4. Receiver design for optical OFDM systems

In this section, we investigate the receiver design for optical OFDM systems. For DCO-OFDM, the receiver is straightforward. For non-DC-biased optical OFDM systems, there exist multiple candidate receivers which will be detailed later. For hybrid systems, since they are constructed mainly based on non-DC-biased ones, the receivers designed for non-DC-biased systems are also applicable for hybrid systems. Moreover, as the duality between non-DC-biased systems, in this chapter we focus on ACO-OFDM. We review the basic receiver design, diversity combining receiver design, and propose an iterative receiver design. All the formulations are based on an AWGN channel model but the results can be readily extended to multipath channels.

#### 4.1. Basic receiver

The basic receiver for ACO-OFDM is very simple. The received signal after CP removal is given by

$$y(n) = x\_c(n) + z(n). \tag{14}$$

In frequency domain, one has

$$Y(k) = X\_c(k) + Z(k). \tag{15}$$

As proved by Ref. [1], the odd subcarriers of XcðkÞ is related to XðkÞ by

$$X\_{\varepsilon}(k) = \frac{1}{2}X(k), \text{ for odd } k. \tag{16}$$

Therefore, the data U, which is only on odd subcarriers of X, can be recovered from Y(k) by

$$\hat{\hat{U}}(k) = \begin{cases} 2Y(2k+1), & \text{ZF} \\ \frac{2Y(2k+1)}{1+\sigma\_n^2}, & \text{MMSE} \end{cases} \qquad k = 0, 1, \ldots, \frac{N}{2} - 1 \tag{17}$$

The receiver diagram is shown in Figure 9.

#### 4.2. Diversity combining receiver

The basic receiver only utilizes the odd subcarriers for signal detection. The even subcarriers, bearing pure clipping noise, are simply discarded. Therefore, half of the received power is wasted. However, the clipping noise has a special inherent signal structure that is dependent on the unclipped signal. This inherent signal structure could be exploited for better detection performance. This is the basic idea of diversity combining receiver and the iterative receiver.

To unveil the relationship between the clipping noise and the unclipped signal, we rewrite the clipped signal as

$$\mathbf{x}\_{\epsilon}(n) = \frac{1}{2} [\mathbf{x}(n) + |\mathbf{x}(n)|] \,\prime \,\forall n. \tag{18}$$

Based on Eqs. (16) and (18), one can see that the clipping noise falls only onto the even subcarriers and has a special form as

Figure 9. Diagram of the basic receiver for ACO-OFDM.

OFDM Systems for Optical Communication with Intensity Modulation and Direct Detection http://dx.doi.org/10.5772/intechopen.68199 95

$$X\_c(k) = \frac{1}{2} FFT\langle |\mathbf{x}(n)|\rangle\_\prime \text{ for even } k,\tag{19}$$

which says that the clipping noise on the even subcarriers is just the FFT of jxðnÞj. Thus, we can generate two signals based on <sup>Y</sup>ðkÞ: the first one is <sup>Y</sup><sup>0</sup> ¼ ½<sup>0</sup> <sup>Y</sup>ð1<sup>Þ</sup> <sup>0</sup> <sup>Y</sup>ð3Þ…<sup>0</sup> <sup>Y</sup>ð<sup>N</sup> � <sup>1</sup>Þ �<sup>T</sup>, and the second one is Y<sup>e</sup> ¼ ½Yð0Þ 0 Yð2Þ 0…YðN � 2Þ 0� <sup>T</sup>. Denoting their time-domain signal by y<sup>0</sup> and ye, respectively, we have

4.1. Basic receiver

In frequency domain, one has

94 Optical Fiber and Wireless Communications

U \_ ðkÞ ¼

The receiver diagram is shown in Figure 9.

4.2. Diversity combining receiver

clipped signal as

and has a special form as

The basic receiver for ACO-OFDM is very simple. The received signal after CP removal is given by

As proved by Ref. [1], the odd subcarriers of XcðkÞ is related to XðkÞ by

XcðkÞ ¼ <sup>1</sup> 2

2Yð2k þ 1Þ, ZF

xcðnÞ ¼ <sup>1</sup> 2

CP removal FFT

Figure 9. Diagram of the basic receiver for ACO-OFDM.

<sup>1</sup> <sup>þ</sup> <sup>σ</sup>n<sup>2</sup> , MMSE

2Yð2k þ 1Þ

8 < :

Therefore, the data U, which is only on odd subcarriers of X, can be recovered from Y(k) by

The basic receiver only utilizes the odd subcarriers for signal detection. The even subcarriers, bearing pure clipping noise, are simply discarded. Therefore, half of the received power is wasted. However, the clipping noise has a special inherent signal structure that is dependent on the unclipped signal. This inherent signal structure could be exploited for better detection performance. This is the basic idea of diversity combining receiver and the iterative receiver. To unveil the relationship between the clipping noise and the unclipped signal, we rewrite the

Based on Eqs. (16) and (18), one can see that the clipping noise falls only onto the even subcarriers

S/P Equaliza�on P/S Demodula�on

N-2 N-1

yðnÞ ¼ xcðnÞ þ zðnÞ: ð14Þ

YðkÞ ¼ XcðkÞ þ ZðkÞ: ð15Þ

k ¼ 0, 1, …,

XðkÞ, for odd k: ð16Þ

N <sup>2</sup> � <sup>1</sup>

½xðnÞþjxðnÞj�, ∀n: ð18Þ

ð17Þ

$$y\_0 = \frac{1}{2}\mathbf{x} + \mathbf{z}\_{0\prime} \tag{20}$$

$$y\_{\varepsilon} = \frac{1}{2}|\mathbf{x}| + z\_{\varepsilon}. \tag{21}$$

Eq. (21) is obtained from Eqs. (16) and (19). A new signal ycðnÞ is generated based on Eqs. (20) and (21):

$$y\_{\varepsilon}(n) = \begin{cases} \ y\_{\varepsilon}(n), \text{if } \quad y\_0(n) \ge 0, \\ -y\_{\varepsilon}(n), \text{if } \quad y\_0(n) < 0. \end{cases} \tag{22}$$

Now, we get two branches of signals that are related to x(n). Thus, diversity combining technique could be used to enhance the detection performance. The diversity combining is performed by

$$
\pi r(n) = a y\_0(n) + (1 - a) y\_c(n), \forall n. \tag{23}
$$

The combining coefficient a is usually a bit larger than 0.5 since ycðnÞ is not as accurate as y0ðnÞ [12]. Based on r(n), the data could be estimated just as in the basic receiver. The whole procedure of diversity combining receiver is shown in Figure 10.

A pairwise-ML receiver based on noise cancellation has been proposed in [13]. It has been proved that this receiver is in fact a special case of diversity combining receiver with a = 0.5 [14].

Figure 10. Diagram of diversity combining receiver for ACO-OFDM.

#### 4.3. Proposed iterative receiver

Although the diversity combining receiver exploits the signal on even subcarriers, it is not performed in an optimal way, resulting in possible performance loss compared to optimal joint detection. Here, we propose an iterative receiver that has a better way to exploit the signal on even subcarriers [14]. The basic idea is to re-estimate the modulated data in a complete mathematical model at each iteration. At the very first iteration, the basic receiver is used for initialization. The details are given as follows.

Define an N by N/2 matrix P<sup>0</sup> whose odd rows form an identity matrix and even rows are all zeros. Similarly, define another N by N/2 matrix P<sup>e</sup> whose even rows form an identity matrix and odd rows are all zeros. Then, we have

$$X = \mathbf{P}\_0 \mathbf{U}, \mathbf{U} = \mathbf{P}\_0^T \mathbf{X} = \mathbf{P}\_0^T \mathbf{P}\_0 \mathbf{U}. \tag{24}$$

In addition, based on Eqs. (16) and (19), we have

$$P\_0 \, ^T X\_c = \frac{1}{2} \, U\_r \tag{25}$$

$$P\_{\varepsilon} \, ^T X\_{\varepsilon} = \frac{1}{2} \, W|\mathbf{x}| = \frac{1}{2} \, W \mathbf{S} \mathbf{x} = \frac{1}{2} \, W \mathbf{S} \mathbf{W}^H \mathbf{X} = \frac{1}{2} \, W \mathbf{S} \mathbf{W}^H \mathbf{P}\_0 \mathbf{U},\tag{26}$$

where W is the FFT matrix and S is a diagonal matrix whose entries on the main diagonal are the signs of x. Then, Eq. (15) could be decomposed to

$$\mathbf{Y}\_{odd} \stackrel{\Delta}{=} \mathbf{P}\_0 \mathbf{^T Y} = \mathbf{P}\_0 \mathbf{^T X}\_c + \mathbf{P}\_0 \mathbf{^T Z} = \frac{1}{2} \mathbf{U} + \mathbf{Z}\_{odd} \tag{27}$$

$$\boldsymbol{Y}\_{\epsilon\epsilon\epsilon\pi} \stackrel{\triangle}{=} \boldsymbol{P}\_{\epsilon} \,^{\mathsf{T}} \boldsymbol{Y} = \boldsymbol{P}\_{\epsilon} \,^{\mathsf{T}} \boldsymbol{X}\_{\epsilon} + \boldsymbol{P}\_{\epsilon} \,^{\mathsf{T}} \boldsymbol{Z} = \frac{1}{2} \boldsymbol{P}\_{\epsilon} \,^{\mathsf{T}} \boldsymbol{W} \boldsymbol{S} \boldsymbol{W}^{H} \boldsymbol{P}\_{0} \boldsymbol{U} + \boldsymbol{Z}\_{\epsilon\epsilon\pi\epsilon} \tag{28}$$

Collecting Eqs. (27) and (28) together, we have

$$
\mathcal{R} = \mathcal{Q}U + V\_{\prime} \tag{29}
$$

where

$$\mathcal{R} = \begin{bmatrix} Y\_{odd} \\ Y\_{even} \end{bmatrix}, \mathcal{Q} = \frac{1}{2} \begin{bmatrix} I \\ \mathbf{P}\_e^{\ \mathbf{T}} WSW^H \mathbf{P}\_\theta U \end{bmatrix}, V = \begin{bmatrix} Z\_{odd} \\ Z\_{even} \end{bmatrix} \tag{30}$$

where I denotes the identity matrix of proper size. Eq. (29) is a complete signal model of the received signal with respect to the information symbol. Therefore, based on Eq. (29), we can readily get the estimation of U by

$$
\widehat{\mathbf{U}} = \begin{cases}
& \left(\boldsymbol{\varrho}^{H}\boldsymbol{\varrho}\right)^{-1}\boldsymbol{\varrho}^{H}\boldsymbol{R}, & \boldsymbol{Z}\boldsymbol{F} \\
& \left(\boldsymbol{\varrho}^{H}\boldsymbol{\varrho} + \sigma\_{\mathfrak{u}}{}^{2}\boldsymbol{I}\right)^{-1}\boldsymbol{\varrho}^{H}\boldsymbol{R}, & \text{MMSE}
\end{cases}
\tag{31}
$$

Note that Q is in fact a function of U due to the component S. However, at each iteration, we assume Q is known by substituting U \_ from previous iteration to get Q. At the first iteration, the basic receiver is used to get the initial estimate of U \_ .

## 5. Performance comparison

#### 5.1. Spectral efficiency

4.3. Proposed iterative receiver

96 Optical Fiber and Wireless Communications

initialization. The details are given as follows.

and odd rows are all zeros. Then, we have

Pe

Yeven ¼ <sup>Δ</sup> Pe

Collecting Eqs. (27) and (28) together, we have

<sup>R</sup> <sup>¼</sup> Yodd Yeven " #

> U \_

readily get the estimation of U by

where

In addition, based on Eqs. (16) and (19), we have

TX<sup>c</sup> <sup>¼</sup> <sup>1</sup> 2

the signs of x. Then, Eq. (15) could be decomposed to

Yodd ¼ <sup>Δ</sup> P<sup>0</sup>

TY <sup>¼</sup> Pe

Although the diversity combining receiver exploits the signal on even subcarriers, it is not performed in an optimal way, resulting in possible performance loss compared to optimal joint detection. Here, we propose an iterative receiver that has a better way to exploit the signal on even subcarriers [14]. The basic idea is to re-estimate the modulated data in a complete mathematical model at each iteration. At the very first iteration, the basic receiver is used for

Define an N by N/2 matrix P<sup>0</sup> whose odd rows form an identity matrix and even rows are all zeros. Similarly, define another N by N/2 matrix P<sup>e</sup> whose even rows form an identity matrix

> TX<sup>c</sup> <sup>¼</sup> <sup>1</sup> 2

where W is the FFT matrix and S is a diagonal matrix whose entries on the main diagonal are

TXc <sup>þ</sup> <sup>P</sup><sup>0</sup>

TZ <sup>¼</sup> <sup>1</sup> 2 Pe

I

where I denotes the identity matrix of proper size. Eq. (29) is a complete signal model of the received signal with respect to the information symbol. Therefore, based on Eq. (29), we can

�1

2IÞ �1

TWSWHP0U " #

QHR, ZF

QHR, MMSE

TX <sup>¼</sup> <sup>P</sup><sup>0</sup>

WSWHX <sup>¼</sup> <sup>1</sup>

TZ <sup>¼</sup> <sup>1</sup> 2 2

TP0U: <sup>ð</sup>24<sup>Þ</sup>

WSWHP0U, <sup>ð</sup>26<sup>Þ</sup>

U þ Zodd, ð27Þ

, ð30Þ

ð31Þ

TWSWHP0<sup>U</sup> <sup>þ</sup> Zeven: <sup>ð</sup>28<sup>Þ</sup>

R ¼ QU þ V, ð29Þ

, <sup>V</sup> <sup>¼</sup> Zodd Zeven " #

U, ð25Þ

X ¼ P0U, U ¼ P<sup>0</sup>

P0

WSx <sup>¼</sup> <sup>1</sup> 2

<sup>W</sup>jxj ¼ <sup>1</sup> 2

TY <sup>¼</sup> <sup>P</sup><sup>0</sup>

, <sup>Q</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>¼</sup> <sup>ð</sup>QHQ<sup>Þ</sup>

(

<sup>ð</sup>QHQþσ<sup>n</sup>

Pe

TXc <sup>þ</sup> Pe

In this section, we give a comparison on the spectral efficiencies of different optical OFDM systems.

For DCO-OFDM, each OFDM symbol only contains N/2 information-bearing complexmodulated symbols. Assuming the modulation order is M, then the spectral efficiency of DCO-OFDM is given by

$$
\eta\_{\rm DCO-OFDM} = \frac{1}{2} \log\_2 \text{M bits/s/Hz}.\tag{32}
$$

For all the non-DC-biased optical OFDM systems, as redundancy (zeros) is used in either frequency domain (ACO-OFDM and PAM-DMT) or time expansion is used (flip-OFDM), the spectral efficiencies are only half of DCO-OFDM:

$$
\eta\_{\rm ACO-OFDM} = \eta\_{\rm PM-DM} = \eta\_{\rm Filp-OFDM} = \frac{1}{4} \log\_2 M \,\text{bits/s/Hz}.\tag{33}
$$

For hybrid systems, things are a little bit complex. There is no general expression but one has to analyze the specific system.

For ADO-OFDM, in addition to a conventional ACO-OFDM transmission on odd subcarriers, a half-rate DCO-OFDM is used on even subcarriers. Therefore, its spectral efficiency is

$$
\eta\_{\text{ADO}-\text{OFDM}} = \eta\_{\text{ACO}-\text{OFDM}} + \frac{1}{2}\eta\_{\text{DCO}-\text{OFDM}} = \frac{1}{2}\log\_2 \text{M bits/s/Hz}.\tag{34}
$$

For HACO-OFDM, a similar expression could be obtained:

$$
\eta\_{\text{HACO}-\text{OFDM}} = \eta\_{\text{ACO}-\text{OFDM}} + \frac{1}{2}\eta\_{\text{PAM}-\text{DMT}} = \frac{3}{8}\log\_2 \text{M bits/s/Hz}.\tag{35}
$$

For eU-OFDM, the spectral efficiency depends on the number of layers. For an L-layer system, the spectral efficiency is given by

$$\eta\_{\rm elI-OFDM}(L) = \sum\_{l=1}^{L} \left(\frac{1}{2}\right)^{l-1} \eta\_{\rm Filp-OFDM} = 2\left(1 - \frac{1}{2^{L-1}}\right) \eta\_{\rm Filp-OFDM}.\tag{36}$$

When L approaches infinity, we have the upper bound of spectral efficiency:

$$
\eta\_{\rm elI-OFDM} = 2\eta\_{\rm Flip-OFDM} = \frac{1}{2}\log\_2 M \,\text{bits/s/Hz}.\tag{37}
$$

There is a tradeoff between the spectral efficiency and decoding complexity with respect to L: a larger L means the spectral efficiency is closer to its upper bound but the decoding complexity, mainly from signal cancellation for decoded layers, will increase linearly with L. In practice, a fairly small L is desired to achieve a balance between the spectral efficiency and decoding complexity, say, for example, L = 5 is a good choice. In fact, when L = 5, we have

$$
\eta\_{elI-OFDM}(5) = 2\left(1 - \frac{1}{2^{5-1}}\right)\eta\_{flip-OFDM} = 0.94\eta\_{elI-OFDM} \tag{38}
$$

which shows that the spectral efficiency is very close to the upper bound.

On summarizing, we can see that DCO-OFDM has the highest spectral efficiency. However, its power efficiency is not very good due to the non-information-bearing DC. On the contrary, the non-DC-biased optical OFDM systems have better power efficiency due to the elimination of DC offset but their spectral efficiency is only half of that of DCO-OFDM. The hybrid systems, especially ADO-OFDM and eU-OFDM, have better balance between the spectral efficiency and power efficiency. With those facts, one can choose a proper implementation form in practice under specific communication requirement and constraint.

#### 5.2. Receiver complexity

The computational complexity of different receivers is analyzed here using the order notation. For the basic receiver, the main computation burden is the FFT and equalization, which have complexities of ΟðNlog2NÞ and ΟðNÞ, respectively. In total, it is just ΟðNlog2NÞ. For the diversity combining receiver, it involves finite number of FFT/IFFT and a final equalization. Therefore, although it is more complex than the basic receiver, there is no difference when considering the order notation, that is, it is still ΟðNlog2NÞ. For the proposed iterative receiver, its main computation burden is the matrix inversion of Eq. (31) and this operation should be repeated at each iteration. Thus, the total complexity is in the order of <sup>Ο</sup>ðTN<sup>3</sup> Þ, where T is the total number of iterations. As we can see, this receiver is the most complicated one among the three receivers. However, as will be shown later, its performance is the best and can be far better than the other two. Thus, it is acceptable considering the performance gains. In addition, with the rapid development of modern signal-processing hardware, the computation burden will not be a limiting factor for these small-scale computations.

#### 6. Simulations

In this section, we compare the average uncoded BER performance of different receivers in VLC channels through simulations. The channels are generated using the method in [15] with the following configurations: an empty room of size 8 · 6 · 4 m with reflection coefficients 0.8, 0.8, and 0.3 for the ceiling, the walls, and the floor, respectively; LEDs are used as the optical source and they are attached 0.1-m below the ceiling. The photodetectors (PDs) are 1 m above the floor with an 80 of field of view (FOV). Both line-of-sight (LOS) and nonline-ofsight (NLOS) channels are tested (the LEDs point straight downward and upward, respectively). Multiple LEDs and PDs are used to enhance the performance and robustness of the communication link, resulting in a multiple-input multiple-output (MIMO) channel model. The receiver design methods described in Section 4 can be easily extended to this channel model by using vector notation. For each MIMO channel realization, the positions of the LEDs and the photodetectors are randomly drawn from their corresponding plains and the channels are normalized to have power NRNT, where NR and NT denote the number of PDs and LEDs, respectively. The ill-conditioned channels are rejected for fair comparison. The number of subcarriers is N = 64. In the legend of figures, "conventional" denotes the basic receiver, "pairwise ML" denotes the receiver in [13], which is a special case of the diversity combining receiver. "Lower bound" denotes the ideal curve of the proposed receiver with perfect estimation of matrix Q. In all receivers, MMSE equalization is used.

<sup>η</sup>eU�OFDM <sup>¼</sup> <sup>2</sup>ηFlip�OFDM <sup>¼</sup> <sup>1</sup>

complexity, say, for example, L = 5 is a good choice. In fact, when L = 5, we have

which shows that the spectral efficiency is very close to the upper bound.

25�<sup>1</sup> 

On summarizing, we can see that DCO-OFDM has the highest spectral efficiency. However, its power efficiency is not very good due to the non-information-bearing DC. On the contrary, the non-DC-biased optical OFDM systems have better power efficiency due to the elimination of DC offset but their spectral efficiency is only half of that of DCO-OFDM. The hybrid systems, especially ADO-OFDM and eU-OFDM, have better balance between the spectral efficiency and power efficiency. With those facts, one can choose a proper implementation form in practice

The computational complexity of different receivers is analyzed here using the order notation. For the basic receiver, the main computation burden is the FFT and equalization, which have complexities of ΟðNlog2NÞ and ΟðNÞ, respectively. In total, it is just ΟðNlog2NÞ. For the diversity combining receiver, it involves finite number of FFT/IFFT and a final equalization. Therefore, although it is more complex than the basic receiver, there is no difference when considering the order notation, that is, it is still ΟðNlog2NÞ. For the proposed iterative receiver, its main computation burden is the matrix inversion of Eq. (31) and this operation should be

total number of iterations. As we can see, this receiver is the most complicated one among the three receivers. However, as will be shown later, its performance is the best and can be far better than the other two. Thus, it is acceptable considering the performance gains. In addition, with the rapid development of modern signal-processing hardware, the computation burden

In this section, we compare the average uncoded BER performance of different receivers in VLC channels through simulations. The channels are generated using the method in [15] with the following configurations: an empty room of size 8 · 6 · 4 m with reflection coefficients 0.8, 0.8, and 0.3 for the ceiling, the walls, and the floor, respectively; LEDs are used as the

repeated at each iteration. Thus, the total complexity is in the order of <sup>Ο</sup>ðTN<sup>3</sup>

will not be a limiting factor for these small-scale computations.

<sup>η</sup>eU�OFDMð5Þ ¼ 2 1 � <sup>1</sup>

under specific communication requirement and constraint.

5.2. Receiver complexity

98 Optical Fiber and Wireless Communications

6. Simulations

2

There is a tradeoff between the spectral efficiency and decoding complexity with respect to L: a larger L means the spectral efficiency is closer to its upper bound but the decoding complexity, mainly from signal cancellation for decoded layers, will increase linearly with L. In practice, a fairly small L is desired to achieve a balance between the spectral efficiency and decoding

log2M bits=s=Hz: ð37Þ

<sup>η</sup>Flip�OFDM <sup>¼</sup> <sup>0</sup>:94ηeU�OFDM, <sup>ð</sup>38<sup>Þ</sup>

Þ, where T is the

Figure 11 shows a sample view of the impulse response of the LOS and NLOS channels with a sampling rate of 300 MHz. It can be seen that the LOS channel is more like a delta function but NLOS channel has relatively longer delay spread, which means it can be viewed as a multipath channel.

First, we compare the BER performance in single-input single-output (SISO) channel. Figure 12 shows the performance with modulation order M = 64. It can be seen that in LOS channel, the

Figure 11. A sample view of LOS and NLOS channels with 300-MHz sampling rate.

Figure 12. BER comparison of SISO ACO-OFDM using modulation sizes of M = 64.

diversity combining receiver and the proposed iterative receiver have similar performance and are much better than the basic receiver. In addition, their performance gap to the lower bound is limited. However, in NLOS channel, things are different: the proposed iterative receiver has the best performance. Compared to the basic receiver, its performance gain is more than 10 dB at high SNR range, which is significant. Even compared to the diversity combining receiver, 1 dB gain could be observed.

Now, we turn to MIMO channels. Figure 13 shows the BER performance of a 4 · 4 MIMO ACO-OFDM using modulation sizes of M = 16. It can be seen that the performance of different receivers has similar behavior as the SISO case. However, compared to the SISO case, the performance gain of the proposed receiver is even larger: compared to the diversity combining receiver, it is more than 6 dB at high SNR regime; compared to the basic receiver, it is much more than 10 dB. Nonetheless, there is still a fairly large performance gap between the proposed receiver and its lower bound, which indicates that more advanced signal processing at the receiver side is desired in the future.

Figure 13. BER comparison of 4 · 4 MIMO ACO-OFDM using modulation sizes of M = 16.

### 7. Conclusions

diversity combining receiver and the proposed iterative receiver have similar performance and are much better than the basic receiver. In addition, their performance gap to the lower bound is limited. However, in NLOS channel, things are different: the proposed iterative receiver has the best performance. Compared to the basic receiver, its performance gain is more than 10 dB at high SNR range, which is significant. Even compared to the diversity combining receiver, 1-

Figure 12. BER comparison of SISO ACO-OFDM using modulation sizes of M = 64.

Now, we turn to MIMO channels. Figure 13 shows the BER performance of a 4 · 4 MIMO ACO-OFDM using modulation sizes of M = 16. It can be seen that the performance of different receivers has similar behavior as the SISO case. However, compared to the SISO case, the performance gain of the proposed receiver is even larger: compared to the diversity combining receiver, it is more than 6 dB at high SNR regime; compared to the basic receiver, it is much more than 10 dB. Nonetheless, there is still a fairly large performance gap between the proposed receiver and its lower bound, which indicates that more advanced signal processing

dB gain could be observed.

100 Optical Fiber and Wireless Communications

at the receiver side is desired in the future.

In this research, we have investigated various forms of optical OFDM systems that are suitable for IM/DD optical channel. Different receivers are described with a proposed iterative one. Spectral efficiencies, computational complexities, as well as BER performance in LOS and NLOS channels of different systems and receivers are given. It is found that DCO-OFDM is more spectrally efficient than the non-DC-biased systems. The hybrid systems achieve a better tradeoff between the spectral efficiency and power efficiency. The proposed iterative receiver has the highest complexity but is far superior than other receivers, especially the basic receiver. Those results reveal the potential of OFDM systems in IM/DD channels for optical communication.

## Acknowledgements

This work is supported by Southeast University 3-Category Academic Programs Project ("Optical Wireless Communication" and "Software-Defined Radio") and Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (no. PPZY2015A035).
