PSK to CSK mapping for hybrid systems involving the radio frequency and the visible spectrum

Abstract

This paper presents an efficient technique to map phase shift keying (PSK) signalling to colour shift keying (CSK) constellation, to establish a full link in hybrid systems involving the radio frequency (RF) and the visible spectrum. It fits in systems combining (first link) wireless communication technologies such as the wireless fidelity (WiFi) or wired communication technologies such as power line communications (PLC) to visible light communications (VLC) technology (second link). On the first link, PSK technique is used to convey the information, while, on the second link, a technique based on colour variation is deployed. WiFi standards targeted are those that employ PSK as sub-carrier modulation techniques (IEEE 802.11a/11g/11n). The PSK complex constellation observed at the output of the first link is converted into colours using the hue-saturation-value/intensity (HSV/I) colour models. The constant lighting required in VLC corresponds with the coordinate I of the HSI and the colour constraint is met by assigning adequate current intensities to the red-green-blue LEDs (RGB-LEDs) used. The design meets the requirements of CSK constellation design outlined in IEEE 802.15.7. The performance of the system is analysed through bit error rate curves obtained by simulations, for binary PSK (BPSK) and quadrature PSK (QPSK), 8PSK and 16PSK constellations. The results show that as the constellation size increases, the performance of the system decreases.

Appendix

1.1 Appendix 1: Power/colour averaging and distance optimisation in CSK design

• Power and colour averaging in CSK design

In CSK constellation design, the average power must be constant during the transmission (see Sect. 4.2). The average power, $P^\mathrm{{avg}}$, is fixed by the chosen lumen value L(lm) by the relation

$$\begin{aligned} P^\mathrm{{avg}}=\frac{L^\mathrm{{avg}}}{\eta }, \end{aligned}$$

(11)

where $\eta$ is the luminous efficacy. The average luminous flux can be represented as function of the symbol and the channel gain by [17]

$$\begin{aligned} L^\mathrm{{avg}}=\langle \overline{g},S_{k} \rangle , \end{aligned}$$

(12)

where $\langle u,v \rangle$ is the inner product of u and v, $S_{k}$ the kth symbol and $\overline{g} = [g_{r},g_{g},g_{b}]$ ($g_{r}$, $g_{g}$ and $g_{b}$ are the optical gain of the red, green and blue channels, respectively). The average colour is given by

$$\begin{aligned} C^\mathrm{{avg}}=\sum _{k=1}^{N} \gamma _{k}S_{k}, \end{aligned}$$

(13)

where $\gamma _{k}$ corresponds to the probability of correctly transmitting the kth symbol. $C^\mathrm{{avg}}$ is defined to avoid large drift between transmitted colours.

• Probabilistic decision and distance optimisation in CSK design

The minimum square Euclidean distance target on the kth colour in CSK is given by

$$\begin{aligned} \varrho = \arg \min \, \left\{ \Vert E_{k}- \mathbf H C^{s}_{k}\Vert ^{2} \right\} \end{aligned}$$

(14)

where $E_{k}$ is the expected colour corresponding to the kth symbol and $\mathbf H$ the channel response (the other parameters are defined in the text). Note that the minimum distance between $C^{s}_{k}$ and $C^{r}_{k}$ is upper-bounded by $Q [(\Vert E_{k}-\mathbf H C^{s}_{k} \Vert )/(2\sqrt{N_{0}/2)}]$, where $(\Vert E_{k}- \mathbf H C^{s}_{k} \Vert )/2$ represents the distance from the two vectors $E_{k}$ and $C^{s}_{k}$ to the decision boundary. The objective function $\Vert \mathbf e _{k}- \mathbf H \mathbf c ^{s}_{k}\Vert ^{2}$ ($\mathbf e _{k}$ being the expectation set and $\mathbf c ^{s}_{k}$ the transmitted set) is rearranged using a decision variable $\phi$ [15, 16], and the optimised detection is given by

$$\begin{aligned} \varrho = min \left\{ \Vert \mathbf e _{k}(\phi )- \mathbf H \mathbf c ^{s}_{k}(\phi ) \Vert ^{2} \right\} , \end{aligned}$$

(15)

where $\mathbf e _{k}$ and $\mathbf c ^{s}_{k}$ are given as function of the variable $\phi$ ($\mathbf e _{k} (\phi )$ and $\mathbf c ^{s}_{k}(\phi )$). We then find a function that will meet the requirements and the constrains of the CSK constellation design. To do this, we write the distance $d_{sr}$:

$$\begin{aligned} d_{sr}^{2} = \Vert C^{r}_{k} - \mathbf H C^{s}_{k} \Vert ^{2}. \end{aligned}$$

(16)

Equation (16), not being differentiable, needs to be approximated [15, 16]. We find a function that approximates (16) and satisfies (14). The following objective function (17) preferably represents the above described objective detection situation [15, 16], and [25].

$$\begin{aligned} \frac{\ln }{\beta }\displaystyle \sum \limits _{r \ne s} exp \left( -\beta \Vert E_{k}-\mathbf H C^{s}_{k} \Vert ^{2}\right) \end{aligned}$$

(17)

The solution f($\phi$) given in (18), which is a function of the decision variable $\phi$ satisfies the maximum detection probability (14) [15].

$$\begin{aligned} f(\phi )= \frac{ln}{\beta }\displaystyle \sum \limits _{e \ne s} exp \left( -\beta \Vert \mathbf e _{k}(\phi )-\mathbf H c^{s}_{k} (\phi ) \Vert ^{2}\right) \end{aligned}$$

(18)