Spectrum-efficient hybrid PAM-DMT for intensity-modulated optical wireless communication

Baolong Li; Baolong Li; Simeng Feng; Wei Xu

doi:10.1364/OE.392127

1. Introduction

Optical wireless communication (OWC) utilizes light waveforms to convey data information, which has gained continuously increasing research attentions due to the abundant and unregulated communication bandwidth [1,2]. Compared to radio frequency (RF) communication, OWC enjoys numerous attractive advantages, such as inherent high security, no electromagnetic pollution, low cost. Given the RF spectrum scarcity and the explosive data growth, OWC is deemed as a promising complementary technique to RF for realizing the ubiquitous wireless coverage [3].

Owing to high spectrum efficiency and ability of resistance to inter-symbol interference, orthogonal frequency division multiplexing (OFDM) has been widely utilized in OWC to achieve ultra high-speed information transmission [4,5]. Different from conventional OFDM schemes which are bipolar and complex in RF communication, real and non-negative OFDM signal is required in OWC since intensity modulated direct detection (IM/DD) is commonly employed [6]. A real OFDM signal can be obtained by using the frequency-domain Hermitian symmetry. In order to guarantee the non-negativity, various optical OFDM (O-OFDM) schemes have been designed in the IM/DD-based OWC, including direct-current-biased O-OFDM (DCO-OFDM), the clipping-based O-OFDM, etc [7].

In DCO-OFDM, a DC bias is directly superimposed on the bipolar signals to yield non-negative signals, which can be easily implemented [8]. However, it leads to relatively poor power efficiency since the additional DC bias results in a waste of energy [9]. Benefit to the properties of Fourier Transform, clipping-based solutions have also been conceived for OWC, such as pulse-amplitude-modulated discrete multitone modulation (PAM-DMT), and asymmetrically clipped O-OFDM (ACO-OFDM) [10,11]. In PAM-DMT and ACO-OFDM, the non-negative signal is generated through the direct clipping operation of negative parts, which averts the additional DC bias introduced in DCO-OFDM. The alternative solutions to ACO-OFDM and PAM-DMT are known as unipolar OFDM (U-OFDM) and flip OFDM (FLIP-OFDM), which produce non-negative signals by exploiting the time-domain symmetry [12,13]. For low-order modulations, these schemes can offer higher power efficiency than DCO-OFDM. However, only half of the frequency/time-domain resources are utilized in these schemes, resulting in spectrum inefficiency [9].

Against this background, some spectrally enhanced schemes have been recently developed based on the clipping-based O-OFDM. One popular scheme is known as hybrid ACO-OFDM (HACO-OFDM), which transmits the time-domain PAM-DMT and ACO-OFDM signals at the same time [14]. At the receiver side, the transmitted information data of the two signal components in HACO-OFDM can be successively detected. Furthermore, the hybrid O-OFDM proposed in [15] realizes interference-free simultaneous transmission of the ACO-OFDM and PAM-DMT signals, which offers the superiority of reduced receiver complexity in comparison with HACO-OFDM. Nevertheless, the subcarrier resources in both schemes are still far from being fully exploited since only the odd subcarriers and the real parts of even subcarriers are activated. In order to make full use of the subcarrier resources, layered ACO-OFDM (LACO-OFDM) superimposes multiple layers of the ACO-OFDM signal for simultaneous transmission [16]. The similar philosophy can be also applied to the cases of PAM-DMT and U-OFDM, leading to the architecture of augmented spectral efficiency DMT (ASE-DMT) and enhanced U-OFDM (eU-OFDM), respectively [17,18]. Superior to HACO-OFDM, they can support a higher spectrum efficiency by using more clipping-based O-OFDM components [19]. Moreover, the layers/depths in these schemes can be flexibly adjusted according to requirements, thus providing a new design freedom for the OWC system [20,21]. However, the multiple superimposed components in these schemes should be successively demodulated in an iterative way, which results in notably increased complexity and latency for reception, especially when the number of layers/depths is enlarged [22].

In this paper, we propose a novel hybrid PAM-DMT (HPAM-DMT) scheme, which transmits only two PAM-DMT signals simultaneously, but is capable of achieving the same spectrum efficiency as LACO-OFDM. To be specific, a PAM-DMT signal utilizing the real part of the subcarriers, termed as real-part PAM-DMT (RPAM-DMT), is superimposed on the classic PAM-DMT signal for simultaneous transmission in HPAM-DMT. In RPAM-DMT, the subcarriers are classified into different groups, and the number of the subcarrier groups activated for data transmission can be adjusted as required, thus maintaining the superiority of LACO-OFDM in flexibility of the subcarrier utilization. Moreover, since only two PAM-DMT components are successively detected at the receiver, HPAM-DMT possesses stable and much lower receiver complexity and processing delay than LACO-OFDM. Simulation results have shown that a better bit-error rate (BER) performance is achieved by the proposed HPAM-DMT compared to LACO-OFDM when the transmitter nonlinearity is considered. Additionally, the proposed HPAM-DMT outperforms the conventional O-OFDM schemes in terms of power efficiency, in a comparatively wide region of bit rate/normalized bandwidth.

2. Proposed hybrid PAM-DMT

The conventional PAM-DMT and ACO-OFDM possess low spectrum efficiency since only half of the subcarrier resources are exploited to transmit information data. To promote the spectrum efficiency, we explore the superposition of two PAM-DMT signals for simultaneous transmission in the proposed HPAM-DMT. The classic PAM-DMT signal is adopted as one component of HPAM-DMT, in which the PAM symbols are loaded on the imaginary parts of subcarriers [11]. The second PAM-DMT component of HPAM-DMT, namely RPAM-DMT, utilizes the real parts of the subcarriers for data transmission. The subcarriers of RPAM-DMT are divided into different groups according to the subcarrier index. Assume that the frequency is divided into $M$ subscarriers. In RPAM-DMT, the $g$-th group includes the $2^{g-1}(2u+1)$-th $\left (u=0,1,\ldots ,M/2^{g}-1\right )$ subcarriers, as demonstrated in Table 1. The real parts of the subcarriers in the first $G$ groups are utilized to transmit the PAM symbols in RPAM-DMT, while the remaining parts are unoccupied. In the classic PAM-DMT, $M-2$ subcarriers are used for data transmission, which can convey $M/2-1$ PAM symbols due to Hermitian symmetry imposed on the frequency domain. In RPAM-DMT, the number of subcarriers in the first $G$ groups is calculated as

(1)$$M_G= \sum\limits_{g = 1}^G {M/2^g} =M-M/2^G.$$

Given Hermitian symmetry, $M/2-M/2^{G+1}$ PAM symbols can be transmitted in the first $G$ groups of RPAM-DMT. Since HPAM-DMT consists of the classic PAM-DMT and RPAM-DMT, a total of $M-M/2^{G+1}- 1$ PAM symbols can be conveyed during one HPAM-DMT symbol.

Table 1. Group of the subcarriers.

View Table

To expound further, let $P_i$ represent the transmitted PAM symbols in HPAM-DMT, where $i=0,1,\ldots ,M-{M}/{2^{G+1}}-2$. Then, the PAM symbols are assigned to the imaginary parts of the subcarrier in the classic PAM-DMT and the real parts of the subcarrier in the first $G$ groups of RPAM-DMT for transmission. Given the property of Hermitian symmetry, the frequency-domain input signals to the inverse fast Fourier transform (IFFT) processor for the classic PAM-DMT and RPAM-DMT can be respectively expressed as

(2)$${X_k} = \left\{ {\begin{array}{ll} {{jP_{k-1},}} & {k = 1,2\cdots,\frac{M}{2}-1,}\\ {{-jP_{M-k-1},}} & {k =\frac{M}{2},\frac{M}{2}+1,\ldots ,M-1,}\\ 0, & {k=0,\frac{M}{2},} \end{array}} \right.$$

and

(3)$${Y_k} = \left\{ {\begin{array}{ll} {{P_{M(1 - {2^{ - g}}) + u - 1}},} & {k = {2^{g - 1}}(2u + 1),}\\ {{P_{M(1 - {2^{ - g}}) + u - 1}},} & {k = M\textrm{ - }{2^{g - 1}}(2u + 1),}\\ 0, & {\textrm{otherwise}.} \end{array}} \right.$$

By executing the IFFT operation of ${X}_k$ and ${Y}_k$, we obtain the time-domain output signals of the two PAM-DMT components, which are given by

(4)$$\begin{aligned} x_m=\frac{1}{\sqrt{M}} \sum\limits_{k=0}^{M-1} {X_k{e}^{j\frac{2\pi m k}{M}}}, m=0,1,\ldots,M-1, \\ y_m=\frac{1}{\sqrt{M}} \sum\limits_{k=0}^{M-1} {Y_k{e}^{j\frac{2\pi m k}{M}}}, m=0,1,\ldots,M-1. \end{aligned}$$

It has been proved that the negative parts of the bipolar signal $x_m$ can be directly clipped at zeros to generate the unipolar PAM-DMT signal while no information is lost [11]. Here, let $x_m^{\textrm {c}}$ represent the clipped signal of $x_n$. In the proposed HPAM-DMT, the unipolar signal $x_m^{\textrm {c}}$ and the bipolar signal $y_m$ are amalgamated for simultaneous transmission, which can be expressed as

(5)$$z_m={x}_m^{\textrm{c}}+{y}_m,~m=0,1,\ldots,M-1.$$

In OWC, the specific nature of the optical modulator restricts the transmitted signal to be non-negative. Unfortunately, $z_n$ remains bipolar, which is not suitable for the OWC transmission. Hence, an additional reconstruction of the signal is further introduced to make it non-negative. To be specific, the reconstructed signal, denoted by $c_m$, is a periodic function with period of $M/2^G$, and it satisfies the symmetry of

(6)$$c_m=c_{\frac{M}{2^G}-m},~m=1,\ldots,{M}/{2^{G+1}}-1.$$

Note that the introduced signal $c_m$ with these properties does not cause any interference contaminating the PAM symbols conveyed by the two PAM-DMT components, which will be fully elaborated in the following section.

Now the amplitude of the reconstructed signal $c_m$ is discussed. Given the periodicity of $c_m$, it is found that the reconstructed signal should be set to the same value for the signal samples $z_{m }, z_{m+\frac {M}{2^G} },\ldots ,z_{m+\frac {(2^G-2)M}{2^G} }, z_{m+\frac {(2^G-1)M}{2^G} }$, where $m=0,1,\ldots ,M/2^G-1$. The minimum amplitude of these signal samples with reversed polarity is calculated as

(7)$$b_m={-}\mathop {\min }_{0 \le i \le 2^G-1}\left\{z_{m+\frac{iM}{2^G}} \right\}, m=0,1,\ldots,M/2^G-1,$$

where $\min \left \{\cdot \right \}$ denotes the minimum in the sequence. In order to ensure the non-negativity, the minimum amplitude of these signal samples with reversed polarity can be selected as the value of the reconstructed signal added to these signal samples, i.e., $c_m=b_m$. Simultaneously, the reconstructed signal should satisfy the symmetry in Eq. (6). Therefore, $c_m$ and $c_{\frac {M}{2^G}-m}$ can be identically set to the maximum of $b_m$ and $b_{\frac {M}{2^G}-m}$ to guarantee the symmetry as well as the non-negativity of the hybrid signal at sampling times $m$ and $\frac {M}{2^G}-m$, where $m=1,\ldots ,{M}/{2^{G+1}}-1$. Consequently, the reconstructed signal $c_m$ for $m=0,1,\ldots ,{M}/{2^{G+1}}-1$ is defined as

(8)$$c_m\!=\!\!\left\{ {\begin{array}{ll} {b_0,} & {m=0,}\\ {\max\left\{b_m,b_{\frac{M}{2^G}-m}\right\},} & {m=1,\ldots,\frac{M}{2^{G+1}}-1,} \end{array}} \right.$$

where $\max \left \{\cdot \right \}$ denotes the maximum value of the sequence. Based on Eq. (8), the reconstructed signal during one HPAM-DMT symbol period can be readily obtained by using the periodicity and symmetry of $c_m$. It is observed that as $G$ increases, the value of $b_m$ is selected from more signal samples. Therefore, $b_m$ trends to be set to a larger value, leading to the increased power of the reconstructed signal.

Furthermore, the introduced reconstructed signal can be directly superposed upon $z_m$ to produce the HPAM-DMT signal $s_m$, which is given by

(9)$$\begin{aligned} s_m= & z_m+c_m\\ = & {x}_m^{\textrm{c}}+{y}_m+c_m ,~m=0,1,\ldots,M-1. \end{aligned}$$

Here, $s_m$ is a real and unipolar signal which is ready for OWC channel. Now we can summarize the transmitter architecture of the proposed HPAM-DMT in Fig. 1, in which the classic PAM-DMT and RPAM-DMT signals are superimposed for simultaneous transmission, and then the reconstructed signal is introduced to guarantee the non-negativity.

Fig. 1. Block diagram of the transmitter for the proposed HPAM-DMT.

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3. Receiver design for HPAM-DMT

In this section, the demodulation process of the proposed HPAM-DMT is presented upon analyzing the frequency-domain structure of the HPAM-DMT signal. To be specific, after the $M$-point FFT operation on the sequence $\{s_m\}_{m=0}^{M-1}$, the frequency-domain signal is obtained as

(10)$$\begin{aligned} S_k= & \frac{1}{\sqrt{M}} \sum_{m=0}^{M-1} {s_m{e}^{{-}j\frac{2\pi m k}{M}}}\\ = & \frac{1}{\sqrt{M}} \sum_{m=0}^{M-1} {\left({x}_m^{\textrm{c}}+{y}_m+c_m \right){e}^{{-}j\frac{2\pi m k}{M}}} \\ = & {X}_k^{\textrm{c}}+{Y}_k+C_k, k=0,1,\ldots,M-1, \end{aligned}$$

where ${X}_k^{\textrm {c}}$ and $C_k$ represent the FFT results of $x^{\textrm {c}}_m$ and $c_m$, respectively. Given the fact that $c_m$ is a periodic function with the period of $M/2^G$, its FFT results can be expressed as

(11)$$\begin{aligned} {C}_k= & \frac{1}{\sqrt{M}} \sum_{m=0}^{{M-1}} {c_m {e}^{{-}j\frac{2\pi m k}{M}}} \\ = & \frac{1}{\sqrt{M}} \sum_{i=0}^{2^G-1} \sum_{m=0}^{\frac{M}{2^G}-1} {c_{m+\frac{iM}{2^G}} {e}^{{-}j\frac{2\pi \left(m+\frac{iM}{2^G}\right) k}{M}}} \\ = & \frac{1}{\sqrt{M}} \sum_{m=0}^{\frac{M}{2^G}-1} {c_m {e}^{{-}j\frac{2\pi m k}{M}}}\sum_{i=0}^{2^G-1}{ {e}^{{-}j\frac{2\pi i k}{2^G} }} . \end{aligned}$$

Since we have

(12)$$\begin{aligned} \left\{ {\begin{array}{ll} {\sum\limits_{i=0}^{2^G-1}{ {e}^{{-}j\frac{2\pi i k}{2^G} }}=2^G,} & {k=q2^G,}\\ {\sum\limits_{i=0}^{2^G-1}{ {e}^{{-}j\frac{2\pi i k}{2^G} }}=0,} & {\textrm{ otherwise},}\\ \end{array}} \right. \end{aligned}$$

where $q=0,1,\ldots ,M/2^{G-1}$, the FFT results of $c_m$ are calculated as

(13)$$\begin{aligned} {C}_k= & \left\{ {\begin{array}{ll} {\frac{2^G}{\sqrt{M}} \sum\limits_{m=0}^{\frac{M}{2^G}-1} {c_m {e}^{{-}j\frac{2\pi m k}{M}}},} & {k=q2^G,}\\ {0,} & {\textrm{otherwise}.}\\ \end{array}} \right. \end{aligned}$$

It is found that the FFT results of $c_m$ at the subcarriers of the first $G$ groups are equal to zeros. Furthermore, due to the symmetry of $c_m$, $C_k$ in the scenario of $k=q2^G$ can be rewritten as

(14)$$\begin{aligned} {C}_k= & \frac{2^G}{\sqrt{M}} \sum\limits_{m=0}^{\frac{M}{2^G}-1} {c_m {e}^{{-}j\frac{2\pi m q}{M/2^G}}}\\ = & \frac{2^G}{\sqrt{M}} \sum\limits_{m=1}^{\frac{M}{2^{G+1}}-1} c_m \left[{e}^{{-}j\frac{2\pi m q}{M/2^G}}+ {e}^{{-}j\frac{2\pi \left(M/2^G-m\right) q}{M/2^G}} \right]+\frac{2^Gc_0}{\sqrt{M}}\\ = & \frac{2^{G+1}}{\sqrt{M}} \sum\limits_{m=1}^{\frac{M}{2^{G+1}}-1} c_m\textrm{cos}\left( {\frac{2\pi m q}{M/2^G}}\right)+\frac{2^Gc_0}{\sqrt{M}}, k=q2^G. \end{aligned}$$

One can clearly observe from Eq. (14) that the FFT results of $c_k$ are located only at the real parts of the $q2^G$-th $(q=0,1,\ldots ,M/2^{G-1})$ subcarriers. Recall that the PAM symbols are transmitted at the imaginary parts of the subcarriers and the real parts of the $2^{g-1}(2u+1)$-th subcarriers, where $u=0,1,\ldots ,M/2^{g}$ and $g=0,1,\ldots ,G$. Therefore, the introduced reconstructed signal does not impose any interference on the transmitted PAM symbols.

As for the component of the classic PAM-DMT, the transmitted symbols are loaded at the imaginary parts of the subcarriers while the clipping noise is imposed on the real parts, which is given by [11]

(15)$$\left\{ {\begin{array}{l} {\textrm{Re}(X_k^{\textrm{c}})=\widehat{X}_k,}\\ {\textrm{Im}(X_k^{\textrm{c}})=\frac{1}{2}\textrm{Im}\left\{X_k\right\},} \end{array}} \right.$$

where $\widehat {X}_k$ denotes the clipping noise, $\textrm {Re}(X)$ and $\textrm {Im}(X)$ denotes the real and imaginary parts of $X$, respectively. Since no clipping operation is performed in RPAM-DMT, the transmitted PAM symbols only occupy the real parts of the subcarriers, i.e., $\textrm {Im}\left \{Y_k\right \}=0$. Therefore, observing the imaginary parts of $S_k$, we have

(16)$$\begin{aligned} \textrm{Im}(S_k)= & \textrm{Im}\left(X_k^c \right) +\textrm{Im}\left(Y_k\right)+ \textrm{Im}\left(C_k\right)\\ = & \frac{1}{2}\textrm{Im}\left(X_k\right),~k=0,1,\ldots,M-1. \end{aligned}$$

Furthermore, the real parts of the $2^{g-1}(2u+1)$-th subcarriers in HPAM-DMT are given by

(17)$$\begin{aligned} \textrm{Re}(S_k)= & \textrm{Re}\left(X_k^c \right) +\textrm{Re}\left(Y_k\right)+ \textrm{Re}\left(C_k\right)\\ = & \widehat{X}_k+Y_k,~k=2^{g-1}(2u+1). \end{aligned}$$

It can be found that the PAM symbols of the classic PAM-DMT component are conveyed at the imaginary parts of subcarriers without any interference in HPAM-DMT. Therefore, the PAM symbols transmitted by the classic PAM-DMT are first detected, and the clipping noise $\widehat {X}_k$ is subsequently regenerated according to the detected symbols, as shown in Fig. 2. After subtracting the clipping noise $\widehat {X}_k$ from the real part of the $2^{g-1}(2u+1)$-th subcarriers, the PAM symbols of the RPAM-DMT component can be recovered. In summary, the receiver architecture for the proposed HPAM-DMT is demonstrated in Fig. 2.

Fig. 2. Block diagram of the receiver for the proposed HPAM-DMT.

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In the conventional superimposed O-OFDM, such as LACO-OFDM, multiple layers of the ACO-OFDM signals are successively detected by an iterative receiver. The transmitted symbols at the $l$-th layer can be recovered only after all of the lower layers have been demodulated. Therefore, the complexity and latency of the receiver significantly increase with the number of layers. By contrast, there are only two PAM-DMT components in the proposed HPAM-DMT. All PAM symbols conveyed by the first $G$ subcarrier groups in RPAM-DMT can be recovered in parallel after the demodulation of the PAM-DMT signals, which leads to a receiver with lower receiver complexity and latency compared to LACO-OFDM.

4. Spectrum efficiency

The spectrum efficiency of the proposed HPAM-DMT is analyzed in this section. There are $M-2$ subcarriers used for data transmission in the classic PAM-DMT. Additionally, the total number of the subcarriers is calculated as $\sum\limits _{i = 1}^G {\frac {M}{{{2^i}}}}$ in the first $G$ groups. Therefore, the spectrum efficiency of the proposed HPAM-DMT is expressed as [17]

(18)$$\begin{aligned} \eta_{\textrm{HPAM}}= \frac{\log_2N_{\textrm{P}}\left(M-2+\sum\limits_{i = 1}^G {\frac{M}{{{2^i}}}} \right)}{2\left(M+M_{\textrm{CP}}\right)}=\frac{\log_2N_{\textrm{P}}\left(2M-2^{{-}G}M-2 \right)}{2\left(M+M_{\textrm{CP}}\right)}, \end{aligned}$$

where $N_{\textrm {P}}$ is the constellation size of PAM, and $M_{\textrm {CP}}$ is the length of cyclic prefix. For comparison, the spectrum efficiency of DCO-OFDM and LACO-OFDM are provided, respectively given by

(19)$$\begin{aligned} & \eta_{\textrm{DCO}}=\frac{\log_2N_{\textrm{Q}}\left(M-2 \right)}{2\left(M+M_{\textrm{CP}}\right)},\\ & \eta_{\textrm{LACO}}=\frac{\log_2N_{\textrm{Q}}\left(M-2^{{-}L}M \right)}{2\left(M+M_{\textrm{CP}}\right)}, \end{aligned}$$

where $N_{\textrm {Q}}$ is the constellation size of QAM, and $L$ is the number of layers adopted in LACO-OFDM. Since the square QAM can be regarded as two quadrature one-dimensional PAM, $N_{\textrm {P}}^2$-QAM achieves the equivalent BER performance to $N_{\textrm {P}}$-PAM at a given signal to noise ratio (SNR) [23]. Therefore, the constellation size of PAM is set to $N_{\textrm {P}}= \sqrt {N_{\textrm {Q}}}$ in Eq. (18) for a fair comparison. Furthermore, since a large value of $M$ is usually adopted to alleviate the degradation of spectrum efficiency caused by cyclic prefix, $\eta _{\textrm {HPAM}}$ can be approximately rewritten as

(20)$$\begin{aligned} \eta_{\textrm{HPAM}}= & \frac{\log_2\sqrt{N_{\textrm{Q}}}\left(2M-2^{{-}G}M-2 \right)}{2\left(M+M_{\textrm{CP}}\right)}=\frac{\log_2N_{\textrm{Q}}\left(M-2^{{-}G-1}M \right)}{2\left(M+M_{\textrm{CP}}\right)}-\frac{\log_2N_{\textrm{Q}}}{2\left(M+M_{\textrm{CP}}\right)}\\ \approx & \frac{\log_2N_{\textrm{Q}}\left(M-2^{{-}G-1}M \right)}{2\left(M+M_{\textrm{CP}}\right)}. \end{aligned}$$

One can find that the proposed HPAM-DMT is able to approach the spectrum efficiency of LACO-OFDM when $L=G+1$.

Since DCO-OFDM fully exploits the subcarrier resource, the spectrum efficiency of DCO-OFDM can be adopted as the benchmark. The ratio between the spectrum efficiency of O-OFDM and the one of DCO-OFDM is utilized for analysis, which is given by

(21)$$\gamma =\frac{\eta}{\eta_{\textrm{DCO}}},$$

where $\eta$ is the spectrum efficiency of O-OFDM. Figure 3 illustrates $\gamma$ of HPAM-DMT and LACO-OFDM at different values of $G$, where $L$ is set to $G+1$. It is clearly seen from Fig. 3 that the curve of HPAM-DMT well matches that of LACO-OFDM, which is consistent with the above analysis. Moreover, the spectrum efficiency of HPAM-DMT is significantly improved and approaches the one of DCO-OFDM when more subcarrier groups are employed. For example, PAM-DMT with $G=4$ can achieve $97.2\%$ of the spectrum efficiency of DCO-OFDM, exhibiting only a marginal loss in spectrum efficiency.

Fig. 3. Spectrum efficiency of HPAM-DMT and DCO-OFDM compared to $\eta _{\textrm {DCO}}$ when $M=256$ is adopted.

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5. Comparison of complexity and processing latency

5.1 Complexity comparison

The complexity analysis of the HPAM-DMT transceiver is presented in this subsection. Since the complexity is mainly determined by the IFFT and FFT operations, we evaluate the complexity in terms of the real multiplications of these dominant operations. To reduce the computational complexity, a radix-2 FFT algorithm is generally applied for the $M$-point FFT operation by partitioning it into $\log _2 M$ stages, which leads to $2M\log _2M$ real multiplications [24]. Note that the $M$-point IFFT has the same complexity as the $M$-point FFT.

In HPAM-DMT, one $M$-point IFFT block is required for the classic PAM-DMT at the transmitter. Since the IFFT block is performed on the imaginary-valued sequence, nearly half of the computations are omitted [25]. Therefore, the number of real multiplications in the classic PAM-DMT can be appropriately set to $M\log _2M$. Additionally, one $M$-point IFFT block performed on the real-valued sequence is required at the transmitter of RPAM-DMT, where the $2^{g-1}(2u+1)$-th $\left (u=0,1,\ldots ,M/2^{g}\right )$ subcarriers are utilized to transmit the PAM symbols while the remaining ones are set to zeros. Since the multiplications corresponding to the zero-valued subcarriers can be excluded, the number of real multiplications for the transmitter of RPAM-DMT based on the decimation-in-time radix-2 FFT algorithm is calculated as [24]

(22)$$N_{RPAM}^{T}=\sum\limits_{g=1}^{G} {\left[\frac{M}{2^g} \log_2\left(\frac{M}{2^{g-1}}\right) + \frac{M}{2^{g}}\right] }.$$

Furthermore, the total number of real multiplications involved in the transmitter of HPAM-DMT is given by

(23)$$\begin{aligned} N_{HPAM}^{T} & =M\log_2M+\sum_{g=1}^{G} {\left[\frac{M}{2^g} \log_2\left(\frac{M}{2^{g-1}}\right) + \frac{M}{2^{g}}\right] } \\ & =2M\log_2M-\frac{M\left(\log_2M-G\right)}{2^G}. \end{aligned}$$

It is observed from Eq. (23) that more real multiplications are required at the transmitter of HPAM-DMT as $G$ increases. In LACO-OFDM, the $\frac {M}{2^{l-1}}$-point IFFT operation is performed on the complex-valued sequence in the $l$-th layer. Given that the even subcarriers are set to zeros in each layer, the number of real multiplications at the transmitter of LACO-OFDM is calculated as

(24)$$\begin{aligned} N_{LACO}^{T} & =\sum_{l=1}^{L} {\left[\frac{M}{2^{l-1}} \log_2\left(\frac{M}{2^{l-1}}\right) + \frac{M}{2^{{l-1}}}\right] } \\ & =2M\log_2M-\frac{M\left(\log_2M-L\right)}{2^{L-1}}. \end{aligned}$$

For $L=G+1$, LACO-OFDM and HPAM-DMT are capable of achieving the same spectrum efficiency. Therefore, substituting $L=G+1$ into Eq. (24), we have

(25)$$\begin{aligned} N_{LACO}^{T} & =2M\log_2M-\frac{M\left(\log_2M-G\right)}{2^{G}}+ \frac{M}{2^{G}}. \end{aligned}$$

Since $M\log _2M$ is generally far greater than $M/2^G$, $N_{LACO}^T$ is approximately equal to $N_{HPAM}^T$, implying that the transmitter of both schemes have the same computational complexity.

At the receiver side of HPAM-DMT, two $M$-point FFT and one $M$-point IFFT are performed to successively demodulate the two PAM-DMT components. Since the FFT and IFFT operations are all executed on the real-valued or imaginary-valued sequence, the receiver of HPAM-DMT requires

(26)$$N_{HPAM}^{R}=3M\log_2M$$

times of the real multiplications. It is observed that the computational complexity of the proposed HPAM-DMT receiver is unrelated to $G$. In LACO-OFDM, one real $M$-point FFT is conducted to detect the transmitted symbols in the first layer, which leads to $M\log _2M$ real multiplications. The detection of the transmitted symbols in the higher layer requires the subtraction of clipping noise induced by the previous layer, which includes an $M/2^{l-1}$-point IFFT and $M/2^{l-1}$-point FFT performing on the complex-valued and real-valued sequence, respectively. Therefore, when considering that LACO-OFDM and HPAM-DMT achieve the same spectrum efficiency with $L=G+1$, we can compute the number of real multiplications involved in the receiver of LACO-OFDM as

(27)$$\begin{aligned} N_{LACO}^{R}= & M\log_2M+ \sum_{l=1}^{L-1} {\left[\frac{M}{2^l} \log_2M -\frac{M(l-2)}{2^l}\right] } + \sum_{l=1}^{L-1} {\frac{M}{2^{l-1}} \log_2\left(\frac{M}{2^{l-1}}\right) } \\ = & 5M\log_2M-\frac{M\left(2\log_2M-2G-1\right)}{2^{G-1}}-2M. \end{aligned}$$

It can be found that $N_{LACO}^{R}$ is an increasing function of $G$, which indicates that the complexity of the LACO-OFDM receiver increases with $G$. For $G=1$, there are $3M\log _2M+M$ real multiplications at the receiver of LACO-OFDM, which is greater than $3M\log _2M$ real multiplications of the HPAM-DMT receiver. Therefore, the proposed HPAM-DMT can provide a much lower computational complexity than LACO-OFDM at the receiver side.

The number of real multiplications required for LACO-OFDM and HPAM-DMT are presented at different values of $G$ in Fig. 4. We observe that LACO-OFDM and HPAM-DMT have the same computational complexity at the transmitter. Moreover, the complexity of the LACO-OFDM receiver significantly increases with respect to $G$, while HPAM-DMT possesses much lower receiver complexity. For $G=4$, the proposed HPAM-DMT offers approximately $35\%$ of complexity reduction compared to LACO-OFDM.

Fig. 4. Computational complexity comparison of HPAM-DMT and LACO-OFDM when $M=256$ is adopted.

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5.2 Processing latency comparison

In the radix-2 FFT algorithm, the $M$-point FFT or IFFT should go through $\log _2M$ stages in the butterfly chart to yield the results [24]. Although there are two $M$-point IFFT blocks at the transmitter of HPAM-DMT, the two blocks are executed concurrently, which requires $\log _2M$ stages in total. On the other hand, since the $L$ layers are calculated in parallel at the transmitter of LACO-OFDM, the processing latency is mainly determined by the maximum-size IFFT block, which needs $\log _2M$ stages. Therefore, both schemes have the same processing latency at the transmitter.

At the receiver of HPAM-DMT, the three $M$-point FFT and IFFT operations are executed sequentially. Therefore, the number of stages required to recover all transmitted data is

(28)$$S_{HPAM}=3\log_2M,$$

which is irrelative of $G$. In LACO-OFDM, the demodulation of the $l$-th layer can only take place after all of the lower layers have been demodulated. In this case, the IFFT and FFT operations are all conducted successively at the receiver of LACO-OFDM. Hence, considering $L=G+1$, the number of stages required to demodulate all layers is calculated as

(29)$$\begin{aligned} S_{LACO} & =\log_2M+ \sum_{l=1}^{L-1} 2\log_2\left(M/2^{l-1}\right)\\ & =(2L-1)\log_2M-(L-2)(L-1),\\ & =(2G+1)\log_2M-G(G-1). \end{aligned}$$

It can be proved that $S_{LACO}$ is an increasing function with respect to $G$. Additionally, since $S_{LACO}$ is equal to $3\log _2M$ for $G=1$, the processing latency of the LACO-OFDM receiver becomes higher than the proposed HPAM-DMT as $G$ increases. Figure 5 illustrates the number of stages required for LACO-OFDM and HPAM, from which one can clearly observed that the proposed HPAM-DMT provides much lower processing latency than LACO-OFDM. For $G=3$, almost half of the processing latency is reduced by HPAM-MDT compared to LACO-OFDM.

Fig. 5. The number of stages required for the receiver of LACO-OFDM and HPAM-DMT with $M=256$ at different values of $G$.

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6. Numerical results

Numerical results of the proposed HPAM-DMT are provided in this section. The channel bandwidth is set to 20 MHz, which is divided to $M=256$ subcarriers. The length of cyclic prefix is set to $M_p=4$. Due to the clipping operation in the classic PAM-DMT, half of the power is allocated to clipping noise. Therefore, the power of the transmitted PAM symbol in the classic PAM-DMT is set to be twice as that of the unclipped RPAM-DMT to guarantee the same BER performance.

The complementary cumulative distribution function (CCDF) of the peak-to-average power ratio (PAPR) for HPAM-DMT using 8-PAM is first presented in Fig. 6. For comparison, the performance of LACO-OFDM and ACO-OFDM using 64-QAM are also provided. In LACO-OFDM, the number of layers $L$ is set to $G+1$ to ensure that HPAM-DMT and LACO-OFDM can provide the same spectrum efficiency. A larger reconstructed signal is required for more activated groups, which increases the average power of the HPAM-DMT signal. Therefore, it is observed from Fig. 6 that the PAPR of the proposed HPAM-DMT is reduced as $G$ increases. Moreover, the PAPR of HPAM-DMT is much lower than that of ACO-OFDM. In the scenario of the same spectrum efficiency, the proposed HPAM-DMT outperforms LACO-OFDM in terms of the PAPR performance. Moreover, the performance gap between HPAM-DMT and LACO-OFDM gradually widens as $G$ increases.

Fig. 6. CCDF of the PAPR for different O-OFDM schemes.

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The BER curves of the classic PAM-DMT and RPAM-DMT using 8-PAM are portrayed in Fig. 7. One can observe from Fig. 7 that the classic PAM-DMT outperforms RPAM-DMT since the error propagation caused by the classic PAM-DMT degrades the performance of RPAM-DMT. The BER performance of the classic PAM-DMT is improved as the bit energy to noise power $E_b/N_0$ increases, which can suppress the influence of error propagation on RPAM-DMT. Theretofore, it is seen from Fig. 7 that the BER of RPAM-DMT approaches the one of the classic PAM-DMT as $E_b/N_0$ increases.

Fig. 7. BER performance of the classic PAM-DMT and RPAM-DMT using 8-PAM at different values of the bit energy to noise power $E_b/N_0$.

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At different values of $E_b/N_0$, Fig. 8 demonstrates BER comparison of HPAM-DMT, LACO-OFDM, and ACO-OFDM. In the simulation, $8$-PAM and $64$-QAM are adopted in HPAM-DMT and LACO-OFDM, respectively. The spectrum efficiencies of ACO-OFDM using 64-QAM and 256-QAM are respectively 1.5 bit/s/Hz and 2 bit/s/Hz while HPAM-DMT utilizing 8-PAM and $G=1$ provides the spectrum efficiency of 2.2 bit/s/Hz. From Fig. 8, it is observed that the performance degradation of HPAM-DMT with $G=1$ and 8-PAM is 1.4 dB compared to ACO-OFDM using 64-QAM. But the spectrum efficiency of the proposed HPAM-DMT is improved by $47\%$. Furthermore, when a large-size constellation of 256-QAM is adopted in ACO-OFDM, the BER performance of ACO-OFDM becomes notably worse than HPAM-DMT using 8-PAM and $G=1$. Moreover, HPAM-DMT still outperforms the conventional ACO-OFDM in terms of spectrum efficiency. It is also seen from Fig. 8 that the same BER performance is achieved by HPAM-DMT and LACO-OFDM for $G=1$ and $G=2$. As $G$ increases, a larger reconstructed signal is required, which leads to the higher power used to make the signal non-negative compared to LACO-OFDM. In this case, the proportion of the signal power used for data transmission decreases. Therefore, it is observed that HPAM-DMT suffers from a slight degradation in BER performance compared to LACO-OFDM when a relatively large value of $G$ is adopted. However, HPAM-DMT enjoys much lower implementation complexity and processing delay than that of LACO-OFDM.

Fig. 8. BER comparison of the proposed HPAM-DMT using 8-PAM, LACO-OFDM using 64-QAM, and ACO-OFDM at different values of $E_b/N_0$.

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Furthermore, the performance of the proposed HPAM-DMT is investigated when the indoor OWC channel in [26,27] is considered. A white LED (Golden DRAGON, ZW W5SG) from OSRAM is used as transmitter. The relevant parameters of the indoor OWC channel and noise are the same as [26]. The BER performance of the proposed HPAM-DMT over the indoor OWC channel is provided in Fig. 9 and Fig. 10, in which the x-axis represents the average electrical OFDM signal power before modulating the LED [26].

Fig. 9. BER comparison of the proposed HPAM-DMT, LACO-OFDM and ACO-OFDM when the indoor OWC channel is considered, where $G=2$ is adopted.

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Fig. 10. BER performance of the proposed HPAM-DMT and LACO-OFDM using different values of $G$ when the indoor OWC channel is considered.

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The BER comparison of HPAM-DMT, LACO-OFDM and ACO-OFDM over the indoor OWC channel is presented in Fig. 9, where $G=2$ is adopted. The spectrum efficiencies of ACO-OFDM using 64-QAM and 256-QAM are respectively 1.5 bit/s/Hz and 2 bit/s/Hz while HPAM-DMT utilizing 4-PAM and 8-PAM provides the spectrum efficiency of 1.7 bit/s/Hz and 2.6 bit/s/Hz, respectively. In the region of the low average signal power, the better BER performance is observed from Fig. 9 as the average signal power increases. The proposed HPAM-DMT achieves the same BER performance as LACO-OFDM for $G=2$, which is consistent with the above analysis. Moreover, HPAM-DMT using 4-PAM and 8-PAM exhibits much better BER performance, while at the same, providing higher spectrum efficiency compared to ACO-OFDM using 64-QAM and 256-QAM, respectively. At the relatively high average signal power, nonlinear distortion occurs as a result of the nonlinearity of LED, which degrades the BER performance. Therefore, it is seen from Fig. 9 that the BER performance gradually deteriorates in the region of the high average power. Since the PAPR of HPAM-DMT is lower than LACO-OFDM and ACO-OFDM, HPAM-DMT is more immune to the transmitter nonlinearity. Therefore, it is observed that the proposed HPAM-DMT outperforms LACO-OFDM and ACO-OFDM at the high average power.

Relying on the indoor OWC channel, Fig. 10 demonstrates the BER performance of HPAM-DMT and LACO-OFDM using different values of $G$. It is seen from Fig. 10 that at the low average signal power, the proposed HPAM-DMT and LACO-OFDM using $G=1$ and $G=2$ achieve the same BER performance while a slight performance degradation is observed for HPAM-DMT with the relatively large value of $G$, due to the increased power of the reconstructed signal. Moreover, thanks to the lower PAPR, a notable BER improvement is observed for HPAM-DMT compared to LACO-OFDM in the region of the high average signal power. It is also observed from Fig. 10 that the BER performance of HPAM-DMT becomes better as $G$ increases at the high average signal power, since the PAPR of HPAM-DMT is reduced with increasing $G$.

In order to comprehensively evaluate the performance, Fig. 11 demonstrates the required $E_b/N_0$ of HPAM-DMT and LACO-OFDM to achieve the BER target of $1\times 10^{-3}$ at various values of bit rate/normalized bandwidth. The performance of the conventional O-OFDM schemes are also provided for comparison, including ACO-OFDM, HACO-OFDM, and DCO-OFDM. HACO-OFDM proposed in [14] combines the time-domain signals of PAM-DMT and ACO-OFDM for simultaneous transmission. In DCO-OFDM, two different levels of the DC bias, i.e., 10 dB and 13 dB, are considered. It is observed from Fig. 11 that for the relatively large value of $G$, HPAM-DMT needs slightly higher $E_b/N_0$ to achieve the target BER compared to LACO-OFDM, which is consistent with the above analysis. From Fig. 11, it is also seen that HPAM-DMT requires much lower $E_b/N_0$ to achieve the BER target than the conventional O-OFDM schemes. Therefore, under the same noise power, the proposed HPAM-DMT is capable of using much lower bit energy $E_b$ to convey the information data at the BER of $1\times 10^{-3}$, which verifies the superiority of HPAM-DMT in terms of the power efficiency.

Fig. 11. Required $E_b/N_0$ of different O-OFDM schemes at different values of bit rate/normalized bandwidth.

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7. Conclusion

A novel HPAM-DMT scheme is proposed in this paper to support a spectrum-efficient OWC link. In HPAM-DMT, the RPAM-DMT signal using the real parts of subcarriers is transmitted simultaneously with the classic PAM-DMT signal utilizing the imaginary parts of subcarriers, which offers the same spectrum efficiency as LACO-OFDM. Meanwhile, the number of the subcarrier groups activated for data transmission in RPAM-DMT can be flexibly adjusted as required, thus providing an extra design freedom for communication. Moreover, superior to LACO-OFDM, the proposed HPAM-DMT has much lower complexity and latency of the receiver since only two PAM-DMT signals are successively detected. Simulation results have shown that the proposed HPAM-DMT achieves a better PAPR performance than LACO-OFDM. Thus, a significant improvement of the BER performance is observed for HPAM-DMT compared LACO-OFDM when the transmitter nonlinearity is considered. Moreover, HPAM-DMT outperforms the conventional O-OFDM schemes in terms of power efficiency, which makes it a competitive candidate for the IM/DD-based optical wireless transmission.

Funding

National Natural Science Foundation of China (61871109, 61941115); Outstanding Youth Foundation of Jiangsu Province of China (BK20190012); Natural Science Foundation of Jiangsu Province (BK20190582); Research Center of Optical Communications Engineering and Technology, Jiangsu Province (ZXF201902); Open Fund of National Mobile Communications Research Laboratory, Southeast University (2019D18); Fundamental Research Funds for the Central Universities (JUSRP11919).

Disclosures

The authors declare no conflicts of interest.

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Group index	Subcarrier index
$1$	$1, 3, 5, 7, \dots, M - 1$
$2$	$2, 6, 10, \dots, M - 2$
$3$	$4, 12, 20, \dots, M - 4$
$⋮$	$⋮$
$\log_{2} M$	$0, M / 2$

Spectrum-efficient hybrid PAM-DMT for intensity-modulated optical wireless communication

Abstract

1. Introduction

2. Proposed hybrid PAM-DMT

3. Receiver design for HPAM-DMT

4. Spectrum efficiency

5. Comparison of complexity and processing latency

5.1 Complexity comparison

5.2 Processing latency comparison

6. Numerical results

7. Conclusion

Funding

Disclosures

References

Cited By

Figures (11)

Tables (1)

Equations (29)

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