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Layered hybrid PAM-DMT for IM/DD OWC systems

Zuhang Geng, Xinke Tang, Xiao-Ping Zhang, and Yuhan Dong

Open Access Open Access

Abstract

Traditional pulse-amplitude-modulated discrete multitone modulation (PAM-DMT) suffers from poor overall performance of spectral and power efficiencies in optical wireless communication (OWC) systems. We propose layered hybrid PAM-DMT (LHPAM-DMT) to utilize more subcarriers to improve the performance. The real part of frequency domain signal is divided into several layers and symmetry biases are added in time domain to generate real-valued and nonnegative signals for intensity modulation with direct detection (IM/DD) OWC systems. By appropriately designing the orthogonality between the signals in lower layers and signals & added biases in higher layers, we further propose an iterative receiver to recover the transmitted information. Theoretical derivation proves that LHPAM-DMT has higher spectral efficiency than PAM-DMT and lower complexity than layered PAM-DMT. Numerical results suggest that LHPAM-DMT is more power efficient than PAM-DMT as well as direct-current (DC) biased optical OFDM (DCO-OFDM), one of the most popular schemes.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Optical wireless communication (OWC) is promising for the wide unregulated bandwidth [1] and orthogonal frequency division multiplexing (OFDM) has been investigated due to its capability of improving spectral efficiency and resisting inter symbol interference [2–4]. Different from radio frequency communication, intensity modulation with direct detection (IM/DD) is generally used in OWC systems and requires real-valued and nonnegative signaling [5]. Hermitian symmetry is adopted in frequency domain to generate real-valued OFDM signals. To ensure nonnegative signals, several optical OFDM (O-OFDM) schemes have been proposed.

Three popular O-OFDM schemes have been proposed to facilitate OWC. Widely used direct-current (DC) biased optical OFDM (DCO-OFDM) simply adds a large DC bias to directly make time-domain signals nonnegative [6], however, suffers from high power cost and bit error rate (BER). Another two O-OFDM schemes of asymmetrically clipped optical OFDM (ACO-OFDM) [7] and pulse-amplitude-modulated discrete multitone modulation (PAM-DMT) [8] only utilize half spectral resources and directly clip the negative values in time domain. ACO-OFDM only utilizes odd subcarriers while PAM-DMT utilizes the imaginary parts of subcarriers. However, both of them suffer from low spectral and power efficiencies and require overall improvement.

To improve either or both efficiencies, one type of approaches is hybrid O-OFDM such as hybrid ACO-OFDM (HACO-OFDM) combining ACO-OFDM and PAM-DMT [9], and hybrid PAM-DMT (HPAM-DMT) employing symmetry biases into PAM-DMT [10]. The other type of approaches is layered O-OFDM including layered ACO-OFDM (LACO-OFDM) [11], [12] using several layers of ACO-OFDM and transmitting time-domain OFDM symbols from these layers simultaneously, and layered PAM-DMT (LPAM-DMT) [13] using several layers of real parts of subcarriers besides the imaginary parts in PAM-DMT and transmitting them simultaneously. Triple-layer hybrid optical OFDM (THO-OFDM) simply combines 2-layer LACO-OFDM and 1-layer PAM-DMT [14]. Nevertheless, in hybrid approaches, HACO-OFDM and THO-OFDM have relatively lower spectral efficiency while HPAM-DMT suffers poor power efficiency due to the relatively large bias. Both layered approaches recover the transmitted information layer by layer and introduce latency.

Motivated by HPAM-DMT and LPAM-DMT, we propose a novel and general efficient layered hybrid PAM-DMT (LHPAM-DMT) scheme in this paper to further improve the overall performance. We aim to make it competent in situations requiring high spectral and power efficiencies and low latency such as real-time high volume signal transmission in unmanned aerial, underwater and ground vehicles (namely UAV, UUV and UGV)-based OWC systems. To combine the superiority of LPAM-DMT and HPAM-DMT and deal with the imposed difficult signal processing, we propose a novel layered method of spectral resource allocation and appropriately design the orthogonality between the signals in lower layers and signals & added biases in higher layers. Then an iterative receiver is proposed to fully recover the desired information. LPAM-DMT and HPAM-DMT are special LHPAM-DMT schemes with the most and fewest layers, respectively. Theoretical derivation proves that LHPAM-DMT requires fewer layers than LACO-OFDM and LPAM-DMT to realize the same spectral efficiency, which implies lower computational complexity. Simulation results suggest that LHPAM-DMT is more power efficient than PAM-DMT and DCO-OFDM at moderate to high bit rates over normalized bandwidth.

2. Proposed layered hybrid PAM-DMT

The proposed LHPAM-DMT for IM/DD OWC systems improves spectral and power efficiencies by utilizing PAM-DMT to transmit the imaginary part of frequency domain and dividing the real part into several layers. At the transmitter side, an adaptive symmetry bias is added in each layer to ensure nonnegative time-domain signals. Thanks to the design that the signals and added symmetric biases in higher layers are orthogonal to signals in lower layers, the proposed iterative receive can recover the desired information layer by layer.

2.1 Transmitter of LHPAM-DMT

Figure 1 shows the block diagram of the transmitter of LHPAM-DMT scheme intending to wrap $L$ layers of HPAM-DMT signals to convey the real part of the frequency-domain signal and employ PAM-DMT signal to transmit the imaginary part of the frequency-domain signal. $X_k$ is the transmitted signal loaded on the $k$-th subcarrier and has the real and imaginary parts denoted as $X_k^{(r)}$ and $X_k^{(i)}$, respectively. As mentioned earlier, $X_k^{(r)}$ is chopped into $L$ layers each containing a real part of a single frequency-domain signal denoted as $X_k^{(rl)}$ in the $l$-th layer. Then in time domain, LHPAM-DMT signal $s_n$ is obtained by $s_n^{(i)}$ denoting time-domain signal of PAM-DMT after power offset [10] and $s_n^{(r)}=\sum _{l=1}^L s_n^{(rl)}$ where $s_n^{(rl)}$, $l=1,\ldots,L$, are time-domain signals.

 figure: Fig. 1.

Fig. 1. Block diagram of the transmitter for LHPAM-DMT.

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The proposed LHPAM-DMT can improve the overall efficiency since its transmitter is capable of adjusting the ratio of untapped subcarriers to total number of available subcarriers in each layer. This ratio is an important parameter indicating spectral efficiency of layered optical OFDM schemes. At the transmitter, an array $\vec {m}$ is set to determine the spectral efficiency and the number of layers $L$. $\vec {m}=(m_1, m_2,\ldots,m_L)$, where $m_l$ implies that the real part of $\frac {m_l-1}{m_l}$ of the available subcarriers is utilized in the $l$-th layer of LHPAM-DMT. For simplicity, we define that $M_l=\prod _{i=1}^l m_i$ and $M_0=1$.

In order to avoid introducing distortion in each layer, $\vec {m}$ is supposed to satisfy the constraint of $\mod (N, M_L)=0$ where mod$(\cdot,\cdot )$ is the modulo operator.

We utilize $\frac {(m_l-1)N}{M_l}$ subcarriers while reserve the remaining $\frac {N}{M_l}$ subcarriers in the $l$-th layer. The reserved and utilized subcarriers in the $l$-th layer are given by

$$X_k^{(rl)}= \begin{cases} 0, & {\rm mod}(k, M_l)= 0,\\ X_k^{(r)}, & {\rm mod}(k, M_{l-1})=0 \ {\rm and}\ {\rm mod}(k, M_l)\neq 0, \\ 0, & {\rm others}. \end{cases}$$

Figure 2 presents the subcarrier allocation principle of PAM-DMT, HPAM-DMT with $G=2$ and LHPAM-DMT with $\vec {m}=(4,2)$ in frequency domain before Hermitian symmetry.

 figure: Fig. 2.

Fig. 2. Subcarrier allocation principle of several O-OFDM schemes.

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After inverse fast Fourier transform (IFFT) operation, the output time-domain signal in the $l$-th layer is generated as

$$\begin{aligned}x_n^{(rl)}&=\frac{1}{\sqrt{N}} \sum\nolimits_{k=0}^{\frac{N}{M_{l-1}}-1} X_{M_{l-1}k}^{(rl)} {\rm exp} \left(j\frac{2 \pi}{N}n M_{l-1} k\right)\\ &=\frac{\sqrt{M_{l-1}}}{M_{l-1}} \frac{1}{\sqrt{\frac{N}{M_{l-1}}}} \sum\nolimits_{k=0}^{\frac{N}{M_{l-1}}-1} X_{k}'^{(rl)} {\rm exp} \left(j \frac{2\pi}{N/M_{l-1}}nk\right)\\ &=\frac{\sqrt{M_{l-1}}}{M_{l-1}} x_{{\rm mod}(n,N/M_{l-1})}'^{(rl)}, \end{aligned}$$
where $n,k\in \{0,1,\ldots,N/M_{l-1}-1\}$, $X_{k}'^{(rl)}=X_{M_{l-1}k}^{(rl)}$ and $x_n'^{(rl)}$ denotes the $\frac {N}{M_{l-1}}$-point IFFT result of $X_{k}'^{(rl)}$. A simple $\frac {N}{M_{l-1}}$-point IFFT operation is sufficient instead of $N$-point IFFT due to periodic $x_n^{(rl)}$.

To make time-domain signals in the $l$-th layer nonnegative, $x_n'^{(rl)}$ is devided into $N/M_l$ groups and each group contains $m_l$ signal samples, i.e., $\left \{x_n'^{(rl)}, x_{n+\frac {N}{M_l}}'^{(rl)},\ldots,x_{n+\frac {(m_l-1)N}{M_l}}'^{(rl)} \right \}, n=0, 1,\ldots, N/M_l-1$. In each group, the same bias is added to each signal sample. The optimized adaptive bias is

$$b_n'^{(rl)}={-}{\rm min}\left\{x_n'^{(rl)}, x_{n+\frac{N}{M_l}}'^{(rl)},\ldots,x_{n+\frac{(m_l-1)N}{M_l}}'^{(rl)} \right\}.$$

To prevent the imaginary parts of $X_k$ from distortion, the symmetry bias in layer $l$ should satisfy the constraint that $c_n^{(rl)}=c_{N-n}^{(rl)}$, $n=0, 1,\ldots, N-1$. Then the symmetry bias in the $l$-th layer $c_n^{(rl)}$ is given by

$$c_n'^{(rl)}= \begin{cases} b_0'^{(rl)}, & n=0\\ {\rm max}\left\{b_n'^{(rl)}, b_{\frac{N}{M_l}-n}'^{(rl)}\right\}, & n=1, 2,\ldots, N/M_l-1, \end{cases}$$
$$c_n^{(rl)}=c_{{\rm mod}(n,N/M_{l})}'^{(rl)},\quad n=0,1,\ldots,N-1.$$

Since $x_n^{(rl)}$ is periodic, the computational complexity decreases evidently by generating $c_n'^{(rl)}$ and repeating $M_l$ times.

Finally, the transmitted time-domain signal $s_n$, $n=0,1,\ldots,N-1$, is given by

$$s_n=s_n^{(i)}+s_n^{(r)}=s_n^{(i)}+\sum\nolimits_{l=1}^L x_n^{(rl)}+\sum\nolimits_{l=1}^L c_n^{(rl)}.$$

2.2 Receiver of LHPAM-DMT

At the reciever side, the received signal $y_n$ and the FFT output $Y_k$ , $\forall n, k\in \{0,1,\ldots,N-1\}$, are expressed as

$$y_n=s_n+w_n, $$
$$Y_k=S_k^{(i)}+X_k^{(r)}+\sum\nolimits_{l=1}^L C_k^{(rl)}+W_k,$$
where the additive white Gaussian noise $w_n$ includes shot noise and thermal noise similarly to [15]. $S_k^{(i)}$, $C_k^{(rl)}$, and $W_k$ in Eq. (8) denote the $N$-point FFT results of $s_n^{(i)}$, $c_n^{(rl)}$, and $w_n$, respectively. The interference in frequency domain is
$$\begin{aligned} C_k^{(rl)}&=\frac{1}{\sqrt{N}} \sum\nolimits_{n=0}^{N-1} c_n^{(rl)} {\rm exp} \left({-}j\frac{2 \pi kn}{N} \right)\\ &=\frac{1}{\sqrt{N}} \sum_{n=0}^{\frac{N}{M_l}-1} c_n^{(rl)} \sum_{i=0}^{M_l-1} {\rm exp} \left({-}j\frac{2\pi k}{N}\left(n+\frac{iN}{M_l}\right)\right)\\ &=\frac{1}{\sqrt{N}} \sum_{n=0}^{\frac{N}{M_l}-1} c_n^{(rl)} {\rm exp} \left({-}j\frac{2 \pi kn}{N} \right) \sum_{i=0}^{M_l-1} {\rm exp} \left({-}j\frac{2\pi ki}{M_l} \right),\\ &k=0, 1,\ldots, N-1, \end{aligned}$$
where the second line of Eq. (9) is obtained due to periodic $c_n^{(rl)}$.

The imaginary parts of $C_k^{(rl)}$ are obtained by

$$\begin{aligned} &{\rm Im}\left(C_k^{(rl)}\right)=\frac{-j}{\sqrt{N}} \sum\nolimits_{n=0}^{N-1} c_n^{(rl)} \sin \left(\frac{2 \pi kn}{N} \right)\\ &=\frac{-j}{2\sqrt{N}} \sum\nolimits_{n=1}^{N-1} c_n^{(rl)} \left(\sin \left(\frac{2 \pi kn}{N} \right)+ \sin \left(-\frac{2 \pi kn}{N} \right) \right)\\ &=0. \end{aligned}$$

If mod$(k,M_l)\neq 0$, we have

$$\sum\nolimits_{i=0}^{M_l-1} {\rm exp} \left({-}j\frac{2\pi ki}{M_l} \right)=0.$$

Substitute Eq. (11) into Eq. (9), we can obtain that

$$C_k^{(rl)}=0.$$

By this appropriate design of Eq. (1) and Eq. (5), the symmetry bias in each layer is orthogonal to the transmitted PAM-DMT signals as shown in Eq. (10) and the symmetry biases added in higher layers are orthogonal to the transmitted signals in lower layers as shown in Eq. (12). We therefore propose an iterative receiver to recover the desired information depicted in Fig. 3.

 figure: Fig. 3.

Fig. 3. Block diagram of iterative receiver for LHPAM-DMT.

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First of all, the receiver detects PAM-DMT symbols and generates $\hat {Y}_k^{(r)}$ by subtracting the estimated PAM-DMT signal $\hat {S}_k^{(i)}$ from $Y_k$. Then the transmitted symbols in the real parts of subcarriers are demodulated layer by layer. In the $l$-th layer, $\hat {Y}_k^{(rl)}$ denotes the frequency-domain signal before demodulation. In order to remove the interference caused by the the symmetry biases added in lower layers, $\hat {Y}_k^{(rl)}$ is obtained by

$$\hat{Y}_k^{(rl)}= \begin{cases} \hat{Y}_k^{(r)},\quad & l=1\\ \hat{Y}_k^{(r)}-\sum_{p=1}^{l-1} \hat{S}_k^{(rp)}, \quad & l\geq2 \end{cases}$$
where $\hat {S}_k^{(rp)}$ denotes the estimated frequency-domain signal in the $p$-th layer. We detect HPAM-DMT symbols and estimate the transmitted signals in each layer, then the frequency-domain signal for demodulation in next layer is generated by Eq. (13). The desired information can be recovered layer by layer by repeating the operations above.

3. Spectral efficiency and complexity analysis

3.1 Spectral efficiency

In LHPAM-DMT, the imaginary parts of $(N-2)$ subcarriers and the real parts of $(N-N/M_L)$ subcarriers are utilized. If the cyclic prefix is simply omitted, the channel capacity and spectral efficiency are respectively presented as

$$C_{\rm LHPAM}=\frac{ 2N-2-N/M_L}{4N}W\log_2 \left(1+\frac{P_x}{P_N}\right), $$
$$\eta_{\rm LHPAM}=\frac{C_{\rm LHPAM}}{W}, $$
where $W$ and $P_N$ denote the bandwidth and noise power, respectively. Especially, given that the added biases contain no information, $P_x$ denotes the power of original time-domain signals without added biases.

We summarize the spectral efficiencies of other O-OFDM schemes in Table 1. The proposed LHPAM-DMT has higher spectral efficiency than PAM-DMT and HACO-OFDM. The layered O-OFDM schemes approximately reaches the same spectral efficiency when $M_{L_1}=2^{L_2}=2^{L_3+1}$. $L_1$, $L_2$, and $L_3$ are layer numbers of LHPAM-DMT, LPAM-DMT, and LACO-OFDM, respectively. Therefore LHPAM-DMT is able to obtain the same spectral efficiency as LPAM-DMT and LACO-OFDM with fewer layers.

Tables Icon

Table 1. Spectral efficiencies of various O-OFDM schemes.

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3.2 Computational complexity

The computational complexity mainly comes from modulation and demodulation operations, bias calculation, and IFFT and FFT operations.

At the transmitter side, O-OFDM schemes with the same spectral efficiency share the same computational complexity from modulation operations. At the receiver side, symbols in the highest layer only need to be demodulated once, however both demodulation and modulation operations are required for other symbols. Given that only a small number of symbols are allocated in the highest layer, which is negligible, layered O-OFDM schemes with the same spectral efficiency usually have similar computational complexity from modulation and demodulation operations.

In terms of bias calculation, we assume an LHPAM-DMT scheme with $\vec {m}=(m_1, m_2,\ldots, m_L)$ and $M_l=\prod _{i=1}^l m_i$ without loss of generality. In the $l$-th layer, the time-domain signal is divided into $\frac {N}{2M_l}$ groups and each group contains $2m_l$ values. Due to the symmetry of time-domain signals, calculating the minimum of $m_l$ values in each group is enough, which requires $(m_l-1)$ times of comparison operation. Therefore, the computational complexity from bias calculation is given by

$$\begin{aligned}C_{BC}&=\sum_{l=1}^{L}\frac{N}{2M_l}\left(m_l-1\right)\\ &=\sum_{l=1}^{L}\left(\frac{N}{2M_{l-1}}-\frac{N}{2M_l}\right)\\ &=\frac{N}{2}\left(1-\frac{1}{M_L}\right), \end{aligned}$$
which implies that layered O-OFDM schemes with the same spectral efficiency and total number of subcarriers share the same computational complexity from bias calculation.

As a result, the computational complexity of layered O-OFDM schemes mainly differs in IFFT and FFT operations and can be evaluated according to the real multiplications at the transmitter and receiver sides. The complexity of each $N$-point IFFT or FFT is approximately ${\rm O}(N\log _2N)$ [16]. Given that the input of IFFT and FFT is either imaginary-valued or real-valued in LHPAM-DMT, half of the real multiplications can be left out.

The transmitter of LHPAM-DMT utilizes an $N$-point IFFT in PAM-DMT and divide other signals into several layers. Then the $l$-th layer employs IFFT and FFT blocks with size $N_{\rm LHPAM}^{(l)}=N$ for $l=1$ and $N_{\rm LHPAM}^{(l)}=\frac {N}{M_{l-1}}$ for $l\geq 2$.

Then, the complexities of transmitter and iterative receiver of LHPAM-DMT are given by

$$\begin{aligned}Q_{\rm TX}&=\frac{1}{2}\left({\rm O}(N\log_2N)+\sum_{l=1}^{L} {\rm O}\left(N_{\rm LHPAM}^{(l)}\log_2 N_{\rm LHPAM}^{(l)}\right)\right)\\ &\approx \left(1+\sum\nolimits_{l=2}^{L} \frac{1}{2M_{l-1}}\right) {\rm O}\left(N\log_2N\right)\end{aligned}$$
$$\begin{aligned}Q_{\rm RX}&=\frac{1}{2}\left({\rm O}\left(N\log_2N\right)+2{\rm O}\left(N\log_2N\right)\right)\\ &+\frac{1}{2}\sum\nolimits_{l=1}^{L-1} 2{\rm O}\left(N_{\rm LHPAM}^{(l)}\log_2 N_{\rm LHPAM}^{(l)}\right)\\&\approx \left(\frac{3}{2}+\sum\nolimits_{l=1}^{L-1} \frac{1}{M_{l-1}}\right) {\rm O}\left(N\log_2N\right), L\geq 2 \end{aligned}$$

Table 2 shows the computational complexities of various O-OFDM schemes including LPAM-DMT with 6 layers and LHPAM-DMT with various $\vec {m}$. For a fair comparison, these O-OFDM schemes are selected with the same spectral efficiency. The computational complexity of the proposed LHPAM-DMT is lower at both transmitter and receiver sides compared with that of the traditional LPAM-DMT and can be further decreased by choosing appropriate configuration.

Tables Icon

Table 2. Computational complexity of LPAM-DMT and LHPAM-DMT.

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4. Numerical results

In this section, we evaluate the proposed LHPAM-DMT as well as the comparison with conventional O-OFDM schemes for IM/DD OWC systems.

Figure 4 shows the BER performance of these O-OFDM schemes. Similar to [17], the optical bit energy to noise power ratio $E_{b({\rm opt})}/N_0$ is normalized for all these schemes. Especially, HPAM-DMT with $G=4$ and LPAM-DMT with $L=4$ can be regarded as special LHPAM-DMT with $\vec {m}=(16)$ and $\vec {m}=(2,2,2,2)$, respectively. It is shown that fewer layers in LHPAM-DMT corresponds to higher BER performance. The reason is that relatively larger symmetry biases added in LHPAM-DMT with fewer layers lead to larger noise power at the same $E_{b({\rm opt})}/N_0$. The error floor of DCO-OFDM with 7 dB bias level is caused by the clipped noise. The proposed LHPAM-DMT outperforms DCO-OFDM in terms of BER performance while it is less desirable compared with PAM-DMT. Considering that LHPAM-DMT has much higher spectral efficiency than PAM-DMT, the disparity in BER performance is acceptable.

 figure: Fig. 4.

Fig. 4. BER performance versus $E_{b({\rm opt})}/N_0$ for DCO-OFDM with 7 dB, 10 dB and 13 dB bias levels, PAM-DMT, HPAM-DMT with $G=4$, LPAM-DMT with $L=4$, and proposed LHPAM-DMT with $\vec {m}=(4,4)$ and $\vec {m}=(2,4,2)$. 64-QAM and 8-PAM are adopted and the number of subcarriers is $N=1024$.

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In Fig. 5, we plot $<E_{b({\rm opt})}/N_0>_{\rm BER}$ versus bit rate over normalized bandwidth for various optical OFDM schemes and give consideration to both BER performance and spectral efficiency, where $<E_{b({\rm opt})}/N_0>_{\rm BER}$ denotes the required $E_{b({\rm opt})}/N_0$ for $10^{-3}$ BER target while bit rate over normalized bandwidth is obtained by the constellations and spectral efficiency. From Fig. 5, it is observed that LHPAM-DMT requires lower $E_{b({\rm opt})}/N_0$ than DCO-OFDM and PAM-DMT to reach the target BER in the regime of bit rate over normalized bandwidth of 4 to 10 bps/Hz. It is shown that PAM-DMT has relatively lower spectral efficiency which results in the larger slope than other optical OFDM schemes. Furthermore, DCO-OFDM with 7 dB and 10 dB bias levels suffer from the clipping noise and can not reach the target BER with large constellation sizes, which leads to several missing points in Fig. 5. In consideration of the facts mentioned above, the proposed LHPAM-DMT has higher power efficiency compared with its conventional counterparts.

 figure: Fig. 5.

Fig. 5. The required $E_{b({\rm opt})}/N_0$ for the target BER of $10^{-3}$ against bit rate over normalized bandwidth for DCO-OFDM with 7 dB, 10 dB and 13 dB bias levels, PAM-DMT, HPAM-DMT with $G=4$, LPAM-DMT with $L=4$, and proposed LHPAM-DMT with $\vec {m}=(4,4)$ and $\vec {m}=(2,4,2)$. 4-, 16-, 64-, 256- and 1024-QAM and 2-, 4-, 8-, 16- and 32-PAM are adopted.

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5. Conclusion

In this paper, we proposed a novel LHPAM-DMT scheme with appropriate signal design for IM/DD OWC systems. By dividing the real parts of subcarriers into several layers, LHPAM-DMT is capable of reaching higher spectral efficiency than conventional PAM-DMT and lower complexity than layered PAM-DMT as theoretical analysis. Numerical results suggest that the proposed LHPAM-DMT is more power efficient than PAM-DMT and DCO-OFDM. It is plausible that LHPAM-DMT OWC systems enjoy the benefits of design flexibility due to the adjustable parameter $m$ in each layer.

Funding

Basic and Applied Basic Research Foundation of Guangdong Province (2022A1515010209); Natural Science Foundation of Shenzhen Municipality (JCYJ20200109143016563).

Acknowledgments

Y. D. thanks the support from Shenzhen Engineering Laboratory of Smart Metropolitan Sensing Networks. Corresponding authors are X. T. and Y. D.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Block diagram of the transmitter for LHPAM-DMT.

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Fig. 2.
Fig. 2. Subcarrier allocation principle of several O-OFDM schemes.

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Fig. 3.
Fig. 3. Block diagram of iterative receiver for LHPAM-DMT.

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Fig. 4.
Fig. 4. BER performance versus $E_{b({\rm opt})}/N_0$ for DCO-OFDM with 7 dB, 10 dB and 13 dB bias levels, PAM-DMT, HPAM-DMT with $G=4$, LPAM-DMT with $L=4$, and proposed LHPAM-DMT with $\vec {m}=(4,4)$ and $\vec {m}=(2,4,2)$. 64-QAM and 8-PAM are adopted and the number of subcarriers is $N=1024$.

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Fig. 5.
Fig. 5. The required $E_{b({\rm opt})}/N_0$ for the target BER of $10^{-3}$ against bit rate over normalized bandwidth for DCO-OFDM with 7 dB, 10 dB and 13 dB bias levels, PAM-DMT, HPAM-DMT with $G=4$, LPAM-DMT with $L=4$, and proposed LHPAM-DMT with $\vec {m}=(4,4)$ and $\vec {m}=(2,4,2)$. 4-, 16-, 64-, 256- and 1024-QAM and 2-, 4-, 8-, 16- and 32-PAM are adopted.

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Tables (2)

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Table 1. Spectral efficiencies of various O-OFDM schemes.

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Table 2. Computational complexity of LPAM-DMT and LHPAM-DMT.

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Equations (18)

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X k ( r l ) = { 0 , m o d ( k , M l ) = 0 , X k ( r ) , m o d ( k , M l − 1 ) = 0   a n d   m o d ( k , M l ) ≠ 0 , 0 , o t h e r s .
x n ( r l ) = 1 N ∑ k = 0 N M l − 1 − 1 X M l − 1 k ( r l ) e x p ( j 2 π N n M l − 1 k ) = M l − 1 M l − 1 1 N M l − 1 ∑ k = 0 N M l − 1 − 1 X k ′ ( r l ) e x p ( j 2 π N / M l − 1 n k ) = M l − 1 M l − 1 x m o d ( n , N / M l − 1 ) ′ ( r l ) ,
b n ′ ( r l ) = − m i n { x n ′ ( r l ) , x n + N M l ′ ( r l ) , … , x n + ( m l − 1 ) N M l ′ ( r l ) } .
c n ′ ( r l ) = { b 0 ′ ( r l ) , n = 0 m a x { b n ′ ( r l ) , b N M l − n ′ ( r l ) } , n = 1 , 2 , … , N / M l − 1 ,
c n ( r l ) = c m o d ( n , N / M l ) ′ ( r l ) , n = 0 , 1 , … , N − 1.
s n = s n ( i ) + s n ( r ) = s n ( i ) + ∑ l = 1 L x n ( r l ) + ∑ l = 1 L c n ( r l ) .
y n = s n + w n ,
Y k = S k ( i ) + X k ( r ) + ∑ l = 1 L C k ( r l ) + W k ,
C k ( r l ) = 1 N ∑ n = 0 N − 1 c n ( r l ) e x p ( − j 2 π k n N ) = 1 N ∑ n = 0 N M l − 1 c n ( r l ) ∑ i = 0 M l − 1 e x p ( − j 2 π k N ( n + i N M l ) ) = 1 N ∑ n = 0 N M l − 1 c n ( r l ) e x p ( − j 2 π k n N ) ∑ i = 0 M l − 1 e x p ( − j 2 π k i M l ) , k = 0 , 1 , … , N − 1 ,
I m ( C k ( r l ) ) = − j N ∑ n = 0 N − 1 c n ( r l ) sin ⁡ ( 2 π k n N ) = − j 2 N ∑ n = 1 N − 1 c n ( r l ) ( sin ⁡ ( 2 π k n N ) + sin ⁡ ( − 2 π k n N ) ) = 0.
∑ i = 0 M l − 1 e x p ( − j 2 π k i M l ) = 0.
C k ( r l ) = 0.
Y ^ k ( r l ) = { Y ^ k ( r ) , l = 1 Y ^ k ( r ) − ∑ p = 1 l − 1 S ^ k ( r p ) , l ≥ 2
C L H P A M = 2 N − 2 − N / M L 4 N W log 2 ⁡ ( 1 + P x P N ) ,
η L H P A M = C L H P A M W ,
C B C = ∑ l = 1 L N 2 M l ( m l − 1 ) = ∑ l = 1 L ( N 2 M l − 1 − N 2 M l ) = N 2 ( 1 − 1 M L ) ,
Q T X = 1 2 ( O ( N log 2 ⁡ N ) + ∑ l = 1 L O ( N L H P A M ( l ) log 2 ⁡ N L H P A M ( l ) ) ) ≈ ( 1 + ∑ l = 2 L 1 2 M l − 1 ) O ( N log 2 ⁡ N )
Q R X = 1 2 ( O ( N log 2 ⁡ N ) + 2 O ( N log 2 ⁡ N ) ) + 1 2 ∑ l = 1 L − 1 2 O ( N L H P A M ( l ) log 2 ⁡ N L H P A M ( l ) ) ≈ ( 3 2 + ∑ l = 1 L − 1 1 M l − 1 ) O ( N log 2 ⁡ N ) , L ≥ 2
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