Blind Channel Estimation in MIMO OFDM Systems with Multiuser Interference Sarod Yatawatta and Athina P. Petropulu Electrical and Computer Engineering Department Drexel University, Philadelphia PA 19104 {sarod, athina}@ece.drexel.edu
Abstract We consider a multi-user OFDM system, where each user transmits using all available subcarriers. We propose a linear non-redundant block precoding scheme that is applied at the input of the OFDM system. The precoding spreads the symbols of each user over all subcarriers, thus increasing multipath diversity. At the same time, it introduces a structure to the transmitted symbols, which is exploited at the receiver to estimate the channel in a blind fashion. The proposed channel estimation approach employs computationally simple cross-correlation operations and yields the channel up to a diagonal ambiguity. It does not require channel length information and is not sensitive to additive stationary noise. The precoding does not increase transmission power and maintains even distribution of power between OFDM blocks. We provide the general description of precoding matrices, and also analytical expressions of symbol error probability and signal to interference ratio, which could be used to obtain optimum precoding schemes.
I. I NTRODUCTION The Multiple-Input Multiple-Output (MIMO) channel estimation problem appears in the context of single user as well as multiple-user systems. In single user systems, it has been shown that multiple transmit and receive antennas, along with coding, can achieve higher rates and diversity than those in a single transmit/receive antenna system [1]. In particular, space-time coding [5] has recently attracted a lot of attention. In a multi-user network, allowing more users to share the communication channel simultaneously can also be viewed as a way to increase throughput [8]. In order to obtain the theoretically achievable high rates, channel state information is needed at the receiver. In static or slowly varying This work was supported by ONE under grant N00014-03-1-0123. Preliminary results of this work were reported in IEEE Statistical Signal Processing Workshop, St. Louis, MO, 2003.
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environments, such information can be obtained using training symbols. In faster varying environments the training overhead can consume a significant amount of bandwidth. Blind channel estimation is a more bandwidth efficient approach in those cases. OFDM systems have recently emerged as good candidates for future generation high-rate wireless systems due to their efficient utilization of bandwidth and robustness to multipath. In this paper we focus on blind channel estimation for OFDM systems. The literature along these lines can be categorized as follows. •
Single-user single antenna OFDM systems- Each symbol is sent over a different carrier, over which it encounters flat fading. Considering blocks of received and transmitted symbols, one can formulate a MIMO problem, where the channel matrix is diagonal. There is a cornucopia of available methods for blind channel estimation in this case [26][28],[27]. A weakness of single antenna OFDM systems is the lack of multipath diversity. Precoding can spread each symbol over multiple carriers thus increasing diversity. Unitary precoding has been shown to provide maximum spectral efficiency [30], however, it does not allow for blind channel estimation. A non-unitary linear precoding scheme was proposed in [6], which trades-off performance for blind channel estimation.
•
Single-user OFDM systems with multiple transmit/receive antennae- Multiple transmit/receive antennas in combination with coding are used to improve diversity and rate. Some representative schemes include the space time block codes (STBC) [5], space time trellis codes, and layered space time [29]. In this case, the channel matrix is non diagonal. A channel estimation scheme for such MIMO systems is the two-input one-output MIMO OFDM system studied in [9], where STBC was applied at the inputs and the system was estimated by exploiting the structure of the codes. This approach relies on transmission redundancy, i.e., each information symbol is transmitted twice in two consecutive time intervals through two different antennas. In [4], the same setup as in [9] was used, and the channel was estimated based on a subspace approach.
•
Multi-user OFDM systems without multi-user interference- Each user is allocated a disjoint set of subcarriers [3]. Such systems are also referred to as orthogonal frequency division multiple access (OFDMA) and have been adopted in IEEE 802.16 standard. In such systems there is no multi-user interference and the channel matrix is diagonal. Thus, any single-user frequency domain channel estimation method can be applied to this scenario.
•
Multi-user OFDM systems with multi-user interference- All users use all available subcarriers independently. Each user can employ one or more antenna. Inevitably, in this case there is multi-user
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interference and the channel matrix is non diagonal. The majority of the single-user OFDM channel estimation methods do not apply to this case. There exists only a few channel estimation schemes for such systems. In [2], an OFDM system with cyclic prefix has been considered. Each block was multiplied with a scalar that varied periodically between blocks. Cyclic statistics of the received blocks, before the removal of the cyclic prefix, were used to yield the channel estimate. This kind of precoding results in variable power between blocks. In [25], a semi blind channel estimation method was proposed for multi-user OFDM systems that employ zero-padding instead of cyclic prefix. We should note here that the channel estimation methods developed for multi-user multi-antenna systems can be applied to most single user multi-antenna systems. At this point one might wonder why we could not just apply to the OFDM case already existing blind MIMO estimation methods that were proposed in different contexts. In fact, efforts along these lines have been reported. In [16] the authors applied the method proposed in [17] for extracting the inputs. Joint diagonalization of the output covariance is required in these methods, which is computationally intensive [18]. Higher-order statistics (HOS) based methods (see [24] and reference therein) could also be applied to solve the MIMO OFDM problem. However, such an approach would require high-complexity and would only work in the case of a small number of carriers. As it was also commented in [2], for large number of carriers, the Discrete Fourier Transform (DFT) performed at the OFDM modulation step would Gaussianize the data, thus the corresponding HOS estimates would be almost zero. In addition, HOS estimates have high computational complexity. Since the key idea behind OFDM systems is simplicity of equalization, one should seek simpler solutions for channel estimation that are consistent with the OFDM goals. Linear block precoding (LBP) has been used extensively for increasing throughput and diversity gains in MIMO OFDM systems [11],[21],[22]. The design of codes to achieve these goals has recently become an area of huge interest and vast potential. For instance, algebraic number theory [14], coding theory [15], and iterative greedy optimization [13] have being used to design optimal codes. However, very little work has been done on developing LBP schemes that allow channel estimation. In this paper we propose a blind estimation method of an MR × MT , (MR ≥ MT ) MIMO OFDM system with MT users and MR receivers. A non redundant linear block precoding is applied at the inputs before they enter the OFDM system, which increases multipath diversity and allows for blind channel estimation at the receiver. The proposed channel estimation approach employs computationally simple cross-correlation operations and yields the channel up to a diagonal ambiguity. It does not require channel length information, and is not sensitive to additive stationary noise. The precoding does not April 25, 2005
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increase transmission power and unlike the method of [2] maintains even distribution of power between OFDM blocks. We provide the general description of precoding matrices. We also provide analytical expressions of symbol error probability and signal to interference ratio, which could be used to obtain optimum precoding schemes. The proposed method can be extended to multi-user STBC systems. The rest of the paper is organized as follows. In Section II we consider a multi-user OFDM system and formulate the channel estimation scheme. We also discuss the general properties of precoding schemes, and provide one specific precoding example. In Section III, we apply the channel estimation method to a multi-user STBC OFDM system. In Section IV, we derive performance metrics for the proposed precoding scheme. In Section V we demonstrate performance and validity of analysis via simulations, and finally conclude in Section VI. A. Notation We will denote all vectors by lower case bold face letters and all matrices by capital bold face letters. All vectors are defined as column vectors. XT denotes matrix transpose, XH denotes matrix Hermitian and X† denotes matrix pseudo inverse. Diag(x) denotes a diagonal matrix with the vector x across its diagonal. ep denotes a vector consisting of all zeros except a one at location p; these vectors will be used for selecting the p-th column of a matrix X via the operation Xep , or selection of the p-th row of X via the operation eTp X. IM denotes an M × M identity matrix (sometimes the subscript M is omitted
for simplicity). 1M denotes a vector of size M × 1 with all elements equal to one. b.c denotes floor operation, while k.k denotes matrix or vector Frobenious norm. II. T HE P ROPOSED
PRECODING AND BLIND CHANNEL ESTIMATION APPROACH FOR MULTI - USER
OFDM
SYSTEMS
A. System setup and assumptions Let us consider a wireless system where MT users are allowed to transmit simultaneously without any bandwidth restrictions. Each user transmits using a single antenna. The receiver, equipped with MR , (MR ≥ MT ) antennas will attempt to recover the symbols of each user. Each user’s stream is OFDM modulated. The OFDM modulation consists of separating the symbols into blocks of length N , applying an N -point inverse DFT (IDFT) on each block, and prepending the last Ncp samples of the IDFT as a cyclic prefix. The augmented blocks, now of length N + Ncp , are transmitted in a serial fashion through the multipath channel. At the receiver, the cyclic prefix is discarded, and an N -point DFT is performed on
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the remainder of each block. Assuming that Ncp is greater or equal to the channel length, no interblock interference will occur at the receiver [12]. In a single user system, assuming that the channel does not change within a block, the k-th symbol within a block is received weighted by the k-th carrier gain. In our MR × MT multi-user OFDM system the received symbol at the m-th receive antenna consists of contributions from all MT users, i.e., i ym (k) =
MX T −1 p=0
Hmp (k)sip (k) + nm (k), k = 0, ..., N − 1, m = 0, ..., MR − 1
(1)
where i is the block index; k is the carrier index; p is the user index; sip (k) is the transmitted symbol over the k-th carrier, which, as will be explained latter, is derived from the source symbols dip (k) for k = 0, ..., N − 1; Hmp (k) is the gain of the k-th carrier between the m-th receive antenna and the p-th
user; nm (k) denotes noise. Let hmp (n) denotes the symbol rate cross-channel between the m-th receive antenna and the p-th user, then it holds: Hmp (k) =
L−1 X n=0
2π
hmp (n)e−j N kn , k = 0, ..., N − 1
(2)
where L (L ≤ Ncp ) is the length of the longest cross-channel. Our goal is to estimate the channel matrix
p=0,...,MT −1 H(k) = {Hmp (k)}m=0,...,M , based on the received symbols. However, due to the fact that L N , R −1
p=0,...,MT −1 in order to recover h(n) = {hmp (n)}m=0,...,M , n = 0, ..., L − 1, we only need to estimate H(k) for R −1
L distinct values of k.
Before we proceed, we make the following assumptions: (A0) We consider a synchronous communication system. (A1) Each user emits i.i.d. symbols from independent digitally modulated sources, with statistics: E{dip (k)(djq (m))? } = δkm δij δpq σ 2 , and E{|dip (k)|4 } = η 4 .
(A2) The noise processes are white, circularly Gaussian, temporally and spatially uncorrelated and are independent of the data. It is assumed that E{np (k)(nq (m))? } = δkm δpq σn2 . (A3) The channel is considered to be slowly varying with the block index i. In particular, it is assumed to be invariant during the number of blocks that are necessary for good channel estimation. (A4) For channel identification, we assume there is a non-empty set of k’s, i.e. L, so that for k ∈ L, H(k) has at least one row full of non-zero elements. For recovery of the transmitted symbols, we
assume the channel matrices H(k) to be full rank for all k. The u-th row of H(k) represents the k-th carrier gains experienced between the u-th receive antenna and all sources. Assumption (A4) requires that for k ∈ L, there is some value of u so that all these April 25, 2005
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gains are different than zero. B. Linear precoding 4
Let dip = [dip (0) ... dip (N − 1)]T denote the i-th block of symbols corresponding to the p-th user before any precoding. During the i-th block, the symbols of the p-th user to be transmitted over carrier k are generated based on an N × 1 code vector, wp (k; i), as: sip (k) = wpH (k; i)dip
(3)
Equivalently, the i-th block of the p-th user is precoded according to: 4
sip = [sip (0) ... sip (N − 1)]T = [wp (0; i), wp (1; i), ...., wp (N − 1; i)]H dip = Wpi dip
(4)
where Wpi is a matrix whose k-th row equals wpH (k; i). The purpose of the precoding matrix is to introduce some correlation structure in the transmitted blocks. Also, by changing the coding matrix between subsequent blocks we will create diversity that will allow us to estimate the channel matrix. Since we will need to obtain autocorrelation estimates, the coding matrices have to stay the same for a number of blocks. Thus, in the following, we will take them to be periodic in i with period M , M ≥ MT . i.e., Wpi = Wpi+M , ∀i, p
(5)
The selection of those values will be discussed in Section II-F.
C. Blind channel estimation For the i-the received block, let us gather the symbols of all users that were received on carrier k in vector yi (k). Then, based on (4) it holds: 4
i yi (k) = [y0i (k) ... yM (k)]T = H(k)si (k) + n(k) R −1
(6)
4
where si (k) = [si0 (k) .... siMT −1 (k)]T . Based on (3), (6) can be written as: yi (k) = H(k)Ψi (k)di + n(k)
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(7)
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where
4
4 i Ψ (k) =
w0H (k; i) 0 .. .
...
0
w1H (k; i) . . . .. .. . .
0 .. .
0
0
0
H . . . wM (k; i) T −1
and di = [(di0 )T , (di1 )T , . . . , (diMT −1 )T ]T .
(8)
For a fixed k, estimating H(k) in (7) is a typical MIMO channel estimation problem. We will next exploit the periodicity of the precoding matrices to estimate the channel in a computationally simple manner. Let us consider a sequence of received symbols that start at block i and are spaced apart by M blocks: ˜ i (k) = {yi (k), yi+M (k), yi+2M (k), . . .} y
(9)
˜ i (k) will be the same. Due to (5), the precoding matrix corresponding to all blocks of y ˜ i (k), i.e., Consider the correlation matrix of y 4
yi (k)˜ yi (l)H }, i = 0, ..., M − 1 Rikl = E{˜
(10)
We present the main result as the following proposition. Proposition 1: Let l ∈ L, and u such that the elements of HH (l)eu are all different than zero. Define w0H (k; 0)w0 (l; 0) ··· w0H (k; M − 1)w0 (l; M − 1) 4 .. .. .. (MT × M ) Ukl = . . . H H wM (k; 0)wMT −1 (l; 0) · · · wM (k; M − 1)wMT −1 (l; M − 1) T −1 T −1 (11) and 4
Cukl =
1 M −1 eu ] [R0 eu , R1kl eu , ..., Rkl σ 2 kl
(MR × M ).
(12)
If Ukl is a full row rank matrix, then it holds: 2
σ 4 ˆ lu (k) = H Cukl U†kl = H(k)Diag(HH (l)eu ) + δk,l n2 eu 1TMT U†kl . σ The proof is given in Appendix I.
(13)
Remarks ˆ lu (k), l ∈ L equals the true channel matrix H(k) within a diagonal The above proposition indicates that H
ambiguity matrix, with a bias term appearing for k = l. In the subsequent section we will propose ways to resolve the diagonal ambiguity. It is interesting to note that the noise contribution affects H(l) only.
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Assuming that N > L, there is enough redundancy in the frequency domain, to recover that particular sample based on the other recovered samples. The full row rank requirement on Ukl guarantees that U†kl exist. A necessary condition for that is M ≥ MT . However, increasing M means increasing the number of correlations to estimate, thus requiring longer data samples. Hence the best possible choice for M would be MT . We should note here that channel length information was not needed for obtaining Hlu (k). If channel length L is known, we can directly estimate the channel in time domain by using the fact that H(k) = PL−1 −j2πpk/N . For l 6= k (13) equals: p=0 h(p)e Cukl U†kl
=
L−1 X
h(p)e−j2πpk/N Diag(HH (l)eu )
(14)
p=0
Selecting a set of frequencies k1 , k2 , ... in the set L we can construct the matrix equation −1 [h(0) h(1) . . . h(L − 1)] = [Cuk1 l U†k1 l . . . Cuk|L| l U†k|L| l ] I ⊗ Diag(HH (l)eu ) † I I ... e−j2πk1 /N I e−j2πk2 /N I ... × .. .. .. . . . e−j2π(L−1)k1 /N I e−j2π(L−1)k2 /N I . . .
(15)
where ⊗ is the Kronecker product.
As long as |L| > L, we have a full rank matrix on the right hand side of (15) and can directly estimate h(n) for n ∈ [0, L − 1]. In practice we could apply (15) even if true L was unavailable, i.e., by using Ncp .
D. Resolving the diagonal ambiguity The diagonal ambiguity involves the l−th carrier gains across all channels between the u-th receive antenna and all sources. There are two approaches that we can use to eliminate this diagonal ambiguity. 1) Resolving ambiguity based on pilots: We can transmit pilots on the l-th carrier, based on which we can estimate H(l) directly. Note that in general, it is not possible to transmit pilots when precoding is done. Hence, we need to transmit pilots in a specific set of blocks corresponding to indices {p1 , p2 , p3 , . . . , pK }, that will not be precoded, neither will be used in the estimation of correlations. Based on these blocks we can estimate H(l) as: ˆ H(l) ≈ [yp1 (l), yp2 (l), . . . , ypK (l)][sp1 (l), sp2 (l), . . . , spK (l)]† . April 25, 2005
(16)
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Finally, based on (13) the ambiguity can be resolved entirely. Moreover, the estimates corresponding to different values of u (see (12)) can be combined together to further improve the channel estimate as follows. For l ∈ L define 4 ˜ kl = C [C0kl , C1kl , . . . , C(MR −1)kl ] H H H = H(k) Diag[H (l)e0 ]Ukl , Diag(H (l)e1 )Ukl , . . . , Diag(H (l)eMR −1 )Ukl
˜ kl . = H(k)U
(17) (18) (19)
˜ kl will have full rank. According to (A4), there will be at least one u with all nonzero HH (l)eu and U
Thus, an estimate of H(k) can be obtained as: b ˜† . H(k) = C˜kl U kl
(20)
2) Resolving ambiguity in a blind fashion for high SNR: We can also reduce the diagonal ambiguity into a unit modulus diagonal ambiguity without the need of pilots when the noise is negligible. Let 4
ˆ † (m)˜ ˜ zilu (m) = H yi (m), m ∈ [0, N − 1] lu
(21)
and the correlation of z˜ilu (m), 4
zilu (m)H } zilu (m)˜ Qilu (m) = E{˜
(22)
for some fixed index m ∈ [0, N − 1], u ∈ [0, MR − 1]. It is shown in Appendix II that for high SNR: i ))1/2 = |Diag(HH (l)eu )| σ(Qilu (m))−1/2 (Diag(wmm
(23)
where 4
i H wkm (k; i)wMT −1 (m; i)]T . = [w0H (k; i)w0 (m; i), ..., wM T −1
(24)
Thus, the magnitude of the diagonal ambiguity matrix of (13) can be computed via (23). Note that the value of |Diag(HH (l)eu )| in (23) is independent of m and i. Hence, we can select any m ∈ [0, N − 1] and i ∈ [0, M − 1] for this evaluation. For better accuracy, this can be evaluated as the
average corresponding to different values of m and i. Finally, the channel matrix can be obtained within a constant unit modulus diagonal ambiguity, i.e., 1 i Cukl U†kl (Qilu (m))1/2 (Diag(wmm ))−1/2 σ
(25)
= H(k)Diag[e−j arg Hu0 (l) , . . . , e−j arg HuMT −1 (l) ].
(26)
4 ˜ H(k) =
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Based on the obtained channel, the transmitted symbols of each user can be recovered within a phase ambiguity, or equivalently, within a rotation. The rotation can be resolved by transmitting at least one training symbol per user. E. Implementation issues The cross correlation Rikl in (10) can be estimated based on I received blocks as follows: ˆi R kl
=
b I−i−1 c M
1 b I−i−1 M c
+1
X
yi+jM (k)(yi+jM (l))H .
(27)
j=0
In the channel estimate given in (13), the diagonal ambiguity involves the cross channels Hu0 (l) to Hu(MT −1) (l), hence l and u must be selected so that those values are different than zero. Since channel
information is not available, the selection of those values will be based on information that is available at the receiver. ˆ i for all possible values of i and l, and let us estimate the H ˆ lu (l) according to Let us consider R ll
Proposition 1 for all values of u ∈ [0, MR − 1]. We have ˆ lu (l) ≈ H(l)Diag[H ? (l), . . . , H ? H uMT −1 (l)] u0
(28)
ˆ lu (l) ≈ [|Hu0 (l)|2 , . . . , |HuMT −1 (l)|2 ]. eTu H
(29)
and subsequently,
ˆ lu (l). Therefore, we can select Thus we can estimate the magnitudes of Hu0 (l) to HuMT −1 (l) using eTu H l and u such that |Hu0 (l)|, ...., |HuMT −1 (l)| are all non zero. If there are more than one choices for those
values, we pick those for which the minimum possible value of |Hu0 (l)|, ....|HuMT −1 (l)| is maximized. We should also note that the freedom in selecting l relies on the matrices Ukl being invertible for all k for which H(k) needs to be estimated i.e., at least L such values if channel length is L. If we do not
have freedom to choose l, we can only vary u. Notwithstanding this, the method will fail if the channel has a deep fade on l, i.e. kHlu (l)k ≈ 0. F. Selection of precoding matrices Let us first consider the criteria for the design of the precoding matrices. •
Identifiability: In order to obtain the estimate of (13), Ukl must be full rank for a given l and k ∈ [0, N − 1]. We see that on any row of Ukl , the elements are derived from the precoding matrices
of one source, and, on any column of Ukl , the elements are derived from precoding matrices of one block number. April 25, 2005
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Hence, in order to have full rank, each source must vary the precoding matrices with the block number such that the variation of the inner product between the k-th row and the l-th row of the precoding matrices is unique. This can be compared with the precoding scheme employed in [2] where the authors use precoding unique in the cyclostationary domain for each antenna. •
Decodability: Once obtaining the channel estimate and equalizing, we need to obtain the original decoded symbols. Hence Wpi , p ∈ [0, MT − 1] must be full rank.
•
Power Conservation: The precoding scheme should keep the power constant as for an uncoded OFDM system. Hence, trace(Wpi (Wpi )H ) =
σo2 N, ∀i ∈ [0, MT − 1] σ2
(30)
where σo2 is the symbol power in an uncoded OFDM system. We should note that there are many possible schemes that satisfy the above conditions. For an MR by MT system with M periodic precoding, we may need at most MT × M precoding matrices. However,
we can reduce this to MT matrices instead, by using the scheme given in Table I, where we have made M = MT , the lowest possible value. TABLE I ASSIGNMENT OF PRECODING MATRICES WITH BLOCK NUMBER AND SOURCE
block i source p
0
1
...
MT − 1
MT
...
0
W0
W1
...
WMT −1
W0
...
1 .. .
WMT −1 .. .
W0 .. .
... .. .
WMT −2 .. .
WMT −1 .. .
... .. .
MT − 1
W1
W2
...
W0
W1
...
In the scheme given in Table I, we have used matrices W0 to WMT −1 between the sources in a cyclic manner. Hence, in this scheme, the matrices Ukl become circulant and for identifiability, the MT point DFT of any row of these matrices should have non-zero coefficients. 1) An example of precoding matrices : Let us consider an extension of the single-user OFDM precoding scheme of [6], [10], where a linear combination of two symbols are transmitted on each carrier. We first select a fixed m from [0, N − 1]. Let us derive W0 based on its rows: αvH (k) + (−1)k βvH (m), k ∈ [0, N − 1], k 6= m w0H (k; i) = γvH (m), k=m
April 25, 2005
(31)
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where vH (k) is the k-th row of an arbitrary unitary matrix V and α, β, γ ∈ C. Let us select the remaining MT − 1 matrices W1 to WMT −1 to be unitary, for instance, we can select them such that Wp = IN , p ∈ [1, MT − 1]
(32)
In this scheme we only precode the first block and the remaining M − 1 blocks of each period are transmitted uncoded. The channel H(k), k = 0, ..., N − 1 βγ ? 0 ... 0 βγ ? k Ukm = (−1) . .. .. . 0 0 ...
For identifiability, we need
can then be estimated according to |γ|2 0 1 0 , k 6= m, Umm = .. . ? 1 βγ
Proposition 1 based on 1 ... ... |γ|2 . . . . . . . (33) .. . 2 . . . . . . |γ|
|γ|2 6= 1, β 6= 0, γ 6= 0,
(34)
and for block power conservation we need |γ|2 + (N − 1)(|α|2 + |β|2 ) = N.
(35)
In spite its simplicity, this scheme does not allow us to choose the value of l at the receiver, since we must take l = m. Hence, if H(m) has no rows with all elements non zero, the method will fail. It is interesting to note that in this scheme, we can directly transmit pilots on the m-th carrier of the precoded blocks because symbols on those carriers are only scaled, i.e. no linear combination of symbols is performed. III. M ULTIUSER OFDM
SYSTEMS WITH SPACE TIME CODING
In this section, we consider adaptation of the proposed method to suit Space-Time Block Coded systems. Starting with the seminal work by Alamouti [5], STBC have been adopted from a symbol level implementation to a block level implementation in numerous ways. In this paper we consider STBC pertaining to OFDM, as given in [9]. Let us first consider a single user STBC OFDM system, without any precoding. The transmitted blocks for each of the two antennae will be as given in Table II (a) for a single user system. At the receiving antenna, after OFDM demodulation, we have y 2i (k) H00 (k) H01 (k) d2i (k) = + n(k) (y 2i+1 (k))? (H01 (k))? −(H00 (k))? d2i+1 (k) April 25, 2005
(36)
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13
which can be considered as a 2 by 2 multi-user OFDM system provided the data vectors d2i and d2i+1 are mutually uncorrelated. Extension of the above scheme for more than one user have appeared in [19] and references therein. Using this result, we can generalize the precoding scheme proposed in section II to a multi-user STBC OFDM system. However, for simplicity, we only consider the two user case. We pursue a combined linear precoding and STBC scheme as given in Table II (b), which is the combined application of the schemes of Tables I and II (a). The precoding cycle here repeats every 8 blocks. TABLE II S PACE - TIME CODING FOR A
SINGLE AND
2
USER SYSTEM
(b) source 1
time
(a)
time
antenna 0
antenna 1
2i
d2i
d2i+1
2i + 1
−(d2i+1 )?
(d2i )?
source 2
s0
s1
s2
s3
0
W0 d00
W3 d10
W2 d01
W1 d11
1
−(W3 d10 )?
(W0 d00 )?
−(W1 d11 )?
(W2 d01 )?
2
W1 d20
W0 d30
W3 d21
W2 d31
3
−(W0 d30 )?
(W1 d20 )?
−(W2 d31 )?
(W3 d21 )?
4
W2 d40
W1 d50
W0 d41
W3 d51
5
−(W1 d50 )?
(W2 d40 )?
−(W3 d51 )?
(W0 d41 )?
6
W3 d60
W2 d70
W1 d61
W0 d71
7
−(W2 d70 )?
(W3 d60 )?
−(W0 d71 )?
(W1 d61 )?
At the receiver, after OFDM demodulation, we have i yST (k) = H(k)siST (k) + n(k), i = 0, 1, . . . , k ∈ [0, N − 1]
(37)
i (k) = [y 2i (k), (y 2i+1 (k))? , y 2i (k), (y 2i+1 (k))? ]T , si (k) = [si (k), si (k), si (k), si (k)]T and where yST 0 1 2 3 0 1 ST 0 1 H00 (k) H01 (k) H02 (k) H03 (k) (H01 (k))? −(H00 (k))? (H03 (k))? −(H02 (k))? . (38) H(k) = H10 (k) H11 (k) H12 (k) H13 (k) (H11 (k))? −(H10 (k))? (H13 (k))? −(H12 (k))?
If di (see (7)) are independent for any given i, (37) can be considered as a 4 by 4 multi-user system
and Proposition 1 can be applied to estimate H(k). However, (38) contains only 8 unknowns. Consider
April 25, 2005
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14
the reduced channel matrix:
˜ H(k) =
H00 (k) H01 (k) H02 (k) H03 (k) H10 (k) H11 (k) H12 (k) H13 (k)
.
(39)
Since we have more equations than unknowns, we can obtain a least squares channel estimate as follows. Let us for some fixed u ∈ [0, 3], define 0 (0, v) . . . R3 (0, v) (R0 (1, v))? . . . (R3 (1, v))? R 4 kl kl kl ˜ v (k, l) = kl , v ∈ [0, 3] R 0 3 0 ? Rkl (2, v) . . . Rkl (2, v) (Rkl (3, v)) . . . (R3kl (3, v))?
(40)
to be a 2 by 16 matrix constructed from the elements of R0kl to R3kl in (10). The u, v element of matrix Rikl is given by Rikl (u, v). Let us also construct the matrices D0 = Diag[(H00 (l))? , (H01 (l))? , (H02 (l))? , (H03 (l))? ], D1 = Diag[H01 (l), −H00 (l), H03 (l), −H02 (l)], D2 = Diag[(H10 (l))? , (H11 (l))? , (H12 (l))? , (H13 (l))? ], D3 = Diag[H11 (l), −H10 (l), H13 (l), −H12 (l)]. Let us assume that H(l) is available through pilots.
In Appendix III we show that 4 ˜ [ ? ? † ˜ ˜ H(k) =R v (k, l) [Dv Ukl PDv Ukl ] = H(k), k 6= l
with
0 −1 0
1 P= 0 0
0 0 0
0
0 . 0 −1 1 0 0
(41)
(42)
The selection of l and u can be done according to section II-E. IV. P ERFORMANCE
ANALYSIS
Since there is ample freedom to select the precoding matrices, it is paramount to derive criteria that will guide us in selecting good precoding schemes. Hence, we derive the analytical BER and the SINR (Signal to Noise Plus Interference Ratio) in this section. One could use these metrics to evaluate different precoding schemes, or even find the optimum ones that will minimize BER and SINR. First, we derive the channel estimation error which will be used in all other derivations. A. Channel estimation error Proposition 2: Consider the channel estimation error on each carrier for some fixed l ∈ L: 4 ˆ ˆ lu (k) − Hlu (k))(Diag(HH (l)eu ))−1 , k ∈ [0, N − 1]. ξ(k) = H(k) − H(k) = (H
April 25, 2005
(43)
DRAFT
15
For σ 2 σn2 it holds: E{ξ(k)} = δk,l
σn2 eu 1TM U†k,l Diag(HH (l)eu )−1 σ2
(44)
and E{ξ(k1 )H ξ(k2 )} = Diag(HH (l)eu )−H (U†k1 l )H Rk1 k2 U†k2 l Diag(HH (l)eu )−1
(45)
where the (i, j) element of Rk1 k2 is given as Ri,j =
δi,j T i e H(l)Ψ (l) trace(Ak1 ,k2 )I u b I−i−1 M c+1
(46)
η 4 − 2σ 4 + Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) ΨiH (l)HH (l) σ4 ! σn2 + 2 δk1 k2 Rill /σ 2 + trace(Ak1 ,k2 )I eu σ
and Ak1 ,k2 = ΨiH (k1 )HH (k1 )H(k2 )Ψi (k2 ). The proof is given in Appendix IV. ˆ If channel length information is available, e.g., L, the obtained channel estimate H(k) , can be improved
by taking an inverse DFT and enforcing the channel length. This process is referred to as denoising. Improvement can also be achieved if an upper bound on the length, instead of the actual length, is used. ˜ Let ξ(k) denote the denoised error corresponding to ξ(k). The relationship between the two can be
obtained using the DFT. Moreover, the error variance after denoising can be given as −L+1 L−1 N −1 N −1 2π 1 X X X X H˜ ˜ E{(ξ(m1 ))H ξ(m2 )}ej N ((m1 −k)n1 +(m2 −k)n2 ) E{(ξ(k)) ξ(k)} = 2 N n =0 n =0 m =0 m =0 1
2
1
(47)
2
where the values on the right hand side are taken from (45). 4
˜ ˜ − 1)), formed using the denoised error. We also define the error variance Let ξ = Diag(ξ(0), . . . , ξ(N 4
as ζ = E{ξ H ξ}. Since ξ is block diagonal, ζ will also be block diagonal. Using this result, we can estimate the mean squared error (MSE) given the channel as N −1 X 1 1 4 H ˆ ˆ trace(ζ) trace E{(H(k) − H(k)) (H(k) − H(k))} = M SE = N MT MR N MT MR
(48)
k=0
Before proceeding, we make several key assumptions about the error ξ for analytical tractability.
•
We assume that E{ξ} = 0, which is a reasonable assumption considering (44) where we have non zero mean only for k = l.
•
We assume that the normalized channel estimation error is small, i.e. kH(k)k kξ(k)k and is independent of noise. This is normally valid for high signal-to-noise ratio (SNR) regimes, and can
April 25, 2005
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be considered as a secondary assumption to the one made in proposition 2 while deriving (45). If the normalized error is not small, it can be either due to the noise being high or due to the channel being not identifiable. In either case, we will have poor performance in terms of the BER and thus the analysis in this section would not be applicable. •
We assume each block on the diagonal of ξ to have i.i.d. entries. In that case, E{ξ H ξ} should be diagonal. Close scrutiny of (45) reveals that if the Ukl matrices are diagonal, ζ becomes diagonal, thus justifying our assumption. Hence, we assume Ukl matrices to be diagonal. This assumption holds in the example precoding scheme for k 6= l.
In the following bit-error probability and SINR analysis, instead of the channel error covariance, ζ = E{ξ H ξ}, we will need an expression of the from E{ξ H Aξ}, where A will be some deterministic
matrix of size N MR × N MR . The latter expression is related to ζ as described in Proposition 3. 4
Proposition 3: Let D(ζ, A) = E{ξ H Aξ} where ζ = E{ξ H ξ} and A is an N MR × N MR deterministic matrix. It holds: D(ζ, A) =
where
1 Diag[trace(A0,0 )trace(ζ(0))I, . . . , trace(AN −1,N −1 )trace(ζ(N − 1))I] MT MR
A=
A0,0
A0,1
...
A0,N −1
A1,0 .. .
A1,1 .. .
... .. .
A1,N −1 .. .
AN −1,0 AN −1,1 . . . AN −1,N −1
and ζ = Diag[ζ(0), . . . , ζ(N − 1)].
(49)
(50)
Proof: Let A(k) be an MR × MR matrix. Let E{ξ(k)H ξ(k)} = ζ(k). If we assume ξ(k) (size
MR × MT ) to have i.i.d. elements with variance ν 2 and zero mean, then ζ(k) = ν 2 MR I. Moreover, E{ξ(k)H A(k)ξ(k)} =
1 trace(A(k))trace(ζ(k))I MT MR
(51)
Let us consider the block diagonal matrix ξ now. Note that A is partitioned into matrices Ap,q , p, q ∈ [0, N − 1] of size MR × MR . E{ξ H Aξ} will be block diagonal with each block as given by (51).
B. Analytical bit error probability For the sake of simplicity, we ignore the block index i, and denote the transmitted symbols at any block index by d. Using (7) and stacking up we can write y = Fd + n April 25, 2005
(52) DRAFT
17
where y = [y(0)T , . . . , y(N − 1)T ]T is an N MR × 1 vector and n = [n(0)T , . . . , n(N − 1)T ]T . The matrix F represents the combined effect of the precoding and the channel. First, the elements of MT sources are precoded by MT matrices. Next, they are permuted into N different subcarriers. Finally,
they are multiplied by the channel matrices on each subcarrier. Hence we have F = HTW
(53)
where H = Diag(H(0), . . . , H(N − 1)) and W = Diag(W0 , . . . , WMT −1 ). The permutation matrix T has ones on row p column bp/MT c + (p mod MT )N , for 0 ≤ p ≤ N MT − 1. The reason behind the special structure of T is explained in [32]. Let G be the equalizer matrix constructed from the channel estimate. Ideally, we will have GF = I, however in practice this will not be the case due to channel estimation errors. Thus, our estimate for d will be ˆ = GFd + Gn ≈ GFd + F† n d
(54)
where, assuming a zero-forcing equalizer, b †. G = W−1 T−1 H
(55)
Note that T−1 is the anti-permutation matrix of T and on the p-th row of T−1 = TH , a one appears on bp/N c + (p mod N )MT for 0 ≤ p ≤ N MT − 1. Let the channel estimation error be ξ such that
Then we have
b = H+ξ H
(56)
G = W−1 T−1 (I + H† ξ)−1 H† ≈ W−1 T−1 (I − H† ξ)H†
(57)
due to normalized error being small, i.e. kHk kξk.
ˆ , i.e. d(p) ˆ , which is the estimate of d(p) in vector d. We assume Let us consider the p-th element in d √ all symbols are drawn from white, identical 4-QAM sources. Let us consider d(p) = (1 + j)σ/ 2, while
all the other symbols in d act as zero-mean, i.i.d. noise sources with variance σ 2 . Since N MT is large, we can apply central limit theorem to model as Gaussian, the additive affect of all other symbols towards ˆ the estimation of d(p). Moreover, since the noise is Gaussian, d(p) will also be Gaussian distributed. ˆ to determine its p.d.f. Hence we need to only estimate the first and second order statistics of d(p)
From (54), we have ˆ = (I − W−1 T−1 H† ξTW)d + W−1 T−1 H† n. d
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(58)
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Let us consider the error ˆ − d(p) = eT d ˆ e(p) = d(p) p − d(p)
(59)
= eTp (I − W−1 T−1 H† ξTW)(d(p)ep + =
−eTp W−1 T−1 H† ξTWd +
NM T −1 X q=0,6=p
d(q)eq ) + eTp W−1 T−1 H† n − d(p)
eTp W−1 T−1 H† n.
We have E{e(p)} = 0 because E{ξ} = E{n} = 0 and moreover, due to the independence of ξ and n, we have σe2 = E{e(p)? e(p)}
(60)
= E{dH WH TH ξ H (H† )H T−H W−H ep eTp W−1 T−1 H† ξTWd} +E{nH (H† )H T−H W−H ep eTp W−1 T−1 H† n} = σ 2 trace(WH TH D(ζ, (H† )H T−H W−H ep eTp W−1 T−1 H† )TW) +σn2 eTp W−1 T−1 H† (H† )H T−H W−H ep
where D(., .) is obtained based on Proposition 3. We should note here that the above analysis concerned a specific block i, the index of which was dropped for notational convenience. Since within M blocks the precoding matrices change, the bit error probability should be computed as the average over M blocks. The bit error probability of p-th symbol in the i-th block, the mean for all symbols of all users in the i-th blocks, and the latter averaged over M blocks are respectively (see Eq. (8.15) of [31]): Pei (p) ≈ Q(
σ 1 ), Pei = σe N MT
NM T −1 X
Pei (p), Pe =
p=0
where the last result is due to averaging over i from 0 to M − 1.
M −1 1 X i Pe M
(61)
i=0
C. SINR ˆ Let us next consider the SINR. Starting from (54), we can calculate the contribution to d(p) from
noise and all the other symbols. The noise contribution will be σn2 eTp GGH ep ≈ σn2 eTp F† (F† )H ep . Let ˆ : us consider the contribution of d(q) to d(p)
2 ? T H H T ˆ E{kd(p)k q } = E{d(q) eq F G ep d(q)ep GFeq }
(62)
˜ −H ep eT W−1 T−1 (I − H† ξ)TWeq } = σ 2 E{eTq WH TH (I − ξ H (H† )H )T−H W p = σ 2 δpq + eTq WH TH D(ζ, (H† )H T−H W−H ep eTp W−1 T−1 H† )TWeq April 25, 2005
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19
where D(., .) is obtained based on Proposition 3. Using superposition, we have for the i−th block SIN Rpi = PN MT −1 q=0,6=p
2} ˆ E{kd(p)k p
(63)
2 } + σ 2 eT (WH TH HH HTW)−1 e ˆ E{kd(p)k p q n p
and the average SINR will be
SIN Ri =
1 N MT
NM T −1 X
SIN Rpi , SIN R =
p=0
M −1 1 X SIN Ri . M
(64)
i=0
Using this result, the BER can be calculated as p Pe (p) ≈ Q( SIN Rp ).
(65)
The expressions derived in this section can be used to compare different precoding schemes and notably, can be used to design optimum precoding schemes. For instance, instead of closed form solutions, numerical optimization schemes could readily be pursued in designing such codes. We are currently pursuing such approaches, and preliminary results can be found in [23]. For instance, we can consider the example precoding scheme given in II-F.1. In [23] we have used genetic algorithms to find optimal parameters α, β, γ for this scheme given the channel. We used expressions (61) and (65) as optimality criteria. We have seen that optimized values indeed perform better than values chosen in an adhoc manner. V. S IMULATION
RESULTS
We next test the performance of the proposed blind channel estimation method via simulations. In all cases we used 4-QAM sources without channel coding, unless stated otherwise. Once the channel was estimated, the symbols were recovered using zero forcing equalizer.
A. Time varying channels We considered 2 channels. The cross-channels had length 3 and were generated based on the modified Jakes Model [20]. This model gives uncorrelated channel taps. For our simulations we considered a normalized Doppler frequency of fd = 10−5 (fast fading). We generated 75 independent channels, and for each channel realization we applied the proposed blind channel estimation method for 10 independent input realizations. The 75 channels were selected so that they all satisfied the non-singularity assumption, i.e., max(cond(H(k))) ≤ 10 1
1
We should note that the
cond(H(k)) = kH(k)ks kH† (k)ks , where k.ks is the spectral norm.
April 25, 2005
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latter condition is not required for channel estimation but rather for symbol recovery. In practice, since forward error control is used, the channels do not have to satisfy that condition. For precoding we used the scheme of Section (II-F.1). We took W1 = I as in (32), and constructed W0 as in (31) with V = I and parameters α = j0.9, β = 0.432, γ = 1.1. Evaluation of the metric of (61)
for α, β in a 10 × 10 grid on the complex plane (α, β ∈ Z , 0 ≤ real(Z) ≤ 1, 0 ≤ imag(Z) ≤ 1) using a set of randomly generated channels indicated that values α = j0.9, β = 0.432, γ = 1.1 was one of the best choices. The index m needed in (31) for precoding was chosen arbitrarily at m = 2 for all channels. The OFDM block length was N = 64 with a cyclic prefix of 8. The blind channel estimation method was applied as described in Proposition 1, followed by (25). In order to determine the magnitude of the diagonal ambiguity, (21) was evaluated for all carriers [0, N − 1] and then averaged. The residual constant phase ambiguity was resolved based on N/2 pilot symbols transmitted before the real data transmission. The obtained channel estimate was improved via denoising where perfect channel length information was assumed. The average BER obtained based on all channel and input realizations is shown in Fig. 1 (a) for different SNR levels. The number of blocks used in the estimation process are shown next to each curve. These numbers resulted in the best performance of the proposed method in each case. We should mention here that, using (20) for resolving the magnitude of the diagonal ambiguity did not change the performance obtained with the blind method much. The advantage of a blind method as opposed to a training-based one is less overhead. To see how much we would save in terms of overhead if we were to use the proposed blind method as opposed to a training based method and achieve the same BER performance, we conducted simulations as follows. For the training based method, we transmitted 2 blocks with known BPSK symbols, followed by J blocks of unknown 4-QAM symbols. We will refer to the J -block segment as one frame. The channel was estimated according to (16). Note that at least two blocks of pilots have to be transmitted for (16) to apply. Denoising was also applied here to improve the channel estimate. Although the channel changes over the frame, the channel estimate obtained based on the first two blocks was used to recover the entire frame. The corresponding BER is shown in Fig. 1 (a). One can see that for the training method to come close to the proposed blind one in terms of BER, pilots need to be transmitted as often as J = 100 blocks. In order to obtain good cross-correlation estimates, the proposed method requires that the channel remains constant over a certain number of OFDM blocks. For the same fast-varying channel considered above and for SNR=30dB, we next show in Fig. 1 (b) the resulting BER as a function of the number April 25, 2005
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21
of blocks used for the estimation (frame length). We see that the BER for the proposed blind method has the best performance for frame length of around 50 to 100 blocks. The BER for the training method keeps increasing for J > 50. For comparison purposes, we also implemented the method of [2]. According to [2], the precoding results in an uneven power distribution among the OFDM blocks. The higher this unevenness is, the better the channel estimate will be, as seen on Fig. 5 of [2]. However, this has diminishing returns because under a constant energy constraint, the unevenness results in some blocks having very poor SNR which inevitably results in poor BER performance. On the other hand, the proposed method does not alter the power distribution between blocks and there is even freedom to vary the power distribution between individual carriers. The BER for the comparison method is shown in In Fig. 1 (b). One can see that although the BER behavior is similar to that of the proposed one, the achieved BER is always higher that that of the proposed method.
B. Time domain estimate We here illustrate the performance of the time domain version of the proposed approach, as given in eq. (15). We considered the following channel (see also [2]): h11 = [0.4851, −0.4851, 0.7276]T , h12 = [0.32, 0.9387, −0.1280]T ,
(66)
h21 = [−0.3676, 0.8823, 0.2941]T , h22 = [0.2182, 0.8729, −0.4364]T
and block size N = 32, cyclic prefix 8 at SNR=30dB. The advantage of the time domain approach is that it requires the estimation of fewer parameters. However, it requires information on channel length. Since such information is not available, we could use the cyclic prefix length as a bound for the channel length. In the following, we present results corresponding to the correct length, L = 3, and also a length equal to Ncp . We performed 100 Monte Carlo simulations, and in each run we randomly selected the set of carriers for precoding. The number of precoded carriers was varied from 3 (equal to the channel length) to the full OFDM block length, i.e., 32. Figure 2 shows the BER corresponding to each case. We can see that we need not precode all the available carriers for best performance. Moreover, we see some degradation of performance in using an overestimate of the channel length in (15), particularly when using few precoded carriers. Thus, not knowing the channel length, the more carriers we precode
April 25, 2005
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the lower the corresponding BER will be, in which case the time domain approach becomes identical to the frequency domain one of (13). C. 2 users with STBC We simulated a system with 2 independent users, who employ space time block coding [5]. Hence we have a system with 4 transmitters and 2 receivers. For comparison, we also simulated the users not employing STBC, i.e. a 2 by 2 system, using the same channels. We again used Jakes model with fd = 10−6 for generation of channels of length 3.
Figure 3 shows the performance of the least squares semi blind channel estimate given in (41). 60 channels were used with 10 Monte Carlo runs for each channel. We used 400 OFDM blocks for channels estimation. We can see that for low SNR, the STBC performs better and this is due to diversity. However, for high SNR, channel estimation error prevents any improvement in the STBC performance at least when using a linear equalizer. On the other hand, the 2 by 2 system has less channel parameters to estimate and hence has less estimation error and outperforms the STBC method. D. Performance metrics First, we compare the analytical performance against simulation results. We have kept the SNR at 30dB and used 40 static channels of length 3. We have calculated the NMSE for each channel as follows. Mc −1 M −1 M −1 PL−1 \2 1 X X 1 X l=0 |hij (l) − hij (l)| (67) N M SE = PL−1 2 Mc M2 |h (l)| ij l=0 m=0 i=0 j=0
where Mc = 20 is the number of Monte Carlo runs and MT = MR = M = 2. We have presented the results in Fig. 5. We have presented the analytical BER results, derived from (61) and (65) in Fig. 4 against simulated results. From this we see a bias in the analytical results. However, there is close agreement in the relative change for different channels. VI. C ONCLUSIONS We have presented a linear precoding scheme for multi-user OFDM systems that enable us to equalize blindly at the receiver. We do not introduce any redundancy in the precoding scheme, and we keep the power evenly distributed among the OFDM blocks. The channel estimation involves solving a set of linear equations, and hence computationally simpler. Figures of merit that can guide the selection of precoding matrices have been given. There is room for improvement of these schemes and design of new codes, which will be tackled in future work. April 25, 2005
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VII. ACKNOWLEDGEMENT We thank the editor and reviewers for their careful review and helpful comments. A PPENDIX I P ROOF
OF
P ROPOSITION 1
From (10) we have i Rikl = σ 2 H(k)Diag(wkl )HH (l) + δk,l σn2 IMR , i = 0, ..., M − 1
(68)
i is given in (24). Let us extract the u-th column of Ri . This can be done by right multiplying where wkl kl
Rikl by the column selection vector eu . Based on (68) it can be seen that: i Rikl eu = σ 2 H(k)Diag[HH (l)eu ]wkl + δk,l σn2 eu
(69)
MT −1 The matrix Cukl is formed based on the u−th columns of matrices R0kl , R1kl , ..., Rkl . It holds:
Cukl =
1 MT −1 [R0 eu , R1kl eu , ..., Rkl eu ] σ 2 kl
MT −1 0 = H(k)Diag[HH (l)eu ][wkl , ..., wkl ] + δk,l
= H(k)Diag[HH (l)eu ]Ukl + δk,l
(70) σn2 eu 1TMT σ2
σn2 eu 1TMT σ2
Post multiplying by of U†kl (assuming Ukl have full row rank) yields (13).
(71) (72)
A PPENDIX II P ROOF
OF
(21)
If σ 2 σn2 , ignoring noise, ˆ † (m)Rimm (H ˆ † (m))H Qilu (m) = H lu lu i = σ 2 (Diag(HH (l)eu ))−1 Diag(wmm )(Diag(HH (l)eu ))−H
(73)
i = σ 2 |Diag(HH (l)eu )|−2 Diag(wmm )
(74)
i in (24). Note that Qilu (m) is a diagonal matrix. where Rimm is given in (10) and wmm
April 25, 2005
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A PPENDIX III P ROOF
OF
(41)
Starting from (68), let us consider a 4 by 4 system (MT = MR = 4). Let the generic channel matrix be
H00 (k) H01 (k) H02 (k) H03 (k)
H10 (k) H11 (k) H12 (k) H13 (k) H(k) = H20 (k) H21 (k) H22 (k) H23 (k) H30 (k) H31 (k) H32 (k) H33 (k)
(75)
Then, the element of on row u column v of Rikl in (68) can be written as i Rkl (u, v) = σ 2
3 X j=0
? wjH (k; i)wj (l; i)Huj (k)Hvj (l), u, v, i ∈ [0, 3]
(76)
However, the channel matrix has a special structure given in (38) due to STBC. Hence, we have ? (k) etc. Substitution of these values into (76) reduces the number H00 (k) = H00 (k) and H11 (k) = −H00
of unknowns. For instance, if we construct a row of 0, 0 elements of the matrices R0kl to R1kl , we have [R0kl (0, 0), R1kl (0, 0), R2kl (0, 0), R3kl (0, 0)] = [H00 (k), H01 (k), H02 (k), H03 (k)]D0 Ukl
(77)
where D0 = Diag[(H00 (l))? , (H01 (l))? , (H02 (l))? , (H03 (l))? ]. Using (77), we can solve for the first row of (39). We can derive similar relationships for the other elements of the matrices R0kl to R1kl and generalize to (41), provided that we know H(l).
A PPENDIX IV P ROOF
OF
P ROPOSITION 2
First, we make the following observation. Let d be a vector of complex, random, i.i.d. elements di , 2 2 4 4 with E{di } = E{d?i } = 0, E{d2i } = E{d?2 i } = 0, E{|di | } = σ , and E{|di | } = η . Then η 4 , i = j, p = q, p = i σ 4 , i = j, p = q, p 6= i ? ? E{di dj dp dq } = σ 4 , i 6= j, p = i, q = j 0, otherwise
(78)
Secondly, let us consider C = E{ddH AddH }, where d is an N by 1 vector with random elements
as described above and, A is an arbitrary deterministic matrix. Then, we can show [32] C = σ 4 A + σ 4 trace(A)I + (η 4 − 2σ 4 )Diag(A(0, 0), A(1, 1), . . . , A(N − 1, N − 1)) April 25, 2005
(79)
DRAFT
25
Using (27) we get c b I−i−1 M
1
ˆi } = E{R kl
b I−i−1 M c+1
X p=0
E{yi+pM (k)(yi+pM (l))H }
(80)
Using (7) we have ˆi } = E{R kl
1 b I−i−1 M c+1
b I−i−1 c M
X p=0
H(k)Ψi+pM (k)E{di+pM (di+pM )H }(Ψi+pM (l))H HH (l) i+pM
+E{n
i+pM
(k)(n
(81)
(l)) } H
Because of the M periodicity in precoding Ψi+pM (k) = Ψi (k), ˆ i } = σ 2 H(k)Ψi (k)(Ψi (l))H HH (l) + σn2 δkl I = Ri + σn2 δkl I E{R kl kl
(82)
where Rikl = σ 2 H(k)Ψi (k)(Ψi (l))H HH (l) is the noiseless estimate. Next we have ˆ i )H R ˆj } = E{(R k1 l k2 l
1 b I−i−1 M c+
c b I−j−1 c b I−i−1 M M
1 1 b I−j−1 M c
X
+1
X
p=0
(83)
q=0
E{yi+pM (l)(yi+pM (k1 ))H yi+qM (k2 )(yi+qM (l))H }
when i 6= j ˆ i )H R ˆ j } = E{(R ˆ i )H }E{R ˆ j } ≈ (Ri )H Rj + σ 2 (δk l Rj + δk l (Ri )H ) E{(R 1 2 k1 l k1 l k1 l n k1 l k2 l k2 l k2 l k2 l
(84)
when i = j ˆ i )H R ˆj } E{(R k1 l k2 l
1 = I−i−1 (b M c + 1)2
b I−i−1 c b I−i−1 c M M
X p=0
X
(85)
q=0,q6=p
E{yi+pM (l)(yi+pM (k1 ))H }E{yi+qM (k2 )(yi+qM (l))H } b I−i−1 c M
+
X
E{y
i+pM
(l)(y
i+pM
H i+pM
(k1 )) y
(k2 )(y
p=0
i+pM
H
!
(l)) }
= (Rik1 l )H Rik2 l + σn2 (δk1 l Rik2 l + δk2 l (Rik1 l )H ) I−i−1
b M X 1 + I−i−1 (b M c + 1)2 p=0
c
E{yi+pM (l)(yi+pM (k1 ))H yi+pM (k2 )(yi+pM (l))H } !
− E{yi+pM (l)(yi+pM (k1 ))H }E{yi+pM (k2 )(yi+pM (l))H }
April 25, 2005
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26
We neglect terms containing powers of σn higher than two and using the fact that noise is complex circular white, get non-zero terms as E{yi (l)(yi (k1 ))H yi (k2 )(yi (l))H } i = H(l)Ψ (l) σ 4 Ak1 ,k2 + σ 4 trace(Ak1 ,k2 )I
(86)
+(η 4 − 2σ 4 )Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) (Ψi (l))H HH (l) i i i 2 2 +σn δk2 l Rlk1 + δk1 l Rk2 l + δk1 k2 Rll + σ trace(Ak1 ,k2 )I
where Ak1 ,k2 = ΨiH (k1 )HH (k1 )H(k2 )Ψi (k2 ) and the result in (79) was used in simplification. Moreover, we have E{yi (l)(yi (k1 ))H }E{yi (k2 )(yi (l))H }
= σ 4 H(l)Ψi (l)Ak1 ,k2 ΨiH (l)HH (l) + σn2 δk2 l Rilk1 + δk1 l Rik2 l
(87)
Substituting (86) and (87) in (85), we get for i = j ˆ j } = (Ri )H Ri + σ 2 (δk l Rj + δk l (Ri )H ) ˆ i )H R E{(R k1 l 2 n 1 k2 l k1 l k1 l k2 l k2 l 1 + I−i−1 H(l)Ψi (l) σ 4 trace(Ak1 ,k2 )I b M c+1 4 4 +(η − 2σ )Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) ΨiH (l)H HH (l) ! +σn2 δk1 k2 Rill + σ 2 trace(Ak1 ,k2 )I
(88)
Finally we formulate an estimate for error. We combine (27) and (43) to form ξ(k) =
1 ˆ0 ˆ M −1 − RM −1 )eu ]U† Diag(HH (l)eu )−1 [(Rkl − R0kl )eu , . . . , (R kl kl kl σ2
(89)
Using the result obtained in (82), we get (44). Next we consider the variance † H −1 ξ H (k1 )ξ(k2 ) = (Diag(HH (l)eu ))−H (U† )H k1 l RUk2 l (Diag(H (l)eu ))
where R = [Ri,j ] with Ri,j =
1 T ˆi ˆ j − Rj )eu eu (Rk1 l − Rik1 l )H (R k2 l k2 l 4 σ
ˆ i − Ri )H (R ˆ j − Rj )} E{(R k1 l k1 l k2 l k2 l
(91) (92)
j iH j iH 2 ˆ iH R ˆj = E{R k1 l k2 l } − Rk1 l Rk2 l − σn δk1 l Rk2 l + δk2 l Rk1 l
April 25, 2005
(90)
DRAFT
27
Hence using (84) and (88), E{Ri,j } =
δi,j T i e H(l)Ψ (l) trace(Ak1 ,k2 )I u b I−i−1 c + 1 M
(93)
η 4 − 2σ 4 + Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) (Ψi (l))H HH (l) σ4 ! σn2 + 2 δk1 k2 Rill /σ 2 + trace(Ak1 ,k2 )I eu σ
which yields (45). R EFERENCES
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29
−1
10
0
10
Blind Training
Training Proposed Comparison
120 blocks −1
BER
BER
10
−2
10
−2
10
100 blocks
−3
10
0
−3
10
0
5
10
15
20
25
30
35
40
50
100
45
SNR/(dB)
(a)
150 Data Blocks (J)
200
250
300
(b)
(a)BER variation with SNR for fast (fd = 10−5 ) varying channels at frame length 100 blocks (b) BER variation with
Fig. 1.
frame length for fast varying and static channels at SNR=30dB.
0
10
With STBC Without STBC −1
10
Prefix Length Actual Length −1
BER
BER
10
−2
10
−2
10
−3
10
0
−3
10
5
10
15
20
25
30
35
40
45
SNR/(dB)
5
10
15 20 No. of selected carriers
25
30
Fig. 2. Variation of BER with the number of selected carriers Fig. 3. for channel estimation using time domain formulation.
BER comparison with and without STBC. Random
channels using the Jakes model with fd = 10−6 were used in this simulation. 400 OFDM blocks were used in channel estimation.
April 25, 2005
DRAFT
30
0
0
10
10
NMSE, Simulated NMSE, Analytical
BER, Simulated BER, Analytical BER, Analytical SINR −1
10
−1
−2
10
NMSE
BER
10
−3
10
−2
10
−4
10
−5
10
−3
10
−6
10
Fig. 4.
0
5
10
15
20 channels
25
30
35
40
April 25, 2005
5
10
15
20
25
30
35
40
channels
Comparison of Analytical and Simulated BER and Fig. 5.
SINR for 40 different channels.
0
Comparison of Analytical and Simulated NMSE for
40 different channels.
DRAFT