiting the available blind symbols to the greatest possible extent, that is, by estimating all but a small fraction 𝑡2 of the total 2𝑟𝑡𝐿 parameters from the blind symbols, one can achieve a much lower MSE performance compared to exclusively pilot-based schemes. Finally, we consider the issue of minimum number of pilot symbols necessary for the estimation of the FIR MIMO channel. Indeed, the answer to this question is inherently related to the question of identifiability of the MIMO FIR channel, and, using the FIM approach, we demonstrate the key result that at least 𝑡 pilot symbols are necessary for the estimation of the MIMO FIR channel.

The rest of the paper is organized as follows. The MIMO FIR estimation problem is formulated in Section 2. In Section 3, we present results on the SB-FIM followed by Sections 4 and 5 which describe the semi-blind scheme for channel estimation and MSE analysis, respectively. Simulation results are presented in Section 7 followed by conclusions in Section 8. In what follows, 𝑖𝑚,𝑛 represents 𝑚𝑖𝑛;𝑖,𝑚,𝑛, where denotes the set of natural numbers, rank() is the rank of a matrix, and 𝒩() represents the null space of a matrix.

2. Problem Formulation

Consider an 𝐿 tap frequency-selective MIMO channel. The system input-output relation can be expressed as 𝐲(𝑘)=𝐿1𝑖=0𝐻(𝑖)𝐱(𝑘𝑖)+𝜂(𝑘),(1) where 𝐲(𝑘),𝐱(𝑘) are the 𝑘th received and transmitted symbol vectors, respectively. 𝜂(𝑘) is spatiotemporally white additive Gaussian noise of variance 𝜎2𝑛, that is, E{𝜂(𝑘)𝜂(𝑙)𝐻}=𝜎2𝑛𝛿(𝑘𝑙)𝐈𝑟, where 𝛿(𝑘)=1 if 𝑘=0 and 0 otherwise. Let 𝑡,𝑟 be the number of transmitters, receivers, and, therefore, 𝐲(𝑘)𝑟×1 and 𝐱(𝑘)𝑡×1. Each 𝐻(𝑖)𝑟×𝑡,𝑖0,𝐿1 is the MIMO channel matrix corresponding to the 𝑖th lag. Also, we assume 𝑟>𝑡, that is, the number of receivers is greater than the number of transmitters. Let {𝐱𝑝(1),𝐱𝑝(2),,𝐱𝑝(𝐿𝑝)} be a burst of 𝐿𝑝 transmitted pilot symbols. The subscript 𝑝 in the above notation represents pilots. Let 𝐇𝑟×𝐿𝑡 be defined as𝐇𝐻(0),𝐻(1),,𝐻𝐿1,𝐻(𝑖)𝐡1(𝑖),𝐡2(𝑖),,𝐡𝑡(𝑖).(2) Let the unknown blind information symbols (which yield only statistical information at the receiver) be stacked as 𝑁>𝐿 transmitted symbol vectors in 𝒳(𝑘) described by the system model given below as 𝐲(𝑘𝑁)𝐲(𝑘𝑁1)𝐲(𝑘1)𝑁+𝐿𝒴(𝑘)=𝐱(𝑘𝑁)𝐱(𝑘𝑁1)𝐱((𝑘1)𝑁+1)𝒳(𝑘)+𝜂(𝑘𝑁)𝜂(𝑘𝑁1)𝜂(𝑘1)𝑁+𝐿,(3) where the matrix (𝑁𝐿+1)𝑟×𝑁𝑡 is the standard block Sylvester channel matrix often employed for the analysis of MIMO FIR channels [14] and is given as 𝐻(0)𝐻(1)𝐻(2)𝐻𝐿100𝐻(0)𝐻(1)𝐻𝐿2𝐻𝐿100𝐻(0)𝐻𝐿3𝐻𝐿2.(4) The input vector 𝒳(𝑘)𝑁𝑡×1 is the data symbol block, where the length of each input block of data is 𝑁 symbols long. Such a stacking of the input/output symbols into blocks results in loss of a small number of output information symbols (𝐿1 symbols per block) due to interblock interference (IBI). This model is frequently employed in FIR system studies such as in [22] and is adapted because eliminating the IBI makes the analysis tractable by yielding simplistic-likelihood expressions. Let the transmitted data symbols 𝐱(𝑘) be spatiotemporally white, that is, E{𝐱(𝑘)𝐱(𝑙)𝐻}=𝜎2𝑠𝛿(𝑘𝑙)𝐈𝑡 and the normalized source power 𝜎2𝑠1. Hence, the covariance of the block input vector 𝒳(𝑘) is given as 𝒳E{𝒳(𝑖)𝒳(𝑖)𝐻}=𝐈𝑁𝑡. The MIMO transfer function of the FIR channel being defined as 𝐻(𝑧)=𝐿1𝑖=0𝐻(𝑖)𝑧𝑖 is assumed to satisfy the following conditions. (A.1)𝐻(𝑧) is irreducible, that is, 𝐻(𝑧) has full column rank for all 𝑧0 (including 𝑧=). It follows that if 𝐻(𝑧) is irreducible, the leading coefficient matrix [𝐡1(0),𝐡2(0),,𝐡𝑡(0)] has full column rank (substitute 𝑧= in 𝐻(𝑧)). (A.2)𝐻(𝑧) is column reduced, that is, the trailing coefficient matrix [𝐡1(𝐿1),𝐡2(𝐿1),,𝐡𝑡(𝐿1)] has full column rank.

In practical applications, since the fading coefficients of the wireless channel matrices arise from the random scattering effects of the ambient propagation environment, the above assumptions are satisfied with very high probability. For a discussion about special scenarios where the above conditions are not satisfied, the reader is referred to works [7, 23]. Next, we present insights into the nature of the above estimation problem.

3. Semiblind Fisher Information Matrix (FIM)

In this section, we formally setup the complex FIM for the estimation of the channel matrix 𝐇 and provide insights into the nature of semi-blind estimation. The parameter vector to be estimated 𝜃2𝐿𝑟𝑡×1 is defined by stacking the complex parameter vector and its conjugate as suggested in [21, 24] as 𝜃𝜃𝐻(0)𝜃𝐻(1)𝜃𝐻(𝐿1),where𝜃𝐻(𝑖)vec(𝐻(𝑖))vec(𝐻(𝑖))2𝑟𝑡×1,(5) and vec() denotes the standard matrix vector operator which represents a column-wise stacking of the entries of a matrix into a single column vector. In what follows, 𝑘0,𝐿1,𝑖1,𝑟𝑡. Observe also that 𝜃(2𝑘𝑟𝑡+𝑖)=𝜃((2𝑘+1)𝑟𝑡+𝑖). Let 𝐿𝑏 blocks of data symbols 𝒳(𝑝),𝑝1,𝐿𝑏 be transmitted. In addition, let the data symbol vectors 𝐱(𝑙),𝑙𝐿𝑝+1,𝑁𝐿𝑏+𝐿𝑝 be Gaussian. Then, 𝒴, the correlation matrix of the output 𝒴 defined in Section 2 is given as 𝒴=E𝒴(𝑙)𝒴(𝑙)𝐻=𝐻+𝜎2𝑛𝐈,(6) where 𝑦(𝑁𝐿+1)𝑟×(𝑁𝐿+1)𝑟. Hence, employing the zero IBI assumption as defined in the section above, the log-likelihood expression for the semi-blind scenario is given by a sum of the blind and pilot log-likelihoods as 𝒴;𝜃=𝑏+𝑝,(7) where 𝑏, the Gaussian log-likelihood of the blind symbols, is given as 𝑏=𝐿𝑏𝑘=1tr𝒴(𝑘)𝐻1𝒴𝒴(𝑘)𝐿𝑏lndet𝒴,(8) and 𝑝, the least-squares log-likelihood of the pilot part, is given as 𝑝=1𝜎2𝑛𝐿𝑝𝑖=1𝐲𝑝(𝑖)𝐿1𝑗=0𝐻(𝑗)𝐱𝑝(𝑖𝑗)2.(9) Hence, the FIM for the sum likelihood is given as 𝐽𝜃=𝐽𝑏+𝐽𝑝,(10) where 𝐽𝑏,𝐽𝑝2𝑟𝑡𝐿×2𝑟𝑡𝐿 are the FIMs for the blind and pilot symbol bursts, respectively, which are defined by the likelihoods 𝑏,𝑝 [21, 25]. This splitting of the FIM into pilot and blind components is similar to the approaches employed in [26, 27] and can be considered as representing the blind and pilot information components available for the estimation of the wireless channel. In the semi-blind scheme, we wish to make complete use of the blind and pilot information for channel estimation, as against an exclusively pilot-based scheme which employs only pilot information. This forms the central basis for the development of semi-blind estimation schemes. Next, we present a general result on the properties of the FIM before we apply it to the problem at hand in the succeeding sections.

3.1. FIM: A General Result

In this section, we present an interesting property of an FIM-based analysis by demonstrating a relation between the rank of the FIM and the number of unidentifiable parameters. Let 𝛼𝑘×1 be the complex parameter vector of interest. As described in [21, 24], for estimation of complex parameters, we employ a stacking of 𝛼 as 𝜃=[𝛼𝑇,𝛼𝐻]𝑇𝑛×1 where 𝑛2𝑘. Let 𝑝(𝜔;𝑔(𝜃)) be the pdf of the observation vector 𝜔 parameterized by 𝑔(𝜃), where 𝑔(𝜃)𝑛×1𝑙×1 is a function of the parameter vector 𝜃. Similar to stacking 𝛼,𝛼, let the function 𝑓(𝜃)𝑛×1𝑚×1,𝑚2𝑙 be defined as 𝑓𝜃=𝑔𝜃𝑔𝜃.(11) Given the log-likelihood (𝜔;𝜃)ln𝑝(𝜔;𝑓(𝜃)), the FIM 𝐽𝜃𝑛×𝑛 is given [25] as 𝐽𝜃E𝜕2𝜔;𝜃𝜕𝜃𝜕𝜃𝐻.(12) Let 𝑓(𝜃) be an identifiable function of the parameter 𝜃, that is, the FIM with respect to 𝑓(𝜃) has full rank. We then have the following result.

Lemma 1. Let 𝑝(𝜔;𝑓(𝜃)), be the pdf of the observation vector 𝜔, and let 𝑓(𝜃)𝑛×1𝑚×1 be a function of the parameter vector 𝜃 satisfying the following conditions. (C.1)Let 𝑓(𝜃) itself be an identifiable function of the parameter 𝜃, that is, the FIM with respect to 𝑓(𝜃) has full rank. (C.2)Let rank(𝒩(𝜕𝑓(𝜃)/𝜕𝜃))=𝑑, or, in other words, the dimension of the null space of 𝜕𝑓(𝜃)/𝜕𝜃 is 𝑑. Under the above conditions, the FIM 𝐽(𝜃)𝑛×𝑛 is rank deficient and, moreover, rank𝐽𝜃=𝑛𝑑.(13)

Proof. Let 𝑝(𝜔;𝑓(𝜃)) be the pdf of the observations 𝜔. The derivative of the log-likelihood with respect to the parameter vector 𝜃 is given as 𝜕𝜕𝜃ln𝑝𝜔;𝑓𝜃=𝜕𝜕𝑓𝜃ln𝑝𝜔;𝑓𝜃𝜕𝑓𝜃𝜕𝜃.(14) The unconstrained FIM for the estimation of the parameter vector 𝜃 is given as𝐽𝜃=E𝜕𝜕𝜃ln𝑝𝜔;𝑓𝜃𝑇𝜕𝜕𝜃ln𝑝𝜔;𝑓𝜃=𝜕𝑓𝜃𝜕𝜃𝑇E𝜕𝜕𝑓𝜃ln𝑝𝜔;𝑓𝜃𝑇𝜕𝜕𝑓𝜃ln𝑝𝜔;𝑓𝜃𝜕𝑓𝜃𝜕𝜃.𝐽𝑓𝜃(15)Hence, from the condition C.2 above, it follows that rank(𝐽(𝜃))=𝑛𝑑.

Thus, the rank of the FIM is deficient by 𝑑, which is the number of unidentifiable parameters. We now provide a deeper insight into the above result that connects the nature of the FIM to the number of unidentifiable parameters. Explicitly, let 𝜃 be reparameterized by the real parameter vector 𝜉[𝜉𝑇1,𝜉𝑇2]𝑇, 𝜉1𝑑×1,𝜉2𝑑×1 as 𝜃(𝜉). Let 𝑓(𝜃(𝜉))𝑚×1 satisfy the property 𝜕𝑓𝜃𝜉𝜕𝜉2=𝜕𝑓𝜃𝜕𝜃𝜕𝜃𝜕𝜉2=𝟎𝑚×𝑑,(16) or, in other words, the function 𝑓(𝜃) remains unchanged as the parameter vector 𝜃 varies over the 𝑑 dimensional constrained manifold 𝜃(𝜉2), and thus 𝜕𝑓(𝜃)/𝜕𝜃 has at least a 𝑑 dimensional null space. The parameter vector 𝜉2 is the unconstrained parameterization of the constraint manifold and represents the unidentifiable parameters. This implies that each parameter 𝜃𝑖 in 𝜃 is identifiable only up to 𝑑 degrees of freedom owing to the unidentifiability of the 𝜉2 component which is of dimension 𝑑. This result has interesting applications, especially in the investigation of identifiability issues in the context of blind and semi-blind wireless channel estimation. The implications of this result in the context of semi-blind MIMO channel estimation are explored in the next section where we examine the rank of the semi-blind FIM and derive further insights into the nature of the estimation problem.

3.2. Blind FIM

We now apply the above result to our problem of MIMO FIR channel estimation. We start by investigating the properties of the blind FIM 𝐽𝑏. Let the block Toeplitz parameter derivative matrix (𝑘)(𝑁𝐿+1)𝑟×𝑁𝑡 be defined employing complex derivatives as (𝑘𝑟𝑡+𝑖)𝜕𝜕𝜃2𝑘𝑟𝑡+𝑖.(17) From the results for the Fisher information matrix of a complex Gaussian stochastic process [28], 𝐽𝑏 defined in (10) above is given as 𝐽𝑏2𝑘𝑟𝑡+𝑖,2𝑙𝑟𝑡+𝑗=𝐽𝑏(2𝑙+1)𝑟𝑡+𝑗,(2𝑘+1)𝑟𝑡+𝑖=𝐿𝑏tr(𝑘𝑟𝑡+𝑖)𝐻1𝒴(𝑙𝑟𝑡+𝑗)𝐻1𝒴,𝐽𝑏2𝑘𝑟𝑡+𝑖,(2𝑙+1)𝑟𝑡+𝑗=𝐽𝑏(2𝑙+1)𝑟𝑡+𝑗,2𝑘𝑟𝑡+𝑖=𝐿𝑏tr(𝑘𝑟𝑡+𝑖)𝐻1𝒴(𝑙𝑟𝑡+𝑗)𝐻1𝒴,(18) where 𝐽𝑏𝑘,𝑙 denotes its (𝑘,𝑙)th element. We can now apply the result in Lemma 1 above to this FIM matrix 𝐽𝑏, and we have the following result on the rank of the blind FIM for the MIMO FIR channel. Under the assumptions above, it is known [15, 20] that 𝐻(𝑧) can be identified up to a constant 𝑡×𝑡 unitary matrix from second-order statistical information. Such a unitary matrix has 𝑡2 real parameters, [21]. Hence, from Lemma 1 above, we have the following result.

Theorem 2. Let the MIMO FIR channel transfer function 𝐻(𝑧) satisfy (A.1) and (A.2) above. Then, the rank upper-bound on the blind FIM 𝐽𝑏2𝑟𝑡𝐿×2𝑟𝑡𝐿 defined in (18) above is given as rank𝐽𝑏=2𝑟𝑡𝐿𝑡2.(19) In fact, a basis for the 𝑡2×1 null space 𝒩(𝐽𝑏) is given by 𝑈(𝐇)2𝑟𝑡𝐿×𝑡2 as 𝑈(𝐇)𝑈(𝐻(0))𝑈(𝐻(1))𝑈𝐻𝐿,(20) where the matrix function 𝑈(𝐻)𝑟×𝑡2𝑟𝑡×𝑡2 for the matrix 𝐻=[𝐡1,𝐡2,,𝐡𝑡] is defined as 𝑈(𝐻)=𝐡1𝐡2𝐡3𝟎𝟎𝟎𝟎𝟎𝐡1𝐡2𝟎𝟎𝟎𝟎𝟎𝐡1𝟎𝟎𝐡2𝟎𝟎𝐡1𝟎𝟎𝐡2𝟎𝟎𝐡1𝟎𝟎.(21)

Proof. See Appendix A.

Thus, it is clear that MIMO FIR impulse response of the channel can be estimated up to an indeterminacy of 𝑡2 real parameters from the statistical or blind information. This result has significant implications for estimation of the MIMO channel. As 𝑟,𝐿 increase, the number of real parameters in the system, which is equal to 2𝑟𝑡𝐿 (the dimension of matrix 𝐽𝜃), increases manyfold. However, the number of parameters that cannot be identified from blind symbols may be as small as 𝑡2 implying that a wealth of data can be identified from the blind symbols without any need for pilots. This reduction in the number of parameters to be estimated from pilots results in a significant decreases in the MSE of estimation, a result which will be rigorously justified in Section 5. Next, we present an algorithm for the SB estimation of the MIMO FIR channel matrix.

4. Semiblind Estimation: Algorithm

As shown above, since only 𝑡2 (of the total 2𝑟𝑡𝐿) parameters cannot be identified from blind data, they can be identified from the pilot symbols. These 𝑡2 parameters correspond to a unitary matrix. More precisely, Let 𝐻(𝑧)𝑟×𝑡(𝑧) be the 𝑟×𝑡 irreducible channel transfer matrix. let the input-output system model be as shown in Section 2. Then, 𝐻(𝑧) can be identified up to a unitary matrix from the blind statistical information, that is, without the aid of any pilots. In the following discussion, let ̂𝜃 denote the estimate of the quantity 𝜃. The matrix 𝐇𝑟×𝐿𝑡 can be expressed as 𝐇=𝐖𝐈𝐿𝑄𝐻,where𝐖𝑊(1),𝑊(2),,𝑊𝐿1.(22) From the above result, the matrices 𝑊(𝑖),𝑖0,𝐿1 can be estimated from the blind second-order statistical information, that is, from the correlation lags 𝑅𝑦(𝑗)E{𝐲(𝑗)𝐲(0)𝐻},𝑗0,𝐿1, without the aid of pilot symbols. In the flat fading channel case (𝐿=1), this can be done by a simple Cholesky decomposition of the instantaneous output correlation matrix 𝑅𝑦(0) (as 𝑅𝑦(0)=𝑊(0)𝑊(0)𝐻). However, for the case of frequency selective channels, estimating the matrices 𝑊(𝑖) is not straight forward and a scheme based on designing multiple delay linear predictors is given in [7] (Set 𝑛𝑎=0,𝑑=𝑛𝑏=𝐿1 and it follows that 𝑊(𝑖)=𝐹𝑖). It thus remains to compute the unitary matrix 𝑄𝑡×𝑡, that is, 𝑄 is such that 𝑄𝑄𝐻=𝑄𝐻𝑄=𝐈 and 𝑄 has very few parameters (𝑡2 real parameters, [21]). In the next section, we present algorithms for the estimation of this unitary 𝑄 indeterminacy from the transmitted pilot symbols.

4.1. Orthogonal Pilot ML (OPML) for 𝑄 Estimation

We now describe a procedure to estimate the unitary matrix 𝑄 from an orthogonal pilot symbol sequence 𝑋𝑝. Let 𝑋𝑝(𝑖),𝑖0,𝐿1 be defined as 𝑋𝑝(𝑖)[𝐱𝑝(𝐿𝑖),𝐱𝑝(𝐿𝑖+1),,𝐱𝑝(𝐿𝑝𝑖)]. Let 𝑋𝑜𝑝 the pilot matrix be defined as 𝑋𝑜𝑝𝑋𝑝(0)𝑋𝑝(1)𝑋𝑝𝐿1.(23) The least squares cost function for the constrained estimation of the unitary matrix 𝑄 can then be written as 𝑌𝑝𝐿1𝑖=0𝑊(𝑖)𝑄𝐻𝑋𝑝(𝑖)2,subjectto𝑄𝑄𝐻=𝐈𝑡.(24) Let the pilot matrix 𝑋𝑜𝑝 be orthogonal, that is, 𝑋𝑜𝑝(𝑋𝑜𝑝)𝐻=𝐿𝑝𝐈𝐿𝑡. The cost minimizing 𝑄 is then given as𝑄=𝑈𝑉𝐻,where𝑈Σ𝑉𝐻=SVD𝐿1𝑖=0𝑋(𝑖)𝑌𝐻𝑊(𝑖).(25)

Proof. It follows from an extension of the result in [21].

Finally, 𝐇 is given as 𝐇𝐖(𝐈𝐿𝑄𝐻). It now remains to demonstrate a scheme to construct the orthogonal pilot matrix 𝑋𝑜𝑝 which is treated next.

4.2. Orthogonal Pilot Matrix Construction

An orthogonal pilot matrix in the context of MIMO FIR channels can be constructed by employing the Paley Hadamard (PH) orthogonal matrix structure shown in Figure 2, and such a scheme has been described in [29]. Another scheme based on signal constellations derived from the roots of unity is presented in [5]. The PH matrix has blocks of shifted orthogonal rows (illustrated with the aid of rectangular boundaries), thus giving it the block Sylvester structure. Each transmit stream of orthogonal pilots for the FIR system can be constructed by considering the “L”-shaped block shown in the figure and removing the prefix of 𝐿 (channel length) symbols at the receiver. Thus, the pilot matrix for orthogonal pilots 𝑋𝑜𝑝 is given as 𝑋𝑜𝑝𝐱𝑝𝐿𝐱𝑝𝐿+1𝐱𝑝𝐿𝑝𝐱𝑝𝐿1𝐱𝑝𝐿𝐱𝑝𝐿𝑝1𝐱𝑝(1)𝐱𝑝(2)𝐱𝑝𝐿𝑝𝐿+1.(26) Orthogonal pilots have shown to be optimally suited for MIMO channel estimation in studies such as [30, 31]. However, the Sylvester structure of FIR pilot matrices further constrains the set of orthogonal pilot symbol streams compared to flat-fading channels. As the number of channel taps increases, employing a PH matrix to construct an orthogonal pilot symbol stream implies choosing a PH matrix with a large dimension. This in turn implies an increase in the length of the pilot symbol sequence and hence a larger overhead in communication. This problem can be alleviated by employing a nonorthogonal pilot symbol sequence which results in slightly suboptimal estimation performance but enables the designer to choose any pilot length desired. This iterative general maximum-likelihood (IGML) scheme for channel estimation using nonorthogonal pilots is described in [21] for flat-fading channels and can be extended to FIR channels in a straight forward manner. Experimental results have shown its performance to be comparable to the above OPML scheme.

758918.fig.002
Figure 2: Paley Hadamard matrix.

5. Semiblind MSE of Estimation

Consider now the asymptotic performance of the semi-blind scheme from an FIM perspective. As the amount of blind information increases (which does not increase the pilot overhead of the system), the variance of estimation of the the covariance matrices 𝑅𝑦(𝑖) progressively decreases to zero, implying that the blindly identifiable parameters (such as the whitening matrix) can be estimated accurately. Thus, intuitively, the SB estimation problem reduces to the constrained estimation of the 𝑡2 blindly unidentifiable parameters from the pilot symbols, a proposition which is rigorously justified below. Hence, the limiting MSE is equal to the MSE for the complex constrained estimation of the 𝑡2 blindly unidentifiable parameters. SB techniques can therefore yield a far lesser MSE of estimation than an exclusively pilot-based scheme as illustrated by the following result.

Theorem 3. Let 𝐽𝑝=(𝐿𝑝/𝜎2𝑛)𝐈2𝑟𝑡𝐿, which is achieved by the orthogonal pilot matrix 𝑋𝑜𝑝. Then, as the number of blind symbol transmissions increases ( 𝐿𝑏), the semi-blind CRB 𝐽𝜃1 approaches the CRB for the exclusive estimation of the 𝑡2 blindly unidentifiable parameters. Further, the semi-blind MSE bound, given by the trace of the CRB matrix, converges to lim𝐿𝑏E𝐇𝐇2𝐹12tr𝐽𝜃1=𝜎2𝑛2𝐿𝑝𝑡2,(27) which depends only on 𝑡2, the number of blindly unidentifiable parameters.

Proof. Given the fact that 𝐽𝑝=(𝐿𝑝/𝜎2𝑛)𝐈2𝑟𝑡𝐿, the semi-blind FIM can be expressed as 𝐽𝜃=𝐿𝑝𝜎2𝑛𝐈2𝑟𝑡𝐿×2𝑟𝑡𝐿+𝐿𝑏̃𝐽𝑏,(28) where ̃𝐽𝑏 is the blind FIM corresponding to a single observed blind symbol block 𝒴 and is given as ̃𝐽𝑏(𝐽𝑏/𝐿𝑏), where the blind FIM 𝐽𝑏 is defined in (18). From Lemma 1, it can be seen that ̃𝐽𝑏 is rank deficient and rank(̃𝐽𝑏)=rank(𝐽𝑏)=2𝑟𝑡𝑡2. Let the eigen-decomposition of ̃𝐽𝑏 be given as ̃𝐽𝑏=𝐸𝑏Λ𝑏𝐸𝑏𝐻, where Λ𝑏(2𝑟𝑡𝑡2)×(2𝑟𝑡𝑡2) is a diagonal matrix. Then, 𝐽𝜃=𝐿𝑝𝜎2𝑛𝐸𝑏,𝐸𝑏𝐸𝑏,𝐸𝑏𝐻+𝐿𝑏𝐸𝑏Λ𝑏𝐸𝐻𝑏=𝐸𝑏,𝐸𝑏𝐿𝑝𝜎2𝑛𝐈+𝐿𝑏Λ𝑏𝟎𝟎𝐿𝑝𝜎2𝑛𝐈𝐸𝑏,𝐸𝑏𝐻.(29) Hence, the CRB 𝐽1(𝜃) is given as 𝐽1𝜃=𝐸𝑏,𝐸𝑏𝐿𝑝𝜎2𝑛𝐈+𝐿𝑏Λ𝑏1𝟎𝟎𝜎2𝑛𝐿𝑝𝐈𝐸𝑏,𝐸𝑏𝐻.(30) As the number of blind symbols 𝐿𝑏, the diagonal matrix ((𝐿𝑝/𝜎2𝑛)𝐈+𝐿𝑏Λ𝑏)1𝟎(2𝑟𝑡𝑡2)×(2𝑟𝑡𝑡2) in the above expression. Thus the semi-blind bound approaches the complex constrained Cramer-Rao bound (CC-CRB) [21] given as lim𝐿𝑏𝐽1𝜃=𝜎2𝑛𝐿𝑝𝐸𝑏𝐸𝑏𝐻.(31) In fact, the bound on the MSE is clearly seen to be given as Ê𝜃𝜃2𝐹𝜎2𝑛𝐿𝑝tr𝐸𝑏𝐸𝑏𝐻2E𝐇𝐇2𝐹𝜎2𝑛𝐿𝑝tr𝐸𝑏𝐸𝑏𝐻E𝐇𝐇2𝐹12𝜎2𝑛𝐿𝑝2𝑟𝑡2𝑟𝑡𝑡2=𝜎2𝑛2𝐿𝑝𝑡2,(32) which is the constrained bound for the estimation of the MIMO channel matrix 𝐇.

Thus, the bound for the MSE of estimation and hence the asymptotic MSE of the maximum-likelihood estimate of the channel matrix 𝐇 with the aid of blind information, are directly proportional to 𝑡2.

Contrast this result with the MSE of estimation exclusively using an exclusively pilot-based scheme (i.e., a scheme which does not leverage the blind data like the semi-blind scheme). This MSE is given as (1/2)tr({𝐽𝑝}1)=(𝜎2𝑛/2𝐿𝑝)2𝑟𝑡𝐿 and is proportional to 2𝑟𝑡𝐿, the total number of real parameters. Hence, the SB estimate which has an asymptotic MSE lower by a factor of 2(𝑟/𝑡)𝐿 can potentially be very efficient compared to exclusive pilot only channel estimation schemes. For instance, in a MIMO system with 𝑟=4,𝑡=2, and 𝐿=2 channel taps, the potential reduction in MSE by employing a semi-blind estimation procedure is 2(𝑟/𝑡)𝐿=9 dB as demonstrated in Section 7. Thus, the SB estimation scheme can result in significantly lower MSE. Finally, it is worth mentioning that the above results, which are derived employing the Gaussian data symbol distribution in Section 3, are in close agreement with the performance of a system employing a discrete signal constellation such as quadrature phase-shift keying (QPSK), as illustrated in the simulation results of Section 7.

6. Pilots and FIM

In this section, we obtain a lower bound for the minimum number of pilot symbols necessary to achieve regularity or a full rank FIM 𝐽𝜃, that is, for the SB identifiability of the MIMO channel 𝐻(𝑧). Recall that {𝐱𝑝(1),𝐱𝑝(2),,𝐱𝑝(𝐿𝑝)} are the 𝐿𝑝 transmitted pilot symbols. Then, the FIM of the pilot symbols 𝐽𝑝 is given as 𝐽𝑝=𝐿𝑝𝑖=1𝐽𝑝(𝑖), where 𝐽𝑝(𝑖) is the FIM contribution from the 𝑖th pilot symbol transmission. Given complex vectors in 𝑡×1, let the matrix function 𝑖𝑉𝑗(𝑡×1,𝑡×1)2𝑟𝑡×2𝑟𝑡 be defined as 𝑖𝑉𝑗𝐱𝑝(𝑖)𝐱𝑝(𝑗)𝐻𝐼𝑟𝟎𝟎𝐱𝑝(𝑖)𝐱𝑝(𝑗)𝑇𝐈𝑟,if𝑖,𝑗>0,(33) and 𝑖𝑉𝑗=𝟎2𝑟𝑡×2𝑟𝑡 if 𝑖0 or 𝑗0. After some manipulations, it can be shown that the FIM contribution 𝐽𝑝(𝑖)2𝑟𝑡𝐿×2𝑟𝑡𝐿 is given as𝐽𝑝(𝑖)=1𝜎2𝑛𝑖𝑉𝑖𝑖𝑉𝑖1𝑖𝑉𝑖𝐿+1𝑖1𝑉𝑖𝑖1𝑉𝑖1𝑖1𝑉𝑖𝐿+1𝑖𝐿+1𝑉𝑖𝑖𝐿+1𝑉𝑖1𝑖𝐿+1𝑉𝑖𝐿+1.(34) The following result bounds the rank of the semi-blind (pilot + blind) FIM 𝐽𝜃.

Theorem 4. Let 𝐿𝑝𝑡 pilot symbols 𝐱𝑝(1),𝐱𝑝(2),,𝐱𝑝(𝐿𝑝) be transmitted and let the matrix 𝐻(0) be full column rank as assumed above. A rank upper bound of the sum (pilot + blind) FIM 𝐽𝜃 defined in (10) above is given as rank𝐽𝜃2𝑟𝑡𝐿𝑡𝐿𝑝2,0𝐿𝑝𝑡(35) or, in other words, a lower bound on the rank deficiency is given as (𝑡𝐿𝑝)2.

Proof. See Appendix B.

The above result gives an expression for the rank upper bound of the MIMO FIR Fisher information matrix for each transmitted pilot symbol. Since identifiability requires a full rank FIM, it thus presents a key insight into the number of pilot symbols needed for identifiability of the MIMO FIR system as shown next.

6.1. Pilots and Identifiability

From the above result, one can obtain a lower bound on the number of pilot symbols necessary for SB identifiability of the MIMO channel. This result is stated below.

Lemma 5. The number of pilot symbol transmissions 𝐿𝑝 should at least equal the number of transmit antennas 𝑡 for the FIM 𝐽𝜃 to be full rank and hence the MIMO FIR system in (1) to be identifiable.

Proof. It is easy to see from (35) that, for 𝐿𝑝<𝑡, rank𝐽𝜃=2𝑟𝑡𝐿𝑡𝐿𝑝2<2𝑟𝑡𝐿,(36) that is, strictly less than full rank. As the number of pilot symbols increases, for 𝐿𝑝=𝑡, rank(𝐽𝜃)2𝑟𝑡𝐿𝑝, where 2𝑟𝑡𝐿𝑝 is the dimension of 𝐽𝜃 and therefore represents full rank. Hence, at least 𝑡 pilot symbols are necessary for the identifiability of the MIMO FIR wireless channel.

Thus at least 𝑡 pilot symbols are necessary for the system to become identifiable. One has to observe that in the case of semi-blind estimation potentially fewer number (𝑡2) of parameters need to be estimated. Hence, even though semi-blind schemes necessitate the transmission of 𝑡 pilot symbols, the accuracy of estimation of such a scheme can be higher owing to the fact that they estimate fewer parameters from the limited pilot symbols. This improvement in MSE performance has been quantified in Section 5 where we presented results about the asymptotic performance of the SB estimator using the above FIM framework. Next, we present results of simulation studies.

7. Simulation Results

In this section, we present results of simulation experiments to illustrate the salient aspects of the work described above. In a majority of our simulations below, we consider a 4×2 MIMO FIR channel with 2 taps, that is, 𝐿=2, 𝑟=4, and 𝑡=2. Each of the elements of 𝐇 is generated as a zero-mean circularly symmetric complex Gaussian random variables of unit variance, that is, a Rayleigh fading wireless channel. The orthogonal pilot sequence is constructed from Paley Hadamard matrices by employing the scheme in Section 4.1. The transmitted symbols, both pilot and blind (data), are assumed to be drawn from a quadrature phase shift keying (QPSK) symbol constellation [32].

Experiment 1. In Figure 3, we plot the rank deficiency of the FIM of a 6×5 MIMO FIR system (𝑟=6,𝑡=5) with 𝐿=5 channel taps as a function of the number of transmitted pilot symbols 𝐿𝑝. The rank was computed for 100 realizations of randomly generated Rayleigh fading MIMO channels, and the rank deficiency observed was precisely [25,16,9,4,1,0] for 𝐿𝑝=[0,1,2,3,4,5] transmitted pilot symbols, respectively. Hence, rank deficiency 25 for 𝐿𝑝=0 verifies that the assumptions (A.1), (A.2) (in turn Lemma 1) hold with overwhelming probability in the case of randomly generated MIMO channels. Further, for 1𝐿𝑝5, the rank deficiency is given as (5𝐿𝑝)2 which additionally verifies the bound in (35) for FIM rank deficiency as a function of number of pilot symbols.

758918.fig.003
Figure 3: Rank deficiency of the complex MIMO FIM versus number of transmitted pilot symbols (𝐿𝑝) for a 6×5 MIMO FIR system of length 𝐿=5.

Experiment 2. In Figure 4, we plot the MSE versus SNR when the whitening matrix 𝑊(𝑧) is estimated from 𝑁𝐿𝑏=1000,5000 blind received symbols employing the linear prediction-based scheme from [7]. The 𝑄 matrix is estimated from 𝐿𝑝=20 orthogonal pilot symbols employing the semi-blind scheme in Section 4.1. For comparison, we also plot the MSE when 𝐇 is estimated exclusively from training using least-squares [21, 25], the asymptotic complex constrained CRB (CC-CRB) given by (27), and the MSE of estimation with the genie-assisted case of perfect knowledge of 𝑊(𝑧). It can be observed that the MSE progressively decreases towards the complex constrained CRB as the number of blind symbols increases. Also observed as illustrated in Theorem 3, the asymptotic SB estimation error is 10log(32/4)=9 dB lower than the pilot-based scheme as illustrated in Section 5.
In Figure 5(a), we plot the MSE performance of the competing estimation schemes above for different transmitted pilot symbol lengths 𝐿𝑝 and 5000 transmitted QPSK data symbols (blind received symbols). As illustrated in Section 4.1, we employ Paley Hadamard matrices to construct the orthogonal pilot sequences. Since such matrices exist only for certain lengths 𝐿𝑝, we plot the performance for 𝐿𝑝=12,20,48,68,140 pilot symbols. The asymptotic semi-blind performance is 9 dB lower in MSE as seen above. Also, for a given number of blind symbols, the performance gap in MSE of performance of the semi-blind scheme with 𝑊(𝑧) estimation and that of the training scheme progressively decreases. This is due to the fact that more blind symbols are required to accurately estimate the whitening matrix 𝑊(𝑧) for the MSE performance of the semi-blind scheme to be commensurate with the performance improvement of the pilot-based scheme. Finally, in Figure 5(b), we plot the performance of the competing schemes for different number of blind symbols in the range 1000–5000 QPSK symbols and 𝐿𝑝=12 pilot symbols. The performance of the SB scheme with 𝑊(𝑧) estimated can be seen to progressively improve as the number of received blind symbols increases.

758918.fig.004
Figure 4: MSE versus SNR in a 4×2 MIMO channel with 𝐿=2 channel taps, 𝐿𝑝 = 20 pilot symbols.
fig5
Figure 5: MSE performance for estimation of a 4×2 MIMO frequency-selective channel. (a) MSE versus 𝐿𝑝 and (b) MSE versus number of blind symbols.

Experiment 3. We compare the symbol error rate (SER) performance of the training and semi-blind channel estimation schemes. At the receiver, we employ a stacking as in (3) of 7 received symbol vectors 𝐲(𝑘) followed by a MIMO minimum mean-square error (MMSE) detector [1] constructed from the MIMO channel matrix . In Figure 6, we plot the SER of detection of the transmitted QPSK symbols versus SNR in the range 2–16 dB. It can be seen that the asymptotic semi-blind estimator has a 1-2 dB improvement in detection performance over the exclusive training-based scheme. The SER drops from around 1×101 at 2 dB to 1×108 at 16 dB. Thus, an SB-based estimation scheme can potentially yield significant throughput gains when employed for the estimation of the wireless MIMO frequency-selective channel.

758918.fig.006
Figure 6: Symbol error rate (SER) versus SNR for QPSK symbol transmission of a 4×2 MIMO frequency selective channel with 𝐿=2 channel taps.

8. Conclusion

In this work, we have investigated the rank properties of the semi-blind FIM of a 𝐿 tap 𝑟×𝑡 (𝑟>𝑡) MIMO FIR channel. The MIMO channel transfer function 𝐻(𝑧) can be decomposed as 𝐻(𝑧)=𝑊(𝑧)𝑄𝐻, where the whitening transfer function 𝑊(𝑧) can be estimated from the blind symbols alone. A constrained semi-blind estimation scheme has been presented to estimate the unitary matrix 𝑄 from pilot symbols along with an algorithm to achieve an orthogonal pilot matrix structure for MIMO frequency selective channels using Paley Hadamard matrices. From an asymptotic MSE analysis, it has been demonstrated that the semi-blind scheme achieves a significantly lower MSE than an exclusively pilot-based scheme. It has also been demonstrated that at least 𝑡 pilot symbol transmissions are necessary to achieve a full-rank FIM (and hence identifiability). Simulation results demonstrate the performance of the proposed semi-blind scheme.

Appendices

A. Proof of Theorem 2

Proof. Consider the result for the simpler case of the flat fading channel, that is, 𝐿=1. Then, 𝐇==𝐻(0)=𝐻𝑟×𝑡. Let the system be parameterized as 𝛼=vec(𝐻) and, hence, from the discussion above, 𝜃vec(𝐻)vec(𝐻).(A.1) From the system model described in (1), the pdf of the observation vector 𝐲𝑟×1 is given by 𝐲𝒩(𝟎,𝑔(𝜃)), where 𝑔(𝜃)𝑟2×1 is the output correlation defined as 𝑔(𝜃)vec(𝐻𝐻𝐻+𝜎2𝑛𝐈). Consider a reparameterization of the channel matrix 𝐻 by the real parameter vector 𝜉 as 𝐻(𝜉)=𝑊(𝜉1)𝑄(𝜉2), where 𝑊(𝜉1)𝑟×𝑡 is also known as a whitening matrix and 𝑄(𝜉2)𝑡×𝑡 is a unitary matrix. 𝑔(𝜃) is now a many to one mapping since 𝑔(𝜃(𝜉))=vec(𝑊(𝜉1)𝑊(𝜉1)𝐻+𝜎2𝑛𝐈) and 𝑓𝜃𝜉=vec𝑊𝜉1𝑊𝜉1𝐻+𝜎2𝑛𝐈vec𝑊𝜉1𝑊𝜉1𝐻+𝜎2𝑛𝐈,(A.2) which is independent of the parameter vector 𝜉2. Hence, it is a many to one mapping since for all unitary matrices 𝑄(𝜉2) we have 𝜕𝑓𝜃𝜉𝜕𝜉2=𝟎𝑟2×𝑡2,(A.3) since 𝜉2𝑡2×1 (𝑡2 is the number of real parameters to characterize a 𝑡×𝑡 unitary matrix). Hence, 𝑑=𝑡2, and, from Lemma 1, the first result follows. The proof for the general frequency selective case follows from the result in [15].
For the second result, let Φ(𝐻𝐻𝐻+𝜎2𝑛𝐈)1. Let the blind FIM 𝐽𝑏 be block partitioned as 𝐽𝑏𝐽𝑏11𝐽𝑏12𝐽𝑏21𝐽𝑏22.(A.4) It can be verified from (18) that 𝐽𝑏21=𝐽𝑏12𝐻 and 𝐽𝑏22=𝐽𝑏11𝑇. The block components of the FIM are given as 𝐽𝑏11=𝐡𝐻1Φ𝐡1Φ11𝐡𝐻1Φ𝐡1Φ21𝐡𝐻1Φ𝐡𝑡Φ𝑟1𝐡𝐻1Φ𝐡1Φ12𝐡𝐻1Φ𝐡1Φ22𝐡𝐻1Φ𝐡𝑡Φ𝑟2𝐡𝐻1Φ𝐡1Φ1𝑟𝐡𝐻1Φ𝐡1Φ2𝑟𝐡𝐻1Φ𝐡𝑡Φ𝑟𝑟𝐡𝐻2Φ𝐡1Φ11𝐡𝐻2Φ𝐡1Φ21𝐡𝐻2Φ𝐡𝑡Φ𝑟1𝐡𝐻2Φ𝐡1Φ12𝐡𝐻2Φ𝐡1Φ22𝐡𝐻2Φ𝐡𝑡Φ𝑟2𝐡𝐻𝑡Φ𝐡1Φ1𝑟𝐡𝐻𝑡Φ𝐡1Φ2𝑟𝐡𝐻𝑡Φ𝐡𝑡Φ𝑟𝑟,(A.5) which can be written succinctly as (𝐻𝐻Φ𝐻)Φ𝑇. Similarly, 𝐽𝑏12 is given as 𝐽𝑏12=𝜒11𝜒11𝜒12𝜒11𝜒11𝜒21𝜒1𝑟𝜒𝑡1𝜒11𝜒12𝜒12𝜒12𝜒11𝜒22𝜒1𝑟𝜒𝑡2𝜒11𝜒1𝑟𝜒12𝜒1𝑟𝜒11𝜒2𝑟𝜒1𝑟𝜒𝑡𝑟𝜒21𝜒11𝜒22𝜒11𝜒21𝜒21𝜒2𝑟𝜒𝑡1𝜒21𝜒12𝜒22𝜒12𝜒21𝜒22𝜒2𝑟𝜒𝑡2𝜒21𝜒1𝑟𝜒22𝜒1𝑟𝜒21𝜒2𝑟𝜒2𝑟𝜒𝑡𝑟𝜒𝑡1𝜒11𝜒𝑡2𝜒11𝜒𝑡1𝜒21𝜒𝑡𝑟𝜒𝑡1𝜒𝑡1𝜒1𝑟𝜒𝑡2𝜒1𝑟𝜒𝑡1𝜒2𝑟𝜒𝑡𝑟𝜒𝑡𝑟,(A.6) where 𝜒𝐻𝐻Φ. It can now be seen that 𝐽𝑈=0, where 𝑈 is as defined in (21). For instance, the top 𝑡 elements of 𝐽𝑈(,1) (where we employ MATLAB notation and 𝑈(,1) denotes the first column of 𝑈) are given as [𝐽𝑏11,𝐽𝑏12]𝑈(,1)=(𝐡𝐻1Φ𝐡1)Φ𝑇𝐡2+(𝐡𝑇2Φ)𝑇(𝐡𝐻1Φ𝐡1)=𝟎, and so on. The structure of the FIM for the most general case of arbitrary 𝐿 is complex, but the result can be seen to hold by employing a symbolic manipulation software tool such as the MATLAB symbolic toolbox package.
It can be seen that the top part of the null space basis matrix 𝑈(𝐻) is 𝑈(𝐻(0)). As assumed earlier, rank(𝐻(0))=𝑡. Now it is easy to see that if 𝑈(𝐻) is rank deficient, 𝑈(𝐻(0)) is rank deficient, and, from its structure, 𝐻(0) is rank deficient violating the assumption. Hence, rank(𝑈(𝐻))=𝑡2.

B. Proof of Theorem 4

Proof. 𝐽𝑏 and 𝐽𝑝(𝑖),𝑖1,𝑘 are positive semidefinite (PSD) matrices. We use the following property: if 𝐴,𝐵 are PSD matrices, (𝐴+𝐵)𝐯=𝟎𝐴𝐯=𝐵𝐯=𝟎. Therefore, 𝐽𝜃𝐯=0𝐽𝑏𝐯=𝐽𝑝(𝑖)𝐯=𝟎2𝑟𝑡𝐿×1,forall𝑖1,𝑘. In other words, 𝒩(𝐽)=𝒩𝐽𝑏𝐿𝑝𝑖=1𝒩(𝐽𝑝(𝑖)).(B.1) Let 𝐯𝒩(𝐽). Then, from the null space structure of 𝐽𝑏 in (21), it follows that 𝐯=𝑈(𝐇)𝐬, where 𝐬𝑡2×1. Also, 𝐽𝑝(𝑖)𝐯=𝐽𝑝(𝑖)𝑈(𝐇)𝐬=𝟎,𝑖1,𝐿𝑝.(B.2) From Lemma 6, this implies that 𝑖𝑉𝑖𝑈(𝐻(0))𝐬=𝟎,forall𝑖1,𝐿𝑝, where 𝑖𝑉𝑖 is as defined in (33). Let the matrix 𝒯(𝐮)𝑡×12𝑡×𝑡2 be defined as 𝒯(𝐮)0𝐮1𝐮2𝐮100𝐮2𝐮1000𝐮1.(B.3) Recall that 𝐻(0) is assumed to have full column rank. Then, from the structure of 𝑈(𝐻(0)) given in (21), it can be shown that the relation above holds if and only if, 𝐬=𝟎, where the matrix 2𝐿𝑝𝑡×𝑡2 is given as 𝒯𝐱𝑝(1)𝒯𝐱𝑝(2)𝒯𝐱𝑝𝐿𝑝.(B.4) It can then be seen that matrix 𝔾2𝑡𝐿𝑝×𝐿𝑝2 forms a basis for the left nullspace of , that is, 𝔾𝑇=𝟎, where 𝔾𝐱𝑝(1)𝐱𝑝(2)𝟎𝐱𝑝(1)𝟎𝐱𝑝(2)𝟎𝟎𝐱𝑝(1)𝟎𝐱𝑝(1)𝟎.(B.5) Thus, rank()2𝑡𝐿𝑝𝐿𝑝2, and, therefore, right nullity (or nullity) of is dim(𝒩())𝑡2(2𝑡𝐿𝑝𝐿𝑝2). And, therefore, rank(𝐽𝜃)2𝑟𝑡𝐿dim(𝒩())=2𝑟𝑡𝐿𝑡2+(2𝑡𝐿𝑝𝐿𝑝2).

Lemma 6. Let 𝐽𝑝(𝑖)𝐯=𝟎,forall𝑖1,𝐿𝑝, where 𝐯=𝑈(𝐇)𝐬. Then, 𝑖𝑉𝑖𝑈(𝐻(0))𝐬=𝟎,forall𝑖1,𝐿𝑝, where 𝑖𝑉𝑖 is as defined in (33).

Proof. Consider 𝐽𝑝(1), the FIM contribution of the first transmitted pilot symbol. It can be seen clearly that 𝐽𝑝(1) is given as 𝐽𝑝(1)=1𝑉1𝟎2𝑟𝑡×(2𝐿2)𝑟𝑡𝟎(2𝐿2)𝑟𝑡×2𝑟𝑡𝟎(2𝐿2)𝑟𝑡×(2𝐿2)𝑟𝑡.(B.6) Hence, 𝐽𝑝(1)𝑈(𝐇)𝐬=𝟎 implies that 1𝑉1𝑈(𝐻(0))𝐬=𝟎. Further, from the properties of the matrix Kronecker product, one has 𝐴𝐵𝐶𝐷=(𝐴𝐶)(𝐶𝐷). Substituting 𝐴=𝐱𝑝(𝑖),𝐵=𝐱𝑝(𝑖)𝐻,𝐶=𝐷=𝐈𝑟, one can then obtain that 𝐾(𝐱𝑝(1))𝑈(𝐻(0))𝐬=𝟎𝐾𝐱𝑝(𝑖)𝐱𝑝(𝑖)𝐻𝐈𝑟𝟎𝟎𝐱𝑝(𝑖)𝑇𝐈𝑟.(B.7) Since 𝐻(0) (and hence 𝐻(0)) is full rank, after some manipulation it can be shown that the above condition implies 𝒯(𝐱𝑝(1))𝐬=𝟎. Now consider the contribution of the second pilot transmission 𝐽𝑝(2). 𝐽𝑝(2)𝑈(𝐇)𝐬=𝟎 implies that 𝐾𝐱𝑝(2)𝑈(𝐻(0))𝐬+𝐾𝐱𝑝(1)𝑈(𝐻(1))𝐬=𝟎.(B.8) Since 𝒯(𝐱𝑝(1))𝐬=𝟎 and 𝑈(𝐻(0)),𝑈(𝐻(1)) have the same structure, it can be shown that 𝐾(𝐱𝑝(1))𝑈(𝐻(1))𝐬=𝟎, and, hence, it follows from the above equation that 𝐾(𝐱𝑝(2))𝑈(𝐻(0))𝐬=𝟎 which in turn implies that 𝒯(𝐱𝑝(2))𝐬=𝟎 and hence 2𝑉2𝑈(𝐻(0))=𝟎, and so on. This proves the lemma.

Acknowledgment

This work was supported by UC Discovery Grant com04-10176.

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