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Abstract

In mobile cooperative orthogonal frequency division multiplexing (OFDM) systems with amplify-and-forward (AF) relays, in which the source and relay terminals are moving, Doppler frequency might become very large due to the formation of double-hop channels, possibly causing very large intercarrier interference (ICI). Therefore, ICI mitigation technique capable of reducing the impact of high Doppler frequency is required. This paper reports the study of ICI mitigation technique that uses frequency domain equalizer (FDE) with improvements in three aspects, namely, better channel estimation using cubic-spline approximation, use of selection combining to utilize cooperative diversity in order to suppress ICI, and simple error correcting code with double interleaving to suppress errors due to residual ICI. By using the proposed ICI mitigation technique, error rate performance of mobile cooperative OFDM systems can be improved significantly. Even at high Doppler frequencies, error floors due to residual ICI can be decreased by up to two decades.

1. Introduction

Orthogonal frequency division multiplexing (OFDM) using a number of mutually orthogonal carriers to transmit data simultaneously has been widely used for high-speed data transmission. Orthogonality between subcarriers allows signal spectra of the OFDM subchannels to be separated without interference or power loss due to leakage to other subcarriers. However, time-varying channel because of large Doppler shift can damage the orthogonality nature, leading to ICI (intercarrier interference) and degradation of the performance of system BER.

Doppler shift can increase if the communicating terminals move faster or if there is a multihop channel with moving amplify-and-forward (AF) relays as in cooperative communication for such applications as vehicular communication network, robotic communications, and communications among distributed moving sensors. The greater the Doppler shift the greater the shift in the carrier frequency and widening spectrum. In this situation, ICI mitigation techniques adopted in OFDM systems must be able to deal with large ICI generated by high Doppler frequencies. Various attempts have been reported to find a way to overcome the ICI. Generally, ICI mitigation techniques can be divided into two types: the first seeks to prevent ICI that will occur and suppress it as small as possible, whereas the second type attempts to overcome ICI that has occurred. Both have their own weaknesses: the first requires higher bandwidth for coding to shape the subcarrier spectra to minimize ICI [1, 2], whereas the second requires more complex signal processing to eliminate ICI. In this paper, we are concerned with the second-type ICI mitigation techniques, considering bandwidth limitation to be an inevitable factor.

The second type of ICI mitigation technique that has been proposed is the one using frequency domain equalizer (FDE) [3–6]. FDE is designed based on channel estimation. In time-varying channels, channel estimation and FDE filter coefficients should be updated at every detection of one OFDM symbol. Therefore, it needs a low-complexity channel estimation technique. A simple channel estimation method with piecewise linear approach that has been proposed in [7] has simple interpolation computational advantages. However, the approach can only work well when the normalized maximum Doppler shift, defined as the maximum Doppler shift normalized to the subcarrier spacing, is less than 0.2. Our paper proposes several techniques to improve the performance of ICI mitigation technique with FDE, so that the technique can still work well on mobile cooperative OFDM system with moving AF relays, in which Doppler frequency tends to be high.

This paper describes some techniques to improve the performance of ICI mitigation based on FDE applied in three different systems. In the first system, block-type arrangement of pilots is used with alternating polarity to suppress midpoint channel estimation error, so that the channel influence can be cancelled by the equalizer. The alternating polarity for pilots in block is used because the pilot's side lobe spectra will cancel each other so that the power leakage on other pilots’ main lobe spectrum can be minimized and ICI distortion due to orthogonality loss will be reduced [8]. Selection combining is selected because when the equalizer fails to reduce ICI in one OFDM symbol, most of the symbols will be in error and should be replaced with the correct ones from the other branch. In this paper, this system is referred to as LI system because it still uses piece-wise approach to linear interpolation.

The second system is obtained by replacing the piecewise linear approximation interpolation with cubic-spline interpolation (third-order polynomial) to anticipate a fast channel change due to high Doppler so that channel estimation error can be suppressed using interpolation. We call this second system CI, which stands for cubic-spline interpolation.

The third system referred to as the CI-CDI is realized by adding a simple error correcting code technique with double interleaving to the CI system. This addition is applied to correct errors bursts due to residual ICI in OFDM symbols that cannot be replaced by symbols from the other branch because the substitutes also experience burst errors.

Hence, there are two main contributions of this paper. Firstly, we evaluate the use of linear and cubic interpolation for channel estimation, coupled with double interleaving and coding in a cooperative network with MM channels. By combining the strengths of the individual techniques, the overall scheme shows an increase in performance. Secondly, we propose and evaluate the use of selection combining between direct and relay link to reduce residual ICI.

Numerical results show that, by exploiting the nature of cooperative diversity with selection combining, performance of ICI mitigation technique can be improved. In addition, at relatively high Doppler frequency, use of pilot with alternating polarity in the LI systems can improve the performance of ICI mitigation technique significantly. With the CI system, the use of cubic-spline interpolation can reduce the error floor by up to one decade. While in the CI-CDI system, combined application of simple error correcting technique and double interleaving can reduce the error floor by at least another decade.

Section 2 of this paper discusses the mobile channel and the system under study. Section 3 focuses on the proposed ICI mitigation techniques, namely, the LI, CI, and CI-CDI system. Simulation results are reported and analyzed in Section 4 and conclusions are given in Section 5.

2. Description of Channel and System

2.1. Mobile Channel Models

In a simple cooperative system with moving source and relay terminals as shown in Figure 1, there are three types of mobile channels, namely, mobile-to-fixed (MF), mobile-to-mobile (MM), and, if such cooperative systems use AF relaying scheme, mobile-to-mobile-to-fixed (MMF). The mobile channel model used in this analysis is derived from the normalized MM channel complex envelope with double-ring scatterers model which is given as [9, 10]

\begin{array}{l} h_{MM} (t) = \sqrt{\frac{2}{P Q}} \sum_{p = 1}^{P} ‍ \sum_{q = 1}^{Q} ‍ e^{j (2 π f_{d_{1}} \cos (α_{p}) t + 2 π f_{d_{2}} \cos (β_{q}) t + ϕ_{p q})} \\ α_{p} = \frac{2 p π - π + θ_{p}}{4 P}; β_{q} = \frac{2 (2 q π - π + ψ_{q})}{4 Q}, \end{array}

(1)

where $f_{d_{i}} = v_{i} / λ$ , $i = 1,2, 3$ , denotes the maximum Doppler shift due to the transmitter $(i = 1)$ or receiver $(i = 2)$ which moves at velocity $v_{i}$ and carrier wavelength λ. Index p is associated with the propagation path from the transmitter to the pth scatterer on the ring surrounding the transmitter, whereas the index q is related to the propagation path from the qth scatterer on the ring centered at the fixed receiver. Herein, P and Q are the number of scatterers in the ring located at the transmitter and receiver. Angle $θ_{p} \in (- π, π]$ is a random angle of departure (AOD) from the pth path, whereas $ψ_{q} \in (- π, π]$ is a random angle of arrival (AOA) from the qth path, each of which is measured with respect to the velocity vector of transmitter (Tx) or receiver (Rx). Also, $ϕ_{p q} \in (- π, π]$ is the phase shift related to the reflection coefficient of the scatterers. All angles are assumed to be independent of each other for all p and q.

Figure 1

Simple cooperative diversity.

For the MF channel, it is assumed that there is only one ring of scatterers which is located around the transmitter, so that (1) is reduced to [11, 12]

\begin{matrix} h_{MF} (t) = \sqrt{\frac{2}{P}} \sum_{p = 1}^{P} ‍ e^{j (2 π f_{d} t \cos α_{p} + ϕ_{p})}, \end{matrix}

(2)

where $f_{d}$ is the maximum Doppler shift due to mobile sources, $ϕ_{p}$ is the phase shift due to the reflection coefficient of scatterers, and $α_{p}$ is defined similarly as in (1). For cooperative systems with an AF relay, the relayed channel is expressed as a MMF channel and is defined as in [9] as

\begin{matrix} h_{MMF}^{SRD} (t) = A h_{MM}^{SR} (t) h_{MF}^{RD} (t), \end{matrix}

(3)

where $h_{MM}^{SR} (t)$ and $h_{MF}^{RD} (t)$ denote the MM channel response of the source-relay link modelled by (1) and MF channel response of the relay-destination link modeled by (2), respectively, as shown in Figure 1. In expressions following (3), the notation $f_{d}$ for the MF channel is changed into $f_{d_{3}}$ , denoting the Doppler shift caused by the AF relay as a transmitter. It is assumed that $f_{d_{3}} = f_{d_{2}}$ . In (3), A is the amplification factor of the relay which is assumed to be fixed and defined as

\begin{matrix} A = \sqrt{\frac{E_{2}}{E_{1} σ_{MM}^{2} + σ_{w}^{2}}}, \end{matrix}

(4)

where $E_{1}$ is the average power of transmitted signal, $E_{2}$ is the average power transmitted by the relay, and $σ_{MM}^{2}$ is the MM channel power gain and $σ_{w}^{2}$ is the AWGN variance. Thus the signal received by the destination in two phases in cooperative communication systems is as follows:

time slot number 1 [13]:

\begin{matrix} Z^{(1)} (k) = H_{MF}^{SD} (k) X (k) + W_{d}^{(1)} (k); \end{matrix}

(5)

time slot number 2 [9, 13]:

\begin{matrix} Z^{(2)} (k) = H_{MMF}^{SRD} (k) X (k) + A H_{MF}^{RD} (k) W_{r} (k) + W_{d}^{(2)} (k), \end{matrix}

(6)

where $H_{MF}^{SD} (k)$ , $H_{MF}^{RD} (k)$ , and $H_{MMF}^{SRD} (k)$ are the MF channel frequency response of source-destination link, the MF channel frequency response of source-relay link, and the MMF channel frequency response of source-relay-destination link, respectively. Signal $X (k)$ is the transmitted QPSK symbol on the kth subcarrier. The discrete time domain OFDM signal samples obtained after applying an inverse discrete Fourier transforms to $X = [X (0), X (1), \dots, X (N - 1)]$ are given by

\begin{matrix} x (n) = \frac{1}{\sqrt{N}} \sum_{k = 0}^{N - 1} ‍ X (k) e^{j 2 π n k / N}, n = 0, \dots, (N - 1), \end{matrix}

(7)

where N is the number of subcarriers, whereas $W_{r} (k)$ and $W_{d}^{(i)} (k)$ , $i = 1,2$ , are the discrete Fourier transforms of zero-mean AWGN with identical variances $σ_{w}^{2}$ which occurs in the source-relay, source-destination, and relay-destination link, respectively. In subsequent analyses, all maximum Doppler shift frequencies in (1), (2), and (3) are expressed in normalized form with respect to frequency spacing between adjacent subcarriers and are denoted by ε.

2.2. System Description

A block diagram of the OFDM system with ICI mitigation technique is given in Figure 2. Output from the fast Fourier transform (FFT) block is expressed as follows:

\begin{matrix} Z_{i} = H_{i, 0} X_{i} + \underset{ICI}{\underset{︸}{\sum_{d = 1}^{N - 1} ‍ H_{i, d} X_{{((i - d))}_{N}}}} + W_{i}, \end{matrix}

(8)

where $W_{i}$ denotes the FFT of AWGN $w_{i}$ , a sampled version of $w (t)$ , $H_{k, 0}$ denotes the channel frequency response corresponding with the information symbol carried on the kth subcarrier, $H_{i, d}$ is the channel frequency response corresponding with the information symbol carried on the $((i - d))_{N}$ th subcarrier that causes ICI on the information symbol on the ith subcarrier, and $X_{i}$ denotes the symbol carried by the ith subcarrier, whereas $((\cdot))$ is the notation for the modulo operation. The second term on the right-hand side of (8) constitutes the ICI.

Figure 2

OFDM system and ICI mitigation.

3. ICI Mitigation Technique Improvement

FDE-based ICI mitigation requires inverse operation of channel frequency response matrix, so that its performance depends on the accuracy of channel estimation. As described in Section 1, for ICI mitigation techniques improvement, there are three systems proposed. Explanation of each system is given as follows.

3.1. LI System

Channel estimation here is done in the time domain. There are two stages in the channel response estimation. The first stage involves channel estimation at each symbol midpoint, the result of which is denoted by $h^{mid}$ . This is the result of average channel estimation for each symbol obtained with the help of pilots. The second stage is channel estimation between the midpoints by interpolation. In the first estimation, the pilot significantly affects the accuracy of channel estimation. Therefore, this paper proposes block-type pilot with alternating polarity to replace the comb type. The alternating polarity is used herein because the pilot's spectrum accumulation will produce total spectrum with lower side lobe so that the ICI term in (8) between adjacent pilots can be suppressed and the $h^{mid}$ estimation error can be minimized. The comb-type pilot is expressed as [7]

\begin{array}{l} X_{k} = {\begin{cases} P_{i}, & k = i \times N_{n} \\ D_{q}, & k = 1 + m \times N_{n} + {((q))}_{N_{n} - 1}, \end{cases} \\ P_{i} = \frac{1 + j}{\sqrt{2}}, \\ m = ⌊ \frac{(q - {((q))}_{N_{n} - 1})}{(N_{n} - 1)} ⌋, \\ 0 \leq i \leq N_{p} - 1; 0 \leq q \leq N - N_{p} - 1; 0 \leq k \leq N - 1, \\ j = \sqrt{- 1}; N_{n} = \frac{N}{N_{p}}, \end{array}

(9)

where $P_{i}$ denotes the complex baseband representation of the ith pilot that is inserted into the kth subchannel, $N_{p}$ is the number of pilots, and $D_{q}$ is the qth information symbol carried by an OFDM symbol and $⌊ ξ ⌋$ is the largest integer not larger than ξ. Whereas the block-type pilot with the proposed alternating polarity can be expressed as

\begin{matrix} X_{k} = {\begin{cases} P_{i}, & k = i \\ D_{q}, & k = N_{p} + q - 1, \end{cases} \end{matrix}

(10)

in (10), $P_{i}$ is defined as

\begin{matrix} P_{i} = {\begin{cases} \frac{1 + j}{\sqrt{2}}, & for i even \\ - \frac{1 + j}{\sqrt{2}}, & for i odd . \end{cases} \end{matrix}

(11)

From these pilots, samples from the channel frequency response at subcarrier frequency points carrying pilot signals, ${\hat{H}}_{i}$ , can be obtained using the following equation:

\begin{matrix} {\hat{H}}_{i} = \frac{Z_{i}}{P_{i}} = H_{i} + \frac{I_{i} - W_{i}}{P_{i}}; 0 \leq i \leq N_{p} - 1, \end{matrix}

(12)

where $I_{i}$ represents ICI that occurs on the ith pilot symbol. Subsequently, the average channel estimation can be obtained from the $N_{p}$ -point IFFT of ${\hat{H}}_{i}$ of length $N_{p}$ as follows:

\begin{matrix} {\hat{h}}_{m}^{ave} = \frac{1}{N_{p}} \sum_{i = 0}^{N_{p} - 1} ‍ {\hat{H}}_{i} e^{j (2 π i m / N_{p})}; 0 \leq m \leq N_{p} - 1 . \end{matrix}

(13)

In this case the midpoint value of the channel response is assumed to be represented by its average value, that is, ${\hat{h}}^{mid} = {\hat{h}}_{0}^{ave}$ [7]. Further estimates of the second stage are obtained by interpolating channel response points between adjacent ${\hat{h}}^{mid}$ , which for piecewise linear interpolation is accomplished by applying the following equation [7]:

\begin{matrix} {\hat{h}}^{(k)} = {\hat{h}}^{mid} + (k + 1 - \frac{N}{2}) \times \hat{μ} \times T_{S}; 0 \leq k \leq N - 1, \end{matrix}

(14)

where $T_{S}$ is the sampling period defined as $T_{S} = T / (N + G)$ , T is the OFDM symbol period, N is the number of IFFT points, and G is the number of guard interval sample points. Whereas $\hat{μ}$ is the linear gradient from ${\hat{h}}^{mid}$ to the next ${\hat{h}}^{mid}$ , namely, ${\hat{h}}^{mid, next}$ , which can be obtained as follows:

\begin{matrix} \hat{μ} = \frac{{\hat{h}}^{mid, next} - {\hat{h}}^{mid}}{T_{S}}, \end{matrix}

(15)

piecewise linear interpolation is quite simple and only requires information of two midpoints to estimate along one symbol duration. The estimation error with this approach is relatively low when the normalized maximum Doppler shift is small, that is, $ε_{d} < 0.2$ . However, for $ε_{d} \leq 0.2$ , the error in channel estimation by this interpolation becomes relatively large [7]. In matrix form, received output signal coming out of the FFT block is defined as

\begin{matrix} Z = F z = F h F^{H} X + F w = H X + F w, \end{matrix}

(16)

where $H = F h F^{H}$ is the channel frequency response matrix, h denotes the $N \times N$ channel response matrix, F is the N-point FFT matrix, $F^{H}$ is the N-point IFFT matrix, $X = [X_{0} \dots X_{N - 1}]^{T}$ , and $w = [w_{0} \dots w_{N - 1}]^{T}$ . By using the FDE with filter coefficients obtained from the inverse of $\hat{H}$ , where $\hat{H} = F \hat{h} F^{H}$ and $\hat{h}$ is a $N \times N$ diagonal matrix, the channel estimate corresponding to the received OFDM symbol obtained from (14) is

\begin{matrix} \hat{h} = {[diag [\begin{bmatrix} {\hat{h}}^{(0)} & {\hat{h}}^{(1)} & \dots & {\hat{h}}^{(N - 1)} \end{bmatrix}]]}_{N \times N} . \end{matrix}

(17)

The symbol vector contained in each OFDM symbol is estimated using the following equation:

\begin{matrix} \hat{X} = {\hat{H}}^{- 1} Z = {\hat{H}}^{- 1} (H X + W), \end{matrix}

(18)

where $Z = [Z_{0} \dots Z_{N - 1}]^{T}$ and $W = F w = [W_{0} \dots W_{N - 1}]^{T}$ are received signal vectors and noise at the output of the FFT block, respectively, while ${\hat{H}}^{- 1}$ is the inverse of $\hat{H}$ .

Performance of mitigation technique can also be improved by adopting cooperative diversity. To maximize the improvement, the most suitable combining technique must be chosen. Because FDE does not work when the channel gain is very low or $\approx 0$ , there is a possibility of ICI mitigation failure in the OFDM symbol in this particular situation. If this is the case, then the number of information symbol errors in one OFDM symbol will be relatively large, and if this condition happens to one branch, then all symbols in that particular OFDM symbol should be replaced with a replica of the symbols from the other branch. Therefore, after the mitigation process at receiver, selection combining is required based on the results of the midpoint channel estimation, that is, $h^{mid}$ , as follows [14]:

\begin{matrix} \hat{D} (ν) = {\begin{cases} {\hat{D}}^{SD} (ν), & if | {\hat{h}}_{SRD}^{mid} (ν) | \geq | {\hat{h}}_{SRD}^{mid} (ν) | \\ {\hat{D}}^{SRD} (ν), & if | {\hat{h}}_{SD}^{mid} (ν) | < | {\hat{h}}_{SRD}^{mid} (ν) |, \end{cases} \end{matrix}

(19)

where vector $\hat{D} (ν) = [{\hat{D}}_{0} (ν) {\hat{D}}_{1} (ν) \dots {\hat{D}}_{N - 1} (ν)]$ is a vector of data bits estimation of the data that are carried in νth OFDM symbol that results from selection combining, while ${\hat{D}}^{SD} (ν)$ and ${\hat{D}}^{SRD} (ν)$ are the data output after pilot removal process from source-destination link and source-relay-destination link, respectively. Block diagram of the proposed LI system is given in Figure 3.

Figure 3

The diagram block of LI system.

3.2. CI System

To get small channel estimation error when Doppler frequency is high, in the second system, the cubic-spline interpolation technique is proposed to replace the piecewise linear approach [15, 16]. Cubic-spline interpolation requires at least 3 midpoint samples of the channel response, ${\hat{h}}^{mid}$ . With those three sample points, the channel impulse response of two OFDM symbols can be estimated as shown in Figure 4.

Figure 4

Cubic-spline interpolation.

For every cubic-spline interpolation, vectors ${\hat{h}}^{mid}$ are formed consisting of three elements taken from the ${\hat{h}}^{mid} (ν)$ sequence, where $ν = 1, \dots, N_{h}$ is defined as follows:

\begin{matrix} {\hat{h}}^{mid} = [\begin{bmatrix} {\hat{h}}^{mid} (κ - 2) & {\hat{h}}^{mid} (κ - 1) & {\hat{h}}^{mid} (κ) \end{bmatrix}] \\ for κ = 2 l + 1, l = 1,2, \dots, \frac{N_{h} - 1}{2} . \end{matrix}

(20)

Each ${\hat{h}}^{mid} (ν)$ is a complex number that can be written as follows:

\begin{matrix} {\hat{h}}^{mid} (ν) = {\hat{h}}_{R}^{mid} (ν) + j {\hat{h}}_{I}^{mid} (ν) . \end{matrix}

(21)

Cubic-spline interpolation is done separately for the real and imaginary parts but the interpolation steps between the real and imaginary parts are identical. Therefore, cubic-spline interpolation steps are described herein for the real part only. The elements of these three-point coordinates in the image above, for the real part, can be expressed in vector t and vector ${\hat{h}}_{R}^{mid}$ , and, for the imaginary part, can be expressed in vector t and vector ${\hat{h}}_{I}^{mid}$ . Each vector is defined as follows:

\begin{matrix} t = [\begin{bmatrix} t (κ - 2) & t (κ - 1) & t (κ) \end{bmatrix}], \\ {\hat{h}}_{R}^{mid} = [\begin{bmatrix} {\hat{h}}_{R}^{mid} (κ - 2) & {\hat{h}}_{R}^{mid} (κ - 1) & {\hat{h}}_{R}^{mid} (κ) \end{bmatrix}], \\ {\hat{h}}_{I}^{mid} = [\begin{bmatrix} {\hat{h}}_{I}^{mid} (κ - 2) & {\hat{h}}_{I}^{mid} (κ - 1) & {\hat{h}}_{I}^{mid} (κ) \end{bmatrix}] . \end{matrix}

(22)

In the above equations, $N_{h}$ is assumed to be odd integer. Otherwise, a zero-valued sample is added to ${\hat{h}}^{mid} (ν)$ sequence so that $N_{h}$ becomes odd integer. Channel impulse response estimates between the three points of ${\hat{h}}^{mid} (ν)$ are divided into two segments [17]: the first segment, called $(q = 1)$ , in the range of $t (κ - 2) \leq t \leq t (κ - 1)$ and the second $(q = 2)$ in the range of $t (κ - 1) \leq t \leq t (κ)$ , each being approximated by a 3rd-order polynomial equation as follows.

For the qth segment, where $q = 1,2$ ,

\begin{matrix} {\hat{h}}^{(q)} (t) = \sum_{l = 0}^{3} ‍ s^{(q l)} {(t - t (κ - 3 + q))}^{l} \\ for t (κ - 3 + q) \leq t \leq t (κ - 2 + q) . \end{matrix}

(23)

To get $s^{(q l)}$ , $q = 1,2$ and $l = 0,1, 2,3$ , we define the following:

\begin{matrix} ρ^{(q)} = t (κ - 2 + q) - t (κ - 3 + q) . \end{matrix}

(24)

Matrix A of size $3 \times 3$ is arranged as follows:

\begin{matrix} A = {[\begin{bmatrix} 1 & 0 & 0 \\ ρ^{(1)} & 2 (ρ^{(1)} + ρ^{(2)}) & ρ^{(2)} \\ 0 & 0 & 1 \end{bmatrix}]}_{3 \times 3} . \end{matrix}

(25)

Then, by using equation

\begin{matrix} d^{(q)} = \frac{{\hat{h}}_{R}^{mid} (κ - 2 + q) - {\hat{h}}_{R}^{mid} (κ - 3 + q)}{ρ^{(q)}}, \end{matrix}

(26)

vector r is defined as follows:

\begin{matrix} r = {[\begin{bmatrix} 0 & 6 (d^{(2)} - d^{(1)}) & 0 \end{bmatrix}]}^{T}, \end{matrix}

(27)

where $[\cdot]^{T}$ is the transpose notation. Furthermore, from matrix A and vector r, vector $u = [u^{(1)} u^{(2)} u^{(3)}]$ can be obtained by using the equation

\begin{matrix} u = A^{- 1} r . \end{matrix}

(28)

The coefficients on the cubic-spline interpolation equation for the first and second segment, namely, $s^{(q l)}$ , $q = 1,2$ , and $l = 0,1, 2,3$ , are defined as follows:

\begin{matrix} s^{(q 0)} = {\hat{h}}_{R}^{mid} (κ - 3 + q), \\ s^{(q 1)} = d^{(q)} - \frac{ρ^{(q)}}{6} (2 u^{(q)} + u^{(1 + q)}), \\ s^{(q 2)} = \frac{u^{(q)}}{2}, \\ s^{(q 3)} = \frac{u^{(1 + q)} - u^{(q)}}{6 ρ^{(q)}} . \end{matrix}

(29)

For the imaginary part, ${\hat{h}}_{R}^{mid}$ in the above equations is replaced with ${\hat{h}}_{I}^{mid}$ . Subsequently, ICI mitigation with FDE is performed on each OFDM symbol, as is done in the LI system.

3.3. CI-CDI System

The performance of the second ICI mitigation technique above, that is, the CI system, can be improved further by incorporating error correcting codes. When ICI mitigation in OFDM symbol fails, the ICI distortion on the symbols contained in OFDM symbol will cause burst errors. To overcome these burst errors, the third system is proposed using CI-CDI system, that is, with the addition of double interleaving technique with simple error correcting code to the CI systems. In the CI-CDI system, simple error correcting codes can be used, such as cyclic code $(n_{c}, k_{c})$ , which is capable of correcting burst errors no more than 2 bits. To overcome the error burst, there are two layers of interleaving techniques used: bit interleaving and symbol interleaving, which will be explained in the following.

3.3.1. Simple Error Correcting Code

The cyclic code $(n_{c}, k_{c})$ is applied to the OFDM with $N_{d}$ subcarrier. The cyclic code has a burst error correction capability of [18]

\begin{matrix} t_{corr} \leq ⌊ \frac{n_{c} - k_{c}}{2} ⌋ . \end{matrix}

(30)

After the error-correction encoding and before the modulation stage, double interleaving is performed in two stages, as described in the following.

3.3.2. Bit Interleaving

Bit interleaving is intended to separate the double-bit errors that may occur when a 4-ary symbol modulation (QPSK) encountered an error. To get a bigger possibility to spread burst errors into a single error in each block, separation between the in-phase and quadrature bits should be far enough. When the number of subcarriers for data is $N_{d}$ , then the interleaver requires a buffer of size $2 \times N_{d} \times N_{d}$ as given in Figure 5(a). As many as $2 \times N_{d} \times N_{d}$ data bits are loaded in sequence into the interleaver matrix starting with the first row from the first column through to the $N_{d}$ th column, continued with the second row with the same order of columns. Next, data are released in sequence starting from the first and second rows of the first column through to the first and second rows of the $N_{d}$ th column. Once the buffer is empty, the next $2 \times N_{d} \times N_{d}$ data bits are inserted and removed from the buffer in the same way. The process is repeated until the last bit of data blocks. In Figure 5, before interleaving, sequence of bits in each block consisting of $2 \times N_{d} \times N_{d}$ bits is arranged as follows: ${b_{1} b_{2} b_{3} \dots b_{2 \times N_{d} \times N_{d}}}$ , and, after getting out from interleaving, the sequence of bits in each block becomes ${b_{1} b_{N_{d} \times N_{d} + 1} b_{2} b_{N_{d} \times N_{d} + 2} \dots b_{N_{d} \times N_{d}} b_{2 \times N_{d} \times N_{d}}}$ . This sequence is subsequently mapped into QPSK-modulated symbols, so that the block made up of $2 \times N_{d} \times N_{d}$ bits is transformed into another consisting of $N_{d} \times N_{d}$ QPSK symbols.

Figure 5

Interleaving mechanism.

3.3.3. Symbol Interleaving

Symbol interleaving is intended to separate the burst errors occurring in the symbols occupying adjacent subcarriers in an OFDM symbol. Burst errors occurring in one OFDM symbol are spread into a single error in one block of OFDM symbols. The symbol interleaving requires a buffer of the size $N_{d} \times N_{d}$ . Each block consists of $N_{d} \times N_{d}$ QPSK symbols loaded and removed from the buffer in the same way as the bit interleaving. Symbol interleaving is illustrated in Figure 5(b). Next, the block of symbols is divided further into smaller blocks, each having $N_{d}$ symbols per block. Each symbol block is then inserted into demultiplexer (S/P) block to be distributed to each subcarrier that will take it. Block diagram of CI-CDI system is given in Figure 6, which is an extension of CI-system block diagram.

Figure 6

The CI-CDI system block diagram.

4. Performance Analysis by Simulation

4.1. Simulation Description

The OFDM system parameters used in the simulation are given in Table 1. Intersymbol interference (ISI) is assumed to be eliminated completely by using a guard interval longer than the maximum delay spread. Simulation parameter observed is the resulting bit error rate (BER) as a function of the ratio between energy per bit and power spectral density $(E_{b} / N_{0})$ . The simulation is observed at the normalized maximum Doppler shift frequency, $ε_{dR} = ε_{d_{1}} + ε_{d_{2}} + ε_{d_{3}} = 0.4$ , for the source-relay-destination link and at $ε_{d_{1}} = 0.2$ for the source-destination link, which are considered relatively large. It is assumed that $ε_{d_{2}} = ε_{d_{3}} = 0.1$ , the equality being assumed based on the fact that both Doppler frequencies are caused by the same relay with relatively the same frequency operation. It is also assumed that time division multiple access (TDMA) is used [9]. As a reference for BER performance comparison, we look into ICI mitigation technique with comb-type pilots without cooperative link.

Table 1

Parameters in simulation.

Parameters	Specifications
FFT size	2
Number of carriers	32
Pilot ratio	$1 / 8$
Subcarrier constellation	QPSK
Channel model	MF NLOS channel with single ring scatterer and MM NLOS channel with double ring scatterer
Cooperative scheme	Classic with single relay
Relaying technique	Amplify and forward (AF) with fixed gain
Mitigation technique	Frequency domain equalizer (FDE) with improvement as is proposed using LI system, CI system, or CI-CDI system

4.2. Simulation Result and Analysis

In Figure 7, ICI mitigation technique that incorporates a change in pilot from the comb type into the block type with alternating polarity indicates considerable performance improvement for both cooperative and noncooperative systems with or without AF relay. The performance of noncooperative system using AF relay is not better than system without relay because the moving relay contributes two maximum Doppler shift components, that is, $ε_{d_{2}} = 0.1$ and $ε_{d_{3}} = 0.1$ , to the overall normalized maximum Doppler shift frequency. In the relay branch, the total normalized maximum Doppler shift becomes 0.4 and has large impact on ICI distortion. If the normalized maximum Doppler shift is higher than 0.2, the error channel estimation using piecewise linear interpolation becomes very large. Hence, by employing cubic-spline interpolation in the CI system, significant performance improvement is obtained on ICI mitigation, as indicated by decrease in error floor by a decade. This error floor is caused by the FDE failure in eliminating ICI when the channel gain is very low (close to zero). The error floor still exists in cooperative systems due to residual ICI that occurs on both branches simultaneously. In this condition, selection combining cannot replace the erroneous OFDM symbols with the correct ones. Channel estimation error minimization is very important to repair FDE-based ICI mitigation technique. In large Doppler frequency condition, characterized herein by the normalized maximum Doppler frequency shift, $ε_{d_{2}} \geq 0.2$ , the benefit of cubic-spline interpolation is significant because when $ε_{d}$ is large, changes in the channel response within one symbol interval occur very fast. If piecewise linear approach is used, large error in channel estimation results. In addition, because, in the cooperative system the combining is done after ICI mitigation, the application of selection combining to reduce the number of burst error blocks is important and absolutely necessary.

Figure 7

BER performance of mobile cooperative OFDM system with ICI mitigation technique for $ε_{d_{1}} = 0.2$ and $ε_{d_{2}} = ε_{d_{3}} = 0.1$ .

For the CI-CDI system, use of two types of cyclic code is evaluated, that is, the (15, 11) cyclic code representing $n_{c} < N_{d}$ and (31, 26) representing $n_{c} > N_{d}$ , in which the number of subcarriers $N_{d} = 28$ . Both cyclic codes prove to improve significantly the performance of ICI mitigation; for example, at BER $1 0^{- 4}$ , the value of $E_{b} / N_{0}$ can be reduced by up to 10 dB. The performance of cyclic code (31, 26) is slightly less than the (15, 11) cyclic code by about 1 dB, but the former has better spectral efficiency. Hence, the application of the CI-CDI system for ICI mitigation in OFDM cooperative system makes the system more robust in channels with large Doppler shift.

Simulation results of the systems performances to investigate the combining techniques between selection combining (SC) as is proposed and maximum ratio combining (MRC) are given in Figure 8. In this figure, MRC combining is slightly better than SC for small $E_{b} / N_{0}$ (<16 dB for system without coding and <14 dB for system with cyclic code (15, 11)). For large $E_{b} / N_{0}$ , SC is better than MRC. This is because the residual ICI tends to occur in the form of burst errors within OFDM symbol. Therefore, if the symbol is combined with another correct symbol, then it is possible that the combination will produce an OFDM symbol with burst error. This condition can be avoided by using selection combining because the combining will choose the correct symbol. If the occurrences of OFDM symbol with error burst can be reduced, then the system performance can be further improved by using simple coding with double interleaving.

Figure 8

BER performance comparison of SC (selection combining) and MRC (maximum ratio combining) in mobile cooperative OFDM system with and without coding $ε_{d_{1}} = 0.2$ and $ε_{d_{2}} = ε_{d_{3}} = 0.1$ .

To investigate the effect of channel parameter on system performance, in Figure 9, the system performances are investigated with various maximum Doppler shifts at relay branch (source-relay-destination link). In this figure, the CI-CDI system is unable to mitigate ICI if the normalized Doppler shift on relay branch is more than 0.5.

Figure 9

BER performance of OFDM cooperative system for various normalized maximum Doppler shift at relay branch, $ε_{dR} = 0.4$ , $0.5$ and $0.6$ .

5. Conclusion

In cooperative OFDM systems, performance of ICI mitigation with FDE can be enhanced by exploiting the diversity nature coupled with selection combining. The improvement of channel estimation is very important, which in this case is achieved by employing cubic-spline interpolation in place of piecewise linear approach. Aside from that, the addition of a simple error correcting code and double interleaving improves the system BER performance significantly, especially at high Doppler shifts. With normalized maximum Doppler shift at source branch = 0.2, using CI-CDI system, the total performance improvement can produce a reduction in the error floor by more than two decades provided that the normalized maximum Doppler shift at relay branch does not exceed 0.4.

Footnotes

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The graduate study of Titiek Suryani is supported by a BPPS Scholarship from the Indonesia Government. The reported work is part of research program funded by Graduate Research Grant BOPTN 2012 from the Indonesian Ministry of National Education.

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