Adaptive Spatial Modulation for Visible Light Communications with an Arbitrary Number of Transmitters

07/26/2018
by   Jin-Yuan Wang, et al.
0

As a power and bandwidth efficient modulation scheme, the optical spatial modulation (SM) technique has recently drawn increased attention in the field of visible light communications (VLC). To guarantee the number of bits mapped by the transmitter's index at each timeslot is an integer, the number of transmitters (i.e., light-emitting diodes) in the SM based VLC system is often set be a power of two. To break the limitation on the required number of transmitters and provide more design flexibility, this paper investigates the SM based VLC with an arbitrary number of transmitters. Initially, a channel adaptive bit mapping (CABM) scheme is proposed, which includes three steps: bit mapping in space domain, bit mapping in signal domain, and the channel adaptive mapping. The proposed CABM scheme allows operation with an arbitrary number of transmitters, and is verified to be an efficient scheme through numerical results. Based on the CABM scheme, the information-theoretical aspects of the SM based VLC are analyzed. The theoretical expression of the mutual information is first analyzed. However, it is very hard to evaluate system performance. To obtain more insights, a lower bound of the mutual information is derived, which is in closedform. Both theoretical analysis and numerical results show that the gap between the mutual information and its lower bound is small. Finally, to further improve the system performance, the precoding scheme is proposed for the SM based VLC. Numerical results show that the system performance improves dramatically when using the proposed precoding scheme.

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I Introduction

Multi-input multi-output (MIMO) technique can significantly increase the transmission rate and link reliability of the system without occupying additional system bandwidth, and is a promising candidate for future 5G wireless communications [1]. However, the MIMO technique suffers from inter-channel interference and synchronization problems [2]. Moreover, multiple radio frequency (RF) chains in MIMO systems result in an increase in computational complexity and cost. Against this background, spatial modulation (SM) technique has been proposed as a solution with low complexity [3].

For SM based communication system, multiple antennas are deployed at the transmitter, but only one of them is active at each timeslot to transmit information and the others are silent. The source data bits are divided into two parts, one part is mapped onto the conventional constellation points in signal domain, and the other part is used to determine the index of the active antenna in space domain [4]. Therefore, the index of the antenna is utilized as an additional dimension to transfer information, which helps to improve the data rate. Moreover, only one RF chain is employed in SM based system, which can effectively reduce the system complexity.

Recently, the SM technique has been extended to the optical wireless communication (OWC) field for both the outdoor and indoor scenarios. For outdoor OWC (also named as free-space optical (FSO) communications), the performance for the SM based system has been investigated. In [5], an analytical framework was provided for both uncoded and coded outdoor SM in FSO communication channels. In [6]

, the average bit error probability (ABEP) was analyzed for SM based FSO system over H-K turbulence channels. In

[7], the constrained capacity was maximized for power-imbalanced optimal SM MIMO system. For the indoor environment, the OWC can be applied to the visible light communications (VLC). The optical SM in VLC was first proposed in [8] and then extended to many scenarios. In [9], the ABEP of SM combining pulse amplitude modulation (PAM) and space shift keying (SSK) in VLC was investigated. In [10], the mutual information of the SM based VLC system was derived. Moreover, ref. [11] proposed an enhancement in SM performance by aligning the light-emitting diodes (LEDs) and the photodiodes (PDs). In [12], the performance of SM was compared with that of optical spatial multiplexing and optical repetition coding. The effect of synchronization error on optical SM was investigated in [13]. In [14], an active-space, collaborative constellation-based generalized SM MIMO encoding scheme was proposed. In [15], a constellation optimization scheme was proposed for indoor SM based VLC. For a complete discussion about the concept of SM and its variants, the readers can refer to [16]. It should be emphasized that the design flexibility in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] is limited by the fact that the number of LEDs must be a power of two to guarantee that the number of bits mapped by the transmit LED index at each timeslot is an integer. That is, when the number of LEDs is not a power of two, the design methods in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] are not available.

In conventional RF wireless communications, some bit mapping schemes have been proposed when the number of transmit antennas is not a power of two. A fractional bit encoded spatial modulation (FBE-SM) was proposed in [17]. Unfortunately, the FBE-SM scheme suffers from the error propagation problem. In [18], a joint mapped spatial modulation scheme was proposed. However, the number of transmit antennas must satisfy an equality constraint. In addition, the bit mapping schemes with an arbitrary number of transmit antennas were also analyzed in [19] and [20]. However, the channel state information at the transmitter (CSIT) was not considered in [17, 18, 19, 20]. By considering the CSIT, link adaptive mapper designs were proposed in [21] and [22]. In [21], the CSIT was employed to design the SSK modulated system. Note that the SSK modulation is a special case of SM, where only antenna index is used to convey information. Therefore, the bit mapping in signal domain was not considered in [21]. In [22], an adaptive brute forth mapper (BFM) was proposed for SM with lightweight feedback overhead. However, the number of transmit antennas must be a power of two.

Motivated by the above literature, this paper investigates the channel adaptive SM for indoor VLC to break the limitation on the required number of LEDs. That is, the CSIT is utilized for bit mapping, and the number of LEDs can be an arbitrary positive number larger than one. The main contributions of this paper are listed as follows:

  • With an arbitrary number of LEDs, a channel adaptive bit mapping (CABM) scheme for SM based VLC is proposed. The proposed CABM scheme is a modification of the schemes in [19] and [20] by using the CSIT, which includes three steps. The first two steps are the bit mappings in space domain and signal domain, respectively. At last, by using CSIT, a constellation optimization is formulated to obtain better modulation combinations. In the CABM scheme, when the number of LEDs is not a power of two, space domain symbols are mapped with different numbers of bits. To keep the same number of transmit bits at each timeslot, the constellations in signal domain are also mapped with different modulation orders.

  • After performing the proposed CABM scheme, the mutual information is analyzed for SM based VLC with an arbitrary number of LEDs. In the system, the finite alphabet and input-dependent noise are employed. According to Shannon’s information theory, the theoretical expression of the mutual information is derived, but is with an integral expression and cannot be easily used in practice. To reduce the complexity, a closed-form expression of the lower bound on the mutual information is obtained.

  • To further improve the system performance, a precoding scheme is proposed for the SM based VLC. The purpose of the precoding is to maximize the minimum distance between any two constellation points in the received signal space. The optimization problem is non-convex and non-differentiable, which is very hard to obtain the optimal solution. Alternatively, an approximation is employed for the minimum function, and the optimization problem is then solved by using the interior point algorithm.

The rest of the paper is organized as follows. In Section II, the system model is described. In Section III, the CABM scheme is proposed. Section IV analyzes the mutual information and its lower bound. A precoding scheme is proposed in Section V. Numerical results are shown in Section VI. Section VII draws conclusions of this paper.

Notations

: Throughout this paper, italicized symbols denote scalar values, and bold symbols denote vectors/matrices/sets. We use

for positive integer, for the inner product, for the expect operator, for the Frobenius norm, for the probability of an event. We use

for a Gaussian distribution with zero mean and variance

, for the number of in set , and

for the probability density function (PDF) of a random variable. We use

for the logarithm with base 2, and for the natural logarithm. We use for the mutual information, for the entropy, and for the Gaussian-Q function.

Ii System Model

Consider an SM based indoor VLC system with transmitters (i.e., LEDs) and one receiver (i.e., PD), as illustrated in Fig. 1. In the system, is an arbitrary positive number larger than one. With finite alphabet ( bits), the source data is divided into two parts by using optical SM. One part is mapped onto the indexes of the LEDs in space domain, and the other part is mapped onto the constellation points in signal domain.

Fig. 1: An SM based VLC system.

By employing SM, only one LED is active at each timeslot. We assume that the -th LED is selected at the current timeslot to transmit the -th symbol. According to [23], the received signal at the PD is given by

(1)

where denotes the -th transmitted optical intensity signal in signal domain. The average power of is denoted as . and are independent of each other, where is the input-independent Gaussian noise, and is the input-dependent Gaussian noise. denotes the ratio of the input-dependent noise variance to the input-independent noise variance. Moreover, represents the real-valued channel gain between the -th LED and the PD, which is given by [24, 25]

(2)

where is the physical area of the PD. is the Lambertian emission order, and is the semi-angle at half-power of the LED. is the distance between the -th LED and the PD. and are the emission angle and the incidence angle from the -th LED to the PD, respectively. denotes the field of view (FOV) of the PD. By using SM, only one element in is non-zero at each timeslot.

Iii Channel Adaptive Bit Mapping

In this paper, the number of LEDs is arbitrary. When , bits can be mapped onto the LEDs’ indexes at each timeslot using the traditional bit mapping scheme [26]. However, when , the traditional bit mapping scheme is not available. In this section, a CABM scheme is provided for bitting mapping.

The principle of the CABM scheme is presented in Fig. 2. The CABM scheme includes three steps. The first two steps are the bit mapping in space domain and the bit mapping in signal domain, respectively. Because the number of the LEDs is not a power of two, the LEDs’ indexes in space domain are mapped with different numbers of bits. To keep the same total bit length at each timeslot (i.e., bits), different modulation modes are employed in signal domain. Assuming that the CSIT is known, the last step is to adaptively select the best modulation order combination in signal domain to get the optimal bit mapping. To facilitate the understanding, the detailed descriptions about the CABM scheme are given in the following three subsections.

Fig. 2: The channel adaptive bit mapping scheme.

Iii-a Bit Mapping in Space Domain

When the number of LEDs is not a power of two, an integer can be found such that . After that, prescribe some information bits transmitted by LEDs with a length of bits and the others with a length of bits. Referring to [19] and [20], the bit mapping in space domain can be implemented by using the following procedures.

  1. Choose such that .

  2. Define be the set of the LEDs’ indexes, and select the former elements from into set . Therefore, .

  3. Divide set into two sets and , where contains the former elements and contains the remainder elements. Then, bits are used to map these elements.

  4. Define set containing the unsettled LEDs. The bit mapping in set is initially the same as that in set .

  5. “0” is appended at the end of each bit mapping for set , and “1” is appended at the end of each bit mapping for set . So far, bits are mapped in sets and bits are mapped in set .

Iii-B Bit Mapping in Singal Domain

After performing the bit mapping in space domain, the bit lengths mapped on each LED are not the same. To ensure the total number of bits transmitted at each timeslot remains the same (i.e., bits), different modulation modes should be employed in signal domain. Without loss of generality, the PAM is employed for the VLC system. Therefore, the signals with higher-order (-ary) PAM are transmitted from the LEDs pertaining to bits and those with lower-order (-ary) PAM are transmitted from the LEDs pertaining to bits. Briefly, the bit mapping in signal domain is performed by using the following procedures.

  1. Choose .

  2. When the LEDs with the index in sets and are activated, -ary PAM is employed to transmit information.

  3. When the LEDs with the index in set are activated, -ary PAM is employed to transmit information.

Iii-C Channel Adaptive Mapping

In [19] and [20], the CSIT is not considered for bit mapping. In this case, LED with bad channel state may employ high-order signal modulation, which will degrade system performance. To improve system performance, the channel adaptive mapping is employed.

In VLC, typical indoor illumination environments offer very high signal-to-noise ratio (SNR) [27]. For the channel in (1), when the -th LED is activated to transmit the symbol , the conditional pairwise error probability (PEP) at high SNR is given by [28]

(3)

where . Note that for a given , and thus . Therefore, the conditional PEP is given by

(4)
Remark 1

It can be observed from (4) that the conditional PEP is a monotonically decreasing function with respect to . Moreover, it can be seen from (2) that is a monotonicity decreasing function with respect to the angle of emission . This indicates that the system error performance degrades with the increase of .

To verify Remark 1, the BER performance in the receive plane of the PD is provided in Fig. 3 when dBm, , and the coordinates of the LED and the PD are (2.5m, 2m, 3m) and (m, m, 0.8m). It can be found that when the PD is located at (2.5m, 2m, 0.8m), the angle of emission is zero, the best BER performance achieves. When the PD moves far away from (2.5m, 2m, 0.8m), the angel of emission increase, and the value of the BER increases, which verifies the conclusion in Remark 1.

Fig. 3: BER performance at the receiver plane.

Moreover, it can be observed from (4) that the system error performance is dominated by the term among constellation points. The minimum value of is given by

(5)

where denotes the set of all possible transmit signals. To improve the system performance, the minimum term should be maximized. Consequently, the modulation order optimization problem can be formulated as

(6)

where are the optimal modulation orders in signal domain on all LEDs. and denotes the total numbers of and in set , respectively.

In the optimization problem (6), the corresponding modulation orders are the optimal solution. However, the problem (6) is an integer optimization problem, which is non-convex and very hard to obtain the optimal solution. From the viewpoint of implementation, computationally efficient algorithms are more preferred. Here, Algorithm 1 is proposed to solve problem (6), which is shown in Fig. 4.


Algorithm 1 (Modulation order optimization algorithm)

Step 1): Given the positions of LEDs and the PD, calculate the channel gains by using (2).
Step 2): Find all possible modulation order combinations , where is the total number of possible combinations, denotes the -th modulation order combination, and denotes the modulation order of the -th LED in the -th modulation order combination.
Step 3): Compute by using (5) for all combinations in .
Step 4): Select the combination with the maximum as the output .

Fig. 4: Modulation order optimization algorithm.

By using Algorithm 1, the best modulation order combination is obtained, and the corresponding set of the LEDs’ indexes is . Sorting the elements in by using decreasing order, and the corresponding new set of the LEDs’s indexes becomes . Then, replace with in Step 2) of Section III-A, and perform the bit mapping in the space and signal domains again, the performance of the bit mapping can be improved.

Iii-D Case Study

To facilitate the understanding, an example is provided in Fig. 5 to compare the proposed CABM scheme with previous schemes in [19] and [20]. In Fig. 5, the number of LEDs is set to be . Assume that the indexes of the LEDs are 1, 2, 3, 4, 5 and 6, and the corresponding channel gains are 0.08, 0.15, 0.13, 0.25, 0.01, 0.22, respectively. For Fig. 5(a), the bit mappings in space and signal domains are performed according to Sections III-A and III-B. In the figure, “**” represents the 2 bits mapped by 4-ary PAM modulation, which can be 00, 01, 10 or 11, and “” stands for the single bit mapped by 2-ary PAM, which can be 0 or 1. However, if the CSIT is considered as in Fig. 5(b), set is first changed to , and then Sections III-A and III-B are performed. Because LED 4 and LED 6 have good channel states, 4-PAM is employed in signal domain. The other LEDs with not so good channel states are using 2-PAM. To show the improvement of system performance, some results will be shown in Section VI-A.

Fig. 5: Comparison between the previous scheme and the proposed CABM scheme when .

Iv Mutual Information Analysis

In the above section, the CABM scheme is proposed. Based on the CABM scheme, the theoretical expression of the mutual information and its lower bound will be derived. In addition, some remarks and insights are also provided.

According to the CABM scheme and the system model, the system diagram can be illustrated as Fig. 6. In Fig. 6, the total number of the source data bits is bits. The source data is divided into two groups by using SM. One group is with (or ) bits to selective the LED (i.e., select the channels) in space domain, while the other one is with (or ) bits to select the constellation point in signal domain. Consequently, the total number of bits transmitted at any timeslot is a constant.

Fig. 6: Principle of bit mapping for SM based VLC.

Assume that the -th LED is selected to transmit signal, and then can be expressed as

(7)

where the total number of elements in sets is , and the number of elements in set is .

When , the space domain includes bits, and the signal domain must include bits to guarantee the total number of transmit bits invariant at each timeslot. In this case, the symbol set in signal domain is denoted as . Similarly, when , the space domain uses bits, and the signal domain must use bits. On this occasion, the symbol set in signal domain is denoted as . Therefore, we have

(8)

and

(9)

Moreover, and can be written, respectively, as

(10)

and

(11)

Then, a unified expression for (10) and (11) can be written as

(12)

According to (1), if (or ), the following conditional PDF is derived by

(13)

Furthermore, we have

(14)

According to (12) and (14), the output PDF is given by

(15)

Iv-a Mutual Information

From [29], the mutual information between the input and the output can be expressed as

(16)

where denotes the conditional mutual information between and when given , is the mutual information between and .

According to the channel model in (1

), and using the probability theory, the mutual information is analyzed in the following theorem.

Theorem 1

For the SM based VLC, the theoretical expression of the mutual information is given by (17) as shown at the top of the next page,

(17)

where , and holds when given and .

See Appendix A.

Remark 2

Define the SNR as . It can be easily proved that in (17) is a monotonic increasing function with . Moreover, when , we have

(18)

Moreover, when , we can get (19) as shown at the top of the next page.

(19)
Remark 3

When is a power of 2 (i.e., ) and , the mutual information in (17) becomes

(20)

Furthermore, when and , (18) and (19) become

(21)

and

(22)

Note that the results in (20)-(22) are consistent with the conclusions in [10].

Iv-B Lower bound on Mutual Information

In (17), it is very difficult to derive a closed-form expression for the mutual information. To reduce computational complexity and obtain more insights, a lower bound on the mutual information is derived in the following theorem.

Theorem 2

For the SM based VLC, a closed-form expression for the lower bound of the mutual information is given by (23) as shown at the top of the next page,

(23)

where .

See Appendix B.

Remark 4

It can be easily proved that in (23) is a monotonic increasing function with . Moreover, when , (23) becomes

(24)

and when , (23) becomes (25) as shown at the top of the next page.