# The diversity of STF-coded MIMO-OFDM systems with a general correlation model

- Mang Liao
^{1}Email authorView ORCID ID profile, - Youguang Zhang
^{2}and - Zixiang Xiong
^{3}

**2016**:39

https://doi.org/10.1186/s13638-016-0527-2

© Liao et al. 2016

**Received: **8 June 2015

**Accepted: **15 January 2016

**Published: **5 February 2016

## Abstract

Owing to insufficient antenna spaces, mobile scenarios, and multipaths in practice, transmission correlations in space, time, and frequency domains are inevitable in wireless communications. This paper studies the effect of general spatial, temporal, and frequency/path correlations on the performance of space-time-frequency (STF)-coded multiple-input, multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems over frequency-selective block-fading channels. Specifically, we first derive an upper bound on the maximum achievable diversity by Hadamard and tensor products and analyze the effect of general spatial, temporal, and frequency/path correlations on it using rank properties of block matrices. We then address STF code designs and give two examples, one traditional STF code and another quasi-SF code, to show that our upper bound on the maximum diversity is achievable. The decoding complexity is considered in the MIMO system with arbitrary correlated fading channels using the traditional STF code. We also identify the newly developed statistical channel models for MIMO LTE and 802.11n as special cases of our STF-coded MIMO-OFDM system by showing that our theoretical diversity results match those simulated from these statistical channel models. Finally, we show that our general diversity result recovers various maximum diversity gains for different special correlation scenarios that have appeared in the literature.

## Keywords

## 1 Introduction

To guarantee reliable transmission, various diversity schemes have been proposed in three physical domains: space, time, and frequency. Previous works on spatial diversity assumed independent links between the transmitter and the receiver [1–3]. However, this assumption is not always valid due to insufficient antenna spaces or scarce scatterers during transmission. Firstly, the probability of error for two-dimensional signal constellations were analyzed in [4]. Then, the effect of space-time (ST) code on the performance of multiple-input multiple-output (MIMO) systems was studied over flat-fading spatial correlation channels in [5, 6]. But different mobile stations or a mobile moving through different geographical locations may experience channel variations in time. The performance of ST-coded MIMO systems was analyzed over temporal correlated Rayleigh fading channels in [7].

For the more interesting case of frequency-selective fading channels, orthogonal frequency division multiplexing (OFDM) has been recognized as an attractive approach to coping with the multipath effect [8].

Recently, several papers have studied that using space-time-frequency (STF) codes across multiple OFDM blocks obtains the full diversity in frequency-selective fading channels [9–11]. Others have analyzed the approaches of decoding for reducing the complexity at the receiver [12, 13]. However, limited attention has been devoted so far to the problem of the influence of correlated fading channels on diversity. The spatial correlation of the fading channels is always as a Kronecker model, in which the transmitter correlation is independent of the receiver correlation. This model requires few scatterers between the transmitter and the receiver. However, the channel measurements indicate that the Kronecker structure does not describe the multipath propagation channel correctly [14, 15]. The maximum diversity of SF-coded MIMO-OFDM systems with arbitrary spatial correlation was studied in [16]. Unfortunately, it considers no frequency correlation. However, in most practical situations, multipath delay could cause the channel correlation in frequency domain. An arbitrary spatial and temporal correlation model for STF-coded MIMO-OFDM was presented in [17]. The assumption in [17] is that the multipath delays are independent and the multipaths are separated. However, in a multipath channel environment, when the scatterers are located far from the transmit antenna arrays in a narrow angular range, multipath signals that bounce from these scatterers can be correlated temporally, causing path correlation among channels. Furthermore, path correlation can also be caused by using a pulse shaping filter at the transmitter or the receiver [18]. In fact, the frequency-selective fading channels could not avoid spatial, temporal, and frequency/path correlations.

Motivated by this problem, in our recent paper [19], considering general spatial, temporal, and frequency/path correlations in wireless communication, we studied the performance of STF-coded block-fading MIMO-OFDM systems. We went beyond the limitations of ideal assumptions such as quasi-static or rapid fading channels, unlimited antenna spaces or abundant scatterers, and separable multipaths between the transmitter and the receiver. Our spatial correlation is arbitrary and affected by multipaths; hence, it is not subjected to the constraint of the Kronecker model in [9]. We derived an upper bound on the maximum achievable diversity by Hadamard and tensor products. We also addressed STF code designs with maximum diversity. In this paper, based on rank properties of block matrices, we re-derive the upper bound on the maximum achievable diversity in greater details and discuss the physical meaning of each part in the expression that is conducive to analyze the influence of the individual correlation into the performance of block-fading MIMO-OFDM systems. For achieving the diversity of the system with the arbitrarily correlated channels, we give two examples of the STF codes: one traditional STF code and another quasi-SF code, which are designed for achieving maximum diversity. The decoding complexity is considered in the MIMO system with arbitrary correlated fading channels using the traditional STF code. We also identify the newly developed statistical channel models for MIMO Long-Term Evolution (LTE) [20] and 802.11n [21] as special cases of our STF-coded MIMO-OFDM system by showing that our theoretical diversity results match those simulated from these statistical channel models. Finally, we show that our general diversity result recovers various maximum diversity gains for different special correlation scenarios that have appeared in the literature.

The rest of this paper is organized as follows. Section 2 describes our system model. Section 3 gives an upper bound on the maximum diversity of block-fading MIMO-OFDM system with general spatial, temporal, and frequency/path correlations. Section 4 presents two STF code design criteria and two corresponding examples to show achievability of our upper bound. Section 5 provides the decoding complexity. Section 6 specializes our maximum diversity to several specific channel models, including MIMO LTE and 802.11n; it also shows that our result recovers existing maximum diversity gains in the existing literature. Section 7 concludes the paper.

Notation-wise, I
_{
N
} denotes the *N*×*N* identity matrix, 1
_{
N×M
} and 0
_{
N×M
}, are the all-one and all-zero *N*×*M* matrices, respectively; (.)^{T} and (.)^{H} represent transpose and conjugate transpose, respectively; and ⊙ and ⊗ signify Hadamard and tensor products, respectively. We use rank(R) and range(R) to denote the rank and range of matrix R, respectively, and null(R) the dimension of the null space of matrix R. In addition, \(\bar h\) and *h* denote symbols in the time and frequency domain, respectively. Finally, in the temporal domain, *k*(1≤*k*≤*K*) is the index of OFDM blocks; in the frequency domain, *n*(1≤*n*≤*N*) is the index of subcarriers; in the multipath domain, *l*(1≤*l*≤*L*) is the index of multipaths; in the spatial domain, *i*(1≤*i*≤*N*
_{
t
}) and *j*(1≤*j*≤*N*
_{
r
}) are the indexes of the transmit and receive antennas, respectively.

## 2 System model

*N*

_{ t }transmit antennas,

*N*

_{ r }receive antennas,

*N*subcarriers, and

*K*OFDM blocks. There are

*L*correlated multipaths between each pair of transmit and receive antennas. The channel impulse response between transmit antenna

*i*and receive antenna

*j*in the

*k*th OFDM block is given by

*τ*

_{ l }and \({\alpha _{i,j,k}[\!l]}\sim \mathcal {CN}\left (0, {\sigma _{l}^{2}}\right)\) are the delay and complex amplitude of the

*l*th path between transmit antenna

*i*and receive antenna

*j*, respectively. The powers of the

*L*paths are normalized such that \(\sum \limits _{l = 0}^{L - 1} {{\sigma _{l}^{2}} = 1}\). We assume that all path delays are located exactly at the sampling instances of the receiver. From (1), the frequency response of the channel is

where \(\mathrm {j} = \sqrt { - 1}\) and *h*
_{
i,j,k
}(*f*) is the Fourier transform of \({\bar h}_{i,j,k}\left (\tau \right) \) in (1).

*N*

_{ t }transmit antennas,

*K*OFDM blocks, and

*N*subcarriers. Each STF codeword is a

*KN*×

*N*

_{ t }

*KN*matrix given by

_{ F }signifying the Frobenius norm. For 1≤

*k*≤

*K*,

*C*

_{ k }is an

*N*×

*N*

_{ t }

*N*matrix that represents the transmitted codes in the

*k*th OFDM block and is constructed by transmitting codes in

*N*subcarriers. That is,

_{ k }[

*n*] is a 1×

*N*

_{ t }vector, representing an STF code in the

*k*th OFDM block and the

*n*th subcarrier from

*N*

_{ t }transmit antennas, that can be written as

with *c*
_{
i,k
}[*n*], 1≤*i*≤*N*
_{
t
}, 1≤*k*≤*K*, 1≤*n*≤*N*, being the transmitted code from the *i*th transmit antenna in the *k*th OFDM block and the *n*th subcarrier.

*j*th receive antenna in the

*k*th OFDM block and the

*n*th subcarrier is given by

*ρ*is the average signal-to-noise ratio (SNR) at each receive antenna. The channel frequency response

*h*

_{ i,j,k }[

*n*] from transmit antenna

*i*to receive antenna

*j*in the

*k*th OFDM block and the

*n*th subcarrier is a uniformly sampled version of

*h*

_{ i,j,k }(

*f*) in (2) and can be expressed as

where \({(W)_{N}} = \frac {1}{\sqrt N}{e^{-\frac {\mathrm {j}2\pi }{N} }}\).

*N*

_{ r }

*KN*×1 vectors, representing the received signals and noises, respectively.

*N*

_{ r }

*N*

_{ t }

*KN*×1 vector, with

*H*

_{ k }, 1≤

*k*≤

*K*, representing the fading channels in the

*k*th OFDM block. The

*N*

_{ r }

*N*

_{ t }

*N*×1 vector

*N*subcarriers, where the

*N*

_{ r }

*N*

_{ t }×1 vector h

_{ k }[

*n*], 1≤

*k*≤

*K*, 1≤

*n*≤

*N*, representing the fading channels

*H*

_{ k }in the

*n*th subcarrier, can be written as

*k*≤

*K*, 1≤

*l*≤

*L*, we denote the

*N*

_{ r }

*N*

_{ t }×1 vector

*k*th OFDM block and the

*l*th path. We also denote the

*N*

_{ r }

*N*

_{ t }

*L*×1 vector

*k*th OFDM block through

*L*paths. Finally, the

*N*

_{ r }

*N*

_{ t }

*KL*×1 vector

is the *N*×*L* FFT matrix.

## 3 An upper bound on maximum diversity

*KN*×

*N*

_{ t }

*KN*matrix Δ as the difference of the transmitted codeword and its corresponding detected codeword, i.e., \({\boldsymbol \Delta } \buildrel \Delta \over = {\boldsymbol C}-{\boldsymbol {\tilde C}}\). This allows us to explore the difference of two codewords in all three domains, i.e., across

*N*

_{ t }transmit antennas,

*K*OFDM blocks, and

*N*subcarriers. Similar to the expression of an STF code in (3), we can write the difference of two STF codewords as

where *Δ*
_{
k
}, 1≤*k*≤*K*, represents the difference of two codewords in the *k*th OFDM block.

*N*

_{ r }

*KN*×1 vector \({ \boldsymbol \Delta } \otimes {\boldsymbol I}_{N_{r}}{ \boldsymbol H }\) for a fixed code realization has a Gaussian distribution with zero mean and

*N*

_{ r }

*KN*×

*N*

_{ r }

*KN*covariance matrix

where rank(R) and *λ*
_{
i
}(R) are the rank and the *i*th eigenvalue of R, respectively. From (21), we see that the diversity order depends on rank(R).

*E*{H H

^{H}} in (20) represents the correlation of frequency-selective block-fading channels in space, time, and frequency/path domains, and from (15), it can be rewritten as

*E*{A A

^{H}} by φ or post-multiplying it by φ

^{H}does not increases its rank, hence from (23), the rank of covariance matrix R satisfies

with equality holding when matrix φ has full column rank.

For the sake of simplicity, in the sequel, we only consider the influence of the channel frequency/path correlation on the channel spatial correlation and assume that the channel temporal correlation is independent of the spatial and frequency/path correlations. The effect of the channel temporal correlation on the channel spatial and frequency/path correlations can be studied in a similar way, and hence is omitted here.

*i*,

*p*≤

*N*

_{ t },1≤

*j*,

*q*≤

*N*

_{ r },1≤

*k*,

*m*≤

*K*,1≤

*l*,

*d*≤

*L*,

where \(r_{l,d}^{L}\) and \(r_{k,m}^{K}\) denote the frequency/path correlation coefficient between the *l*th path and the *d*th path, and the temporal correlation coefficient between the *k*th OFDM block and the *m*th OFDM block, respectively.

*E*{A A

^{H}} can be expressed as

*K*×

*K*Hermitian matrix R

^{ K }and the

*L*×

*L*Hermitian matrix R

^{ L }represent the channel temporal and path/frequency correlations, respectively, and they can be expressed as

*N*

_{ r }

*N*

_{ t }×

*N*

_{ r }

*N*

_{ t }Hermitian matrix \({\boldsymbol R}^{\boldsymbol {S}}_{l,d}\) in (27) denotes the channel spatial correlation between the

*l*th path and the

*d*th path. For 1≤

*l*,

*d*≤

*L*, it can be written as

*N*

_{ r }

*N*

_{ t }×

*N*

_{ r }

*N*

_{ t }

*L*block matrix

*l*with 1≤

*l*≤

*L*. Then, the covariance matrix

*E*{A A

^{H}} in (27) can be rewritten as

To compute the rank of covariance matrix *E*{A
A
^{H}}, we first give the following lemma that states the relationship between the rank of a block matrix and those of its submatrices.

###
**Lemma**
**1**.

*p*,

*p*

^{′}≤

*P*, 1≤

*q*,

*q*

^{′}≤

*Q*,

*p*≠

*p*

^{′},

*q*≠

*q*

^{′}, then

*p*≤

*P*−1, 1≤

*q*≤

*Q*−1 then

###
*Proof*.

See Appendix A.

*E*{A A

^{H}} in (31) can be computed as

Thus, from (25) and (36), we have the following theorem.

###
**Theorem**
**1**.

Furthermore, under the assumption of *N*≥*N*
_{
t
}
*L*, equality in (37) holds if the matrix φ in (24) has full column rank, i.e., rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*.

In (37), the \(\sum _{l = 1}^{L}\text {{null}}\left ({\boldsymbol r}_{l}^{\mathrm {T}} \right) - \text {{null}}\left ({\left [ {\boldsymbol r}_{1}^{\mathrm {T}}, \ldots, {\boldsymbol r}_{L}^{\mathrm {T}} \right ]}\right)\) part represents the effect of the frequency/path correlation on the performance, while the \(\sum _{l = 1}^{L}\sum _{d=1}^{L}{\text {{null}}\left (r^{L}_{l,d} {\boldsymbol R}^{\boldsymbol {S}}_{l,d} \right)} - \sum _{l = 1}^{L}{\text {{null}}\left ({{\boldsymbol r}_{l}} \right)}\) part signifies the influence of multipaths on the spatial correlation.

Theorem 1 indicates that our upper bound on the maximum diversity of MIMO-OFDM systems depends on the correlation of fading channels, and the STF coding structure does not affect the upper bound. In addition, the maximum diversity in multiple domains is the product of the maximum diversities in each separated domain. Finally, the channel correlations in each domain, the correlation between the spatially correlated channels, and the frequency/path correlated channels affect our upper bound.

*l*,

*l*

^{′}≤

*L*,

*l*≠

*l*

^{′}, then according to Lemma 1, our upper bound on the maximum achievable diversity in Theorem 1 can be simplified as

This result implies that an upper bound on the maximum diversity is the sum of the maximum diversities in each separated path, and the maximum diversities in each path are affected by the spatial correlation of the links between the transmitter and the receiver across different paths and the channel temporal correlation.

*l*≤

*L*, according to Lemma 1, we have

In this case, the maximum diversity of block-fading MIMO-OFDM system is the product of the time diversity and the sum of maximum space diversities in each path.

## 4 STF code designs with maximum diversity

From (23), if matrix φ has full column rank, i.e., rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*, equality in (37) holds, which means that our upper bound on the maximum diversity in Theorem 1 could be achieved. This prompts us to study the STF code design criteria for maximum diversity. We start with addressing the rank of matrix φ.

*k*≤

*K*, \({\Delta }_{k} {\boldsymbol T}_{N,L}\otimes {{\boldsymbol I}_{N_{t}}}\phantom {\dot {i}\!}\) can be rewritten as

where the *N*×1 vector \({\boldsymbol W}_{N}^{l}\phantom {\dot {i}\!}\) is the *l*th column of the FFT matrix T
_{
N,L
} in (16). That is, for 1≤*l*≤*L*, \({\boldsymbol W}_{N}^{l} \buildrel \Delta \over = \left [ {(W)}_{N}^{l}, { (W)}_{N}^{2l}, \ldots, { (W)}_{N}^{Nl} \right ]^{\mathrm {T}}\).

*l*th and the

*l*

^{′}th columns of the FFT matrix, respectively, then due to the property of the FFT matrix, \(\text {range}\left ({\boldsymbol W}_{N}^{l}\right) \cap \text {range}\left ({\boldsymbol W}_{N}^{l'}\right) = \{0\}\phantom {\dot {i}\!}\), 1≤

*l*,

*l*

^{′}≤

*L*,

*l*≠

*l*

^{′}. According to Lemma 1, from (41) to (43), we have

Thus, to guarantee that rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*, we have the following proposition.

###
**Proposition**
**1**.

Under the assumption of *N*≥*N*
_{
t
}
*L*, in each OFDM block, if the differences of two codewords are independent from different transmit antennas across *N* subcarriers, and if the differences from *N*
_{
t
} antennas in each subcarrier are not all zeros, then the upper bound on the maximum diversity in Theorem 1 can be achieved.

###
*Proof*.

where \({\Delta }_{k} {\boldsymbol 1}_{N \times 1}\otimes {\boldsymbol I}_{N_{t}}\phantom {\dot {i}\!}\) is an *N*×*N*
_{
t
} matrix, and the elements in its *i*th column represent the differences of two codewords from the *i*th antenna across *N* subcarriers in the *k*th OFDM block.

If in each OFDM block, the differences of two codewords are independent from different transmit antennas across *N* subcarriers (\({\text {range}\left ({\Delta }_{k}[\!n]\!\right)}\bigcap {\text {range}\left ({\Delta }_{k}[m]\!\right)} =0,\) 1≤*n*,*m*≤*N*,*n*≠*m*) and are not all zeros from *N*
_{
t
} antennas in each subcarrier (the elements in each row of matrix *Δ*
_{
k
} are not all zero), then matrix \({\Delta }_{k} {\boldsymbol 1}_{N \times 1}\otimes {\boldsymbol I}_{N_{t}}\phantom {\dot {i}\!}\) will have full column rank, i.e., \(\text {rank}\left ({\Delta }_{k} {\boldsymbol 1}_{N \times 1}\otimes {\boldsymbol I}_{N_{t}}\right) = N_{t}\phantom {\dot {i}\!}\). Hence, from (44) matrix φ has full column rank matrix with rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*.

*K*×

*K*diagonal matrix, with non-zero elements, satisfying the energy constraint \(E||{\mathbb {D}}||_{F}^{2} = K\). Similar to a traditional SF code, the

*N*×

*N*

_{ t }

*N*matrix

*N*

_{ t }×1 vector

denoting the transmitted codes in the *n*th subcarriers across *N*
_{
t
} antennas.

From (49), we see that the resulting STF code is constructed by repeating an SF code *K* times over *K* OFDM blocks.

Because of special structure of quasi-SF code in (46), we have the following proposition in terms of designing quasi-SF code with maximum diversity.

###
**Proposition**
**2**.

Given *N*≥*N*
_{
t
}
*L*, if the differences of two quasi-SF codewords are independent from different transmit antennas across *N* subcarriers, and if the differences are not all zeros from *N*
_{
t
} transmit antennas in each subcarrier, then the upper bound on the maximum diversity of quasi-SF-coded MIMO-OFDM system in Theorem 1 can be achieved.

###
*Proof*.

where \({\Delta ^{\text {SF}}} = {C^{\text {SF}}} - {{\tilde C}}^{\text {SF}}\).

Given *N*≥*N*
_{
t
}
*L*, if the differences of two quasi-SF codewords are independent from different transmit antennas across *N* subcarriers (\({\text {range}\left ({\Delta }^{\text {SF}}[\!n]\!\right)}\bigcap {\text {range}\left ({\Delta }^{\text {SF}}[\!m]\!\right)} = 0,\) 1≤*n*,*m*≤*N*,*n*≠*m*) and are not all zeros from *N*
_{
t
} transmit antennas in each subcarrier (the elements in each row of matrix *Δ*
_{
k
} are not all zero), \({\Delta }^{\text {SF}} {\boldsymbol 1}_{N \times 1}\otimes {\boldsymbol I}_{N_{t}}\) has full column rank, i.e., \(\text {rank}\left ({\Delta }^{\text {SF}} {\boldsymbol 1}_{N \times 1}\otimes {\boldsymbol I}_{N_{t}}\right) = N_{t}\). Hence, from (44), matrix φ has full column rank as well, i.e., rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*. Consequently, the upper bound on the matrix diversity in Theorem 1 is achieved.

We thus see that achievability of our upper bound on the maximum diversity of MIMO-OFDM system in Theorem 1 is independent of the fading channel correlation. By employing an appropriate STF code design, we can achieve the maximum diversity of our MIMO-OFDM systems. There are two design requirements: (1) In each OFDM block, the differences of two codewords should be independent from different transmit antennas and (2) in each subcarrier, the differences of two codewords are not all zeros. In addition, compared with a quasi-SF code satisfying the conditions of Proposition 2, to achieve our upper bound on the maximum diversity in Theorem 1, STF code schemes in general can be different across OFDM blocks.

### 4.1 STF code examples

We now give two STF code examples: one STF code and another quasi-SF code that meet the conditions of our Propositions 1 and 2, respectively, to prove the achievability of our upper bound on the maximum diversity of block-fading MIMO-OFDM system with arbitrary correlations.

For simplicity, we assume that there are two OFDM blocks with two subcarriers in a 2×2 MIMO system. The number of path between the transmitter and the receiver is only one to guarantee *N*≥*N*
_{
t
}
*L*. The modulation scheme is BPSK.

### 4.2 Example 1: a traditional STF code

with *C*
_{
k
}, *k*=1,2, given in (3). Because the structure of this STF code satisfies the conditions of Proposition 1, rank(*Δ*
_{
k
}
1
_{2×1}⊗I
_{2})=*N*
_{
t
}=2, *k*=1,2.

where \({\tilde x}_{i}\), 1≤*i*≤4, denotes the detected code. The φ matrix has full column rank, i.e., rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*=8. So using this STF code, the upper bound on maximum diversity in Theorem 1 can be achieved. In addition, the code rate in this case is 1 bits/s/Hz.

### 4.3 Example 2: a quasi-SF code

with *C*
^{SF} given in (49). This quasi-SF code satisfies the conditions of Proposition 2, i.e., rank(*Δ*
^{SF}
1
_{2×1}⊗I
_{2})=*N*
_{
t
}=2.

^{SF}in (49) can be written as

where \({\tilde x}_{1}\) and \({\tilde x}_{2}\) are the detected codes of *x*
_{1} and *x*
_{2}, respectively. The matrix φ has full column rank with rank(φ)=*N*
_{
r
}
*N*
_{
t
}
*KL*=8. Therefore, the upper bound on the maximum diversity of MIMO-OFDM system with general correlation in Theorem 1 can also be achieved by using this special quasi-SF code, whose rate is 1/2 bits/s/Hz. Compared to the STF code in Example 1, because this quasi-SF code is constructed by repeating an SF code over different OFDM blocks, it does not utilize the temporal resource, hence has lower code rate.

Note that this quasi-SF code was used to achieve an upper bound on the maximum diversity of MIMO-OFDM system with independent fading channels in the spatial domain in [25]. Here, we show that the same code achieves the maximum diversity for MIMO-OFDM systems with general spatial, temporal, and frequency/path correlations.

## 5 Decoding complexity

We know that if the ML decoder at the receiver chooses *K* complex information symbols *x*
_{
k
} and the modulation scheme is q-PSK, the decoding complexity cannot exceed *q*
^{
K
} metric computations.

With quasistatic and frequency-flat i.i.d fading channels, using orthogonal space-time block code (OSTBC), the decoding complexity is linear. However, on time-varying and frequency-selective channels, the STF block codes lose their reduced complexity decoding. We assume the fading channels are independent identically distributed with *K* OFDM blocks and *N* subcarriers. If we use STF block codes (such as example 1), the decoding complexity in each group is linear and the decoding complexity of these *KN* groups is *q*
^{
KL
}, in which *L* is the number of separated multipaths. If the fading channels are correlated, the decoding complexity will be further increased.

where *N*
_{
r
}×*N*
_{
t
} matrix \({\bar {\boldsymbol H}}\) denotes the fading channels, *N*
_{
t
}×1 vector \({\bar {\boldsymbol x}}\) denotes the transmitted signals, and *N*
_{
r
}×1 vector \({\bar {\boldsymbol y}}\) is the received signals.

A QR decomposition of the matrix \({\bar {\boldsymbol H}}\) can be obtained by applying the Gram-Schmidt procedure to the columns of \({\bar {\boldsymbol H}} =\, [{\bar {\boldsymbol h}}_{1}, \ldots, {\bar {\boldsymbol h}}_{N_{t}}]\) to obtain \({\bar {\boldsymbol H}} = {\bar {\boldsymbol Q}}{\bar {\boldsymbol R}}\), where the columns of \({\bar {\boldsymbol Q}} =\, [{\bar {\boldsymbol q}}_{1}, \ldots, {\bar {\boldsymbol q}}_{N_{t}}]\) are an orthonormal basis for the subspace spanned by \({\bar {\boldsymbol H}}\), and \(\bar {\boldsymbol R}\) is upper triangular with nonnegative real diagonal elements. In [26], using OSTBC, the decoding complexity is *q*
^{
M
}, where \(M = \frac {N_{R} + 2N_{C}}{4}\), *N*
_{
R
} is the number of off-diagonal elements of \(\bar {\boldsymbol R}\) which are equal to a real number but not zeros, and *N*
_{
C
} is the number of off-diagonal elements of \(\bar {\boldsymbol R}\) which are equal to a complex number but not zeros.

*QR*, i.e.,

Using STF block codes, if the *L* multipaths are independent, the decoding complexity is \(q^{\sum \limits _{i = 1}^{L} M_{i}}\), where \(M_{i} = \frac {N_{R}(i) + 2N_{C}(i)}{4}\), *N*
_{
R
}(*i*) is the number of off-diagonal elements of \(\bar {\boldsymbol R}_{\textit {ii}}\) which are equal to a real number but not zeros, and *N*
_{
C
}(*i*) is the number of off-diagonal elements of \(\bar {\boldsymbol R}_{\textit {ii}}\) which are equal to a complex number but not zeros. Otherwise, if there exists the path correlation, the decoding complexity is \(q^{\sum \limits _{i,j = 1}^{i,j = L}{M_{\textit {ij}}}}\), where \(q^{M_{\textit {ij}}}\phantom {\dot {i}\!}\) is the complexity through the fading channels corresponding to one sub-matrix \({{{\bar {\boldsymbol H}}_{\textit {ij}}}}\).

Further, in the correlated frequency-selective fading channels with the *K* independent OFDM blocks, the fading channels and the transmitted codes are divided into these *K* groups and in each groups using STF block codes, the decoding complexity is \(q^{\sum \limits _{i,j = 1}^{i,j = L}{M_{\textit {ij}}}}\). Because the codes of different groups are not orthometric, the decoding complexity of the *K* independent OFDM blocks should be \(q^{K\sum \limits _{i,j = 1}^{i,j = L}{M_{\textit {ij}}}}\). If there exists temporal correlation of the fading channels, the complexity will be increased and the calculation is similar to the one of correlated paths.

## 6 Special cases

We simplify our general diversity result for some special correlation scenarios of practical interests. In the process, we identify the newly developed statistical channel models for MIMO LTE [20] and 802.11n [21] as special cases by showing that our theoretical diversity results match those simulated from these statistical channel models; we also recover several existing results in the literature. Note that in this section, the bit error rate (BER) performance of uncoded Raleigh channel with the diversity is depicted by the function *berfading* in Matlab. Expanding the Alamouti code to a standard STF code by tensor product, we use these STF codes in the simulation as the one in example 1 last section.

### 6.1 Case 1: STF-coded MIMO-OFDM system over temporal, frequency, and separable spatial correlation channels

*l*th path, the spatial correlation can be expressed as

From Fig. 1, the BER performance of 2×2 MIMO system with two blocks and three paths are better than the ones of 2×2 MIMO system, 2×2 MIMO system with three paths, and 2×2 MIMO system with two blocks. It illustrates that the effect of reducing one dimension on the BER performance is higher than the effect of correlation of one dimension. Furthermore, compared to the system with two separated blocks or with three separated paths, the diversities of the system with two correlated blocks or with three correlated paths are decreased, respectively. That explains the effect of temporal and multipath correlations on the performance. In addition, the BER performance of 2×2 MIMO system with two blocks, three paths, and separated multipath and temporal correlations is better than the one with the completely correlated fading channels. That illustrates that the correlation between two dimensions could lead to reduce the performance.

Both covariance matrices \({\boldsymbol R}^{\boldsymbol {N_{t}}}_{l,l}\) and \({\boldsymbol R}^{\boldsymbol {N_{r}}}_{l,l}\) have rank two. From (65), the theoretical achievable maximum diversity in this case is 36. It is seen from Fig. 2 that when compared to the BER performance of uncoded Rayleigh fading channels with diversity order 36, the slope of the curve depicting the performance of the 2×2 LTE system with nine separable paths is the same (when SNR is larger than 8, the BER of the LTE system with nine separated multipaths is too small to obtain by Matlab). Thus, the diversity of the 2×2 LTE system is also 36, which matches the theoretical result.

Take 802.11n channel model B for example [21]. It is a 2×2 MIMO system with nine Rayleigh-fading paths which have a bell Doppler spectrum.

Further, we let these nine multipaths be separable and the transmitted correlation \({\boldsymbol R}^{\boldsymbol N_{t}}_{1,1}\) and the received correlation \({\boldsymbol R}^{\boldsymbol N_{r}}_{1,1}\) be the same with the standard 802.11n channel model B.

From Fig. 3, the BER performance is much better than the one of 802.11n channel model B. The reason is that under the conditions of separable multipaths, the achievable diversity is linearly increased by the number of multipaths. That is consistent with the theoretical result.

### 6.2 Case 2: STF-coded MIMO-OFDM system over frequency and temporal correlation channels

which agrees with [25].

### 6.3 Case 3: SF-coded MIMO-OFDM system over only spatial correlation channels

^{ K })=1 into the result of case 1 and have

which is consistent with the result in [16]. It indicates that the diversity order of the system is equal to the number of degrees of freedom offered by independent scatterers.

### 6.4 Case 4: SF-coded MIMO-OFDM system over only frequency correlation channels

*K*= 1) and the fading channels in the spatial domain are independent. Hence,

which agrees with [28].

### 6.5 Case 5: ST-coded MIMO system over only temporal correlation channels

Because of only one path during the transmitter and the receiver, the covariance matrix *E*{A
A
^{H}} in (27) degrades to a tensor product of a temporal channel covariance matrix R
^{
K
} and a spatial channel covariance matrix \({{{\boldsymbol R}_{1,1}^{\boldsymbol S} }}\).

which was given in [7].

### 6.6 Case 6: ST-coded MIMO system over spatial correlation channels with Kronecker model

^{ K }and R

^{ L }degenerate, leading to

which is the result obtained in [27].

### 6.7 Case 7: independent fading channels

which is consistent with the result in [29].

## 7 Conclusions

In this paper, we have studied the performance of STF-coded MIMO-OFDM system with arbitrary spatial, temporal, and frequency/path correlations. Our analysis is based on a general transmitted correlation model that goes beyond limitations of ideal assumptions such as quasi-static or rapid fading channels, channel independence in different antennas, and separable multipaths between the transmitter and the receiver. Our channel spatial correlation covers both the Kronecker and non-Kronecker models. We derive an upper bound on the maximum achievable diversity of this system using Hadamard and tensor products. Based on rank properties of block matrices, we also analyze the effect of the general channel correlation on the performance of block-fading MIMO-OFDM systems. Furthermore, achievability of our upper bound is proved via two code design examples: one traditional STF code and another quasi-SF code. The decoding complexity is considered in the MIMO system with arbitrary correlated fading channels using the traditional STF code. By identifying the newly developed statistical channel models for MIMO LTE and 802.11n as special cases of our STF-coded MIMO-OFDM system, we can directly use our theoretical diversity results without resorting to simulations. Finally, our theoretical result for general correlation scenarios subsumes those in the existing literature that only deal with different special cases.

## 8 Appendix A: proof of Lemma 1

###
*Proof*.

_{1},B

_{2},…,B

_{ P }] and \(\left [ {{\boldsymbol D}_{1}^{\mathrm {T}}}, {{\boldsymbol D}_{2}^{\mathrm {T}}}, \ldots, {{\boldsymbol D}_{Q}^{\mathrm {T}}} \right ]^{\mathrm {T}}\) can be calculated, respectively, as

*p*,

*p*

^{′}≤

*P*,

*p*≠

*p*

^{′}, and \(\text {range}\left ({\boldsymbol D}_{q}^{\mathrm {T}}\right)\cap \text {range}\left ({\boldsymbol D}_{q'}^{\mathrm {T}}\right) = \{0\}\), 1≤

*q*,

*q*

^{′}≤

*Q*,

*q*≠

*q*

^{′}, then

hence (34) holds.

_{ p })⊆range(B

_{ P }) and \(\text {range}\left ({\boldsymbol D}_{q}^{\mathrm {T}}\right) \subseteq \text {range}\left ({\boldsymbol D}_{Q}^{\mathrm {T}}\right) \), 1≤

*p*≤

*P*−1, 1≤

*q*≤

*Q*−1, we have

then (35) follows from (83) according to the rank properties of block matrices [31].

## Notes

## Declarations

### Acknowledgments

This work was supported by the National Science Foundation for Innovative Research Groups of China (61521091).

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- JC Guey, MP Fitz, MR Bell, WY Kuo, Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels. IEEE Trans. Commun.
**47:**, 527–537 (1999).View ArticleGoogle Scholar - V Tarokh, N Seshadri, AR Calderbank, Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans. Inform. Theory.
**44:**, 744–765 (1998).View ArticleMathSciNetMATHGoogle Scholar - V Tarokh, A Naguib, N Seshadri, AR Calderbank, Space-time codes for high data rate wireless communications: performance criteria in the presence of channel estimation errors, mobility, and multiple paths. IEEE Trans. Commun.
**47:**, 199–207 (1999).View ArticleMATHGoogle Scholar - JW Craig, in
*Proc. IEEE Military Communications Conference, 1991. MILCOM ’91, Conference Record, Military Communications in a Changing World*. A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations (IEEEMcLean, VA, USA, 1991), pp. 571–575.Google Scholar - V Veeravalli, On performance analysis for signaling on correlated fading channels. IEEE Trans. Commun.
**49:**, 1879–1883 (2001).View ArticleMATHGoogle Scholar - M Damen, A Abdi, M Kaven, in
*Proc. IEEE VTC Fall 2001*, 1. On the effect of correlated fading on several space-time coding and detection schemes (IEEEAtlantic City, New Jersey, USA, 2001), pp. 13–16.Google Scholar - W Su, Z Safar, KJR Liu, Diversity analysis of space-time modulation over time-correlated Rayleigh fading channels. IEEE Trans. Inform. Theory.
**50:**, 1832–1840 (2004).View ArticleMathSciNetMATHGoogle Scholar - H Bolcskei, M Borgmann, A Paulraj, Impact of the propagation environment on the performance of space-frequency coded MIMO-OFDM. IEEE J. Selected Areas Commun.
**21:**, 427–439 (2003).View ArticleGoogle Scholar - F Riera-Palou, G Femenias, A unified view of diversity in multiantenna-multicarrier systems: analysis and adaptation strategies. EURASIP. J. Wireless Commun. Netw.
**1:**, 1–14 (2012).View ArticleGoogle Scholar - T Bao, Y Liang, in
*Proc. IEEE International Conference on Signal Processing Communication and Computing (ICSPCC) 2012*. Improved space-time-frequency block code for MIMO-OFDM wireless communications (IEEEHong Kong, China, 2012), pp. 538–541.View ArticleGoogle Scholar - G Owojaiye, F Delestre, Y Sun, Differential distributed quasi-orthogonal space-time-frequency coding. IEEE Wireless Advanced (WiAd), 115–120 (2012).Google Scholar
- M Shahabinejad, S Talebi, Full-diversity space-time-frequency coding with very low complexity for the ML decoder. IEEE Commun. Lett.
**16**(5), 658–661 (2012).View ArticleGoogle Scholar - C Huang, Y Guo, MH Lee, in
*Proc. IEEE International Conference on Computer Science and Electronics Engineering (ICCSEE), 2012*, 1. High-rate full-diversity space-time-frequency codes with partial interference cancellation decoding (IEEEHangzhou, Zhejiang, China, 2012), pp. 235–240.View ArticleGoogle Scholar - H Ozcelik, M Herdin, W Weichselberger, J Wallace, E Bonek, Deficiencies of ‘Kronecker’ MIMO radio channel model. Electron Lett.
**39:**, 1209–1210 (2003).View ArticleGoogle Scholar - N Costa, H Simon, Multiple-Input Multiple-Output Channel Models: Theory and Practice, vol. 65 (John Wiley & Sons, 2010).Google Scholar
- AK Sadek, W Su, KJ Ray Liu, in
*Proc. IEEE Global Telecommunications Conference (GLOBECOM) 2004*, 4. Maximum achievable diversity for MIMO-OFDM systems with arbitrary spatial correlation (IEEEDallas, TX, USA, 2004), pp. 2664–2668.View ArticleGoogle Scholar - A Sadek, W Su, KJR Liu, Diversity analysis for frequency-selective MIMO-OFDM system with general spatial and temporal correlation model. IEEE Trans. Commun.
**54:**, 878–888 (2006).View ArticleGoogle Scholar - LM Davis, IB Collings, RJ Evans, in
*Proc. IEEE Workshop on Statistical Signal and Array Processing (SSAP)*. Maximum likelihood delay-Doppler imaging of fading mobile communication channels (IEEEPocono Manor, PA, USA, 2000), pp. 151–155.Google Scholar - M Liao, Y Zhang, Z Xiong, in
*Proc. IEEE WCNC 2013*. Diversity analysis for space-time-frequency (STF) coded MIMO system with a general correlation model (IEEEShanghai, China, 2013), pp. 2661–2666.Google Scholar - 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception (Release 10). 3GPP TS 36 (2010): V10.Google Scholar
- V Erceg, Z Wireless, et al, IEEE P802.11 Wireless LANs TGn Channel Models, doc.: IEEE 802.11-03/940r4, 1–45 (2004).Google Scholar
- D Gesbert, H Bolcskei, D Gore, A Paulraj, Outdoor MIMO wireless channels: models and performance prediction. IEEE Trans. Commun.
**50:**, 1926–1934 (2002).View ArticleGoogle Scholar - A Mathai, S Provost,
*Quadratic Forms in Random Variables*(Marcel Dekker, New York, 1992).MATHGoogle Scholar - SM Alamouti, A simple transmit diversity technique for wireless communications. IEEE J. Select Areas Commun.
**16:**, 1451–1458 (1998).View ArticleGoogle Scholar - W Su, Z Safar, KJR Liu, in
*Proc. 5th Eur. Wireless Conf*. Diversity analysis of space-time-frequency coded broadband OFDM systems (IEEEBarcelona, Spain, 2004). Vol. 2, No. 2, pp. 1–5.Google Scholar - MO Sinnokrot,
*Space-time block codes with low maximum-likelihood decoding complexity*(Georgia Institute of Technology, Doctoral dissertation, 2009).Google Scholar - H Bolcskei, A Paulraj, in
*Proc. IEEE 34th Asilomar Conf. Signals, System and Computers*, 1. Performance of space-time codes in the presence of spatial fading correlation (IEEECA, Pacific Grove, 2000), pp. 687–693.Google Scholar - W Su, Z Safar, M Olfat, KJ Ray Liu, Obtaining full-diversity space-frequency codes from space-time codes via mapping. IEEE Trans. Signal Proc.
**51:**, 2905–2916 (2003).View ArticleGoogle Scholar - W Zhang, X Xia, P Ching, High-rate full-diversity space-time frequency codes for broadband MIMO block fading channels. IEEE Trans. Commun.
**55:**, 25–34 (2007).View ArticleGoogle Scholar - CD Meyer,
*Matrix Analysis and Applied Linear Algebra (Society for Industrial and Applied Mathematics Philadelphia*(PA, USA, 2000).View ArticleGoogle Scholar - G Marsaglia, GPH Styan. Equalities and inequalities for ranks of matrices.Linear Multilinear Algebra. 2:, 269–292 (1974).Google Scholar