Simplified Frequency Offset Estimation for MIMO OFDM Systems

TL;DR

This paper proposes a simplified and efficient frequency offset estimator for MIMO OFDM systems that leverages Chu sequences and factor decomposition, achieving reduced complexity and confirmed by simulation results.

Contribution

It introduces a novel simplified CFO estimation method for MIMO OFDM systems using Chu sequences and derivative-based factor decomposition, reducing computational complexity.

Findings

Estimator achieves low mean-squared error in simulations

Training-assisted approach improves estimation accuracy

Method demonstrates good performance over frequency selective channels

Abstract

This paper addresses a simplified frequency offset estimator for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems over frequency selective fading channels. By exploiting the good correlation property of the training sequences, which are constructed from the Chu sequence, carrier frequency offset (CFO) estimation is obtained through factor decomposition for the derivative of the cost function with great complexity reduction. The mean-squared error (MSE) of the CFO estimation is derived to optimize the key parameter of the simplified estimator and also to evaluate the estimator performance. Simulation results confirm the good performance of the training-assisted CFO estimator.

Equations

$$\mathbf{y}$$

$${\mathbf{h}}_{\nu }$$

$$\mathbf{h}$$

$${\mathbf{D}}_{\bar{N}}(\tilde{\epsilon })$$

$$\mathbf{y}=\sqrt{N}{e}^{j2\pi \tilde{\epsilon }{N}_g/N}\{{\mathbf{I}}_{N_r}\otimes [{\mathbf{D}}_N(\tilde{\epsilon })\mathbf{S}]\}\mathbf{h}+\mathbf{w},$$

$$\mathbf{S}$$

$$\bar{\mathbf{F}}$$

$$\breve{\mathbf{F}}$$

$$\rho(\iota)=\left\{{\begin{array}[]{*{20}l}{2\iota{\Re\{(\sum\nolimits_{\mu=0}^{N_{t}-1}{z_{\mu}^{2\iota}})(\sum\nolimits_{\mu=0}^{N_{t}-1}{z_{\mu}^{-\iota}})^{2}\}/{|\sum\nolimits_{\mu=0}^{N_{t}-1}{z_{\mu}^{\iota}}|^{2}}},}&{1\leq\iota\leq Q/2}\\ {2(Q-\iota){\Re\{(\sum\nolimits_{\mu=0}^{N_{t}-1}{z_{\mu}^{2\iota}})(\sum\nolimits_{\mu=0}^{N_{t}-1}{z_{\mu}^{\iota}})^{2}\}/{|\sum\nolimits_{\mu=0}^{N_{t}-1}{z_{\mu}^{\iota}}|^{2}}},}&{Q/2<\iota\leq Q-1}\\ \end{array}}\right.$$

$${\mathbf{b}}_{\mu }$$

$$\mathbf{B}(\tilde{\epsilon })$$

$$\mathbf{Y}=\mathbf{B}(\tilde{\epsilon })\mathbf{X}+\mathbf{W},$$

$$\mathbf{X}$$

$${\mathbf{X}}_{\nu }$$

$${\mathbf{x}}^{(\nu ,\mu )}$$

$$\ln p(\mathbf{Y}|\mathbf{B}(\epsilon ),\mathbf{X})=-{\sigma }_w^{-2}\mathrm{Tr}\{[\mathbf{Y}-\mathbf{B}(\epsilon )\mathbf{X}][\mathbf{Y}-\mathbf{B}(\epsilon )\mathbf{X}{]}^H\}.$$

$$\ln p(\mathbf{Y}|\epsilon )=\mathrm{Tr}[{\mathbf{B}}^H(\epsilon ){\hat{\mathbf{R}}}_{\mathbf{Y}\mathbf{Y}}\mathbf{B}(\epsilon )],$$

$$\begin{gathered} f(z)={\mathbf{c}}^T\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu })\right]\odot \mathbf{b}(z)\right\} \\ +{\mathbf{c}}^H\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu }^{-1})\right]\odot \mathbf{b}(z^{-1})\right\}, \end{gathered}$$

$$\begin{gathered} f^{\prime }(z)=z^{-1}\{{\mathbf{c}}^T\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu })\right]\odot \mathbf{b}(z)\odot \mathbf{q}\right\} \\ -{\mathbf{c}}^H\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu }^{-1})\right]\odot \mathbf{b}(z^{-1})\odot \mathbf{q}\right\}\}, \end{gathered}$$

$$\begin{gathered} {\mathbf{c}}^H\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu }^{-1})\right]\odot \mathbf{b}(z^{-1})\odot \mathbf{q}\right\} \\ =z^{-Q}\kappa (\iota )\cdot {\mathbf{c}}^T\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu })\right]\odot \mathbf{b}(z)\odot \mathbf{q}\right\}, \end{gathered}$$

$$f^{\prime }(z)=z^{-(Q+1)}[z^Q-\kappa (\iota )]\cdot {\mathbf{c}}^T\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu })\right]\odot \mathbf{b}(z)\odot \mathbf{q}\right\}.$$

$${\mathbf{c}}^T\left\{\left[\sum _{\mu =0}^{N_t-1}\mathbf{b}(z_{\mu })\right]\odot \mathbf{b}(\tilde{z})\odot \mathbf{q}\right\}>0,f^{\prime }(\tilde{z})=0.$$

$$\hat{\epsilon }=\underset{\epsilon \in \{{\epsilon }_q{\}}_{q=0}^{Q-1}}{\mathrm{argmax}}\{f(z)|z=e^{j2\pi \epsilon /Q}\},$$

$${\hat{\mathbf{R}}}_{\mathbf{Y}\mathbf{Y}}\doteq N_{r}P{\sigma }_x^2\mathbf{B}(\tilde{\epsilon }){\mathbf{B}}^H(\tilde{\epsilon })+{\hat{\mathbf{R}}}_{\mathbf{Y}\mathbf{W}}+{\hat{\mathbf{R}}}_{\mathbf{W}\mathbf{W}},$$

$$\begin{gathered} \mathrm{E}\{[{\mathbf{h}}^{(\nu ,\mu )}{]}_l^{\ast }[\mathbf{W}{]}_{i,j}\}=0,\mathrm{E}\{[{\mathbf{h}}^{(\nu ,\mu )}{]}_l^{\ast }[{\mathbf{h}}^{({\nu }^{\prime },{\mu }^{\prime })}{]}_{l^{\prime }}\}=0, \\ \forall (\nu ,\mu )\ne ({\nu }^{\prime },{\mu }^{\prime }). \end{gathered}$$

$$\mathrm{E}[[{\hat{\mathbf{R}}}_{\mathbf{Y}\mathbf{W}}{]}_{i,j}]=\mathrm{E}[[{\hat{\mathbf{R}}}_{\mathbf{W}\mathbf{W}}{]}_{i,j}]=\mathrm{E}[[{\hat{\mathbf{R}}}_{\mathbf{Y}\mathbf{W}}{]}_{i,j}^{\ast }[{\hat{\mathbf{R}}}_{\mathbf{W}\mathbf{W}}{]}_{i^{\prime },j^{\prime }}]=0,$$

$$\mathrm{E}[|[{\hat{\mathbf{R}}}_{\mathbf{Y}\mathbf{W}}{]}_{i,j}{|}^2]=2N_t{N}_{r}P{\sigma }_x^2{\sigma }_w^2,\mathrm{E}[|[{\hat{\mathbf{R}}}_{\mathbf{W}\mathbf{W}}{]}_{i,j}{|}^2]=N_{r}P{\sigma }_w^4.$$

$$[\mathbf{c}{]}_{\iota }=N_{r}P{\sigma }_x^2(Q-\iota ){\tilde{z}}^{-\iota }\sum _{\mu =0}^{N_t-1}{z}_{\mu }^{-\iota }+{\alpha }_{\iota }+{\beta }_{\iota },$$

$$\kappa (\iota )=\frac{{\tilde{z}}^{\iota }+{\zeta }_{\iota }}{{\tilde{z}}^{-Q+\iota }+{\eta }_{\iota }}=\frac{{\tilde{z}}^Q+{\xi }_{\iota }}{|{\tilde{z}}^{-Q+\iota }+{\eta }_{\iota }{|}^2},$$

$${\zeta }_{\iota }$$

$${\eta }_{\iota }$$

$${\xi }_{\iota }$$

$$\mathrm{var}\{{\zeta }_{\iota }\}=\frac{2N_t{\gamma }^{-1}+{\gamma }^{-2}}{N_{r}P(Q-\iota ){\left|\sum _{\mu =0}^{N_t-1}{z}_{\mu }^{\iota }\right|}^2},\mathrm{var}\{{\eta }_{\iota }\}=\frac{2N_t{\gamma }^{-1}+{\gamma }^{-2}}{N_{r}P\iota {\left|\sum _{\mu =0}^{N_t-1}{z}_{\mu }^{\iota }\right|}^2},$$

$$\mathrm{var}\{{\zeta }_{\iota }\}\gg \mathrm{var}\{{\zeta }_{\iota }{\eta }_{\iota }^{\ast }\},\mathrm{var}\{{\eta }_{\iota }\}\gg \mathrm{var}\{{\zeta }_{\iota }{\eta }_{\iota }^{\ast }\}.$$

$$\displaystyle\mathrm{var}\{\xi_{\iota}\}\$$

Full text

I Introduction

Carrier frequency offset (CFO) estimation is an important issue for both single-antenna and multiple-antenna orthogonal frequency-division multiplexing (OFDM) systems [1, 2, 3, 4, 5, 6, 7]. Numerical calculation of the maximum likelihood (ML) CFO estimation is computationally complicated since it requires a large point discrete Fourier transform (DFT) operation and a time consuming line search. Therefore, many papers have proposed reduced-complexity algorithms [2, 3, 5, 6, 7]. Especially, the search-free approaches were proposed in [3] [6] [7], where the polynomial rooting is exploited to estimate the CFO. The solution proposed in [3] is based on computing the roots from the derivative of the cost function whereas the solutions proposed in [6] [7] are based on computing the roots directly from the cost function. However, both solutions still need the complicated polynomial rooting operation, which is hard to be implemented in practical OFDM systems [8].

In this paper, by further investigating the above search-free approaches, a simplified CFO estimator is developed for MIMO OFDM systems over frequency-selective fading channels. With the aid of the training sequences generated from the Chu sequence [9], we propose to estimate the CFO via a simple polynomial factor. Thus, the complicated polynomial rooting operation is avoided. Correspondingly, the CFO estimator can be implemented via simple additions and multiplications. To optimize the key parameter of the simplified CFO estimator and also to evaluate the estimator performance, the mean-squared error (MSE) of the CFO estimation is derived.

Notations: $(\cdot)_{P}$ denotes the remainder of the number within the brackets modulo $P$. $\otimes$ and $\odot$ denote the Kronecker product and Schur-Hadamard product, respectively. $\Re(\cdot)$ and $\Im(\cdot)$ denote the real and imaginary parts of the enclosed parameters, respectively. $\bm{x}^{(m)}$ denotes the $m$-cyclic-down-shift version of $\bm{x}$. $\bm{F}_{N}$ and $\bm{I}_{N}$ denote the $N\times N$ unitary DFT matrix and identity matrix, respectively. $\bm{e}_{N}^{k}$ denotes the $k$-th column vector of $\bm{I}_{N}$. Unless otherwise stated, $0\leq\mu\leq N_{t}-1$, $0\leq\nu\leq N_{r}-1$, $0\leq p\leq P-1$ and $0\leq q\leq Q-1$ are assumed, where $Q=N/P$ with $(N)_{2P}=0$.

II Signal Model

Consider a MIMO OFDM system with $N_{t}$ transmit antennas and $N_{r}$ receive antennas and $N$ subcarriers. The training sequences for CFO estimation are the same as in [6] [7]. Let $\bm{s}$ denote a length-$P$ Chu sequence [9]. Then, the $P\times 1$ pilot sequence vector at the $\mu$-th transmit antenna is generated from $\bm{s}$ as follows $\tilde{\bm{s}}_{\mu}=\sqrt{Q/N_{t}}\bm{F}_{P}\bm{s}^{(\mu M)}$, where $M=\lfloor P/N_{t}\rfloor$. Define $\bm{\Theta}_{q}=[\bm{e}_{N}^{q},\bm{e}_{N}^{q+Q},\cdots,\bm{e}_{N}^{q+(P-1)Q}]$. Then, the $N\times 1$ training sequence vector at the $\mu$-th transmit antenna is constructed as follows $\tilde{\bm{t}}_{\mu}=\bm{\Theta}_{i_{\mu}}\tilde{\bm{s}}_{\mu}$, where $0\leq i_{\mu}\leq Q-1$ and $i_{\mu}=i_{\mu^{\prime}}\ \mathrm{iff}.\ \mu=\mu^{\prime}$. For convenience, we refer to $\{\tilde{\bm{t}}_{\mu}\}_{\mu=0}^{N_{t}-1}$ as the Chu sequence based training sequences (CBTS).

Let $\bm{y}_{\nu}$ denote the $N\times 1$ received vector at the $\nu$-th receive antenna after CP removal. Let $\bm{h}^{(\nu,\mu)}$ denote the $L\times 1$ channel impulse response vector with $L$ being the maximum channel length. Assume that $L$ is shorter than the length of cyclic prefix (CP) $N_{g}$. Let $\tilde{\varepsilon}$ denote the frequency offset normalized by the subcarrier frequency spacing. Define

[TABLE]

Then, the cascaded received vector $\bm{y}$ over the $N_{r}$ receive antennas can be written as [6] [7]

[TABLE]

where

[TABLE]

and $\bm{w}$ is an $N_{r}N\times 1$ vector of uncorrelated complex Gaussian noise samples with mean zero and equal variance of $\sigma_{w}^{2}$.

III Simplified CFO Estimator for MIMO OFDM Systems

By exploiting the periodicity property of CBTS, $\bm{y}$ can be stacked into the $Q\times N_{r}P$ matrix $\bm{Y}=[\bm{Y}_{0},\bm{Y}_{1},\cdots,\bm{Y}_{\nu},\cdots,\bm{Y}_{N_{r}-1}]$ with its element given by $[\bm{Y}_{\nu}]_{q,p}=[((\bm{e}_{N_{r}}^{\nu})^{T}\otimes\bm{I}_{N})\bm{y}]_{qP+p}$. Define

[TABLE]

Then, $\bm{Y}$ can be expressed in the following equivalent form [6] [7]

[TABLE]

where

[TABLE]

and $\bm{W}$ is the $Q\times N_{r}P$ matrix generated from $\bm{w}$ in the same way as $\bm{Y}$.

According to the multivariate statistical theory, the log-likelihood function of $\bm{Y}$ conditioned on $\bm{B}(\varepsilon)$ and $\bm{X}$ with $\varepsilon$ denoting a candidate CFO can be obtained as follows

[TABLE]

Exploit the condition $i_{\mu}=i_{\mu^{\prime}}\ \mathrm{iff}.\ \mu=\mu^{\prime}$. Then, after some straightforward manipulations, we can obtain the reformulated log-likelihood function conditioned on $\varepsilon$ as follows

[TABLE]

where $\hat{\bm{R}}_{\bm{Y}\bm{Y}}=\bm{Y}\bm{Y}^{H}$. Direct grid searching from (4) yields the ML estimate, however, this approach is computationally quite expensive. In order to compute the CFO efficiently, we will propose a simplified CFO estimator for MIMO OFDM systems subsequently.

Define $z=e^{j2\pi\varepsilon/Q}$, $z_{\mu}=e^{j2\pi i_{\mu}/Q}$, $\bm{b}(z)=[1,z,\cdots,z^{q},\cdots,z^{Q-1}]^{T}$. Then, by exploiting the Hermitian property of $\hat{\bm{R}}_{\bm{Y}\bm{Y}}$, the log-likelihood function in (4) can be transformed into the following equivalent form

[TABLE]

where $\bm{c}$ is a $Q\times 1$ vector with its $q$-th element given by $[\bm{c}]_{q}=\sum\nolimits_{j-i=q}{[\hat{\bm{R}}_{\bm{Y}\bm{Y}}]_{i,j}}$. It can be seen from its definition that the $q$-th element of $\bm{c}$ corresponds to the summation of the $q$-th upper diagonal elements of $\hat{\bm{R}}_{\bm{Y}\bm{Y}}$. Taking the first-order derivative of $f(z)$ with respect to $z$ yields

[TABLE]

where $\bm{q}=[0,1,\cdots,q,\cdots,Q-1]^{T}$. By letting the derivative of the log-likelihood function $f^{\prime}(z)$ be zero, the solutions for all local minima or maxima can be obtained. Put these solutions back into the original log-likelihood function $f(z)$ and select the maximum by comparing all the solutions obtained in the previous stage. The improved blind CFO estimator exploiting the above mathematical rule has been addressed for single-antenna OFDM systems in [3]. Although the search-free approach has a relatively lower complexity, it still requires a complicated polynomial rooting operation, which is hard to be implemented in practical OFDM systems. With the aid of the CBTS training sequences, we will show in the following that the polynomial rooting operation can be avoided for the training aided CFO estimation in MIMO OFDM systems.

Assume that $P\geq L$, the channel taps remain constant during the training period, and the channel energy is mainly concentrated in the first $M$ taps with $M<L$. Then, we have (see Appendix I for the details)

[TABLE]

where $\kappa(\iota)={\iota[\bm{c}]^{*}_{\iota}}/{[(Q-\iota)[\bm{c}]_{Q-\iota}]}$ with $1\leq\iota\leq Q-1$, and the parameter $\iota$ denotes the index of the upper diagonal of $\hat{\bm{R}}_{\bm{Y}\bm{Y}}$. From (7), it follows immediately that $f^{\prime}(z)$ can be decomposed as follows

[TABLE]

Define $\tilde{z}=e^{j2\pi\tilde{\varepsilon}/Q}$. Assume $N_{t}<Q$. Then, with (29) as shown in Appendix I, we have (see Appendix II for the details)

[TABLE]

It follows from (8) and (9) that $z=\tilde{z}$ is one of the roots of both $f^{\prime}(\tilde{z})=0$ and $z^{Q}-\kappa(\iota)=0$. Unlike $f^{\prime}(\tilde{z})=0$, the roots of $z^{Q}-\kappa(\iota)=0$ can be calculated without the polynomial rooting operation. Therefore, by solving the simple polynomial equation $z^{Q}-\kappa(\iota)=0$, the CFO estimate can be obtained efficiently as follows

[TABLE]

where $\varepsilon_{q}=\mathrm{arg}\{\kappa(\iota)\}/{(2\pi)}+q-{Q}/{2}$. It can be calculated that the main computational complexity of the simplified CFO estimator is $4N_{r}NQ+8Q^{2}$. Compared with the CFO estimator in [6] [7], whose main computational complexity is $4N_{r}N\log_{2}N+9Q^{3}+64/3(Q-1)^{3}$, the complexity of the simplified CFO estimator is generally lower. Furthermore, since the polynomial rooting operation is avoided, the simplified CFO estimator can be implemented via simple additions and multiplications, which is more suitable for practical OFDM systems. Note that $\iota$ is a key parameter for the proposed CFO estimator. We will show in the following how to determine the optimal $\iota$.

IV Performance Analysis and Parameter Optimization

To optimize $\iota$ and also to evaluate the estimation accuracy, we first derive the MSE of the simplified CFO estimator. Invoking the definition of $\hat{\bm{R}}_{\bm{Y}\bm{Y}}$, we can readily obtain

[TABLE]

where $\sigma_{x}^{2}=\mathrm{E}[|[\bm{x}^{(\nu,\mu)}]_{p}|^{2}]$, $\hat{\bm{R}}_{\bm{Y}\bm{W}}=\bm{B}(\tilde{\varepsilon})\bm{X}\bm{W}^{H}+\bm{W}\bm{X}^{H}\bm{B}^{H}(\tilde{\varepsilon})$, $\hat{\bm{R}}_{\bm{W}\bm{W}}=\bm{W}\bm{W}^{H}$. Assume

[TABLE]

Then, for $\ i\neq j$ and $\ i^{\prime}\neq j^{\prime}$, it can be concluded directly from their definitions that

[TABLE]

[TABLE]

Invoking the definition of $\bm{c}$, we have

[TABLE]

where $\alpha_{\iota}=\sum\nolimits_{j-i=\iota}{[\hat{\bm{R}}_{\bm{Y}\bm{W}}]_{i,j}}$, $\beta_{\iota}=\sum\nolimits_{j-i=\iota}{[\hat{\bm{R}}_{\bm{W}\bm{W}}]_{i,j}}$. It follows immediately from its definition that $\kappa(\iota)$ can be expressed as

[TABLE]

where

[TABLE]

From (15), we can see that the MSE of the estimated CFO is highly related to the variances of $\zeta_{\iota}$, $\eta_{\iota}$ and $\xi_{\iota}$. By invoking their definitions, the variances of $\zeta_{\iota}$ and $\eta_{\iota}$ can be directly calculated as follows

[TABLE]

where $\gamma=\sigma_{x}^{2}/\sigma_{w}^{2}$. When $\gamma\gg 1$, i.e., the signal-to-noise ratio (SNR) is large enough, we have

[TABLE]

Accordingly, the variance of $\xi_{\iota}$ can be approximated as follows

[TABLE]

where $\rho(\iota)$ is shown in (19) at the bottom of the page. Note that $\rho(\iota)$ is a nonlinear function with respect to $\iota$ and $z_{\mu}$. When $\mathrm{var}\{\xi_{\iota}\}\ll 1$, which is a reasonable assumption for the practical systems, it follows immediately from (15) that

[TABLE]

Then, the MSE of the estimated CFO can be readily obtained as follows

[TABLE]

It can be seen from (21) that $\mathrm{MSE}\{\hat{\varepsilon}\}$ depends on $\iota$ for fixed $\{i_{\mu}\}_{\mu=0}^{N_{t}-1}$, $N_{t},N_{r},P,Q$ and $\gamma$. To obtain a better estimator performance, we can optimize the parameter $\iota$ based on (21).

V Simulation Results

Numerical results are provided to verify the analytical results and also to evaluate the performance of the proposed CFO estimator. The considered MIMO OFDM system is of bandwidth $20$MHz and carrier frequency $5$GHz with $N=1024$ and $N_{g}=80$. Each of the channels is with $6$ independent Rayleigh fading taps, whose relative average-powers and propagation delays are $\{0,-0.9,-4.9,-8.0,-7.8,-23.9\}$dB and $\{0,4,16,24,46,74\}$ samples, respectively. The other parameters are as follows: $P=64$, $Q=16$, $N_{t}=3$, $N_{r}=2$, $\tilde{\varepsilon}\in(-Q/2,Q/2)$.

Figs. 1 and 2 present the MSE of the proposed CFO estimator as a function of $\iota$ with $\{i_{\mu}\}_{\mu=0}^{N_{t}-1}=\{3,5,11\}$ and $\{3,7,14\}$, respectively. The solid and dotted curves are the results from analysis and Monte Carlo simulations, respectively. It can be observed that the results from analysis agree quite well with those from simulations except when the actual MSE of the estimate is very large. It can also be observed that $\mathrm{MSE}\{\hat{\varepsilon}\}$ achieves its minimum for $\iota=6,8,10$ with $\{i_{\mu}\}_{\mu=0}^{N_{t}-1}=\{3,5,11\}$ and for $\iota=7,9$ with $\{i_{\mu}\}_{\mu=0}^{N_{t}-1}=\{3,7,14\}$. These observations imply that we can obtain the optimum value of the parameter $\iota$ from the analytical results after $\{i_{\mu}\}_{\mu=0}^{N_{t}-1}$ is determined.

Depicted in Fig. 3 is the performance comparison between the proposed CFO estimator ($\iota=7$, $\{i_{\mu}\}_{\mu=0}^{N_{t}-1}=\{3,7,14\}$) and the estimator in [6] [7] and [10]. In order to substantiate that the training sequences generated from the Chu sequence do help to improve the estimation accuracy, the performance of the proposed estimator with random sequences (RS), whose elements are generated randomly, is included. Also included as a performance benchmark is the extended Miller and Chang bound (EMCB) [11] [12], which is obtained by averaging the snapshot Cramer-Rao bound (CRB) over independent channel realizations as follows

[TABLE]

where $\bm{\mathcal{X}}=\bm{I}_{N_{r}}\otimes\bm{S}$, $\bm{\mathcal{B}}=\bm{I}_{N_{r}}\otimes\mathrm{diag}\{[N_{g},N_{g}+1,\cdots,N_{g}+N-1]^{T}\}$. We resort to Monte Carlo simulation for its evaluation. It can be observed that the performance of the proposed estimator with CBTS is far better than that in [10] and slightly worse than that in [6] [7], and its performance also approaches the EMCB which verifies its high estimation accuracy. It can also be observed that the performance of the proposed CFO estimator with CBTS is far better than that with RS, which should be attributed to the good correlation property of CBTS.

VI Conclusions

In this paper, we have presented a low complexity CFO estimator for MIMO OFDM systems with the training sequences generated from the Chu sequence. The MSE of the CFO estimation has been developed to evaluate the estimator performance and also to optimize the key parameter. By exploiting the optimized parameter from the estimation MSE, our CFO estimator with CBTS yields good performance.

Appendix A

This appendix presents the proof of (7). It follows immediately that the polynomials on the two sides of (7) are both $(Q-1)$-degree. Therefore, in order to validate the relationship (7), we only need to prove that the corresponding polynomial coefficients are pairwise equal.

Ignoring the noise items, $\hat{\bm{R}}_{\bm{Y}\bm{Y}}$ can be expressed as follows

[TABLE]

where $\hat{\bm{R}}_{\bm{X}\bm{X}}=\bm{X}\bm{X}^{H}$. Define

[TABLE]

Then, with the assumptions that $P\geq L$ and the channel taps remain constant during the training period, we can readily obtain

[TABLE]

where $[\bm{A}^{(\mu,\mu^{\prime})}]_{l,l^{\prime}}=(\bm{s}_{\mu}^{(l)})^{T}\bm{D}_{P}(\varpi_{\mu,\mu^{\prime}}Q)(\bm{s}_{\mu^{\prime}}^{(l^{\prime})})^{*}$. With the aim of complexity reduction, $[\hat{\bm{R}}_{\bm{X}\bm{X}}]_{\mu,\mu^{\prime}}$ is replaced with its expected value. Exploiting the good correlation property of CBTS, which is inherited from the Chu sequence, we obtain

[TABLE]

Let $[\bm{s}]_{p}=e^{j\pi\upsilon p^{2}/P}$ with $\upsilon$ being coprime with $P$. Define $p_{\mu,l}=\mu M+l$. Then, we have

[TABLE]

It follows immediately that

[TABLE]

With the assumption that the channel energy is mainly concentrated in the first $M$ taps, \bigl{|}[\hat{\bm{R}}_{\bm{X}\bm{X}}]_{\mu,\mu^{\prime}}\bigl{|}_{\mu\neq\mu^{\prime}} can be made very small with CBTS, which yields

[TABLE]

Then, we have $\hat{\bm{R}}_{\bm{X}\bm{X}}\doteq N_{r}P\sigma_{x}^{2}\bm{I}_{N_{t}}$, where $\sigma_{x}^{2}=\mathrm{E}[|[\bm{x}^{(\nu,\mu)}]_{p}|^{2}]$. It follows immediately that $\hat{\bm{R}}_{\bm{Y}\bm{Y}}\doteq N_{r}P\sigma_{x}^{2}\bm{B}(\tilde{\varepsilon})\bm{B}^{H}(\tilde{\varepsilon})$. By invoking the definition of $[\bm{c}]_{q}$, we can further obtain

[TABLE]

where $\tilde{z}=e^{j2\pi\tilde{\varepsilon}/Q}$. By substituting the above result into (7), the polynomial coefficient corresponding to $z^{-q}$ at both sides of (7) can be calculated to be

[TABLE]

where we have utilized the following property $z_{\mu}^{Q}=1$. This completes the proof.

Appendix B

This appendix presents the proof of (9). With CBTS and (29), we can readily obtain

[TABLE]

where $\bm{\Psi}=[\Psi(0),\Psi(1),\cdots,\Psi(q),\cdots,\Psi(Q-1)]^{T}$, $\Psi(q)=e^{-j2\pi qi_{0}/Q}\sum\nolimits_{\mu=0}^{N_{t}-1}{e^{j2\pi qi_{\mu}/Q}}$. From the definition of $\bm{\Psi}$, we have

[TABLE]

where $\bm{\Phi}=[\bm{\phi}_{0},\bm{\phi}_{1},\cdots,\bm{\phi}_{N_{t}-1}]$, $\bm{\phi}_{\mu}=[1,e^{j2\pi(i_{\mu}-i_{0})/Q},\cdots,e^{j2\pi q(i_{\mu}-i_{0})/Q},\cdots,e^{j2\pi(Q-1)(i_{\mu}-i_{0})/Q}]^{T}$. Since $\bm{\Phi}$ is a Vandermonde matrix, it is of full-rank (rank $N_{t}$) with $N_{t}<Q$. Consequently, $\bm{\Psi}$ cannot be the all zero vector and then $\bm{c}^{T}\left\{\left[\sum\nolimits_{\mu=0}^{N_{t}-1}{\bm{b}(z_{\mu})}\right]\odot\bm{b}(\tilde{z})\odot\bm{q}\right\}>0$. From (6), it follows immediately that $f^{\prime}(\tilde{z})=0$. This completes the proof.