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- Open Access

# Fuzzy MIMO detector for MC-CDMA systems with carrier frequency offset over multipath fading channels

- Juinn-Horng Deng
^{1}Email author and - Shu-Min Liao
^{1}

**2012**:197

https://doi.org/10.1186/1687-1499-2012-197

© Deng and Liao; licensee Springer. 2012

**Received:**3 November 2011**Accepted:**21 May 2012**Published:**21 June 2012

## Abstract

A multistage fuzzy minimum output energy (MOE) detector is proposed for multiple input multiple output (MIMO) multi-carrier code-division-multiple access (MC-CDMA) uplink systems with carrier frequency offset (CFO) over multipath fading channels. The first stage of the receiver uses a novel MIMO receiver model with offset symbols to achieve a fuzzy CFO-constrained MOE detector that suppresses multiple access interference (MAI) and minimizes cancellation of the desired signal. To suppress noise and to enhance signal reception after the fuzzy MOE detector, a signal subspace projection and minimum mean square error weight combiner is proposed to enhance signal-to-interference-plus-noise ratio and bit error rate performances. Simulation results show that the proposed MIMO detector outperforms conventional MOE detectors and achieves the ideal receiver performance against MAI and CFO effects.

## Keywords

- Multi-carrier CDMA (MC-CDMA)
- Multiple input multiple output (MIMO)
- Multiple access interference (MAI)
- Carrier frequency offset (CFO)
- Minimum output energy (MOE)

## Introduction

Multi-carrier code-division-multiple access (MC-CDMA) systems [1–4] are currently under consideration for application in WiMAX and WiFi systems in future broadband wireless communication systems. The system has recently attracted attention because it combines high spectral efficiency with a robust channel equalizer against multipath fading channels. Like a direct-sequence CDMA (DS-CDMA) system, MC-CDMA inherently overcomes the intersymbol interference caused by multipath propagation. However, its performance is limited by multiple access interference (MAI) and carrier frequency offset (CFO) [5]. The CFO can cause loss of subcarrier orthogonality, which results in intercarrier interference (ICI). To minimize the CFO effect, the receiver in most uplink MC-CDMA systems [6] must first accurately estimate and compensate for CFO to maintain the orthogonality between multi-carriers. Additionally, the rotated constellation of the desired signal degrades the performance of MC-CDMA systems [7, 8]. Popular frequency offset estimators are widely reported in the literature [9–13]. For example, the maximum likelihood estimator is the optimal estimator under additive white Gaussian noise. Existing frequency offset estimators typically assume little or no interference. Unfortunately, frequency offset cannot be estimated accurately when MAI is strong. Therefore, an MC-CDMA receiver must efficiently suppress MAI before performing frequency offset compensation. More importantly, MAI suppression must be independent of frequency offset.

The efficient minimum output energy (MOE) multiuser detector developed earlier for use in suppressing MAI in MC-CDMA systems [14] is designed to minimize the entire output energy of the receiver by constraining the spreading code sequence. However, the ICI effect of the CFO in MC-CDMA systems produces a mismatched spreading code signature, which substantially deteriorates MOE detector performance due to the signal cancellation effect [15]. One proposed solution is the use of subspace MOE detectors [16, 17], which are designed to project the weight vector of the MOE scheme onto the signal subspace of the received observation data. However, the subspace MOE still involves the problem of the desired signal cancellation to degrade the system performance. Besides, a well-known technique for improving the link quality for wireless communication in the multiple input multiple output (MIMO) scheme is space–time block coding (STBC) [18]. For example, the WiMAX system [19] adopts the spatial multiplexing and diversity schemes of the matrices A and B MIMO schemes [20] used for user uplink communication. Therefore, this study proposes the use of MIMO STBC MC-CDMA system with multiuser uplink communication for the CFO and multipath fading channel environments. Additionally, since the conventional space–time block decoding scheme in [18] cannot be used for the MAI, ICI, and multipath interference scenarios, the proposed MIMO STBC receiver with fuzzy constrained MOE detector suppresses interference and effectively detects desired signals.

Moreover, to acquire spatial diversity, this study develops a robust MIMO fuzzy MOE detector to suppress MAI, to prevent signal cancellation caused by the CFO effect, and to acquire the multipath and spatial diversity gains. The receiver design procedure is the following. First, a novel MIMO received data model with offset symbol is proposed to prevent the CFO from causing symbol detection failure in the STBC MC-CDMA system. Second, the offset MIMO receiver uses a simple spectral peak search scheme for roughly estimating CFO. Although the CFO estimate need not be accurate, it should lead to the correct CFO region. Next, a MIMO fuzzy MOE detector is then proposed for identifying the novel eigen-based signature with MIMO fuzzy CFO constraint used to suppress MAI. Finally, to maximize suppression of noise and reception of signals after the MIMO fuzzy MOE detector, use of the signal subspace projection and minimum mean square error (MMSE) techniques are proposed to enhance signal-to-interference-plus-noise ratio (SINR) and bit error rate (BER) performances. Simulation results confirm that the proposed MIMO detector has sufficient robustness to suppress CFO and MAI effects and that it outperforms conventional detectors. The rest of this article is organized as follows: Section 2 presents the data model and the ideal receiver. Section 3 then presents the robust MIMO multistage fuzzy MOE detector. Section 4 analyzes the computational complexity of the proposed MIMO multistage fuzzy MOE detector. Next, Section 5 summarizes the simulation results. Finally, Section 6 concludes the study.

### Notation

The following notation is used. Bold uppercase letters, e.g., **A**, are used to denote matrices, bold lowercase letters, e.g., **a**, are used to denote column vectors, and non-bold letters in italics, e.g., *a* or *A*, are used to denote scalar values. The symbols **A**^{
H
}, **A**^{
T
}, **A**^{*}, and tr(**A**) denote the transpose-conjugate, the transpose, the complex conjugate, and the trace operations of **A**, respectively. The diag(**a**) represents a diagonal matrix whose diagonal entries are the elements of the vector **a**.

## Data model for MIMO MC-CDMA systems

*N*

_{ R }receiver antennas, the proposed scheme can simultaneously be processed by each receiver. Next, the combiner can then collect the equalized signal from different receivers, which enhances detection performance. In the transmitter, assume a

*K*active user transmission in the uplink MIMO MC-CDMA system, each user is assigned a unique spreading code in the frequency domain such that the complex transmitted signal vectors with

*N*× 1 dimension $({\mathbf{s}}_{k}^{(m)}(i),{\mathbf{s}}_{k}^{(m)}(i+1))\text{,}\phantom{\rule{1em}{0ex}}m=1,2$ of the

*k*th user for the

*i*th and (

*i*+ 1)th data symbol after the STBC can be expressed by

**c**

_{ k }is an

*N*× 1 vector with the spreading code of length

*N*, where

**c**

_{1}is the spreading code of the desired user 1. The

**Q**

^{ H }denotes the

*N*×

*N*orthogonal inverse fast Fourier transform matrix. The

*d*

_{ k }(

*i*) and

*d*

_{ k }(

*i*+ 1) are the

*i*th and (

*i*+ 1)th data symbols, respectively. Next, to prevent interblock interference, a cyclic prefix (CP) longer than the multipath channel response is then inserted into each transmitted data block. For fixed wireless communication, the channel impulse response (CIR)

*L*

_{ k }× 1 vector of user

*k*from the

*n*

_{ T }th transmit antenna to the

*n*

_{ R }th receive antenna can be modeled as ${h}_{k}^{({n}_{R},{n}_{T})}={[{h}_{k}^{({n}_{R},{n}_{T})}(0){h}_{k}^{({n}_{R},{n}_{T})}(1)\cdots {h}_{k}^{({n}_{R},{n}_{T})}({L}_{k}-1)]}^{T}$,

*n*

_{ T }

*, n*

_{ R }= 1 or 2, where

*L*

_{ k }is the channel length. The assumptions are that CIR ${h}_{k}^{({n}_{R},{n}_{T})}(i)\text{,}\phantom{\rule{0.25em}{0ex}}i=0,\dots ,{L}_{k}-1$ is an independent, identically distributed (i.i.d.) complex Gaussian random variable with zero-mean and unit-variance and that ${\mathbf{h}}_{k}^{({n}_{R},{n}_{T})}$ is a quasi-static channel that remains constant during each packet and is independent between packets. Next, let $\Delta {\epsilon}_{k}=\Delta {f}_{k}{T}_{k}$ denote the unknown normalized CFO of user

*k*between the transmitter and receiver, let Δ

*f*

_{ k }denote the CFO of the

*k*th user, and let

*T*

_{ k }denote the MC-CDMA symbol duration of the

*k*th user [21].

*i*th and (

*i*+ 1)th post-FFT received signal blocks with

*N*× 1 dimension from the first and second receiver antennas in the presence of CFO can be expressed by

*n*

_{ T }

*, n*

_{ R }= 1 or 2, denotes the

*N*×

*N*frequency domain channel matrix from the

*n*

_{ T }th transmit antennas to the

*n*

_{ R }th receive antenna. The

*N*×

*N*CFO effect matrix in (2) is modeled by ${e}^{j{\varphi}_{k}(q)}{\mathbf{G}}_{k}(\Delta {\epsilon}_{k})$,

*q*=

*i*– 1,

*i*, with ${\mathbf{G}}_{k}(\Delta {\epsilon}_{k})=\mathbf{Q}\text{diag}\left\{{e}^{j2\pi \Delta {\epsilon}_{k}{N}_{g}/N},\dots ,{e}^{j2\pi \Delta {\epsilon}_{k}({N}_{g}+N-1)/N}\right\}{\mathbf{Q}}^{H}$ and ${\varphi}_{k}=j2\pi \Delta {\epsilon}_{k}({N}_{g}+N)/N$. The ${\mathbf{v}}^{({n}_{R})}(i)$ and ${\mathbf{v}}^{({n}_{R})}(i+1)$,

*n*

_{ R }= 1, 2, denote the

*N*× 1 complex white Gaussian noise with zero mean and covariance matrix ${\sigma}_{n}^{2}{\mathbf{I}}_{N\times N}$. Without loss of generality, the signal model in (2) then assumes that user 1 is the desired user and that the others are MAI. Thus, the received signals of the

*i*th and (

*i*+ 1)th symbol blocks in (2) can be rewritten as

*n*

_{ R }= 1, 2, where ${\overline{\mathbf{C}}}_{1}$ is the

*N*×

*L*

_{1}frequency shift spreading code matrix of user 1 due to CFO and multipath delay time, i.e., ${\overline{\mathbf{C}}}_{1}={\mathbf{G}}_{1}(\Delta {\epsilon}_{1}){\mathbf{C}}_{1}$ with the

*N*×

*L*

_{1}composite spreading code matrix ${\mathbf{C}}_{1}=[{\mathbf{c}}_{1,0}\cdots {\mathbf{c}}_{1,l}\cdots {\mathbf{c}}_{1,{L}_{1}-1}]$ and ${\mathbf{c}}_{1,l}=\text{diag}({\mathbf{c}}_{1}){\mathbf{q}}_{l}$ with the

**q**

_{ l }being the

*l*th

*N*× 1 column vector of

**Q**matrix. Although the original spreading code

**c**

_{1,0}is used in the frequency domain in the MC-CDMA system, note that the original spreading code in the CFO and multipath environment is distorted by the CFO and different time delay of multipath effect in the frequency domain. Next, the channel response gain vectors with

*N*× 1 dimensions are ${\overline{\mathbf{h}}}_{1}^{({n}_{R},{n}_{T})}=1/\sqrt{2}{\sigma}_{1}{\mathbf{h}}_{1}^{({n}_{R},{n}_{T})}$,

*n*

_{ T },

*n*

_{ R }= 1 or 2. The

*N*× 1 MAI vectors are denoted by ${\mathbf{i}}^{({n}_{R})}(i)$ and ${\mathbf{i}}^{({n}_{R})}(i+1)$,

*n*

_{ R }= 1, 2. The conventional STBC system can then be used to construct the orthogonal signal structure and to acquire the diversity gain of the transmitter antenna over the single user in the perfect synchronization scenario. However, in (3) with MAI and CFO effects, the conventional STBC receiver design [18] cannot be used for MAI and CFO scenarios when constructed directly by ${\mathbf{y}}^{{n}_{R}}(i)$ and ${\mathbf{y}}^{{n}_{R}}(i+1)$, ${n}_{R}=1,2$. Therefore, the conventional receiver design detects the signal in (3) by first compensating for the frequency offset effect and then using linear weight combiner to suppress the MAI interference. That is, after an ideal frequency offset compensation, the post-FFT received signal vector in (3) can be rewritten as

*n*

_{ R }= 1, 2. The post-FFT received signal in (4) for the

*i*th and (

*i*+ 1)th symbols can also be cascaded as $\tilde{\mathbf{y}}(i)$ with 4

*N*× 1 vector size:

*N*× 1 composite channel vectors, i.e.,

*N*× 1 composite MAI and noise vectors, respectively. To recover

*d*

_{1}(

*i*) and

*d*

_{1}(

*i*+ 1) from $\tilde{\mathbf{y}}(i)$, a linear receiver can be used to calculate

**w**

_{1}and

**w**

_{2}are the 4

*N*× 1 weight vectors. The ideal maximum SINR (MSINR) weight vector can be chosen by [22], i.e.,

*N*× 4

*N*covariance matrices are

Thus, the desired signals *d*_{1}(*i*) and *d*_{1}(*i* + 1) can be detected by *x*_{1}(*i*) and *x*_{1}(*i +* 1), respectively.

It is noted that the ideal MSINR receiver in (6) is implemented by artificially knowing the desired signal components *d*_{1}(*i*) and *d*_{1}(*i* = 1) in the data, and using the true composite channel vector ${\tilde{\mathbf{h}}}_{1}$ and ${\tilde{\mathbf{h}}}_{2}$ to obtain (7). That is, the ideal MSINR receiver is designed to perform the optimum bound of the proposed MIMO MC-CDMA system, which cannot be realized since the desired signal and the true composite channel are the ideal known assumptions. Therefore, in this article, we will propose a robust fuzzy MIMO MC-CDMA receiver with MAI suppression over CFO and multipath fading channel environment.

## Robust MIMO MC-CDMA receiver

Figure 1b shows a block diagram of the multistage fuzzy MIMO MC-CDMA receiver. In the first stage of the receiver, a novel received data structure is proposed to overcome the failure of the STBC MC-CDMA system due to CFO. Next, a robust fuzzy CFO-constrained MIMO MOE method with coarse CFO estimation and signal subspace projection is designed to suppress MAI. Finally, the post-MOE output data are constructively combined by MMSE weight vector to extract the symbols transmitted by the desired user.

### Novel MIMO receiver model with offset symbols

*n*

_{ R }= 1, 2, where ${\mathbf{b}}_{1}(i)={[{d}_{1I}(i)j{d}_{1Q}(i+1)]}^{T}$ and ${\overline{\mathbf{b}}}_{1}(i)={[j{d}_{1Q}(i){d}_{1I}(i+1)]}^{T}$ are the 2 × 1 reconstructed transmission signal vectors.The reconstructed 4

*N*× 1 data vector $\stackrel{\u2322}{y}(i)$ over the

*i*th and (

*i*+ 1)th symbols can then be cascaded by

*N*× 4

*L*

_{1}dimension and the 4

*L*

_{1}× 2 MIMO channels are ${\mathbf{H}}_{1}={\left[{\overline{\mathbf{h}}}_{1}^{{(1,1)}^{T}},\phantom{\rule{0.25em}{0ex}},{\overline{\mathbf{h}}}_{1}^{{(1,2)}^{T}},\phantom{\rule{0.25em}{0ex}},{\overline{\mathbf{h}}}_{1}^{{(2,1)}^{T}},\phantom{\rule{0.25em}{0ex}},{\overline{\mathbf{h}}}_{1}^{{(2,2)}^{T}},{\overline{\mathbf{h}}}_{1}^{{(1,2)}^{T}},\phantom{\rule{0.25em}{0ex}},{\overline{\mathbf{h}}}_{1}^{{(1,1)}^{T}},\phantom{\rule{0.25em}{0ex}},{\overline{\mathbf{h}}}_{1}^{{(2,2)}^{T}},\phantom{\rule{0.25em}{0ex}},{\overline{\mathbf{h}}}_{1}^{{(2,1)}^{T}}\right]}^{T}$ and ${\mathbf{H}}_{2}={\left[\begin{array}{l}{\overline{\mathbf{h}}}_{1}^{{(1,1)}^{T}}-{\overline{\mathbf{h}}}_{1}^{{(1,2)}^{T}}\phantom{\rule{0.25em}{0ex}}{\overline{\mathbf{h}}}_{1}^{{(2,1)}^{T}}-{\overline{\mathbf{h}}}_{1}^{{(2,2)}^{T}}\\ {\overline{\mathbf{h}}}_{1}^{{(1,2)}^{T}}-{\overline{\mathbf{h}}}_{1}^{{(1,1)}^{T}}\phantom{\rule{0.25em}{0ex}}{\overline{\mathbf{h}}}_{1}^{{(2,2)}^{T}}-{\overline{\mathbf{h}}}_{1}^{{(2,1)}^{T}}\end{array}\right]}^{T}$. The$\stackrel{\u2322}{i}(i)$ and $\stackrel{\u2322}{n}(i)$ are the MAI and the noise vectors with 4

*N*× 1 dimension, respectively. In (10), the 4

*N*× 4

*N*data correlation matrix ${\mathbf{R}}_{\stackrel{\u2322}{y}\stackrel{\u2322}{y}}$ can be derived by

*N*× 4

*N*MAI-plus-noise correlation matrix. Notably, the two reconstructed symbols (

**b**

_{1}(

*i*),${\overline{\mathbf{b}}}_{1}(i)$) in (11) involve the orthogonal cross-correlation and the same covariance property, i.e.,

**R**

_{ s }with 4

*N*× 4

*N*dimension can be rewritten as

Since **H**_{1} and **H**_{2} are CFO-independent, **R**_{
s
} can only be affected by the frequency shift matrix ${\tilde{\mathbf{C}}}_{1}(\Delta \epsilon )$. Therefore, this feature can be used for coarse estimation of CFO as described below.

### Coarse CFO estimator

**C**

_{1}is given, the CFO shift spreading code matrix ${\tilde{\mathbf{C}}}_{1}(\Delta \epsilon )$ with 4

*N*× 4

*L*

_{1}dimension can be constructed by the different frequency offset, i.e.,

**G**

_{1}(Δε) is an

*N*×

*N*matrix,

**C**

_{1}is an

*N*×

*L*

_{1}matrix, and Δε is uniformly increased to search for the coarse CFO. Next, based on the reconstructed data model in (13) and the CFO shift spreading code matrix in (14), a simple CFO estimator is determined as the solution to the following spectral search problem:

where the 4 *N* × 2 *K* matrix **U**_{
s
} contains 2 *K* dominant eigenvectors of ${\mathbf{R}}_{\stackrel{\u2322}{y}\stackrel{\u2322}{y}}$ in (11), which span the signal subspace. Note that the signal subspace **U**_{s}, which is obtained from the signal covariance matrix **R**_{s} in (11) and (13), can be used to provide the desired signal subspace and to limit the interference subspace to converge the desired coarse CFO estimate for the spectral search in (15). That is, in (15), for the estimated frequency offset Δ*ε*_{1} approaching the CFO of desired user, ${\tilde{\mathbf{C}}}_{1}(\Delta \epsilon )$ produces a spectral lobe at Δ*ε*_{1} and suppresses MAI outside Δ*ε*_{1}. Therefore, if the spectral peak error in the spectrum satisfies the condition $\left|{S}_{i}(\Delta {\epsilon}_{1})-{S}_{i-1}(\Delta {\epsilon}_{1})\right|/{S}_{i}(\Delta {\epsilon}_{1})<0.1$, the Δ*ε*_{1} is varied with the *i* th searching iteration by using Δ*ε*_{1} as the coarsely estimated CFO of the desired user.

### Fuzzy CFO-constrained MIMO MOE detector

**i**

_{ i }is the

*i*th column of the identity matrix

**I**and

*D*is the constraint size. Matrix

**E**

_{1}, which is the MIMO fuzzy CFO-constrained signature matrix with coarse CFO $\Delta {\widehat{\epsilon}}_{1}$ initiation, is then chosen from the dominant eigenvectors of the distorted signature matrix with a dense set of CFO ${\widehat{\epsilon}}_{n}=\Delta {\widehat{\epsilon}}_{1}+{\epsilon}_{n}$,

*n*= 1, 2,…,

*N*

_{ t }, i.e.,

**{**λ

_{ i }

**}**and

**{e**

_{ i }

**}**are eigenvalues and 4

*N*× 1 eigenvectors of the 4

*N*× 4

*N*R

_{ ε }matrix. The shift matrix ${\tilde{\mathbf{C}}}_{1}({\widehat{\epsilon}}_{n})=\text{diag}\{{\mathbf{G}}_{1}({\widehat{\epsilon}}_{n}){\mathbf{C}}_{1}\text{,}\phantom{\rule{0.25em}{0ex}}{e}^{j{\widehat{\varphi}}_{1}}{\mathbf{G}}_{1}({\widehat{\epsilon}}_{n}){\mathbf{C}}_{1}\text{,}\phantom{\rule{0.25em}{0ex}}{\mathbf{G}}_{1}({\widehat{\epsilon}}_{n}){\mathbf{C}}_{1}\text{,}\phantom{\rule{0.25em}{0ex}}{e}^{j{\widehat{\varphi}}_{1}}{\mathbf{G}}_{1}({\widehat{\epsilon}}_{n}){\mathbf{C}}_{1}\}$ involves the phase shift ${\widehat{\varphi}}_{1}=j2\pi {\widehat{\epsilon}}_{n}({N}_{g}+N)/N$. Since the

*D*is the dominant mode with large eigenvalues in (17) and

*N*

_{ t }is the number of frequencies in the dense set of frequencies ${\widehat{\epsilon}}_{n}$, the

*N*

_{ t }shift matrices ${\tilde{\mathbf{C}}}_{1}({\widehat{\epsilon}}_{n})$ span an effective rank of

*D*for the fuzzy CFO-constrained matrix

**E**

_{1}with 4

*N*×

*D*dimension. The 4

*N*× 1 weight vector

**w**

_{ i }of the MIMO fuzzy CFO-constrained MOE detector in (16) can be determined by

where **W** is the 4 *N* × *D* weight matrix of fuzzy MOE detector. Notably, the MIMO fuzzy CFO constraint enables the proposed MIMO MOE detector to avoid the problem of sensitivity to signature mismatch arising in the conventional MOE [7], which the signature mismatch effect can significantly degrade system performance. Moreover, in (17), the dominant eigenvectors can span the desired signal subspace containing the mismatch signatures induced by CFO effect. Therefore, the new constrained signature **E**_{1} in (16) can consist of the dominant eigenvectors to collect the desired signal energy and to suppress other user interference.

### Signal subspace projection of MOE weight

*N*×

*D*projected weight matrix

**W**

_{s}can be expressed by

where **U**_{
s
} is the signal subspace of ${\mathbf{R}}_{\stackrel{\u2322}{y}\stackrel{\u2322}{y}}$ as in (15).

### MMSE weight combiner

**z**

_{1,l},

*l*= 1, 2,…,

*D*. These output data ${z}_{1,l}$ collected according to MMSE criteria are therefore used to extract the desired signal vector

**x**(

*i*) with 2 × 1 dimension. Let

**V**be the

*D*× 2 weight matrix for the combining procedure, i.e.,

**z**(

*i*) is a

*D*× 1 vector

**b**

_{1}(

*i*) and ${\overline{\mathbf{b}}}_{1}(i)$ in (9) are the training symbol sequences. This leads to the well-known Wiener solution

where ${}_{\mathbf{R}}^{\mathit{zz}}=E\left\{\mathbf{z}(i){\mathbf{z}}^{H}(i)\right\}$ is the *D* × *D* covariance matrix of the post-MOE data and ${\tilde{\mathbf{H}}}_{1}=E\left\{\mathbf{z}(i){\mathbf{b}}_{1}^{H}(i)\right\}$ and ${\tilde{\mathbf{H}}}_{2}=E\left\{\mathbf{z}(i){\overline{\mathbf{b}}}_{1}^{H}(i)\right\}$ are the estimated composite channel matrices with *D* × 2 dimensions.

## Complexity analysis

**W**of the MIMO fuzzy MOE detector in (18) is computed as

**E**

_{1}are the 4

*N*× 4

*N*and 4

*N*×

*D*matrices, respectively. Next, the number of complex multiplications for signal subspace projection in (19) is computed as

According to (30), the computational complexity of the proposed multistage detector depends on the spreading code length *N*, the number of multipath channels *L*_{1}, the number of active users *K*, and the size *D* of the dominant eigenvalues of the fuzzy constrained matrix **E**_{1}. The equation also shows that the computational load is approximately *O*(*N*^{3}). Note that the computational load of the proposed multistage detector is determined mainly by the weight matrix inversion calculated by the fuzzy MOE detector in (18).

Various existing methods are then used to evaluate the computational complexity of the conventional receivers, e.g., MOE and subspace MOE detectors are considered for comparison. The conventional MOE detector with ${\mathbf{R}}_{\stackrel{\u2322}{y}\stackrel{\u2322}{y}}^{-1}$ as reported in [14] is considered first. The total number of multiplication operations is about *O*(*N*^{3}) due to the major computations needed for inversion of ${}_{\mathbf{R}}^{\stackrel{\u23de}{y}\stackrel{\u23de}{y}}$. Next, the algorithm for the conventional subspace MOE detector in [17] resembles that for the projected weight matrix in (19). The major computations still require calculation of MOE weight by ${\mathbf{R}}_{\stackrel{\u2322}{y}\stackrel{\u2322}{y}}^{-1}$. Therefore, the number of complex multiplication is about *O*(*N*^{3}). In summary, the complexity analysis shows that the total number of complex multiplication operations required by the proposed multistage detector, the conventional MOE, and subspace MOE detectors is about *O*(*N*^{3}). That is, the MOE calculation results in a similar computational complexity in the three MOE calculation methods.

Although the proposed multistage method requires additional computational operations, mainly because of the coarse CFO estimator and MMSE combiner, the computational load is still smaller than that of the MOE detector *O*(*N*^{3}). As shown above, the proposed multistage detector has higher computational complexity compared to the conventional MOE and subspace MOE detectors. However, the simulation results confirm that proposed method outperforms the conventional MOE detector and confirms its robust performance over serious CFO environment.

## Computer simulations

Simulations are performed to confirm the performance of the proposed MIMO MC-CDMA system over uplink multipath channels. For all users, the QPSK-modulated signals are spread by orthogonal gold codes of length 32 (*N* = 32). The channel profile assumes *L*_{
k
} = 4 independent equal power Rayleigh fading paths with the time delays chosen from {0,3*T*_{s}}, which is smaller than the guard interval *T*_{G} =8*T*_{s} where *T*_{s} is the sampling period. In this simulation, the selected *D* is *D* = 2 *L*_{1}, i.e., two degrees of freedom are used to retain the desired user signal and to suppress MAI. The quasi-static multipath fading channel in the simulations is assumed to be constant during each packet and independent between packets. The fading gains are the i.i.d. complex Gaussian random variables with zero mean and unit variance. The output SINR, which is used as a performance index, is defined as the ratio of the signal power in **x**(*i*) to the MAI-puls-noise power in **x**(*i*) at the MMSE combiner output. The input SNR is also defined as $\text{SNR}=10{log}_{10}({\sigma}_{1}^{2}/{\sigma}_{n}^{2})$, and the near-far ratio (NFR) is defined as $\text{NFR}=10{log}_{10}({\sigma}_{k}^{2}/{\sigma}_{1}^{2})$, *k* ≠ 1. The analysis is also simplified by assuming all MAI powers are identical. Executing 10,000 Monte-Carlo trials with different fading gains and data/noise sequences in each trial obtains one output SINR and BER value. The normalized CFOs of all users are also assumed to be in the range $-0.5<\Delta {\epsilon}_{k}<0.5$. The performance comparison included the results obtained by conventional MOE, subspace MOE, and optimal MSINR detectors. The optimal MSINR receiver is implemented by using perfect CFO compensation and by artificially removing the desired user signal component in the data, and then using the ideal composite channel vectors ${\tilde{\mathbf{h}}}_{1}$ and ${\tilde{\mathbf{h}}}_{2}$ to obtain (7). Finally, the following “standard” parameters are used throughout the section unless otherwise mentioned: SNR = 10 dB, NFR = 0 dB, *K* = 10, Δ*ε*_{1} = 0.25.

*K*. Figure 3 shows that, again, the MIMO multistage MOE detector successively approaches the ideal MIMO MSINR receivers and outperforms the conventional MOE and subspace MOE detectors. Moreover, the output SINRs of the conventional MOE receiver and first stage MOE receiver increase as the number of users increases. Because the power of MAI with different CFOs increases as the number of users increases, the MOE effectively suppress the MAI of overall CFO region. However, the output SINR performances of the conventional MOE and first stage MOE receivers are severely degraded in comparison with the proposed multistage and optimal receivers. Moreover, as the number of users increases, the proposed full-stage detector does not severely degrade the output SINR performance due to lack of degree of freedom effect, which is overcome by the larger code length.

*ε*

_{1}). Within the entire range $-0.5<\Delta {\epsilon}_{1}<0.5$, the output SINR of the proposed receiver is fairly constant. However, the conventional MOE and subspace MOE detectors exhibit severely degraded output SINR because of their extreme sensitivity to inaccuracies in frequency synchronization, which cause the desired signal cancellation. Therefore, Figure 4 indicates that the proposed multistage MOE detector with coarse CFO estimate, fuzzy CFO constraint, signal subspace projection, and MMSE combiner achieves the ideal performance and performs robustly against CFO by effectively collecting the desired signal and by canceling MAI interference. Notably, the performance at CFO = 0 case is better than other CFO cases. It is due to no ICI interferences, which will be easily to suppress the MAI and get more degrees-of-freedom to enhance the output SINR performance.

**h**is a random vector with the entries being i.i.d. complex Gaussian random variables. Note that the entries of

**h**

_{1}are i.i.d. complex Gaussian random variables with variance ${\sigma}_{h1}^{2}$. Figure 5 shows the output SINR versus ${\sigma}_{\Delta}/{\sigma}_{h1}$. The simulation results confirm that the proposed MIMO MOE receiver performs reliably with relative error limited to 20% in channel estimation. Figure 5 also confirms that the proposed receiver outperforms other receivers in the channel estimation error scenario.

## Conclusions

The novel robust MIMO MC-CDMA transceiver proposed in this study is suitable for uplink systems because of its high-quality performance over multipath fading channels and CFO effects. In the multiuser receiver, a MIMO fuzzy CFO-constrained MOE detector incorporating the rough CFO estimator suppresses MAI and overcomes the degradation problem of the conventional MOE detector with CFO effect. Next, by using signal subspace projection, the proposed detector further reduces noise with its fuzzy MOE filter. Finally, an MMSE weight combiner is proposed to capture the signal multipath diversity gain. Simulation results confirm that the proposed MIMO receiver performs comparably to the ideal receiver, outperforms conventional MOE detectors, and performs reliably over strong MAI and large frequency offsets.

## Declarations

### Acknowledgments

This study was sponsored by the National Science Council, R.O.C., under NSC contract 100-2220-E-155-006. The authors express the deepest gratitude to Prof. Tsui-Tsai Lin for his kind help and enthusiastic guidance. The authors also thank the Editor and anonymous reviewers for their helpful comments and suggestions in improving the quality of this article. Ted Knoy is appreciated for his editorial assistance.

## Authors’ Affiliations

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