Joint CFO and DOA estimation for multiuser OFDMA uplink

Abstract

In this article, we develop a new subspace-based multiuser joint carrier frequency offset (CFO) and direction-of-arrival (DOA) estimation scheme for orthogonal frequency division multiple access uplink transmissions. We leverage multi-antenna at the receiver and consider that the signals transmitted by each user arrive at the receiving antenna array from multiple DOAs after bouncing from both surrounding and far scatterers. The rank reduction approach is then exploited to estimate the multiple CFOs and DOAs. Specifically, for each user, after the CFO estimation from one-dimensional search, its multiple DOAs can be obtained simultaneously via polynomial rooting. The proposed method supports generalized subcarrier assignment scheme and fully loaded transmissions. Both performance analysis and numerical results are provided to corroborate the proposed studies.

Introduction

As has widely been studied in recent years [1–3], orthogonal frequency division multiple access (OFDMA) is deemed as a promising technique for next-generation multiuser wireless communications. The performance of OFDMA, however, is sensitive to multiple carrier frequency offsets (CFOs) introduced by the mismatch of the transceiver oscillators or the Doppler effect. In multiuser scenarios, the non-zero CFOs lead to both inter-carrier interference and multiple-access interference, which could severely degrade the system performance.

The CFO estimation scheme for OFDMA uplink transmissions has intensively been investigated in the past few years. Using the frequency domain embedded pilot symbols, an iterative CFO estimation approach was described in [4] for tile structure-based OFDMA transmission [5]. The CFOs can also be estimated from the maximum likelihood (ML) approach by transmitting training sequences from each user, but with very high complexity. The alternating-projection algorithm was introduced in [6] to replace the multi-dimensional search with a sequence of one-dimensional (1D) searches. An improved approach was later proposed in [7, 8] to further reduce the complexity of [6] by using the divide-and-update frequency estimator. An interesting alternative to avoid the ML multi-dimensional search is to use the mean likelihood estimator combined with the importance sampling technique [9, 10]. Another complexity-reduced CFO estimator was reported in [11] by approximating the inverse of a CFO-dependent matrix with that of a predetermined matrix.

Blind CFO estimation methods, on the other hand, were also developed to improve the bandwidth efficiency. The CFOs can be computed by looking for the position of null subcarriers within the signal bandwidth in the system for subband subcarrier assignment scheme (SAS) [12]. A frequency estimation scheme for uplink OFDMA with interleaved SAS that exploits the periodic structure of the signals from each user has been reported in [13], where the subspace estimation theory was utilized, which makes the scheme similar to the multiple signal classification technique [14]. Based on the observation of [13], several advancements have been proposed later [15, 16]. Despite their good performance, both [13] and its variations [15, 16] are only applicable for interleaved SAS and cannot be used for generalized SAS. Moreover, they must reserve null subcarriers or a much longer cyclic prefix (CP) to construct the noise space, which reduces the bandwidth efficiency.

More recently, several CFO estimation schemes have been developed for the OFDMA systems by leveraging multi-antenna at the receiver. For instances, a CFO estimation scheme for interleaved OFDMA/space division multiple access uplink systems was developed in [17] to support spatially separated users and to maximize the channel throughput. Another several schemes were proposed in [18, 19] to support generalized SAS as well as fully loaded transmissions. They adopted the estimation of signal parameters via rotational invariance technique (ESPRIT)-like approach and exploited the direction-of-arrival (DOA) information to separate the signals from different users .

In this article, we develop a new subspace-based multiuser joint CFO and DOA estimation scheme for OFDMA uplink transmissions. We leverage multi-antenna at the receiver and consider that the signals transmitted by each user arrive at the receiver's antenna array from multiple DOAs, after bouncing from both surrounding and far scatterers [20]. The multiple CFOs and DOAs are then derived by a rank-reduction approach. Specifically, for each user, after the CFO estimation using 1D search, its multiple DOAs can also be obtained simultaneously by polynomial rooting, which is one unique property of our scheme. In summary, the main contributions of this article include the following:

• 1. With the consideration of multi-cluster channels, we design a new joint CFO and DOA estimation method for multiuser OFDMA uplink. The proposed method supports generalized SAS and fully loaded transmissions.

• 2. We provide the theoretical performance analysis of our method in terms of both CFO and DOA estimation.

• 3. Compared with [18, 19], the simulation results demonstrate that our method not only has the advantage of being applicable to multi-cluster channels, but also can obtain much better performance in single cluster channels.

Notations: Superscripts (·)*, (·) ^T , (·) ^H , [·]^†, and E[·] represent conjugate, transpose, Hermitian, pseudo inverse, and expectation, respectively; $\mathbf{j}=\sqrt{-1}$is the imaginary unit; || X|| denotes the Frobenius norm of X, and diag(·) is a diagonal matrix with main diagonal (·); The kronecker product is denoted by ⊗; The component-wise product is denoted by °; I _N denotes the N × N identity matrix and 1 _N denotes the 1 × N matrix with all entries being 1; Matlab matrix representations are adopted, for example, X(r₁ : r₂, c₁ : c₂) denotes the submatrix of X with the rows from r₁ to r₂ and the columns from c₁ to c₂.

System model

We consider a multiuser OFDMA system with K users, N subcarriers. The base station (BS) is equipped with a uniform linear array (ULA) with M antennas, which is elevated above the rooftop. All subcarriers are sequentially indexed with {0, 1, . . . , N - 1}. Assume that the channel between each user and the receiver is composed of N_cl clusters (N_cl ≥ 1). The multipath components in each cluster exhibit similar DOAs. Among the total N_cl clusters, one cluster is called surrounding cluster that corresponds to the scatterers located around each user, and the remaining N_cl - 1 clusters, called far clusters, correspond to high-rise buildings in urban environments and hills/mountains in rural environments [20, 21]. As the BS is deployed above its surrounding scatterers, following [18, 19], we further approximate that the multipath components from one cluster have a single DOA. Here we should note that the works in [18, 19] considered only the surrounding cluster, but ignored the existence of far clusters. However, as has been reported in [20], in the typical Urban environment, the fractions of the cases with two and three clusters are 9 and 4%, respectively. The fractions are even higher in the bad Urban environment, which are given by 28 and 45%, respectively. We should note that the methods developed in [18, 19] may not be applicable to these multi-cluster scenarios.

Denote χ = d/λ¸, where d is the antenna spacing of the ULA, and λ is the radio wavelength. Assume that in the g th block, the multipath channel components between the i th cluster of the k th user and the reference antenna (1st) of ULA can be modeled by a length-L _p vector

$${\mathbf{h}}_{1,i,g}^{\left(k\right)}={\left[h_{i,g}^{\left(k\right)}\left(0\right),h_{i,g}^{\left(k\right)}\left(1\right),\dots ,h_{i,g}^{\left(k\right)}\left(L_p-1\right)\right]}^T.$$

(1)

We assume that the entries of ${\mathbf{h}}_{1,i,g}^{\left(k\right)}$ are independent Gaussian variables with variance 1/L _p such that the expectation of the channel vector norm is 1, i.e., $E\left[\parallel {\mathbf{h}}_{1,i,g}^{\left(k\right)}\parallel \right]=1.$

Let ${\phi }_i^{\left(k\right)}$ denote the DOA of the i th cluster of the k th user and then the multipath channel components between the i th cluster of the k th user and the m th antenna of ULA can be expressed as

$${\mathbf{h}}_{m,i,g}^{\left(k\right)}=a_{m,i}^{\left(k\right)}\cdot {\mathbf{h}}_{1,i,g}^{\left(k\right)}$$

(2)

where $a_{m,i}^{\left(k\right)}={\mathsf{\text{e}}}^{\mathbf{j}2\pi \chi \left(m-1\right)\text{cos}{\phi }_i^{\left(k\right)}}$. Correspondingly, its frequency domain channel is given by

$${\mathbf{H}}_{m,i,g}^{\left(k\right)}=\sqrt{N}\mathbf{F}\cdot {\left[{\left({\mathbf{h}}_{m,i,g}^{\left(k\right)}\right)}^T,{\mathbf{0}}_{1\times \left(N-L_p\right)}\right]}^T=a_{m,i}^{\left(k\right)}\cdot {\left[H_{i,g}^{\left(k\right)}\left(0\right),H_{i,g}^{\left(k\right)}\left(1\right),\dots ,H_{i,g}^{\left(k\right)}\left(N-1\right)\right]}^T,$$

(3)

where F stands for the N × N DFT matrix with its (i, j) entry $\mathbf{F}\left(i,j\right)=\frac{1}{\sqrt{N}}{\mathsf{\text{e}}}^{-\mathbf{j}2\pi \frac{\left(i-1\right)\left(j-1\right)}{N}}$. We denote the normalized CFO of the k th user by ξ^(k)= Δf^(k)/Δf, where Δf is the subcarrier spacing and Δf^(k)is the CFO of the k th user. We assume ξ^(k)∈ (-0.5, 0.5). Denote the number and the index set of the subcarriers allocated to the k th user by N _k and C^(k), where

$$C^{\left(k\right)}=\left\{c_1^{\left(k\right)},c_2^{\left(k\right)},\dots ,c_{N_k}^{\left(k\right)}\right\},{\sum }_{k=1}^K{N}_k=N_{\mathsf{\text{sum}}}\le N.$$

(4)

Let ${\mathbf{s}}_g^{\left(k\right)}={\left[s_{1,g}^{\left(k\right)},s_{2,g}^{\left(k\right)},\dots ,s_{N_k,g}^{\left(k\right)}\right]}^T$be the modulated symbols of the k th user in the g th block. In the noise-free environment, the received time-domain signal components after removing CP from the k th user at the m th antenna can be expressed as

$${\gamma }_{m,g}^{\left(k\right)}\left(n\right)=\frac{1}{\sqrt{N}}\sum _{p=1}^{N_k}{\mathsf{\text{e}}}^{\mathbf{j}\frac{2\pi }{N}\left(c_p^{\left(k\right)}+{\xi }^{\left(k\right)}\right)n}\left(\sum _{i=1}^{N_{\mathsf{\text{c}}1}}{a}_{m,i}^{\left(k\right)}{H}_{i,g}^{\left(k\right)},\left(c_p^{\left(k\right)}\right)\right)s_{p,g}^{\left(k\right)},$$

(5)

where the term in the bracket stands for the composition frequency-domain channel response at the $c_p^{\left(k\right)}$th subcarrier of the k th user resulting from total N_cl clusters. Then the overall received signal from K users at the m th antenna can be expressed as

$${\gamma }_{m,g}\left(n\right)=\sum _{k=1}^K{\gamma }_{m,g}^{\left(k\right)}\left(n\right)=\frac{1}{\sqrt{N}}\sum _{k=1}^K\sum _{p=1}^{N_k}\sum _{i=1}^{N_{c1}}{a}_{m,i}^{\left(k\right)}{X}_{i,p,g}^{\left(k\right)}{\mathsf{\text{e}}}^{\mathbf{j}2\pi {\Theta }_p^{\left(k\right)}n}$$

(6)

where $X_{i,p,g}^{\left(k\right)}=H_{i,g}^{\left(k\right)}\left(c_p^{\left(k\right)}\right)s_{p,g}^{\left(k\right)}$ and ${\Theta }_p^{\left(k\right)}=\frac{c_p^{\left(k\right)}+{\xi }^{\left(k\right)}}{N}$ denote the effective CFO on the $c_p^{\left(k\right)}$ th subcarrier of the k th user.

Stacking the received signals from all M antenna elements at the n th sample, we obtain the following space-domain snapshot vector

$${\gamma }_{n,g}={\left[{\gamma }_{1,g}\left(n\right),{\gamma }_{2,g}\left(n\right),\dots ,{\gamma }_{M,g}\left(n\right)\right]}^T.$$

(7)

Define the Vandermonde vector

$${\mathbf{a}}_i^{\left(k\right)}={\left[a_{1,i}^{\left(k\right)},a_{2,i}^{\left(k\right)},\dots ,a_{M,i}^{\left(k\right)}\right]}^T$$

(8)

which reflects the DOA of the i th cluster of the k th user, and obtain the corresponding Vandermonde matrix

$${\mathbf{a}}^{\left(k\right)}=\left[{\mathbf{a}}_1^{\left(k\right)},{\mathbf{a}}_2^{\left(k\right)},\dots ,{\mathbf{a}}_{N_{\mathsf{\text{c}}1}}^{\left(k\right)}\right]$$

(9)

by collecting N_cl Vandermonde vectors. Considering the noise item, we can then rewrite γ _ng in the following matrix form

$${\gamma }_{n,g}=\frac{1}{\sqrt{N}}\mathbf{\Omega }{\mathbf{\Phi }}^n{\mathbf{X}}_g+{\mathbf{n}}_{n,g},$$

(10)

where

$$\begin{array}{c}\mathrm{\Omega }=\left[{\mathbf{1}}_{N_1}\otimes a^{\left(1\right)},{\mathbf{1}}_{N_2}\otimes a^{\left(2\right)},...,1_{N_K}\otimes a^{\left(K\right)}\right],{\mathbf{X}}_{p,g}^{\left(k\right)}={\left[X_{1,p,g}^{\left(k\right)},X_{2,p,g}^{\left(k\right)},...,X_{Nc1,p,g}^{\left(k\right)}\right]}^T,\\ {\mathbf{X}}_g^{\left(k\right)}={\left[{\left({\mathbf{X}}_{1,g}^{\left(k\right)}\right)}^T,{\left({\mathbf{X}}_{2,g}^{\left(k\right)}\right)}^T,\dots ,{\left({\mathbf{X}}_{N_k,g}^{\left(k\right)}\right)}^T\right]}^T,{\mathbf{X}}_g={\left[{\left({\mathbf{X}}_g^{\left(1\right)}\right)}^T,{\left({\mathbf{X}}_g^{\left(2\right)}\right)}^T,\dots ,{\left({\mathbf{X}}_g^{\left(K\right)}\right)}^T\right]}^T,\\ {\mathbf{\Phi }}^{\left(k\right)}=\mathsf{\text{diag(}}{\mathsf{\text{e}}}^{\mathbf{j}2\mathbf{\pi }{\Theta }_1^{\left(k\right)}}{\mathsf{\text{e}}}^{\mathbf{j}2\pi {\Theta }_2^{\left(k\right)}},\dots ,{\mathsf{\text{e}}}^{\mathbf{j}2\pi {\Theta }_{N_k}^{\left(k\right)}})\otimes I_{N_{c\mathsf{\text{1}}}},\mathbf{\Phi }=\mathsf{\text{diag(}}{\mathbf{\Phi }}^{\left(1\right)},{\mathbf{\Phi }}^{\left(2\right)},...,{\mathbf{\Phi }}^{\left(K\right)}),\end{array}$$

and n _n,g is a length-M additive white Gaussian noise (AWGN) vector with variance matrix ${\sigma }_n^2{\mathbf{I}}_M$ at the n th sample in the g th block.

Joint CFO and DOA estimation

Properties of the subspace

Stacking L (L ≤ N) continuous space-domain snapshot vectors from the n th to the (n+L- 1)th sample time, we obtain

$${\gamma }_g{|}_n^{n+L-1}={\left[{\left({\gamma }_{n,g}\right)}^T,{\left({\gamma }_{n+1,g}\right)}^T,\dots ,{\left({\gamma }_{n+L-1,g}\right)}^T\right]}^T=\mathbf{A}{\mathbf{\Phi }}^n{\mathbf{X}}_g+{\mathbf{N}}_g{|}_n^{N+L-1}$$

(11)

where

$$\mathbf{A}=\frac{1}{\sqrt{N}}{\left[{\left(\mathbf{\Omega }\right)}^T,{\left(\mathbf{\Omega }\mathbf{\Phi }\right)}^T,\dots ,{\left(\mathbf{\Omega }{\mathbf{\Phi }}^{L-1}\right)}^T\right]}^T,{\mathbf{N}}_g{|}_n^{n+L-1}={\left[{\mathbf{n}}_{n,g}^T,{\mathbf{n}}_{n+1,g}^T,\dots ,{\mathbf{n}}_{n+L-1,g}^T\right]}^T.$$

The effect of the parameter L will be discussed later. Afterwards, by defining

$${\mathbf{b}}_p^{\left(k\right)}=\frac{1}{\sqrt{N}}{\left[1,{\mathsf{\text{e}}}^{\mathbf{j}2\pi {\Theta }_p^{\left(k\right)}},\dots ,{\mathsf{\text{e}}}^{\mathbf{j}2\pi \left(L-1\right){\Theta }_p^{\left(k\right)}}\right]}^T,{\mathbf{B}}^{\left(k\right)}=\left[{\mathbf{b}}_1^{\left(k\right)},{\mathbf{b}}_2^{\left(k\right)},\dots ,{\mathbf{b}}_{N_k}^{\left(k\right)}\right],$$

we can rewrite A as

$$\mathbf{A}=\left[{\mathbf{B}}^{\left(1\right)}\otimes {\mathbf{a}}^{\left(1\right)},{\mathbf{B}}^{\left(2\right)}\otimes {\mathbf{a}}^{\left(2\right)},\dots ,{\mathbf{B}}^{\left(K\right)}\otimes {\mathbf{a}}^{\left(K\right)}\right].$$

(12)

We obtain the correlation matrix of ${\gamma }_g{|}_n^{n+L-1}$ as follows

$${\mathbf{R}}_{\gamma }=E\left[{\gamma }_g{|}_n^{n+L-1}{\left({\gamma }_g{|}_n^{n+L-1}\right)}^H\right]=\mathbf{A}{\mathbf{R}}_{XX}{\mathbf{A}}^H+{\sigma }_n^2{\mathbf{I}}_{ML},$$

(13)

where ${\mathbf{R}}_{XX}=\frac{1}{N-L+1}{\sum }_{n=0}^{N-L}{\mathbf{\Phi }}^{n}E\left[{\mathbf{X}}_g{\mathbf{X}}_g^H\right]{\left({\mathbf{\Phi }}^n\right)}^H={\sigma }_s^2{\mathbf{I}}_{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}$ with ${\sigma }_s^2$ being the average power of the transmitted signals. Then, we have ${\mathbf{R}}_{\gamma }={\sigma }_s^2\mathbf{A}{\mathbf{A}}^H+{\sigma }_n^2{\mathbf{I}}_{ML}$. In practice, using successive L _s OFDMA blocks, this correlation matrix can be approximatd by

$${\widehat{\mathbf{R}}}_{\gamma }=\frac{1}{(N-L+1)L_s}{\displaystyle \sum _{g=1}^{L_s}{\displaystyle {\sum }_{n=0}^{N-L}{\gamma }_g{|}_n^{n+L-1}{({\gamma }_g{|}_n^{n+L-1})}^H}}.$$

(14)

Afterwards, we assume the matrix A is tall and has full column rank, and the corresponding discussion will be presented later. Performing singular value decomposition (SVD) on R _γ gives.

$${\mathbf{R}}_{\gamma }=\left[{\mathbf{U}}_{\gamma },{\mathbf{V}}_{\gamma }\right]{\Sigma }_{\gamma }{\left[{\mathbf{U}}_{\gamma },{\mathbf{V}}_{\gamma }\right]}^H,$$

(15)

where U _γ and V _γ represent the (N_clN_sum)-dimensional signal space and (ML - N_clN_sum)-dimensional noise space matrices, respectively. We define the following length-L parameterized Vandermonde vector with respect to ξ:

$$B_p^{\left(k\right)}\left(\xi \right)=\frac{1}{\sqrt{N}}{\left[1,{\mathsf{\text{e}}}^{\mathbf{j}2\pi \frac{c_p^{\left(k\right)}+\xi }{N}},\dots ,{\mathsf{\text{e}}}^{\mathbf{j}2\pi \frac{\left(L-1\right)\left(c_p^{\left(k\right)}+\xi \right)}{N}}\right]}^T,$$

(16)

where ξ ∈ (-0.5, 0.5). Clearly, there holds ${\mathbf{b}}_p^{\left(k\right)}=B_p^{\left(k\right)}\left({\xi }^{\left(k\right)}\right)$. For notational convenience, we denote 0 as the all-zero matrix with appropriate dimension. Lemma 1 gives the key properties to design our joint estimator:

Lemma 1: When the matrix $\underset{-}{\mathbf{A}}$ has full column rank, then for a non-zero length-M vector ω, there holds

$${\left({\mathbb{B}}_p^{\left(k\right)}\left({\xi }^{\left(k\right)}\right)\otimes \mathbf{\omega }\right)}^H{V}_{\gamma }=\left\{\begin{array}{c}\hfill =0,\mathbf{\omega }\in \mathsf{\text{Span(}}{a}^{\mathsf{\text{(}}k\mathsf{\text{)}}}\mathsf{\text{),}}\hfill \\ \hfill \ne 0,\mathbf{\omega }\notin \mathsf{\text{Span(}}{a}^{\mathsf{\text{(}}k\mathsf{\text{)}}}\mathsf{\text{),}}\hfill \end{array}\right.$$

(17)

and

$${\left({\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes \mathbf{\omega }\right)}^H{\mathbf{V}}_{\mathbf{\gamma }}\ne 0,\xi \ne {\xi }^{\left(k\right)},$$

(18)

where $\underset{-}{\mathbf{A}}$ is the first M(L- 1) rows of A which can be expressed as

$$\begin{array}{c}\underset{}{\mathbf{A}}=\left[{\underset{}{\mathbf{B}}}^{\left(1\right)}\otimes {\mathbf{a}}^{\left(1\right)},{\underset{}{\mathbf{B}}}^{\left(2\right)}\otimes {\mathbf{a}}^{\left(2\right)},\dots ,{\underset{}{\mathbf{B}}}^{\left(K\right)}\otimes {\mathbf{a}}^{\left(K\right)}\right],\\ {\underset{}{\mathbf{B}}}^{\left(k\right)}=\left[{\underset{}{\mathbf{b}}}_1^{\left(k\right)},{\underset{}{\mathbf{b}}}_2^{\left(k\right)},\dots ,{\underset{}{\mathbf{b}}}_{N_k}^{\left(k\right)}\right],{\underset{}{\mathbf{b}}}_p^{\left(k\right)}=\frac{1}{\sqrt{N}}{\left[1,{\mathsf{\text{e}}}^{\mathbf{j}2\pi {\Theta }_p^{\left(k\right)}},\dots ,{\mathsf{\text{e}}}^{\mathbf{j}2\pi \left(L-2\right){\Theta }_p^{\left(k\right)}}\right]}^T.\end{array}$$

Proof. See Appendix 1. □

Parameters estimation

CFO estimation

For any non-zero length-M vector ω, we know

$$\begin{array}{ll}\hfill {\sum }_{p=1}^{N_k}\left(\right.{\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes \mathbf{\omega }{}^H{\mathbf{V}}_{\gamma }{\mathbf{V}}_{\gamma }^H\left(\right.{\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes \mathbf{\omega }\left)\right.& =\sum _{p=1}^{N_k}{\mathbf{\omega }}^H{\left({\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M\right)}^H{\mathbf{V}}_{\gamma }{\mathbf{V}}_{\gamma }^H\left(B_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M\right)\mathbf{\omega }\\ ={\mathbf{\omega }}^H{\mathbf{\Pi }}^{\left(k\right)}\left(\xi \right)\mathbf{\omega }\end{array}$$

where

$$\begin{array}{l}\hfill {\mathbf{\Pi }}^{\left(k\right)}\left(\xi \right)={\sum }_{p=1}^{N_k}{\left({\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M\right)}^H{\mathbf{V}}_{\gamma }{\mathbf{V}}_{\gamma }^H\left({\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M\right).\end{array}$$

(19)

Lemma 1 tells us that when A has full column rank,

• (1) the matrix ∏ ^(k)(ξ) is singular at ξ = ξ ^(k). Meanwhile, ∏ ^(k)( ξ ^(k)) has N _cl zero eigenvalues;

• (2) the matrix ∏ ^(k)(ξ) should be positive definite when ξ ≠ ξ ^(k).

It implies the matrix ∏^(k)(ξ) drops rank if and only if ξ = ξ^(k). Based on above observations, we design the CFO estimation as follows. Select a trial ξ from (-0.5, 0.5) and compute the M eigenvalues of the matrix ∏^(k)(ξ), denoted by ${\kappa }_1^{\left(k\right)}\left(\xi \right),{\kappa }_2^{\left(k\right)}\left(\xi \right),\dots ,{\kappa }_M^{\left(k\right)}\left(\xi \right)$in ascending order. The CFO for the k th user can be obtained from 1D search by minimizing the following cost function:

$${\widehat{\xi }}^{\left(k\right)}=\text{arg}\underset{\xi }{\text{min}}\sum _{l=1}^{N_{\mathsf{\text{c}}1}}{\kappa }_l^{\left(k\right)}\left(\xi \right).$$

(20)

DOA estimation

We denote ${\epsilon }_l^{\left(k\right)}\left(\xi \right)$ as the eigenvector of matrix ∏^(k)(ξ) corresponding to its l th eigenvalue ${\kappa }_l^{\left(k\right)}\left(\xi \right)$. Notice that the first N_cl eigenvectors ${\epsilon }_l^{\left(k\right)}\left({\xi }^{\left(k\right)}\right)$, l = 1, 2, . . . , N_cl, correspond to the N_cl zero eigenvalues of ∏^(k)(ξ^(k)). From Lemma 1, the N_cl column vectors of Vandermonde matrix a^(k)constitute the same column space of $\left[{\epsilon }_1^{\left(k\right)}\left({\xi }^{\left(k\right)}\right),{\epsilon }_2^{\left(k\right)}\left({\xi }^{\left(k\right)}\right),\dots ,{\epsilon }_{N_{\mathsf{\text{cl}}}}^{\left(k\right)}\left({\xi }^{\left(k\right)}\right)\right]$, which implies that a^(k)should be orthogonal to the other M - N_cl eigenvectors, i.e.,

$${\left({\mathbf{a}}^{\left(k\right)}\right)}^H{\epsilon }_l^{\left(k\right)}\left({\xi }^{\left(k\right)}\right)=\mathbf{0},l=N_{\mathsf{\text{c}}1}+1,\dots ,M.$$

(21)

Thereby, after the CFO estimation for the k th user, we can further derive the N_cl DOAs for the k th user by finding the N_cl minimum point of the following cost function:

$${\widehat{\phi }}_i^{\left(k\right)}=\text{arg}\underset{\phi }{\text{min}}\sum _{l=N_{\mathsf{\text{c}}1}+1}^M{∥{\left(\alpha \left(\phi \right)\right)}^H{\epsilon }_l^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)∥}^2=\text{arg}\underset{\phi }{\text{min}}{g}^{\left(k\right)}\left(\phi \right),$$

(22)

i = 1, 2, . . . , N_cl, where $\alpha \left(\phi \right)={\left[1,{\mathsf{\text{e}}}^{\mathbf{j}2\pi \chi \text{cos}\phi },\dots ,{\mathsf{\text{e}}}^{\mathbf{j}2\pi \chi \left(M-1\right)\text{cos}\phi }\right]}^T$. Note that the polynomial rooting approach can be used to implement this minimization problem. The basic idea is first obtaining all local minimum/maximum solutions by setting the derivative of the cost function to be zero, and then putting these solutions back to the original cost function and selecting the minimum after comparison [22]. Specifically, denoting ${\Theta }^{\left(k\right)}\left(\xi \right)={\sum }_{l=N_{\mathsf{\text{c}}1}+1}^M{\epsilon }_l^{\left(k\right)}\left(\xi \right){\left({\epsilon }_l^{\left(k\right)}\left(\xi \right)\right)}^H$ and Ψ = diag(0, 1, . . . , M - 1), we obtain

$$\frac{\partial g^{\left(k\right)}\left(\phi \right)}{\partial \phi }=\mathbf{j}2\pi \chi \text{sin}\phi \cdot {\left(\mathit{\alpha }\left(\phi \right)\right)}^H\left(\mathbf{\Psi }{\mathbf{\Theta }}^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)-{\mathbf{\Theta }}^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)\mathbf{\Psi }\right)\alpha \left(\phi \right).$$

(23)

According to [22], we know $z={\mathsf{\text{e}}}^{\mathbf{j}2\pi \chi \text{cos}{\widehat{\phi }}_i^{\left(k\right)}}$, i = 1, 2, . . . , N_cl, is one of the roots for the polynomial $Q\left(z\right)={\sum }_{m=-M+1}^{M-1}{b}_m{z}^m$ where

$$b_m=\sum _{q-p=m}{\left[\mathbf{\Psi }{\mathbf{\Theta }}^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)-{\mathbf{\Theta }}^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)\mathbf{\Psi }\right]}_{pq}.$$

(24)

By putting the roots on the unit circle of Q(z) back to the original cost function g^(k)(φ) and selecting the N_cl minimum points after comparisons, the solution to (22) is obtained. Afterwards, for $\underset{-}{\mathbf{A}}$ with full column rank, we obtain Lemma 2:

Lemma 2: Assume all the K users have distinct DOAs. According to the number of subcarriers allocated, we arrange the user in descending order, as follows

$$e_1,e_2,\dots ,e_K,$$

such that $N_{e_k}\ge N_{e_{k+1}}$, then when

$$\left\{\begin{array}{c}\hfill L\ge N_{e_1}+1,v\ge K,\hfill \\ \hfill L\ge N_{e_1}+1+{\sum }_{k=v+1}^K{N}_{e_k},v<K,\hfill \end{array}\right.$$

(25)

where $v=⌊\frac{M}{N_{\mathsf{\text{c}}1}}⌋\ge 2$ with ⌊·⌋ denoting the integer floor operation, the matrix $\underset{-}{\mathbf{A}}$ has full column rank.

Proof. See Appendix 2. □

Note that when $\underset{-}{\mathbf{A}}$ has full column rank, the matrix A will be tall and also has full column rank, which guarantees the validity of the SVD operation in (15). Following Lemmas 1 and 2, we could make the following important observation; that is when M ≥ 2N_cl, i.e., the number of antennas at the receivers is not less than two times of the number of channel clusters from each user, the validity of our method is guaranteed with L satisfying the condition in (25).

Performance analysis

In this section, we provide theoretical performance analysis for both the CFO and DOA estimation performance of our proposed method. Bearing in mind that ${\mathbf{V}}_{\gamma }{\mathbf{V}}_{\gamma }^H=I_{ML}-U_{\gamma }{U}_{\gamma }^H$,

we can rewrite Π^(k)(ξ) as follows

$$\begin{array}{ll}{\mathbf{\Pi }}^{(k)}(\xi )\hfill & ={\displaystyle \sum _{p=1}^{N_k}\left(\frac{L}{N}{\mathbf{I}}_M-{({\mathbb{B}}_p^{(k)}(\xi )\otimes {\mathbf{I}}_M)}^H{\mathbf{U}}_{\gamma }{\mathbf{U}}_{\gamma }^H({\mathbb{B}}_p^{(k)}(\xi )\otimes {\mathbf{I}}_M)\right)}=\frac{N_{k}L}{N}{\mathbf{I}}_M-{\mathbf{\Xi }}^{(k)}(\xi )\hfill \end{array}$$

(26)

where

$$\begin{array}{l}\hfill {\mathbf{\Xi }}^{\left(k\right)}\left(\xi \right)={\sum }_{p=1}^{N_k}{\left({\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M\right)}^H{\mathbf{U}}_{\gamma }{\mathbf{U}}_{\gamma }^H\left({\mathbb{B}}_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M\right).\end{array}$$

(27)

Denote the M eigenvalues of matrix Ξ^(k)(ξ) by ${\lambda }_1^{\left(k\right)}\left(\xi \right)$, ${\lambda }_2^{\left(k\right)}\left(\xi \right),\dots ,{\lambda }_M^{\left(k\right)}\left(\xi \right)$ in ascending order. The corresponding eigenvectors are denoted by ${\nu }_1^{\left(k\right)}\left(\xi \right)$, ${\nu }_2^{\left(k\right)}\left(\xi \right),...,{\nu }_M^{\left(k\right)}\left(\xi \right)$. We can readily verify the following relationships: ${\lambda }_l^{\left(k\right)}\left(\xi \right)=\frac{N_{k}L}{N}-{\kappa }_{M-l+1}^{\left(k\right)}\left(\xi \right)$ and ${\nu }_l^{\left(k\right)}\left(\xi \right)={\epsilon }_{M-l+1}^{\left(k\right)}\left(\xi \right)$.

Then, (20) and (22) can be rewritten as follows:

$${\widehat{\xi }}^{\left(k\right)}=\text{arg}\underset{\xi }{\text{max}}\sum _{l=M-N_{\mathsf{\text{c}}1}+1}^M{\lambda }_l^{\left(k\right)}\left(\xi \right)=\text{arg}\underset{\xi }{\text{max}}{G}^{\left(k\right)}\left(\xi \right),$$

(28)

$${\widehat{\phi }}_i^{\left(k\right)}=\text{arg}\underset{\phi }{\text{min}}\sum _{l=1}^{M-N_{\mathsf{\text{c}}1}}{∥{\left(\alpha \left(\phi \right)\right)}^H{\nu }_l^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)∥}^{\mathsf{\text{2}}}\mathsf{\text{ = }}\text{arg}\underset{\phi }{\text{min}}{D}^{\left(k\right)}\left(\phi \right).$$

(29)

CFO estimation performance

We rewrite (15) as

$${\mathbf{R}}_{\gamma }={\displaystyle \sum _{i=1}^{ML}{\mathit{\sigma }}_i^2{e}_i{e}_i^H}$$

(30)

where ${\sigma }_i^2$ and e _i , i = 1, 2, ..., N_clN_sum, denote the eigenvalues and eigenvectors corresponding to signal space, respectively, while the remaining ${\sigma }_i^2={\sigma }_n^2$ and e _i , i = N_clN_sum + 1, ..., ML, correspond to the noise space.

Let ${\mathit{\eta }}_i^{\Re }=\Re \left\{{\mathit{e}}_i\right\}$ and ${\mathit{\eta }}_i^{\Im }=\Im \left\{{\mathit{e}}_i\right\}$, i.e., ${\mathit{e}}_i={\mathit{\eta }}_i^{\Re }+j{\mathit{\eta }}_i^{\Im }$.Denote ${\eta }_i={[{({\eta }_i^{\Re })}^T,{({\eta }_i^{\Im })}^T]}^T$, and $\mathit{\eta }={\left[{\mathit{\eta }}_1^T,{\mathit{\eta }}_2^T,\dots ,{\mathit{\eta }}_{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}^T\right]}^T$. Let $\widehat{x}$ stand for the estimated value for x. The CFO estimation variance of the k th user can be given by [23]:

$$Cov\left\{{\widehat{\xi }}^{\left(k\right)}\right\}={\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial {\xi }^2}\right]}^{-1}\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]Cov\left\{\widehat{\mathit{\eta }}\right\}{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]}^T{\left.{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial {\xi }^2}\right]}^{-1}\right|}_{\xi ={\xi }^{\left(k\right)}}.$$

(31)

We denote ${\mathbf{\Gamma }}_p^{\left(k\right)}\left(\xi \right)=B_p^{\left(k\right)}\left(\xi \right)\otimes {\mathbf{I}}_M$ for short. Then, there is

$$\frac{\partial {\mathbf{\Gamma }}_p^{\left(k\right)}\left(\xi \right)}{\partial \xi }=\mathit{D}\cdot {\mathbf{\Gamma }}_p^{\left(k\right)}\left(\xi \right)$$

(32)

Where $D=\frac{\text{j}2\pi }{N}\text{diag(0,1,}\mathrm{...}\text{, }L-1)\otimes {\mathbf{I}}_M$. In the following, we omit the parameterized notation (ξ) for presentation clarity. Then, we have

$$\frac{\partial G^{(k)}}{\partial \xi }={\displaystyle \sum _{l=M-N_{cl}+1}^M{({\mathbf{\nu }}_l^{(k)})}^H\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial \xi }{\mathbf{\nu }}_l,}$$

(33)

$$\frac{{\partial }^2{G}^{(k)}}{\partial {\xi }^2}={\displaystyle \sum _{l=M-N_{c1}+1}^M\left[{({\mathbf{\nu }}_l^{(k)})}^H\frac{{\partial }^2\mathbf{\Xi }(k)}{\partial {\xi }^2}{\mathbf{\nu }}_l^{(k)}+2\Re \left\{{({\mathbf{\nu }}_l^{(k)})}^H\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial \xi }\left(\frac{\partial {\mathbf{\nu }}_l^{(k)}}{\partial \xi }\right)\right\}\right]}$$

(34)

where

$$\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial \xi }={\displaystyle \sum _{p=1}^{N_k}\left({({\mathbf{\Gamma }}_p^{(k)})}^H{\mathit{D}}^H{\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{\Gamma }}_p^{(k)}+{({\mathbf{\Gamma }}_p^{(k)})}^H{\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H\mathit{D}{\mathbf{\Gamma }}_p^{(k)}\right)},$$

(35)

$$\frac{{\partial }^2{\mathbf{\Xi }}^{(k)}}{\partial {\xi }^2}={\displaystyle \sum _{p=1}^{N_k}\left({({\mathbf{\Gamma }}_p^{(k)})}^H{\mathit{D}}^H{\mathit{D}}^H{\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{\Gamma }}_p^{(k)}+2{({\mathbf{\Gamma }}_p^{(k)})}^H{\mathit{D}}^H{\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H\mathit{D}{\mathbf{\Gamma }}_p^{(k)}+{({\mathbf{\Gamma }}_p^{(k)})}^H{\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H\mathit{D}\mathit{D}{\mathbf{\Gamma }}_p^{(k)})\right)},$$

(36)

$$\frac{\partial {\mathbf{\nu }}_l^{(k)}}{\partial \xi }={\displaystyle \sum _{z\ne l}^M\frac{{({\mathbf{\nu }}_z^{(k)})}^H\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial \xi }{\mathbf{\nu }}_l^{(k)}}{{\lambda }_l^{(k)}-{\lambda }_z^{(k)}}}{\mathbf{\nu }}_z^{(k)}.$$

(37)

Next, we obtain

$$\frac{{\partial }^2{G}^{(k)}}{\partial \xi \partial {\mathit{\eta }}_i^{\Re }}={\displaystyle \sum _{l=M-N_{c1}+1}^M{({\mathit{\nu }}_l^{(k)})}^H\frac{{\partial }^2\mathbf{\Xi }(k)}{\partial \xi \partial {\mathit{\eta }}_i^{\Re }}}{\mathit{\nu }}_l^{(k)}+2\Re \left\{{\displaystyle \sum _{l=M-N_{c1}+1}^M{({\mathit{\nu }}_l^{(k)})}^H\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial \xi }\frac{\partial {\mathit{\nu }}_l^{(k)}}{\partial {\mathit{\eta }}_i^{\Re }}}\right\}$$

(38)

where

$$\frac{{\partial }^2{\mathbf{\Xi }}^{(k)}}{\partial \xi \partial {\eta }_i^{\Re }}={\displaystyle \sum _{p=1}^{N_k}\left({({\mathbf{\Gamma }}_p^{(k)})}^H{\mathit{D}}^H\frac{\partial ({\mathbf{U}}_{\gamma }{\mathbf{U}}_{\gamma }^H)}{\partial {\eta }_i^{\Re }}{\mathbf{\Gamma }}_p^{(k)}+{({\mathbf{\Gamma }}_p^{(k)})}^H\frac{\partial ({\mathbf{U}}_{\gamma }{\mathbf{U}}_{\gamma }^H)}{\partial {\eta }_i^{\Re }}\mathit{D}{\mathbf{\Gamma }}_p^{(k)}\right)},$$

(39)

$$\begin{array}{l}\frac{\partial {\mathbf{\nu }}_l^{(k)}}{\partial {\eta }_i^{\Re }}={\displaystyle \sum _{z\ne l}^M\frac{{\mathbf{\nu }}_z^{(k)}}{{\lambda }_l^{(k)}-{\lambda }_z^{(k)}}}\left({({\mathbf{\nu }}_z^{(k)})}^H\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial {\eta }_{\text{i}}^{\Re }}{\mathbf{\nu }}_l^{(k)}\right)\\ ={\displaystyle \sum _{z\ne l}^M\frac{{\mathbf{\nu }}_z^{(k)}}{{\lambda }_l^{(k)}-{\lambda }_z^{(k)}}}\left({\displaystyle \sum _{p=1}^{N_k}{({\mathbf{\nu }}_z^{(k)})}^H}{({\mathbf{\Gamma }}_p^{(k)})}^H\frac{\partial ({\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H)}{\partial {\eta }_i^{\Re }}{\mathbf{\Gamma }}_p^{(k)}{\mathbf{\nu }}_l^{(k)}\right).\end{array}$$

(40)

Using

$${\omega }_1^H\frac{\partial \left({\mathbf{U}}_{\mathit{\gamma }}{\mathbf{U}}_{\mathit{\gamma }}^H\right)}{\partial {\mathit{\eta }}_i^{\Re }}{\omega }_2=e_i^H{\omega }_2{\omega }_1^H+{\omega }_1^H{e}_i{\omega }_2^T$$

(41)

and skipping some algebraic steps, we can simplify (38) as

$$\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial {\mathit{\eta }}_i^{\Re }}=2\Re \left\{{\mathit{e}}_i^H{\mathbf{Q}}^{\left(k\right)}\right\}$$

(42)

where

$${\mathbf{Q}}^{\left(k\right)}={\tilde{\mathbf{Q}}}^{\left(k\right)}+{\left({\tilde{\mathbf{Q}}}^{\left(k\right)}\right)}^H,$$

(43)

$$\begin{array}{l}{\tilde{\text{Q}}}^{(k)}={\displaystyle \sum _{l=M-N_{c1}{}_{+1}}^M{\displaystyle \sum _{p=1}^{N_k}({\mathbf{\Gamma }}_p^{(k)}{\mathbf{\nu }}_l^{(k)}{({\mathbf{\nu }}_l^{(k)})}^H{({\mathbf{\Gamma }}_p^{(k)})}^H{\mathit{D}}^H}}\\ +{\displaystyle \sum _{z=1}^{M-N_{\text{c}1}}\frac{{({\mathbf{\nu }}_l^{(k)})}^H(\frac{\partial {\mathbf{\Xi }}^{(k)}}{\partial \xi }){\mathbf{\nu }}_z^{(k)}}{{\lambda }_{{}_l}^{(k)}-{\lambda }_{{}_z}^{(k)}}}{\mathbf{\Gamma }}_p^{(k)}{\mathbf{\nu }}_l^{(k)}{({\mathbf{\nu }}_z^{(k)})}^H{({\mathbf{\Gamma }}_p^{(k)})}^H).\end{array}$$

(44)

Likewise, we can also obtain

$$\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial {\mathit{\eta }}_i^{\Im }}=-2\Im \left\{{\mathit{e}}_i^H{\mathbf{Q}}^{\left(k\right)}\right\}.$$

(45)

Furthermore, using

$$\mathsf{\text{Cov}}\left\{{\widehat{\mathit{\eta }}}_i,{\widehat{\mathit{\eta }}}_j\right\}=\frac{1}{2}\left[\begin{array}{c}\Re \left\{\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j\right\}+\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j^{*}\right\}\right\}\Im \left\{--\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j\right\}+\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j^{*}\right\}\right\}\\ \Im \left\{\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j\right\}+\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j^{*}\right\}\right\}\Re \left\{--\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j\right\}-\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j^{*}\right\}\right\}\end{array}\right],$$

(46)

we can further obtain

$$\begin{array}{c}\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\mathsf{\text{Cov}}\left\{\widehat{\mathit{\eta }}\right\}{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]}^T=2\Re \left\{\sum _{i=1}^{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}\sum _{j=1}^{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}{\left(e_i^H\mathbf{Q}\right.}^{\left(k\right)}\mathsf{\text{Cov}}\left\{{ê}_i,{ê}_j\right\}{\mathbf{Q}}^{\left(k\right)}{e}_j\right.\\ \left.\left.+e_i^H{\mathbf{Q}}^{\left(k\right)}\mathsf{\text{Cov}}\left\{{ê}_i,{ê}_j^{*}\right\}{\left({\mathbf{Q}}^{\left(k\right)}\right)}^{*}{e}_j^{*}\right)\right\}.\end{array}$$

(47)

Based on the results from [24], we know

$$\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j\right\}=\sum _{q=1,q\ne i}^{ML}\sum _{f=1,f\ne j}^{ML}\left[\frac{1}{L_s{\left(N-L+1\right)}^2}\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}\frac{{\mathit{e}}_q^H{\mathbf{R}}_{t,r}{\mathit{e}}_f{\mathit{e}}_j^H{\mathbf{R}}_{r,t}{\mathit{e}}_i}{\left({\sigma }_i^2-{\sigma }_q^2\right)\left({\sigma }_j^2-{\sigma }_f^2\right)}\right]{\mathit{e}}_q{\mathit{e}}_f^H,$$

(48)

$$\mathsf{\text{Cov}}\left\{{\widehat{\mathit{e}}}_i,{\widehat{\mathit{e}}}_j^{*}\right\}=\sum _{q=1,q\ne i}^{ML}\sum _{f=1,f\ne j}^{ML}\left[\frac{1}{L_s{\left(N-L+1\right)}^2}\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}\frac{{\mathit{e}}_q^H{\mathbf{R}}_{t,r}{\mathit{e}}_j{\mathit{e}}_f^H{\mathbf{R}}_{r,t}{\mathit{e}}_i}{\left({\sigma }_i^2-{\sigma }_q^2\right)\left({\sigma }_j^2-{\sigma }_f^2\right)}\right]{\mathit{e}}_q{\mathit{e}}_f^T$$

(49)

where

$${\mathbf{R}}_{t,r}=E\left[{\mathbf{\gamma }}_g{|}_t^{t+L-1}{\left({\mathbf{\gamma }}_g{|}_r^{r+L-1}\right)}^H\right]={\sigma }_s^2\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H+{\sigma }_n^2{\mathbf{J}}_{t,r},$$

(50)

and J _tr is a submatrix of ${{\mathbf{I}}_N}_{{}_{cl}{N}_{sum}}$ which is given by ${\mathbf{J}}_{t,r}={\mathbf{I}}_{N_{\mathsf{\text{cl}}}{N}_{\mathsf{\text{sum}}}}(1+\left(t-1\right)M:1+\left(t+L-2\right)M$, 1+(r - 1)M : 1+(r+ L- 2)M).

Substituting (48) and (49) into (47) and after some algebraic steps, we arrive at

$$\begin{array}{c}\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]\mathsf{\text{Cov}}\left\{\widehat{\mathit{\eta }}\right\}{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]}^T\\ =\frac{2}{L_s{\left(N-L+1\right)}^2}\Re \left[\sum _{i=1}^{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}\sum _{j=1}^{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}\sum _{q=N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}+1}^{ML}\right.\\ \sum _{f=N_{c1}{N}_{\mathsf{\text{sum}}}+1}^{ML}\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}\left({\mathit{e}}_i^H{Q}^{\left(k\right)}\right.\frac{{\mathit{e}}_q^H{\mathbf{R}}_{t,r}{\mathit{e}}_f{\mathit{e}}_j^H{\mathbf{R}}_{r,t}{\mathit{e}}_i}{\left({\sigma }_i^2-{\sigma }_n^2\right)\left({\sigma }_j^2-{\sigma }_n^2\right)}{\mathit{e}}_q{\mathit{e}}_f^H{\mathbf{Q}}^{\left(k\right)}{\mathit{e}}_j\\ +{\mathit{e}}_i^H{\mathbf{Q}}^{\left(k\right)}\frac{{\mathit{e}}_q^H{\mathbf{R}}_{t,r}{\mathit{e}}_j{\mathit{e}}_f^H{\mathbf{R}}_{r,t}{\mathit{e}}_i}{\left({\sigma }_i^2-{\sigma }_n^2\right)\left({\sigma }_j^2-{\sigma }_n^2\right)}\left.\left.{\mathit{e}}_q{\mathit{e}}_f^T{\left({\mathbf{Q}}^{\left(k\right)}\right)}^{*}{\left({\mathit{e}}_j\right)}^{*}\right)\right].\end{array}$$

(51)

By defining

$$\begin{array}{c}{\mathbf{\Omega }}_{UV}^{\left(k\right)}={\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{Q}}^{\left(k\right)}{\mathbf{V}}_{\mathit{\gamma }},{\mathbf{\Omega }}_{VU}^{\left(k\right)}={\mathbf{V}}_{\mathit{\gamma }}^H{\mathbf{Q}}^{\left(k\right)}{\mathbf{U}}_{\mathit{\gamma }},{\Delta }_{t,r,UU}^{\left(k\right)}={\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{R}}_{t,r}{\mathbf{U}}_{\mathit{\gamma }},\\ {\Delta }_{t,r,VV}^{\left(k\right)}={\mathbf{V}}_{\mathit{\gamma }}^H{\mathbf{R}}_{t,r}{\mathbf{V}}_{\mathit{\gamma }},{\Delta }_{t,r,UV}^{\left(k\right)}={\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{R}}_{t,r}{\mathbf{V}}_{\mathit{\gamma }},{\Delta }_{t,r,VU}^{\left(k\right)}={\mathbf{V}}_{\mathit{\gamma }}^H{\mathbf{R}}_{t,r}{\mathbf{U}}_{\mathit{\gamma }}\end{array}$$

we can further rewrite (51) into the following more compact form

$$\begin{array}{ll}\hfill \left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]\mathsf{\text{Cov}}\left\{\widehat{\mathit{\eta }}\right\}{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]}^T& =\frac{2}{L_s{\left(N-L+1\right)}^2}\Re \left[\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}{d}_{\sigma }^T\right.\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{\Delta }}_{t,r,VV}^{\left(k\right)}{\mathbf{\Omega }}_{VU}^{\left(k\right)}\right){\left({\mathbf{\Delta }}_{t,r,UU}^{\left(k\right)}\right)}^{*}\\ \left.\left.+\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{\Delta }}_{t,r,VU}^{\left(k\right)}\right)\left({\left({\mathbf{\Delta }}_{t,r,UV}^{\left(k\right)}\right)}^{*}{\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}\right)}^T\right)\right)d_{\sigma }\right]\end{array}$$

(52)

where ${\mathit{d}}_{\sigma }={\left[\frac{1}{{\sigma }_1^2-{\sigma }_n^2},\frac{1}{{\sigma }_2^2-{\sigma }_n^2},\dots ,\frac{1}{{\sigma }_{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}^2-{\sigma }_n^2}\right]}^T$.

Based on the fact that ${\mathbf{V}}_{\mathit{\gamma }}^H\mathbf{A}=\mathbf{0}$, we further obtain

$$\begin{array}{c}{\mathbf{\Delta }}_{t,r,UU}^{\left(k\right)}={\sigma }_s^2{\mathbf{U}}_{\mathit{\gamma }}^H\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H{\mathbf{U}}_{\mathit{\gamma }}+{\sigma }_n^2{\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{J}}_{t,r}{\mathbf{U}}_{\mathit{\gamma }}\simeq {\sigma }_S^2{\mathbf{U}}_{\mathit{\gamma }}^H\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H{\mathbf{U}}_{\mathit{\gamma }},\\ {\mathbf{\Delta }}_{t,r,VV}^{\left(k\right)}={\sigma }_n^2{\mathbf{V}}_{\mathit{\gamma }}^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\mathit{\gamma }},{\mathbf{\Delta }}_{t,r,UV}^{\left(k\right)}={\sigma }_n^2{\mathbf{U}}_{\mathit{\gamma }}^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\mathit{\gamma }},{\mathbf{\Delta }}_{t,r,VU}^{\left(k\right)}={\sigma }_n^2{\mathbf{V}}_{\mathit{\gamma }}^H{\mathbf{J}}_{t,r}{\mathbf{U}}_{\mathit{\gamma }}.\end{array}$$

Then, we have

$$\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{\Delta }}_{t,r,VV}^{\left(k\right)}{\mathbf{\Omega }}_{VU}^{\left(k\right)}\right){\left({\mathbf{\Delta }}_{t,r,UU}^{\left(k\right)}\right)}^{*}={\sigma }_s^2{\sigma }_n^2\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{V}}_{\gamma }^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\gamma }{\mathbf{\Omega }}_{VU}^{\left(k\right)}\right){\left({\mathbf{U}}_{\gamma }^H\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H{\mathbf{U}}_{\gamma }\right)}^{*},$$

(53)

$$\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{\Delta }}_{t,r,VU}^{\left(k\right)}\right)\left({\left({\mathbf{\Delta }}_{t,r,UV}^{\left(k\right)}\right)}^{*}{\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}\right)}^T\right)={\sigma }_n^4\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{V}}_{\gamma }^H{\mathbf{J}}_{t,r}{\mathbf{U}}_{\gamma }\right)\left({\left({\mathbf{U}}_{\gamma }^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\gamma }\right)}^{*}{\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}\right)}^T\right).$$

(54)

Bearing in mind that (54) is infinitesimal under high signal-to-noise ratio (SNR) as compared to (53), we can then approximate (52) as

$$\begin{array}{c}\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]\mathsf{\text{Cov}}\left\{\widehat{\mathit{\eta }}\right\}{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]}^T\\ \simeq \frac{2{\sigma }_s^2{\sigma }_n^2}{L_s{\left(N-L+1\right)}^2}\Re \left[\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}{d}_{\sigma }^T\left(\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{V}}_{\gamma }^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\gamma }{\mathbf{\Omega }}_{VU}^{\left(k\right)}\right){\left({\mathbf{U}}_{\gamma }^H\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H{\mathbf{U}}_{\gamma }\right)}^{*}\right)d_{\sigma }\right].\end{array}$$

(55)

Let ${\tilde{\sigma }}_i^2$, i = 1, 2, ... , ML, be the eigenvalues of matrix AA^H . Then, there holds ${\sigma }_i^2={\sigma }_s^2{\tilde{\sigma }}_i^2+{\sigma }_n^2$. By defining ${\tilde{\mathit{d}}}_{\sigma }={\left[\frac{1}{{\tilde{\sigma }}_1^2},\frac{1}{{\tilde{\sigma }}_2^2},\dots ,\frac{1}{{\tilde{\sigma }}_{N_{\mathsf{\text{c}}1}{N}_{\mathsf{\text{sum}}}}^2}\right]}^T$, we have ${\tilde{\mathit{d}}}_{\sigma }={\mathit{d}}_{\sigma }/{\sigma }_s^2$. Then, we can rewrite (55) as

$$\begin{array}{c}\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]\mathsf{\text{Cov}}\left\{\widehat{\mathit{\eta }}\right\}{\left[\frac{{\partial }^2{G}^{\left(k\right)}}{\partial \xi \partial \mathit{\eta }}\right]}^T\\ \simeq \frac{2{\sigma }_n^2}{{\sigma }_s^2{L}_s{\left(N-L+1\right)}^2}\Re \left[\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}{\tilde{\mathit{d}}}_{\sigma }^T\left(\left({\mathbf{\Omega }}_{UV}^{\left(k\right)}{\mathbf{V}}_{\gamma }^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\gamma }{\mathbf{\Omega }}_{VU}^{\left(k\right)}\right){\left({\mathbf{U}}_{\gamma }^H\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H{\mathbf{U}}_{\gamma }\right)}^{*}\right){\tilde{\mathit{d}}}_{\sigma }\right].\end{array}$$

(56)

On the other side, we can also rewrite $\frac{{\partial }^2{G}^{\left(k\right)}}{\partial {\xi }^2}$ as follows

$${\left.\frac{{\partial }^2{G}^{\left(k\right)}}{\partial {\xi }^2}\right|}_{\xi ={\xi }^{\left(k\right)}}=2\sum _{l=M-N_{\mathsf{\text{c}}1}+1}^M\left[-{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^H{\mathbf{\Lambda }}_1^{\left(k\right)}{\mathit{\nu }}_l^{\left(k\right)}+\sum _{z=1}^{M-N_{\mathsf{\text{c}}1}}\frac{|{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^H{\mathbf{\Lambda }}_2^{\left(k\right)}{\mathit{\nu }}_z^{\left(k\right)}{\mathsf{\text{|}}}^{\mathsf{\text{2}}}}{{\lambda }_l^{\left(k\right)}-{\lambda }_z^{\left(k\right)}}\right],$$

(57)

where

$${\mathbf{\Lambda }}_1^{(k)}={\displaystyle \sum _{p=1}^{N_k}{({\mathbf{\Gamma }}_p^{(k)})}^H{\mathbf{D}}^H{\mathbf{V}}_{\mathit{\gamma }}{\mathbf{V}}_{\mathit{\gamma }}^H\mathit{D}{\mathbf{\Gamma }}_p^{(k)},}{\mathbf{\Lambda }}_2^{(k)}={\displaystyle \sum _{p=1}^{N_k}{({\mathbf{\Gamma }}_p^{(k)})}^H}{\mathit{D}}^H{\mathbf{V}}_{\mathit{\gamma }}{\mathbf{V}}_{\mathit{\gamma }}^H{\mathbf{\Gamma }}_p^{(k)}.$$

Finally, substituting (55) and (57) into (31), we arrive at

$$\begin{array}{c}\mathsf{\text{Cov}}\left\{{\widehat{\xi }}^{\left(k\right)}\right\}\\ {\left.\simeq \frac{{\sigma }_n^2}{2{\sigma }_s^2{L}_s{\left(N-L+1\right)}^2}\cdot \frac{\Re \left[\sum _{t=0}^{N-L}\sum _{r=0}^{N-L}{\tilde{\mathit{d}}}_{\sigma }^T\left(\left({\mathbf{\Omega }}_{\mathbf{U}V}^{\left(k\right)}{\mathbf{V}}_{\gamma }^H{\mathbf{J}}_{t,r}{\mathbf{V}}_{\gamma }{\mathbf{\Omega }}_{\mathbf{V}U}^{\left(k\right)}\right){\left({\mathbf{U}}_{\gamma }^H\mathbf{A}{\mathbf{\Phi }}^{t-r}{\mathbf{A}}^H{\mathbf{U}}_{\gamma }\right)}^{*}\right){\tilde{\mathit{d}}}_{\sigma }\right]}{{\left[{\sum }_{l=M-N_{\mathsf{\text{c}}1}+1}^M-{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^H{\mathbf{\Lambda }}_1^{\left(k\right)}{\mathit{\nu }}_l^{\left(k\right)}+{\sum }_{z\mathsf{\text{ = 1}}}^{M-N_{\mathsf{\text{c1}}}}\frac{|{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^{\mathsf{\text{H}}}{\mathbf{\Lambda }}_{\mathsf{\text{2}}}^{\left(\mathsf{\text{k}}\right)}{\mathit{\nu }}_z^{\left(k\right)}{|}^{\mathsf{\text{2}}}}{{\lambda }_{\mathsf{\text{l}}}^{\left(\mathsf{\text{k}}\right)}-{\lambda }_{\mathsf{\text{z}}}^{\left(\mathsf{\text{k}}\right)}}\right]}^2}\right|}_{\xi ={\xi }^{\left(k\right)}}\end{array}$$

(58)

From (58), we make the following observations: First, it is seen that, the CFO estimation variance is decreased when SNR, i.e., ${\sigma }_s^2/{\sigma }_n^2$, increases. Second, increasing the number of blocks, i.e., L _s , also improves the CFO estimation performance. Third, the relationship between the value of L and the CFO estimation variance is quite complicated. We see that, decreasing the value of L, on the one hand, will decrease the value of $\frac{1}{{\left(N-L+1\right)}^2}$; on the other hand, it will also decrease the eigenvalues ${\tilde{\sigma }}_i^2$, resulting in larger entries of ${\tilde{\mathit{d}}}_{\sigma }$ In fact, from (11) and (14), we see that, the effect of L resembles the smoothing technique in array signal processing. That is, decreasing the value of L, on the one hand, will reduce the fluctuations of the estimated correlation matrix ${\widehat{\mathit{R}}}_{\mathbf{\Gamma }}$; on the other hand, it also decreases the potential for higher resolution of the subspace algorithm. We will later investigate the impact of L on the estimation performance via numerical simulations.

DOA estimation performance

Denote ${\mathit{\mu }}_l^{\left(k\right)}={\left[\Re \left\{{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^T\right\},\Im \left\{{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^T\right\}\right]}^T$, and ${\mathit{\mu }}^{\left(k\right)}={\left[{\left({\mathit{\mu }}_1^{\left(k\right)}\right)}^T,{\left({\mathit{\mu }}_2^{\left(k\right)}\right)}^T,\dots ,{\left({\mathit{\mu }}_{M-N_{c1}}^{\left(k\right)}\right)}^T\right]}^T$. Likewise, the DOA estimation variance of the k th user is given by

$$\mathsf{\text{Cov}}\left\{{\widehat{\phi }}_i^{\left(k\right)}\right\}={\left[\frac{{\partial }^2{D}^{\left(k\right)}}{\partial {\phi }^2}\right]}^{-1}\left[\frac{{\partial }^2{D}^{\left(k\right)}}{\partial \phi \partial {\mathit{\mu }}^{\left(k\right)}}\right]Cov\left\{{\widehat{\mathit{\mu }}}^{\left(k\right)}\right\}{\left[\frac{{\partial }^2{D}^{\left(k\right)}}{\partial \phi \partial {\mathit{\mu }}^{\left(k\right)}}\right]}^T{{\left[\frac{{\partial }^2{D}^{\left(k\right)}}{\partial {\phi }^2}\right]}^{-1}}_{\phi ={\phi }_i^{\left(k\right)}}.$$

(59)

We have

$$\frac{\partial D^{\left(k\right)}}{\partial \phi }=2\Re \left\{{\left(\alpha \left(\phi \right)\right)}^H\left({\mathit{\nu }}^{\left(k\right)}{\left({\mathit{\nu }}^{\left(k\right)}\right)}^H\right){\mathbf{\Psi }}_1\alpha \left(\phi \right)\right\},$$

(60)

$$\frac{{\partial }^2{D}^{\left(k\right)}}{\partial {\phi }^2}=2\Re \left\{{\left(\alpha \left(\phi \right)\right)}^H{\mathbf{\Psi }}_1^H\left({\mathit{\nu }}^{\left(k\right)}{\left({\mathit{\nu }}^{\left(k\right)}\right)}^H\right){\mathbf{\Psi }}_1\alpha \left(\phi \right)+{\left(\alpha \left(\phi \right)\right)}^H\left({\mathit{\nu }}^{\left(k\right)}{\left({\mathit{\nu }}^{\left(k\right)}\right)}^H\right){\mathbf{\Psi }}_1\left({\mathbf{\Psi }}_1+{\mathbf{\Psi }}_2\right)\alpha \left(\phi \right)\right\}$$

(61)

where

$$\begin{array}{c}{\mathit{\nu }}^{\left(k\right)}=\left[{\mathit{\nu }}_1^{\left(k\right)},{\mathit{\nu }}_2^{\left(k\right)},\dots ,{\mathit{\nu }}_{M-N_{c1}}^{\left(k\right)}\right],{\mathbf{\Psi }}_1=-\mathbf{j}2\pi \chi \text{sin}\left(\phi \right)\mathsf{\text{diag}}\left(0,1,\dots ,M-1\right),\\ {\mathbf{\Psi }}_2=-\mathbf{j}2\pi \chi \text{cos}\left(\phi \right)\mathsf{\text{diag}}\left(0,1,\dots ,M-1\right).\end{array}$$

We further obtain

$$\begin{array}{c}\left[\frac{{\partial }^{2}D}{\partial \phi \partial {\mathit{\mu }}^{\left(k\right)}}\right]\mathsf{\text{Cov}}\left\{{\widehat{\mathit{\mu }}}^{\left(k\right)}\right\}{\left[\frac{{\partial }^{2}D}{\partial \phi \partial {\mathit{\mu }}^{\left(k\right)}}\right]}^T=\sum _{l=1}^{M-N_{\mathsf{\text{c}}1}}\sum _{z=1}^{M-N_{\mathsf{\text{c}}1}}2\Re \left\{{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^H{T}^{\left(k\right)}\mathsf{\text{Cov}}\left\{{\widehat{\mathit{\nu }}}_l^{\left(k\right)},{\widehat{\mathit{\nu }}}_z^{\left(k\right)}\right\}{T}^{\left(k\right)}\right.{\mathit{\nu }}_z^{\left(k\right)}\mathsf{\text{ + }}\\ \left.{\left({\mathit{\nu }}_l^{\left(k\right)}\right)}^H{T}^{\left(k\right)}Cov\left\{{\widehat{\mathit{\nu }}}_l^{\left(k\right)},{\left({\widehat{\mathit{\nu }}}_z^{\left(k\right)}\right)}^{\mathsf{\text{*}}}\right\}{\left(T^{\left(k\right)}\right)}^{\mathsf{\text{*}}}{\left({\widehat{\mathit{\nu }}}_z^{\left(k\right)}\right)}^{\mathsf{\text{*}}}\right\}\end{array}$$

(62)

where

$$T^{\left(k\right)}={\mathbf{\Psi }}_1\alpha \left(\phi \right){\left(\alpha \left(\phi \right)\right)}^H+\alpha \left(\phi \right){\left(\alpha \left(\phi \right)\right)}^H{\mathbf{\Psi }}_1^H.$$

(63)

Let Δ x denote the perturbation of x, i.e., $\mathbf{\Delta }\mathit{x}=\widehat{\mathbf{x}}-\mathbf{x}$. Bearing in mind that ${\mathit{\nu }}_l^{\left(k\right)}$, l = 1, 2, . . . , M, denotes the eigenvector of matrix ${\Xi }^{\left(k\right)}\left({\widehat{\xi }}^{\left(k\right)}\right)$, its perturbation is introduced by both the CFO estimation error of the k th user and the perturbations of the eigenvectors e _i , i = 1, 2, . . . , N_clN_sum. Hence, using the first-order approximation, the perturbation of ${\mathit{\nu }}_l^{(k}$