Performance Analysis of Blind Subspace-Based Signature Estimation Algorithms for DS-CDMA Systems with Unknown Correlated Noise

We analyze the performance of two popular blind subspace-based signature waveform estimation techniques proposed by Wang and Poor and Buzzi and Poor for direct-sequence code division multiple-access (DS-CDMA) systems with unknown correlated noise. Using the ﬁrst-order perturbation theory, analytical expressions for the mean-square error (MSE) of these algorithms are derived. We also obtain simple high SNR approximations of the MSE expressions which explicitly clarify how the performance of these techniques depends on the environmental parameters and how it is related to that of the conventional techniques that are based on the standard white noise assumption. Numerical examples further verify the consistency of the obtained analytical results with simulation results.


INTRODUCTION
In recent years, extensive research efforts have been devoted to develop different strategies for multiuser detection in DS-CDMA systems [1].One of the most challenging problems in multiuser detection is that of the effect of unknown multipath channel which may result in a significant mismatch between the actual user signature and its presumed value used in the multiuser detection algorithms.Since signature mismatches may cause substantial degradation in the symbol detection performance [2][3][4][5], considerable attention has been paid to designing accurate signature estimation techniques at the receiver.These techniques may be classified into the training-based [6,7] and blind [3,4,[8][9][10][11][12][13] methods.In the training-based approaches, each user transmits a sequence of pilot symbols which is known at the receiver where the user signature is estimated by computing the correlation between the received data and this sequence.In nonstationary environments, a reliable signature estimate requires periodic transmission of the pilot sequence.This may cause a considerable reduction of the bandwidth efficiency [2,3] and has been a strong motivation to develop alternative blind estimation approaches which do not require transmission of the pilot sequence.A promising trend among this type of methods is the subspace-based techniques [3,[8][9][10][11].The latter techniques exploit the facts that the user signals occupy a lowdimensional subspace in the observation space, and that the signature of each particular user belongs to a subspace defined by its associated spreading code.A typical assumption used in these techniques is that the additive ambient noise is temporally white, and, hence, the signal subspace can be extracted using eigendecomposition of the received data covariance matrix.However, in practice this assumption may be violated [14,15].It is well known that in the presence of correlated noise, the signal subspace cannot be identified from the subspace spanned by the eigenvectors associated with the largest eigenvalues of the data covariance matrix.Therefore, some alternative approaches should be employed to identify the signal subspace in the correlated noise case.
One of such approaches has been proposed by Wang and Poor [15].Their technique is based on the assumption that the receiver contains two well-separated antennas so that the receiver noise is spatially white.Using this fact, the signal subspace can be obtained from the crosscorrelation between the received antenna data.Hereafter, we refer to this technique by Wang and Poor as the WP algorithm.
Using a single antenna at the receiver, another technique that addresses the problem of correlated noise has been proposed by Buzzi and Poor [16].It is based on the assumption that the noise is a circular Gaussian process while the transmitted symbols are noncircular BPSK signals.In such a case, it has been shown in [16] that the signal subspace can be directly identified using the singular value decomposition of the data pseudocovariance matrix.Hereafter, we refer to the technique by Buzzi and Poor as the BP algorithm.
Although the performance of the conventional (white noise assumption-based) signature waveform estimation techniques has been well studied in the literature [9,10,[17][18][19], only a little effort has been made to analyze the performance of the estimation algorithms proposed for the unknown correlated noise case.In this paper (see also [20]), we use the first-order perturbation theory to derive approximate expressions for the MSE of the channel vector estimates obtained by the WP and BP algorithms.Under several mild assumptions, simple high SNR approximations of these MSE expressions are also obtained.The derived MSE expressions clarify how the performance of the algorithms depends on the parameters such as the number of data samples, the received power of the user of interest, and the noise covariance matrix.The effect of the spreading factor and the channel length on the performance of the algorithms is also studied.It is shown that the performance of the algorithms depends not only on SNR but also on the direction of the eigenvectors of the noise covariance matrix.To clarify this fact, we fix the eigenvalues of the noise covariance matrix and find the sets of eigenvectors which maximize (minimize) the MSEs of the channel vector estimates.Moreover, over all noise covariance matrices with fixed trace, we obtain those which correspond to the extremal values of the MSEs.It is shown that both the maximum and the minimum values of the MSEs are obtained when the noise covariance matrix is rank deficient.As the trace of the noise covariance matrix is equal to the average noise power, the latter observation shows that the performance of the algorithms may be more sensitive to a low-rank interference than to a full-rank noise with the same average power.We also show that in the presence of white noise, the performances of the WP and BP algorithms are identical to that of the conventional Liu and Xu (LX) algorithm [9] that was developed for the white noise case.
Assuming that the SNR is high and the WP algorithm is used to estimate the channel vector between the user of interest and the first antenna, it is proved that the estimation performance is independent from the noise covariance matrix and the user received power at the second antenna.We use the latter property to show that when the receiver is equipped with multiple antennas, the second antenna can be arbitrarily chosen at high SNRs.
The rest of this paper is organized as follows.In Section 2, we introduce the signal model.A brief overview of the LX, WP, and BP algorithms is provided in Section 3. Section 4 presents our main theoretical results on the performance of the WP and BP algorithms.Simulation results validating our analysis are presented in Section 5. Conclusions are drawn in Section 6.

SIGNAL MODEL
Consider a K-user synchronous DS-CDMA system. 1 The received continuous-time baseband signal can be modelled as [3] x where T s is the symbol period, v(t) is the zero-mean additive random noise process, and A k , b k (m), and w k (t) denote the received signal amplitude, the mth data symbol, and the signature waveform of the kth user, respectively.Note that b k (m) can be drawn from a complex constellation, and, hence, in the general case x(t) is complex valued.Throughout the paper, we use the following common assumptions.
(A1) The chip sequence period is equal to the symbol period, that is, the short spreading code is considered [22].(A2) The user channels are quasistatic, that is, the corresponding impulse channel responses do not change during the whole observation period [9].(A3) The duration of the channel impulse response of each user is much shorter than the symbol period T s , so that the effect of intersymbol-interference (ISI) can be neglected [9,22].(A4) The transmitted symbols and noise are zero-mean random variables.Moreover, transmitted symbols of each user are unit-variance i.i.d.variables, independent from those of the other users, and also independent from the noise [9].
Note that (A1) is common for many multiuser techniques proposed for DS-CDMA systems as most of these algorithms require the received signal x(t) to be cyclostationary.This, in turn, necessitates the use of short spreading codes [23].
Let L c be the spreading factor and let T denote the discrete spreading sequence associated with the kth user where (•) T stands for the transpose and c k [i] can be either real or complex valued.According to assumptions (A1) and (A2), the signature waveform of this user can be expressed as [9] where h k (t) is the channel impulse response of the kth user and T c = T s /L c is the chip period.
Let us assume that h k (t) is zero outside the interval [0, αT c ], where L − 1 ≤ α < L and L is a positive integer.
From assumption (A3), it follows that L L c .Sampling (1) in the interval corresponding to the nth transmitted symbol of each user and ignoring the first L − 1 samples that are contaminated by ISI, the ISI-free received sampled data vector can be written as [9] x where Note that the similar data model also holds when the effects of chip waveform at the transmitter and chip matched filtering at the receiver are taken into account [21].Using (2), we have that the signature vector w k can be written as where As the spreading code of the user of interest is known at the receiver, if the channel vector h k is estimated, then w k can be obtained from (4).Hence, throughout this paper we consider the problem of channel vector estimation rather than that of the signature vector estimation.For the sake of consistency, we also assume without any loss of generality that h k is a unit Euclidean norm vector ( h k = 1) [9], that is, the normalization factor is absorbed in A k .One can present (3) in a more compact form as [9] x(n where

The LX algorithm
The LX algorithm assumes that the noise is white.In such a case, from assumption (A4) and ( 5) we have [9] where σ 2 v I is the noise covariance matrix, I is the identity matrix, and σ 2 v = E{|v(t)| 2 } is the noise variance.The matrix (6) can be eigendecomposed as where U s consists of the eigenvectors associated with the K largest eigenvalues which are the diagonal elements of Ω s + σ 2 v I, and Ω s is a diagonal matrix whose diagonal elements are the signal subspace eigenvalues.Due to the fact that the noise is white, range(W) = range(U s ), or, equivalently, U s and U n span the signal and noise subspaces, respectively.Without any loss of generality, we assume that h 1 is the channel vector of interest.As any column of U n is orthogonal to all vectors in range(U s ), we have [9] where (8), it follows that T 1 is not full rank.Assuming that rank(T 1 ) = L− 1, the null space of T 1 is spanned by h 1 , and, therefore, up to an arbitrary phase rotation, h 1 can be uniquely determined as a nontrivial solution to (8) where dim{•} stands for the dimension of a subspace.Equation ( 9) is the necessary and sufficient condition of signature identifiability using the LX algorithm [9].
In practical scenarios, the data covariance matrix R is not known exactly and can be estimated as As a result, U n is estimated as U n that consists of the eigenvectors associated with the smallest L c − L+1−K eigenvalues of R. Substituting U n in lieu of U n in (8) and solving the obtained equation in the least square (LS) sense, we have that the estimated channel vector h 1 is given by [9] where M{•} stands for the normalized eigenvector associated with the smallest (minor) eigenvalue.Using the firstorder perturbation theory, the mean-square of the encountered estimation error δh 1 = h 1 − h 1 can be approximately written as [18] E δh where T † 1 is the pseudoinverse of T 1 and • F stands for the Frobenius norm of a matrix.Assuming that the signatures of different users are orthogonal to each other, that is, where δ i j stands for the Kronecker delta, the MSE expression ( 12) can be significantly simplified.Note that due to multipath effects, the orthogonality assumption of the signature vectors does not perfectly hold in practice.However, CDMA codes are deliberately designed so that even after passing through a frequency selective channel, the cross correlations between different user signatures are as small as possible.
Hence, in most practical scenarios, ( 13) is an acceptable assumption [1].It directly follows from (13) that Substituting ( 14) into (12) and using ( 13) yields Equation ( 16) can be considered as a reasonable approximation of ( 12) in the high SNR regime.Note that an expression equivalent to ( 16) has been derived for the MSE of the estimated signature, C 1 h 1 , in [9].

WP algorithm
It is well known that if the white noise assumption does not hold, then the signal subspace is not identical to the subspace spanned by the eigenvectors associated with the K largest eigenvalues of R and, consequently, the LX algorithm cannot be directly applied to obtain a reliable estimate of h 1 .To deal with this problem, the WP algorithm assumes that the receiver is equipped with two well-separated antennas such that the noise is spatially uncorrelated between them.Similar to (5), the sampled received data vectors are given by where i is the antenna index, k are the received amplitude and the signature vector of the kth user at the ith antenna, respectively.The covariance matrix corresponding to the sampled received data vector at each antenna is given by [15] where As the noise is uncorrelated between the antennas, we have [15] R (12) E x (1) (n)x (2)  (1)   s U (1)   n Ω (12) where the right-hand side of ( 19) is the singular value decomposition (SVD) of R (12) .It is clear that range(U (1)  s ) = range(W (1) ) and range(U (2)  s ) = range(W (2) ).For the sake of simplicity but without any loss of generality, let us consider only the channel vector between the first user and the first antenna.Then, we have [15] U (1)   n H w (1)  1 = T (1)  1 h (1)  1 = 0, (20) where T (1)   1 1 ) = L − 1, then up to an arbitrary phase rotation, h (1)  1 is the unique nontrivial solution to (20) subject to h (1)   1 = 1 [15].In practice, R (12) can be estimated as x (1) (n)x ( 2) H (n) (21) which results in the following estimate of h (1)  1 [15] h (1) where U (1)  n consists of the left singular vectors associated with the L c − L + 1 − K smallest singular values of R (12) .

BP algorithm
Another approach to solve the problem of channel estimation in presence of unknown correlated noise has been proposed in [16].Without requiring the second antenna, this algorithm is based on the assumption that the transmitted symbols are drawn from the BPSK constellation (b k (n) = ±1) and the noise is a circular Gaussian process.It directly follows from the latter assumption that Let R E{x(n)x T (n)} be the pseudocovariance matrix of the sampled received data.Using (5) along with (23), we have [16] where Ω s is a diagonal matrix whose diagonal elements are the nonzero singular values of R and the columns of U s are the corresponding left singular vectors.It is easy to show that range( U s ) = range(W) [16], and, hence, where and the unique identification of h 1 from (25) requires that rank( T 1 ) = L − 1 [16].In practice, similar to the LX and WP algorithms, h 1 can be estimated by where U n is the matrix containing the left singular vectors associated with the L c − L + 1 − K least singular values of R, and is the sample estimate of R.

WP algorithm
In order to evaluate the performance of the WP algorithm, we use the first-order perturbation theory to prove the following theorem.
Theorem 1. Assume that h (1)  1 is estimated using (22).Then, the first-order perturbation theory-based approximation of the MSE of the estimation error δh (1)  1 = h (1)  1 − h (1)  1 is given by where tr(•) stands for the trace of a matrix and Ψ U (1)  n T (1)   1 † H Moreover, if the following conditions hold: then (28) reduces to where λ max (•) stands for the maximum eigenvalue.
Proof.See Appendix A.
Note that the average received power of the first user at the second antenna is equal to the right-hand side of (31), while the average noise power at the same antenna is lower bounded by the left-hand side because E v (2) Hence, if SNR at the second antenna is reasonably high, it is guaranteed that (31) holds.Using this observation along with the fact that (30) approximately holds in most practical scenarios, we can view (32) as a simple approximation of (28) in the high SNR regime.It explicitly clarifies the MSE of the estimated channel vector in terms of the environmental parameters such as the received power of the user of interest at the first antenna, the number of data samples as well as the noise covariance matrix Σ (1)  v .
Note that both the MSE expressions ( 28) and (32) depend on Σ (1)  v only through tr Σ (1)  v Ψ .To study the parameters which have impact on the value of tr Σ (1)  v Ψ , we first should note that if the channel vector is uniquely identifiable, then rank(T (1)  1 ) = rank(Ψ) = L − 1.Moreover, we have where null(•) stands for the null-space of a matrix. 2 The effects of different parameters on the value of tr Σ (1)  v Ψ are separately clarified in the following discussion.

Effects of L c and L
As Σ (1)  v and Ψ are positive (semi-) definite matrices, it follows that tr(Σ (1)  v Ψ) is real and nonnegative.Note that the projection of Σ (1)  v onto null(Ψ) does not have any effect on the value of tr(Σ (1)  v Ψ) which depends only on the projection of Σ (1)   v onto range(Ψ).Therefore, the larger the projection of Σ (1)   v onto null(Ψ), the smaller the value of tr(Σ (1)  v Ψ).Using the latter fact, the effect of the spreading factor and the channel length on tr(Σ (1)  v Ψ), and, consequently, on the performance of the WP algorithm can be explained as follows.From (34) it can be observed that if either the spreading factor L c increases or the channel length L decreases, then dim{null(Ψ)} increases.In the latter case, the projection of the columns of Σ (1)  v onto null(Ψ) becomes larger, and, therefore, their contribution to the value of tr(Σ (1)  v Ψ) becomes smaller.
Effect of the eigenvectors of Σ (1)   v The directions of the eigenvectors of Σ (1)  v with respect to the eigenvectors of Ψ have a considerable impact on the value of tr(Σ (1)  v Ψ).To show this, let us eigendecompose Ψ as where whose columns are the orthonormal eigenvectors associated with the decreasingly-ordered positive eigenvalues of Ψ that are the diagonal elements of Θ = diag{θ 1 , θ 2 , . . ., θ L−1 }.In contrary to rank(Ψ), m rank(Σ (1)  v ) may not be known.In fact, rank(Σ (1)  v ) may vary from m = 1 for the case of coherent interference to m = L c − L + 1 for the case of full-rank noise.Let us consider an arbitrary value of m and eigendecompose Σ (1)  v as where U v is an L c − L + 1 × m matrix whose orthonormal columns are the eigenvectors associated with the decreasingly-ordered positive eigenvalues of Σ (1)  v which are the diagonal elements of The value of tr(Σ (1)  v Ψ), and, hence, the MSE expressions ( 28) and (32) critically depend on the direction of the columns of U v relative to the columns of Π.To explain this fact, let us fix the matrix Γ v and find the matrices U v max and U v min which maximize and minimize tr Σ (1)  v Ψ , respectively.It can be shown [24,25] that max Uv tr Σ (1)  v and U v max is given by where Π ⊥ l is an L c − L + 1 × l matrix whose l ≤ τ columns are arbitrarily chosen from a set of τ orthonormal vectors in null(Ψ).According to (38), for a fixed Γ v , the MSE expressions ( 28) and (32) are maximal if the first τ 1 columns of U v and Π coincide.In turn, we have [24,25] min Uv tr Σ (1)  v and U v min is given by According to (40), the necessary condition to minimize the MSE expressions ( 28) and ( 32) is that the first τ 2 min{m, τ} columns of U v are in null(Ψ).Note that the matrix Σ v min = U v min Γ v U H vmin has the maximum projection onto null(Ψ), that is, the space spanned by the eigenvectors associated with the τ 2 largest eigenvalues of Σ v min is in null(Ψ).
Assuming that the average noise power at the first antenna is given by e o , that is, we can also obtain the extremal values of the MSE expressions ( 28) and (32) as follows.Since for any pair of positive (semi-) definite matrices Σ (1)  v and Ψ we have [25] tr Σ (1)  v Ψ ≤ λ max (Ψ)tr Σ (1)  v , it directly follows that tr Σ (1) where, assuming that the largest eigenvalue of Ψ is unique, (43) holds with equality if and only if Moreover, it is obvious that among all noise covariance matrices with m i=1 γ i = e o , those in the form of result in the MSE expressions ( 28) and (32) equal to zero.It is interesting to observe from (44) and (45) that, given the average noise power at the first antenna, both the maximal and the minimal values of the MSE of the channel vector estimate are obtained when the noise covariance matrix is rank deficient.As a rank deficient covariance matrix can be attributed to a narrow-band interference, it follows that the performance of the WP algorithm can be more sensitive to a narrow-band interference than a full-rank colored noise.Now, let us consider two important particular scenarios in which the WP algorithm may be used and discuss the pertaining results.
White noise scenario: if the noise at the first antenna is white, that is, Σ (1)  v = σ (1) which is equal to the derived MSE of the LX algorithm in (16).Hence, even though the WP algorithm is proposed to estimate the channel vector in the presence of unknown correlated noise, it is also applicable to the white noise scenario.
In the latter case, the performance of the WP algorithm is identical to that of the LX algorithm.
Multiple antenna systems: it follows from (32) that if the SNR at the second antenna is high enough so that (31) holds, then the MSE of the channel vector estimate between the user of interest and the first antenna is independent of Σ (2)  v and the received power of this user at the second antenna.Let us consider a receiver with M > 2 antennas which are spatially separated so that the noises between the first antenna and all the other antennas are uncorrelated.Moreover, assume that the SNR is high enough: and that we aim to estimate the channel vector between the first user and the first antenna using the WP algorithm.Since this algorithm is based on processing of the data crosscorrelation matrix between the first antenna and another well-separated auxiliary antenna, we have to choose the auxiliary antenna among the M − 1 available antennas.However, it directly follows from (32) that if the aforementioned assumptions hold, the performance of the channel vector estimate is insensitive to the choice of such an antenna, that is, the auxiliary antenna can be chosen arbitrarily.

BP algorithm
The following theorem quantizes the performance of the BP algorithm.
Theorem 2. Assume that the channel vector is estimated using the BP algorithm.Then, the first-order perturbation theorybased approximation of the MSE of the estimation error where Moreover, if (13) holds and then (48) reduces to Proof.See Appendix B.
As can be observed from (53), in the high SNR regime the MSE of the channel vector estimate of the BP algorithm can be expressed in terms of the noise covariance matrix, power of the received signal, and the number of data samples.
Note that if the channel vector is uniquely identifiable from the BP algorithm, we have rank( Ψ) = L−1.Comparing (53) with (32), it can be readily shown that the effect of the spreading factor and the channel length on both the WP and BP algorithms are similar.Moreover, following a discussion similar to that of Section 4.1, we can obtain the extremal values of tr(Σ v Ψ), and, consequently, those of the MSE expression (53).Let us first eigendecompose Ψ as where contains the orthonormal eigenvectors associated with the positive eigenvalues of Ψ and Θ = diag{ θ 1 , θ 2 , . . ., θ L−1 } is the diagonal matrix that contains the decreasingly-ordered positive eigenvalues.Let us denote q rank(Σ v ) and eigendecompose Σ v as where U v contains the orthonormal eigenvectors associated with the positive eigenvalues of Σ v which are ordered decreasingly as the diagonal elements of Γ v = diag{ γ 1 , γ 2 , . . ., γ q }.Denoting Π ⊥ l as an L c − L + 1 × l matrix whose columns are orthonormal vectors in null( Ψ), we have where the matrix U v which maximizes tr(Σ v Ψ) is where the matrix U v which minimizes tr(Σ v Ψ) is Comparing ( 56)-( 59) with ( 37)-( 40), it can be observed that the conclusions which follow ( 37)-( 40) can be easily extended to the BP algorithm, and, hence, we do not repeat them for the sake of brevity.
Let us also consider the case that the average noise power is given by e o , that is, tr(Σ v ) = q i=1 γ i = e o .In such a case, assuming that the largest eigenvalue of Ψ is unique, the noise covariance matrix which maximizes tr Σ v Ψ is given by (60) Moreover, over all noise covariance matrices Σ v with q i=1 γ i = e o , the value of tr(Σ v Ψ) and, consequently, that of the MSE expression (53) is zero if and only if (61) Similar to the WP algorithm, it follows from (60) and ( 61) that the performance of the BP algorithm can be more sensitive to the narrow-band interference than to the full-rank noise.
If noise is white, that is, Hence, the performances of the BP and the LX algorithms are identical in the white noise scenario.Therefore, the BP algorithm can also be applied to estimate the channel vector in the white noise case without any estimation performance loss as compared to the conventional LX algorithm.Another interesting relationship between the WP and BP algorithms follows from comparing (32) and (53).Let the users transmit BPSK modulated symbols and let the receiver be equipped with two well-separated antennas such that noise is spatially uncorrelated between them.Also, let (30) and (31) hold and Then, the MSE expressions (32) and ( 53) can be readily verified to coincide in the following two cases: when h (1)  1 is estimated using the WP algorithm with both antennas, and when h (1)  1 is estimated using the BP algorithm with only the first antenna.

SIMULATIONS
In this section, we validate our analytical results via computer simulations.In all the examples, we consider K = 7 synchronous CDMA users that transmit BPSK-modulated symbols.Each point of the simulation curves is the result of averaging over 200 Monte-Carlo realizations of the noise and transmission data sequences.In Figures 1-8, Gold codes of length L c = 31 are employed as the user spreading sequences and channel vectors of length L = 4 are independently drawn from a zero-mean white complex Gaussian process and then are scaled to become unit-norm vectors.The ambiguity in the phase of the channel vector estimate is resolved by assuming that the phase of the first tap of the channel vector is known at the receiver.In Figures 1-5 and 9, the received noise at each antenna is considered to be a circular Gaussian process such that [Σ v ] i j , the (i, j)th entry of its covariance matrix, is equal to 0.7 |i− j| .In the figures where the MSE of

MSE
Σ v drawn randomly, q = 1 Σ v drawn randomly, q = 5 Σ v drawn randomly, q = 15 Σ v drawn according to (60) Σ v drawn according to (61), q = 1 Σ v drawn according to (61), q = 5 Σ v drawn according to (61), q = 15   the channel estimate is drawn versus SNR, it is assumed that N = 80 data samples are used to estimate the channel.
Figures 1-3 illustrate the accuracy of our analytical results derived for the WP algorithm.In Figure 1, it is assumed that SNRs of all users at both antennas are identical and h (1)   1 is estimated according to (22).The solid curve represents the MSE resulting from this estimate.This curve is compared with our analytical results given by ( 28) and (32).It can be observed that both theoretical curves follow the experimental MSE curve with a good precision.Note that when the SNR is very low, the channel vector estimation error is quite large and, hence, it could not be reliably predicted using the firstorder perturbation theory.In such a condition, the analytical MSE curves obtained from ( 28) and (32) show a considerable discrepancy with the experimental MSE curve.Figure 2 depicts the experimental and the analytical MSE curves versus the number of data samples N. In this figure, it is assumed that the received signal power from each user at each of the two antennas is equal to 10 dB.Due to the fact that SNR is reasonably high, the theoretical curve (28) and its high SNR approximation (32) are almost indistinguishable from each other and they follow the experimental MSE curve with a good accuracy.It can be observed from Figure 2 that, when the number of data samples N is small, the small perturbation assumption is violated, and, hence, the accuracy of the analytical MSE curves decreases.
Figure 3 shows the MSE of the estimated channel h (1)  1 versus SNR at the first antenna (SNR (1) ) for 6 different values of SNR at the second antenna (SNR (2) ).As expected from Section 4.1, the performance of the channel estimation is almost independent from the exact value of SNR (2) , unless SNR (2) is very low.
Figures 4-7 and 9 show the performance of the BP algorithm and compare it to our analytical results.In Figure 4, the experimental MSE curve is plotted versus SNR and is compared with the theoretical curves obtained from (48) and (53).As can be observed from the figure, the two theoretical MSE curves are very close to each other and also closely follow the experimental MSE curve for the SNRs higher than 0 dB.
Figure 5 shows the experimental and the theoretical curves drawn versus the number of data samples N for SNR equal to 10 dB.As the figure shows, the theoretical curve (48) is precisely followed by its high SNR approximation (53) and both of them are very close to the experimental MSE curve.
Figure 6 shows the experimental MSE curves versus SNR for noise covariance matrices with identical Γ v = diag{20, 5, 3} and different matrices of eigenvectors U v .Three random realizations of U v as well as U vmax and U vmin are drawn and then using (55) the corresponding noise covariance matrices are obtained.The BP algorithm is used to estimate the channel vector in the presence of a correlated noise with the so-obtained noise covariance matrices.Figure 6 confirms our theoretical results in Section 4.2 which state that the worst and the best MSE performances are obtained when U v = U vmax and U v = U vmin , respectively.Note that if U v = U vmin , then, unlike the MSE expression (53), the experimental MSE performance is not equal to zero.It is due to the fact that the MSE expression (53) is obtained using the first-order perturbation theory and even in the high SNR regime this expression has a slight difference with the experimental MSE.
Figure 7 plots the experimental MSE curves versus SNR for noises with identical average energy of e o = L c −L+1 = 28 and different covariance matrices.For each value of q = 1, 5, and 15, one noise covariance matrix is drawn randomly and another one is obtained according to (61).A rank-one noise covariance matrix is also derived according to (60).Then, the BP algorithm is used to estimate the channel vector in the presence of correlated noise with the so-obtained noise covariance matrices.Our analytical results in Section 4.2 are validated by observing that the worst and the best MSE performances are obtained when the noise covariance matrix follows (60) and (61), respectively.
In Figure 8, the performances of the LX, WP, and BP algorithms are tested in the white noise environment.As predicted by our analysis in Section 4, all three methods have a nearly identical performance.
Figure 9 shows the experimental and the theoretical MSE curves of the BP algorithm versus the channel length L for two different values of the spreading factors L c = 40 and L c = 80.In this example, we use random spreading codes rather than the optimized Gold codes.The entries of these codes are randomly drawn from the set {−1, +1}.From Figure 9 we see that, as predicted in Section 4, the estimation performance decreases with increasing L. When L c = 80, the MSE of the channel vector estimate is significantly lower than that for L c = 40.It can be observed that the curves corresponding to (48) and (53) are quite close to each other and, therefore, the use of the random spreading codes instead of the Gold codes retains the accuracy of (53).

CONCLUSIONS
In this paper, analytical expressions for the MSE of the signature waveform estimation techniques of [15,16] have been derived.Assuming that different user signature vectors orthogonal, the simplified versions of these expressions have been also obtained for the high SNR regime.The effect of the correlated noise on the performance of both algorithms has been studied.It has been shown that the direction of the eigenvectors of the noise covariance matrix has a significant effect on the performance of both algorithms.In particular, assuming that the eigenvalues of the noise covariance matrix are fixed, the noise covariance matrix eigenvectors corresponding to the extremal values of the MSEs have been obtained.Over all noise covariance matrices with identical average noise power, the extremal values of the MSEs have been derived and it has been shown that both the maximal and the minimal values of the MSEs are achieved when the noise covariance matrix is rank deficient.Moreover, it has been shown that at high SNRs and in the presence of white noise, the performance of these two techniques is identical to that of the conventional white noise-based technique of [9].
In the high SNR regime, it has been proved that the performance of the technique proposed in [15] is independent from the noise covariance matrix and the user received power at the second auxiliary antenna.This property has been generalized to the multiple antenna systems and it has been shown that for such systems the choice of the auxiliary antenna is arbitrary at high SNRs.

B. PROOF OF THEOREM 2
According to (24), we have Using the procedure similar to that in Appendix A, it can be readily shown that where the second line of (B.10) can be derived using the same steps as in (A.15).To obtain Φ 2 , it can be easily shown from (B.9) that where [•] k is the kth entry of a vector and the time index i has been dropped from v(i) for the sake of simplicity.Since v is a multivariate circular Gaussian random vector, we have [28] E To prove (53), we note that (13)  Noting that both Σ v and Ψ are positive (semi-) definite matrices, it is easy to find an upper-bound for α 2 and α 3 as (B.21) Hence, if (52) holds, both α 2 and α 3 are negligible comparing to α 1 .Substituting (B.20) into (B.18)directly yields (53).This completes the proof.

8 EURASIPFigure 1 :Figure 2 :
Figure 1: The MSE of the estimated channel versus SNR.The WP algorithm.

Figure 3 :
Figure 3: The MSE of the estimated channel versus SNR at the first antenna for different values of SNR at the second antenna.The WP algorithm.

Figure 4 :
Figure 4: The MSE of the estimated channel versus SNR.The BP algorithm.

Figure 5 :Figure 6 :
Figure 5: The MSE of the estimated channel versus number of data samples.The BP algorithm.

Figure 7 :
Figure 7: MSEs of the estimated channel versus SNR for e o = 28 and different matrices Σ v .The BP algorithm.

Figure 8 :
Figure 8: MSEs of the estimated channel versus SNR in the white noise environment.The LX, WP, and BP algorithms.

10 EURASIPFigure 9 :
Figure 9: MSEs of the estimated channel versus L for L c = 40 and L c = 80.The BP algorithm.