Impact of Cooperative Space-Time/Frequency Diversity in OFDM Based Wireless Sensor Systems over Mobile Multipath Channels

Abstract

Cooperative linear dispersion coding (LDC) can support arbitrary configurations of source nodes and destination nodes in virtual multi-input multi-output (MIMO) systems. In this paper, we investigate two spatial diversity applications of cooperative LDC for orthogonal frequency division multiplexing (OFDM) based wireless sensor systems in order to achieve space-time/frequency (ST/SF) diversity gains when transmitting over time-/frequency-selective fading channels. Cooperative LDC-aided ST/SF-OFDM is flexible in configuring various numbers of cooperative source nodes and time-slots or frequency-tones. Our results show that the ST-OFDM scheme is sensitive to exploiting diversity gains, subject to the impact of varying channel Doppler spreads; while the performance of SF-OFDM is mainly subject to delay spread. Specifically, when the system involves more than two cooperative nodes, the cooperative LDC-aided ST/SF-OFDM outperforms the cooperative orthogonal block codes (e.g. Tarokh’s codes) aided ST/SF-OFDM, when communicating over a higher Doppler/delay spread.

I. Introduction

Wireless sensor networks have become one of the most popular new technologies for assisting the enhancement of both personal and industrial applications. Multiple sensor devices which employ wireless transceivers may be distributed in the form of ad hoc networks in wireless communications for smart grid, public safety applications, healthcare, etc. [1]–[5].

Multi-input multi-output (MIMO) [6] is a most attractive multi-antenna technique that has been adopted by many emerging wireless communication standards, such as IEEE 802.11n and 3GPP LTE, owing to the achievable antenna array, multiplexing, and diversity gain. In order to improve link reliability, the diversity gain enabled at transmission can be exploited by space-time coding, while the diversity gain achieved at the receiver may benefit from maximum ratio combining (MRC) [7]–[10]. These gains are obtained without increasing the transmission power by employing multiple transmit and/or receive antennas. Particularly, Hassibi’s linear dispersion codes (LDC) [9], [11], [12] allow arbitrary configurations in space-time coding for high-rate MIMO transmissions.

However, due to constraints of size and/or cost, it is impractical to implement the mobile device with multiple antennas which are sufficiently far away to achieve independent fading and generate spatial diversity gain. Therefore, distributed single-antenna terminals, which are properly configured in a virtual MIMO transmission, may enable the so-called cooperative communication [13] in order to obtain the equivalent diversity gain by sharing resources with their cooperative partners. Hence the transceiver schemes that are used in co-located MIMO systems can be used in cooperative/distributed wireless sensor systems as well [4].

Broadband communication plays an increasingly important role in meeting the growing demand for high-speed multi-media transmissions in our daily lives. However, when the bandwidth of a signal exceeds the coherent bandwidth of the wireless channel, the small-scale fading imposed on the signal becomes frequency-selective rather than frequency-flat. Such a fading time-dispersion incurs inter-symbol interference (ISI) to the air-interface and degrades the link performance [14]. Orthogonal frequency division multiplexing (OFDM) [15] is one of the transceiver techniques designed to combat ISI. When the number of subcarriers in OFDM is sufficiently larger than the number of taps, the frequency-selective channel can be decomposed into mutually independent frequency-flat fading channels on each subcarrier with the aid of OFDM transmission. Moreover, a center length of cyclic prefix (CP) or zero-padding (ZP) should be inserted between any two adjacent OFDM blocks to mitigate inter-block interference (IBI) incurred by multi-path fading [16]. As a result, each received signal can be recovered at the low-complexity single-tap equalizer without inter-carrier interference (ICI), thanks to the orthogonality between adjacent subcarriers with flat fading.

For transmit diversity aided MIMO systems, a simple time-reversed space-time block coding scheme [17] was proposed in the context of a broadband MIMO channel to combat ISI. Such a large time-reversal frame requires slow channel varying, which is not suitable for mobile wireless communications associated with long delay spreads. With the aid of a multi-antenna employed at the transmitter, many transmit diversity schemes have been invented to combat the frequency-selective fading incurred by the high-speed data rate and also achieve diversity gain at the same time [18]–[20]. Specifically, space-time block coding (STBC) schemes that were used in frequency-flat fading channels may be applied to each subcarrier to achieve space-time diversity and combat channel time-dispersion [21]. Rather than exploiting the achievable diversity by crossing spatial antennas and time-slots, the alternative approach may benefit from OFDM’s multi-carrier feature relying on so-called space-frequency block coding (SFBC) schemes by exploiting the diversity crossing transmit antennas and subcarriers [22].

In the context of wireless sensor networks, space-time codes and space-frequency codes have been investigated in [23]–[25], while only space-frequency codes are studied with OFDM systems. A further combined version of the above two schemes is known as space-time-frequency block coding, which can exploit diversity in all three domains [26]–[29]. Furthermore, the authors in [29] and [30] propose various LDC-aided OFDMs to achieve space-frequency diversity for the constant fading channel within a single OFDM block; while in [31] the LDC is designed to obtain diversity from the time and frequency domain rather than the spatial domain. In [32]–[34], cooperative/distributed space-time, space-frequency, and space-time-frequency coded OFDM systems were proposed, respectively. However, to the best of our knowledge, there has not been a comprehensive investigation assessing distributed or cooperative LDC for an OFDM system for wireless sensor networks in terms of space-time (ST) and space-frequency (SF) diversity gain impacts with varying channel coherent times and bandwidths when communicating over mobile multipath channels.

In this paper, we investigate transmit diversity for OFDM system that invokes cooperative LDC to take into account the trade-off between ST and SF diversities in the presence of Doppler and delay spreads, respectively. For the sake of simplicity, we assume the channels among cooperative nodes are perfect, hence the distributed MIMO are considered as the equivalent model of co-located ones. Our contributions are highlighted as follows:

• We unify the analyzing structure of cooperative diversity aided block codes in order to compare cooperative LDC with the corresponding Alamouti’s code and Tarokh’s code [7], [8] in diverse MIMO configurations.

• The cooperative LDC, Alamouti’s and Tarokh’s codes are applied to OFDM in both the ST and SF approaches.

• An LDC-aided ST-/SF OFDM system is capable of supporting arbitrary configurations of transmit nodes and receive antenna for cooperative wireless sensor networks, when combined with arbitrary modulation schemes.

• Our results show that when the channel is constant within the coherent time/bandwidth, the cooperative ST-OFDM or SF-OFDM is capable of achieving full diversity gain in space-time or space-frequency domains, respectively.

• We quantify the performance impact with varying Doppler spreads. Results show that the ST-OFDM scheme is sensitive to exploit diversity gains subject to the effect upon varying channel Doppler spreads.

• In parallel, we examine the performance impacts owing to varying numbers of paths in terms of delay spread. As a result, the performance of cooperative SF-OFDM is mainly subject to delay spreads.

• Compared to fixed orthogonal block codes, the cooperative LDC-aided ST/SF-OFDM is flexible to configure various numbers of transmit antennas and time-slots or frequency-tones.

• When the cooperative cluster employs more than two nodes, the performance of cooperative LDC-aided OFDM schemes is less impacted by channel Doppler/delay spreads, as compared with orthogonal block codes.

The rest of this paper is structured as follows. We firstly elaborate on the transceiver system model of cooperative MIMO-OFDM in Section II. The ST- and SF-oriented OFDM schemes that achieve cooperative diversity will be studied in Section III in both the Alamouti’s g2, Tarokh’s g4 and a LDC cases. We will present the simulation results in Section IV, followed by closing remarks in Section V.

II. Multi-Source Cooperative System Model

A. Multi-Source Information Exchange

As shown in Fig. 1, the multi-source cooperative sensor cluster includes $K$ source nodes, each of which employs single-antenna aided OFDM transmitters of Fig. 2. Specifically, the $N_{\mathrm{b}}/K$-length binary source data bit stream ${\mathit{b}}_k={\left[{\mathit{b}}_0^T,{\mathit{b}}_1^T,\cdots ,{\mathit{b}}_{N_{\mathrm{s}}/K-1}^T\right]}^T$ for $k=\mathrm{0,1},\cdots ,K$ is fed into the $ℳ$-ary Gray labeled phase-shift keying (PSK) mapper transmitting $𝒬$ bits per symbol, where we have $N_{\mathrm{s}}=N_{\mathrm{b}}/𝒬$ and $ℳ=2^{𝒬}$.

Fig. 1.

Multi-source cooperation aided wireless sensor systems

Fig. 2.

Transmitter block diagram for cooperative ST/SF-OFDM based wireless sensor node

Moreover, the modulated symbols $s_k=$ ${\left[s_0,s_1,\cdots ,s_{N_{\mathrm{s}}/K-1}\right]}^T$ are exchanged among $K$ cooperative nodes in terms of multi-way relaying in [35] and [36] over the idea channel. Specifically, as shown in Fig. 1, at phase 1 the $(K-1)$ nodes transmit their data to an appropriately selected node that acts as a cluster-head, which then broadcasts all the data back to the $(K-1)$ nodes at phase 2. All the exchanged symbols are perfectly detected (error and interference free) and reformed to an $N_{\mathrm{s}}$-length symbol sequence $\mathit{s}={\left[{\mathit{s}}_{\mathrm{0,0}}^T,{\mathit{s}}_{\mathrm{1,0}}^T,\cdots ,{\mathit{s}}_{K-\mathrm{1,0}}^T,{\mathit{s}}_{\mathrm{0,1}}^T,{\mathit{s}}_{\mathrm{1,1}}^T,\cdots ,{\mathit{s}}_{K-\mathrm{1,1}}^T,\cdots \right]}^T$ at each node. At phase 3, the sequence $\mathit{s}$ is further mapped into $K$ nodes and $N_{\mathrm{T}}$ time-slots or $N_{\mathrm{C}}$ subcarriers in terms of ST/SF coding employed at each node, respectively. We will it further in detail in Section III.

B. Transmitted Signal at the Cooperative Source Nodes

At the $k$-th sensor node as shown in Fig. 2, the $U$-symbol block ${\mathrm{x}}_k={\left[{\mathrm{x}}_{k,0},{\mathrm{x}}_{k,1},\cdots ,{\mathrm{x}}_{k,(U-1)}\right]}^T$ to be transmitted is then converted from serial-to-parallel (S/P) corresponding to $U$ orthogonal subcarriers in the F-domain. These $U$-symbols in ${\mathbf{x}}_k$ are transformed by $U$-point inverse discrete Fourier transform (IDFT) operation matrix ${𝓕}_U^H$ [37] into T-domain at the $t$-th time-slot for $t=\mathrm{0,\; 1},\cdots ,T-1$, expressed by

$$\begin{gathered} {\mathit{x}}_k\left[t\right]={𝓕}_U^H{\mathbf{x}}_k\left[t\right] \\ ={\left[x_{k,0}\left[t\right],x_{k,1}\left[t\right],\cdots ,x_{k,(U-1)}\left[t\right]\right]}^T, \end{gathered}$$ (1)

The CP is inserted at the beginning of ${\mathit{x}}_k\left[t\right]$ by copying the last $L_{\mathrm{C}\mathrm{P}}$ elements of the ${\mathit{x}}_k\left[t\right]$, which results in the $\left(U+L_{\mathrm{C}\mathrm{P}}\right)$-element transmitted OFDM symbol block ${\tilde{\mathit{x}}}_k\left[t\right]$ at the $t$-th time-slot via the $k$-th cooperative node.

C. Signal Representation at the BS Receiver

The multi-antenna aided OFDM receiver is shown in Fig. 3. By satisfying the channel order $L<L_{\mathrm{C}\mathrm{P}}$, after removing the CP at the receiver, the equivalent $U$-element T-domain signal block received at the $t$-th time-slot may be expressed as:

$${\mathit{y}}_{n_{\mathrm{R}\mathrm{x}}}\left[t\right]=\sum _{k=0}^{K-1}{\mathit{H}}_{n_{\mathrm{R}\mathrm{x}},k}\left[t\right]{\mathit{x}}_k\left[t\right]+{\mathit{n}}_{n_{\mathrm{R}\mathrm{x}}}\left[t\right],$$ (2)

where ${\mathit{H}}_{n_{\mathrm{R}\mathrm{x}},k}$ denotes the $U\times U$-element T-domain circulant matrix [37] holding the channel impulse response (CIR) between source node $k$ and receive antenna $n_{\mathrm{R}\mathrm{x}}$. In Eq. (2), ${\mathit{n}}_{n_{\mathrm{R}\mathrm{x}}}$ is the noise imposed at the $n_{\mathrm{R}\mathrm{x}}$-th receiver antenna, each element of which has a power of ${𝒩}_u={\sigma }_{\mathrm{N}}^2$.

Fig. 3.

Receiver block diagram for transmit diversity aided OFDM

Hence, after the $U$-point discrete Fourier transform (DFT) transforming the signal ${\mathit{y}}_{n_{\mathrm{R}\mathrm{x}}}$ into the F-domain, we have the equivalent symbol block given by

$${\mathbf{y}}_{n_{\mathrm{R}\mathrm{x}}}\left[t\right]={𝓕}_U{\mathit{y}}_{n_{\mathrm{R}\mathrm{x}}}\left[t\right]=\sum _{k=0}^{K-1}{\mathbf{H}}_{n_{\mathrm{R}\mathrm{x}},k}\left[t\right]{\mathbf{x}}_k\left[t\right]+{\mathbf{n}}_{n_{\mathrm{R}\mathrm{x}}}\left[t\right].$$ (3)

Since we have ${\mathit{H}}_{n_{\mathrm{R}\mathrm{x}},k}={𝓕}_U^H{\mathbf{H}}_{n_{\mathrm{R}\mathrm{x}},k}{𝓕}_U$ according to [37], the ${\mathbf{H}}_{n_{\mathrm{R}\mathrm{x}},k}$ in Eq. (3) is a diagonal matrix with entries ${\mathrm{h}}_{u,u}(u=0,1,\cdots ,U-1)$ representing the corresponding F-domain channel transfer function on $U$ subcarriers, leading to a low-complexity one-tap channel equalization method. In Eq. (3), we have ${\mathbf{n}}_{n_{\mathrm{R}\mathrm{x}}}={𝓕}_U{\mathit{n}}_{n_{\mathrm{R}\mathrm{x}}}$ with ${𝒩}_u={\sigma }_{\mathrm{N}}^2$.

Furthermore, we reshape Eq. (3) into an $N_{\mathrm{R}\mathrm{x}}$-length multi-antenna received symbol vector for the $u$-th subcarrier at time-slot $t$ expressed by

$${\check{\mathbf{y}}}_u\left[t\right]={\check{\mathbf{H}}}_u\left[t\right]{\check{\mathbf{x}}}_u\left[t\right]+{\check{\mathbf{n}}}_u\left[t\right],$$ (4)

where ${\check{\mathbf{H}}}_u\left[t\right]$ is a $\left(N_{\mathrm{R}\mathrm{x}}\times K\right)$-size MIMO-channel matrix at time-slot $t$, in which the $\left(n_{\mathrm{R}\mathrm{x}},k\right)$-th entry denotes the F-domain coefficients of ${\mathbf{H}}_{n_{\mathrm{R}\mathrm{x}},k}$ on the $u$-th subcarrier; ${\check{\mathbf{x}}}_u\left[t\right]={\left[{\mathrm{x}}_{0,u}\left[t\right],{\mathrm{x}}_{1,u}\left[t\right],\cdots ,{\mathrm{x}}_{(K-1),u}\left[t\right]\right]}^T$ is $K$-antenna transmitted symbol vector in the $F$-domain before IDFT at time-slot $t$. Additionally, ${\check{\mathbf{n}}}_u\left[t\right]={\left[{\mathrm{n}}_{0,u}\left[t\right],{\mathrm{n}}_{1,u}\left[t\right],\cdots ,{\mathbf{n}}_{(K-1),u}\left[t\right]\right]}^T$ is $N_{\mathrm{R}\mathrm{x}}$-antenna noise component added at receiver.

III. Cooperative MIMO-OFDM Schemes

In this section, we elaborate two cooperative diversity aided OFDM schemes, namely cooperative ST coded OFDM and cooperative SF coded OFDM, respectively.

A. Cooperative Space-Time Coded OFDM

In order to achieve the space- and time-diversity, the OFDM may be ST-encoded in a subcarrier-by-subcarrier basis as shown in Fig. 4. Specifically, a $N_{\mathrm{s}}$-length symbol frame $s$ is divided into $U$ segments, and the $u$-th segment for $u=\mathrm{0,1},\cdots ,U-1$ contains $Q$ symbols in ${\mathit{s}}_u$ for the input of ST encoder. The encoder employs specific ST algorithms procuding the ouput frame for the $k$-th antenna (node¹) having $T>1$ consecutive OFDM blocks ${\mathrm{x}}_k\left[t\right]$ over time-slots $t=\mathrm{0,1},\cdots ,T-1$ in F-domain. For instance, the Alamouti’s g2 ST [7] encoded OFDM blocks for $K=2,Q=2$ and $T=2$ may be expressed as

$$\left\{\begin{array}{l}{\mathrm{x}}_{k=0}[t=0]={\left[s_0,s_2,\cdots ,s_{2U-2}\right]}^T,\\ {\mathrm{x}}_{k=1}[t=0]={\left[s_1,s_3,\cdots ,s_{2U-1}\right]}^T,\\ {\mathrm{x}}_{k=0}\left[t=1\right]=-{\mathrm{x}}_1^{*}\left[0\right],\\ {\mathrm{x}}_{k=1}\left[t=1\right]={\mathrm{x}}_0^{*}\left[0\right],\end{array}\right.$$ (5)

where the $u$-th element in symbol vector ${\mathbf{x}}_k\left[t\right]$ is conveyed onto $u$-th subcarriers at time-slot $t$ and emitted via antenna $k$. Alternatively, when the Tarokh’s g4 ST [8] code is invoked in OFDM for $K=4,Q=4$ and $T=8$, we have:

$$\begin{gathered} \left\{\begin{array}{l}{\mathrm{x}}_0\left[0\right]=[s_0,s_4,\cdots ,s_{4U-4}{]}^T,\\ {\mathrm{x}}_1\left[0\right]=[s_1,s_5,\cdots ,s_{4U-3}{]}^T,\\ {\mathrm{x}}_2\left[0\right]=[s_2,s_6,\cdots ,s_{4U-2}{]}^T,\\ {\mathrm{x}}_3\left[0\right]=[s_3,s_7,\cdots ,s_{4U-1}{]}^T,\end{array}\hspace{1em}\left\{\begin{array}{l}{\mathrm{x}}_0\left[1\right]=-{\mathrm{x}}_1\left[0\right],\\ {\mathrm{x}}_1\left[1\right]=-{\mathrm{x}}_0\left[0\right],\\ {\mathrm{x}}_2\left[1\right]=-{\mathrm{x}}_3\left[0\right],\\ {\mathrm{x}}_3\left[1\right]={\mathrm{x}}_2\left[0\right],\end{array}\right.\right. \\ \left\{\begin{array}{l}{\mathrm{x}}_0\left[2\right]=-{\mathrm{x}}_2\left[0\right],\\ {\mathrm{x}}_1\left[2\right]={\mathrm{x}}_3\left[0\right],\\ {\mathrm{x}}_2\left[2\right]={\mathrm{x}}_0\left[0\right],\\ {\mathrm{x}}_3\left[2\right]=-{\mathrm{x}}_1\left[0\right],\end{array}\hspace{1em}\left\{\begin{array}{l}{\mathrm{x}}_0\left[3\right]=-{\mathrm{x}}_3\left[0\right],\\ {\mathrm{x}}_1\left[3\right]=-{\mathrm{x}}_2\left[0\right],\\ {\mathrm{x}}_2\left[3\right]={\mathrm{x}}_1\left[0\right],\\ {\mathrm{x}}_3\left[3\right]={\mathrm{x}}_0\left[0\right],\end{array}\hspace{1em}\left\{\begin{array}{l}{\mathrm{x}}_0\left[4\right]={\mathrm{x}}_0^{*}\left[0\right],\\ {\mathrm{x}}_1\left[4\right]={\mathrm{x}}_1^{*}\left[0\right],\\ {\mathrm{x}}_2\left[4\right]={\mathrm{x}}_2^{*}\left[0\right],\\ {\mathrm{x}}_3\left[4\right]={\mathrm{x}}_3^{*}\left[0\right],\end{array}\right.\right.\right. \\ \left\{\begin{array}{l}{\mathrm{x}}_0\left[5\right]={\mathrm{x}}_0^{*}\left[1\right],\\ {\mathrm{x}}_1\left[5\right]={\mathrm{x}}_1^{*}\left[1\right],\\ {\mathrm{x}}_2\left[5\right]={\mathrm{x}}_2^{*}\left[1\right],\\ {\mathrm{x}}_3\left[5\right]={\mathrm{x}}_3^{*}\left[1\right],\end{array}\hspace{1em}\left\{\begin{array}{l}{\mathrm{x}}_0\left[6\right]={\mathrm{x}}_0^{*}\left[2\right],\\ {\mathrm{x}}_1\left[6\right]={\mathrm{x}}_1^{*}\left[2\right],\\ {\mathrm{x}}_2\left[6\right]={\mathrm{x}}_2^{*}\left[2\right],\\ {\mathrm{x}}_3\left[6\right]={\mathrm{x}}_3^{*}\left[2\right],\end{array}\hspace{1em}\left\{\begin{array}{l}{\mathrm{x}}_0\left[7\right]={\mathrm{x}}_0^{*}\left[3\right],\\ {\mathrm{x}}_1\left[7\right]={\mathrm{x}}_1^{*}\left[3\right],\\ {\mathrm{x}}_2\left[7\right]={\mathrm{x}}_2^{*}\left[3\right],\\ {\mathrm{x}}_3\left[7\right]={\mathrm{x}}_3^{*}\left[3\right].\end{array}\right.\right.\right. \end{gathered}$$ (6)

Fig. 4.

Schematic diagram of space-time coded OFDM

Furthermore, the cooperative LDC encoded OFDM block for the $k$-th antenna on the $u$-th subcarrier at time-slot $t=\mathrm{0,\; 1},\cdots ,T-1$ before IFFT operation is given by

$${\mathrm{x}}_k\left[t\right]={\left[{\left[{𝓑}_k{\mathit{s}}_0\right]}_t,{\left[{𝓑}_k{\mathit{s}}_1\right]}_t,\cdots ,{\left[{𝓑}_k{\mathit{s}}_{U-1}\right]}_t\right]}^T,$$ (7)

where ${𝓑}_k$ is the linear dispersion matrix defined in [11] of $k$-th antenna and ${\mathit{s}}_u={\left[s_{uQ},s_{uQ+1},\cdots ,s_{(u+1)Q-1}\right]}^T$ is the $u$-th input symbol segment having a legnth of $Q$ for $u=0,1,\cdots ,U-1$. Then, by using Eq. (1), the ST-OFDM symbols may be transmitted.

The detection of ST-OFDM receiver is also operated by subcarrier-by-subcarrier basis. After the signal transformed into F-domain by Eq. (4), we consider on the symbol blocks over all $T$ time-slots, having an equivalent F-domain signal expression as

$${\underset{\_}{y}}_u={\underset{\_}{H}}_u{\underset{\_}{x}}_u+{\underset{\_}{n}}_u,$$ (8)

where each component vector ${\underset{\_}{\mathbf{y}}}_u,{\underset{\_}{\mathbf{x}}}_u$ and ${\underset{\_}{\mathbf{n}}}_u$ may be expressed by ${\underset{\_}{\mathbf{a}}}_u={\left[{\check{\mathbf{a}}}_u^T\left[0\right],{\check{\mathbf{a}}}_u^T\left[1\right],\cdots ,{\check{\mathbf{a}}}_u^T[T-1]\right]}^T$; while the channel component matrix is given by ${\underset{\_}{\mathbf{H}}}_u=$ ${\left[{\check{\mathbf{H}}}_u^T\left[0\right],{\check{\mathbf{H}}}_u^T\left[1\right],\cdots ,{\check{\mathbf{H}}}_u^T[T-1]\right]}^T$.

B. Cooperative Space-Frequency Coded OFDM

As shown in Fig. 5, another method to exploit the diversity in both space and frequency is to invoke cooperative SF coding in OFDM system [22]. Specifically, each consecutive $Q$ elements of frame $\mathit{s}$ are SF-encoded having $K$ output blocks, each of which is converyed into $M\le U$ subcarriers within a single time-slot, i.e. $t=0,T=1$. Hence, each OFDM block requires $N=U/M$-set consecutive $Q$-symbol inputs in order to crossing $U$ subcarriers. For example, the Alamouti’s g2 style cooperative SF-encoded OFDM blocks for $K=2,Q=2$ and $M=2$ may be expressed as

$$\left\{\begin{array}{l}{\mathbf{x}}_{k=0}[t=0]={\left[s_0,-s_1^{*},s_2,-s_3^{*},\cdots ,s_{U-2},-s_{U-1}^{*}\right]}^T,\\ {\mathbf{x}}_{k=1}[t=0]={\left[s_1,s_0^{*},s_3,s_2^{*},\cdots ,s_{U-1},s_{U-2}^{*}\right]}^T,\end{array}\right.$$ (9)

where the $u$-th element in symbol vector ${\mathbf{x}}_k\left[t\right]$ is conveyed onto $u$-th subcarriers at time-slot $t$ and emitted via antenna $k$. When the Tarokh’s g4 based cooperative SF code with $K=4,Q=4$ and $M=8$ is employed in OFDM, we have:

$$\left\{\begin{array}{c}{\mathbf{x}}_0\left[0\right]=\left[s_0,-s_1,-s_2,-s_3,s_0^{*},-s_1^{*},-s_2^{*},-s_3^{*},\cdots ,\right.\\ {\left.s_{U-4}^{*},-s_{U-3}^{*},-s_{U-2}^{*},-s_{U-1}^{*}\right]}^T,\\ {\mathbf{x}}_1\left[0\right]=\left[s_1,-s_0,-s_3,-s_2,s_1^{*},-s_0^{*},-s_3^{*},-s_2^{*},\cdots ,\right.\\ {\left.s_{U-3}^{*},s_{U-4}^{*},s_{U-1}^{*},-s_{U-2}^{*}\right]}^T\\ {\mathbf{x}}_2\left[0\right]=\left[s_2,-s_3,-s_0,-s_1,s_2^{*},-s_3^{*},-s_0^{*},-s_1^{*},\cdots ,\right.\\ {\left.s_{U-2}^{*},-s_{U-1}^{*},s_{U-4}^{*},s_{U-3}^{*}\right]}^T\\ {\mathbf{x}}_3\left[0\right]=\left[s_3,-s_2,-s_1,-s_0,s_3^{*},-s_2^{*},-s_1^{*},-s_0^{*},\cdots ,\right.\\ {\left.s_{U-1}^{*},s_{U-2}^{*},-s_{U-3}^{*},s_{U-4}^{*}\right]}^T,\end{array}\right.$$ (10)

Fig. 5.

Schematic diagram of space-frequency coded OFDM

Moreover, the cooperative LDC encoded OFDM block for the $k$-th antenna on the $u$-th subcarrier at time-slot $t=0$ (before IFFT operation) is given by

$${\mathrm{x}}_k\left[t=0\right]={\left[{\left[{𝓑}_k{\mathit{s}}_0\right]}^T,{\left[{𝓑}_k{\mathit{s}}_1\right]}^T,\cdots ,{\left[{𝓑}_k{\mathit{s}}_{N-1}\right]}^T\right]}^T,$$ (11)

where ${\mathit{s}}_u={\left[s_{uQ},s_{uQ+1},\cdots ,s_{(u+1)Q-1}\right]}^T$ is the $u$-th input symbol segment having a legnth of $Q$ for $u=0,1,\cdots ,U-1$. Consequently, by using Eq. (1), the SF-OFDM symbols can be formed.

At the receiver side, the detection of SF-OFDM operates in subcarrier group-by-group basis. Unlike the Section III–A, after the signal transformed into F-domain by Eq. (4), the symbol blocks for the $n$-th subcarrier group which cross over subcarriers from $nM$ to $(n+1)M-1$ for $n=\mathrm{0,1},\cdots ,N-1$ at time-slot $t=0$ can be expressed by

$${\underset{\_}{\mathbf{y}}}_n\left[0\right]={\underset{\_}{\mathbf{H}}}_n\left[0\right]{\underset{\_}{\mathbf{x}}}_n\left[0\right]+{\underset{\_}{\mathbf{n}}}_n\left[0\right],$$ (12)

where each component vector ${\underset{\_}{\mathbf{y}}}_n\left[0\right],{\underset{\_}{\mathbf{x}}}_n\left[0\right]$ and ${\underset{\_}{\mathbf{n}}}_n\left[0\right]$ can be expressed by ${\underset{\_}{\mathbf{a}}}_n\left[0\right]={\left[{\check{\mathbf{a}}}_{nM}^T\left[0\right],{\check{\mathbf{a}}}_{nM+1}^T\left[0\right],\cdots ,{\check{\mathbf{a}}}_{(n+1)M-1}^T\left[0\right]\right]}^T;$ while the channel component matrix is given by ${\underset{\_}{\mathbf{H}}}_n\left[0\right]={\left[{\check{\mathbf{H}}}_{nM}^T\left[0\right],{\check{\mathbf{H}}}_{nM+1}^T\left[0\right],\cdots ,{\check{\mathbf{H}}}_{(n+1)M-1}^T\left[0\right]\right]}^T$.

C. Maximum-Likelihood Detection

Based on Eqs. (8) and (12), the equivalent F-domain system model for detection can be represented by [6]

$$\stackrel{-}{\mathbf{y}}=\stackrel{-}{\mathbf{H}}\Xi {\mathit{s}}_n+{\stackrel{-}{\mathbf{n}}}_n,$$ (13)

where² $\stackrel{-}{\mathbf{y}}=\mathrm{v}\mathrm{e}\mathrm{c}\left(\underset{\_}{\mathbf{y}}\right),\stackrel{-}{\mathbf{H}}=\mathit{I}\otimes \underset{\_}{\mathbf{H}}$ is an equivalent channel matrix with a size of $\left(N_{\mathrm{R}\mathrm{x}}T\times KT\right)$-element for ST-OFDM and $\left(N_{\mathrm{R}\mathrm{x}}M\times KM\right)$-element for SF-OFDM. Most importantly, $\mathbf{\Xi }$ is referred to as the dispersion character matrix (DCM) [6], defined by $\mathbf{\Xi }=\left[\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathit{B}}_0\right)\right],\left.\left.\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathit{B}}_1\right)\right],\cdots ,\mathrm{v}\mathrm{e}\mathrm{c}\left({\mathit{B}}_{Q-1}\right)\right].{\mathit{s}}_n={\left[{\mathit{s}}_0,s_1,\cdots ,s_{Q-1}\right]}^T$ is the $n$-th segment of transmit signal frame $\mathit{s}$ in Eq. (1). Additionally, ${\stackrel{-}{\mathbf{n}}}_n=\mathrm{v}\mathrm{e}\mathrm{c}\left({\underset{\_}{\mathbf{n}}}_n\right)$.

Therefore, we obtain the estimated symbol vector ${\stackrel{\mathit{ˆ}}{\mathit{s}}}_n$ by maximum likelihood (ML) detection expressed as [6]:

$$\stackrel{\mathit{ˆ}}{\mathit{s}}=\mathrm{a}\mathrm{r}\mathrm{g}\left\{\mathrm{m}\mathrm{i}\mathrm{n}\left(\parallel \stackrel{-}{\mathbf{y}}-\stackrel{-}{\mathbf{H}}\Xi \mathbf{a}{\parallel }^2\right)\right\},$$ (14)

where a denotes all possible combinations of the $Q$ transmitted symbols in ${\mathit{s}}_n$.

IV. Simulation Results and Discussions

In this section, we evaluated the performance achieved by the varying sets of simulation parameters [13], [36], [38], which are summarized in Table I.

TABLE I.

Simulation Parameters

Channel model	Time/freq.-selective Rayleigh fading
Bits per symbol	𝒬 = 1
Norm. Doppler freq.	fND = 0.01, … , 0.1
No. of CIR paths	L = 1, 2, 4,8, 16
No. of subcarriers	U = 128
No. of Tx Antennas	K = 2,4
No. of Rx Antennas	NRx= 1
No. of time-slots per code	T = 2, 4, 8
No. of freq.-tone per code	M = 2, 4, 8
No. of symbols per code	Q = 2, 4

The bit error ratio (BER) performance results of cooperative ST and SF-oriented OFDM invoking LDC (2122)³ or LDC (4144) upon varying the number of normalized Doppler frequency $f_{\mathrm{N}\mathrm{D}}$ are shown in Fig. 6(a) and Fig. 6(b), respectively. Both cooperative LDC (2122) aided ST-OFDM having $K=T=Q=2$ and LDC (4144) aided ST-OFDM associated with $K=4,T=4,Q=4$ achieve the best performance for $f_{\mathrm{N}\mathrm{D}}=0.01$ when $L=4$. At the same time the performance degrades when $f_{\mathrm{N}\mathrm{D}}$ increases to 0.06. This means the channel becomes even more time-selective with the duration of whole cooperative ST-OFDM symbol period for $T=2$ and 4. By contrast, the performance of cooperative orthogonal STBC (g4) aided OFDM for $K=4,T=8,Q=4$ is decreased when $L=4$, due to doubling the time slots for transmitting in comparison to the LDC (4144), upon varying the $f_{\mathrm{N}\mathrm{D}}$ from 0.01 to 0.06.

Fig. 6.

BER performance of cooperative MIMO-OFDM experiencing time-selective fading in terms of varying Doppler spread.

Meanwhile, Fig. 6 also demonstrate that both the cooperative LDC and orthogonal code aided SF-OFDM are capable of achieving a constant performance in low delay spreads with the number of CIR taps $L=4$ without the impact of increasing $f_{\mathrm{N}\mathrm{D}}$.

Furthermore, the performance of cooperative ST- and SF-OFDM invoking LDC (2122,4144) upon varying $L$ is shown in Fig. 7. As seen in this figure, the performance of cooperative SF-OFDM is not impacted by the varying delay spreads for a given $f_{\mathrm{N}\mathrm{D}}=0.01$. However, cooperative SF-OFDM benefits frequency-diversity for neighboring $M$ subcarriers having correlated fading coefficients associated with low frequency-selectivity with $L=\mathrm{1,2},4$ for $K=2$ and $L=\mathrm{1,2}$ for $K=4$. Note that, the performance degrades when increasing $L$. Particularly, cooperative LDC (4144) aided SF-OFDM having $M=4$ outperforms cooperative orthogonal SFBC (g4) aided OFDM with $M=8$ when communicating over the frequency-selective fading channel of $L=\mathrm{2,4}$.

Fig. 7.

BER performance of cooperative MIMO-OFDM experiencing frequency-selective fading in terms of varying delay spread (multi-path).

Timing and frequency offsets are significant issues in implementing a cooperative MIMO-OFDM system. As discussed in [32], timing offsets are mainly caused by geometrical separation in the transmit mobile terminals, leading to different propagation delays over the links. Meanwhile, since different mobile terminals are driven by individual local oscillators with nonidentical characteristics, frequency offsets exist not only in the multiple carriers, but also among cooperative partners. Therefore, the synchronization techniques developed for conventional MIMO-OFDM systems may be not applicable to cooperative systems. Specifically, a frequency synchronization algorithm for a cooperative MIMO-OFDM system utilizes the principle of the cooperation protocol. The authors in [32], [39] investigate a variety of equalization and detection schemes for cooperative space-time/frequency coded systems, in the presence of carrier frequency offsets. Synchronization among cooperative diversity aided ad hoc nodes is discussed in [4], [40]. However, in our work we assume that timing and frequency synchronization are perfect.

A high peak-to-average power ratio (PAPR) may be exhibited in OFDM systems due to the superposition of a high number of modulated subcarrier signals [13], [15]. The PAPR problem imposes substantial challenges on the practical design of power amplifiers that have a limited linear range. Diverse techniques of PAPR reduction have been investigated not only for conventional OFDM [41], but also for ST-/SF-aided MIMO-OFDM systems [42]–[47].

As declared in [12], LDC supports arbitrary transmit and receive antenna configurations for MIMO systems, combined with arbitrary modulation schemes. By contrast, the Alamoutis g2 and orthogonal codes have their specific code matrix and design criteria which means switching the configuration is not as feasible as LDC’s. Therefore, cooperative LDC-based space-time/frequency OFDM systems are suitable for forming the wireless sensor network, since the mobile sensor nodes access to the ad hoc network may be dynamic in terms of application requirement.

V. Conclusions

In this contribution, we investigated cooperative ST- and SF-diversity oriented OFDM systems invoking LDC, in order to study the advantages and disadvantages of cooperative diversity based MIMO transmission over time-/frequency-selective fading channels. An LDC-aided ST-/SF OFDM system is capable of supporting arbitrary configurations of transmit nodes and receive antenna for cooperative wireless sensor networks. Our results demonstrate that when the channel is constant within the coherent time/bandwidth, cooperative ST- or SF-OFDM is capable of achieving full diversity gain in ST or SF domains. The cooperative ST-OFDM scheme is sensitive to exploiting diversity gains subject to the impact of varying channel Doppler spreads; while the performance of cooperative SF-OFDM is mainly subject to delay spread. Moreover, compared with the cooperative orthogonal STBC/SFBC (g4), the cooperative LDC-aided ST/SF-OFDM is flexible in configuring various numbers of cooperative nodes and time-slots or frequency-tones. When the multi-source cooperation employs more than two nodes, the performance of cooperative LDC-aided ST/SF-OFDM schemes is less impacted by channel Doppler/delay spreads, as compared with orthogonal block codes.