# Multiuser Transmit Precoding Design for Dimming Compatible Visible Light Communications
In particular, [18] also included an optimal TPC scheme under the max-min fairness criterion. In [19], two kinds of linear precoding schemes were developed for coordinated VLC broadcasting architecture with the objective of minimizing the sum mean-squared error (MSE) given an illumination level and retaining a required performance at the minimal illumination level, respectively. Instead of employing the MSE criterion, the work in [20] proposed the optimal TPC design that maximizes the achievable sum rate of multiuser VLC systems. A recent work [21] considered a leakage-based TPC in VLC to suppress inter-user interference. All the above exciting contributions focused on TPC design using zero-mean modulation in VLC. Note that since the zero-mean modulation does not affect the average optical power, these works can be extended for supporting dimming control by directly adjusting the direct current (DC) bias. However, these works are not applicable to modulation schemes associated with the non-zero mean. This inherently limits their applications, since some modulation schemes like pulse position modulation (PPM) and on-off keying (OOK) are popular in VLC, but have the non-zero mean. Numerous challenges emerge in the multiuser TPC design when using the general modulations with the non-zero mean since these modulations will lead to more complicated optical power constraints, as well as a more intractable design objective function than the special case of zero-mean modulation. To the best of our knowledge, there is no general solution to the multiuser TPC design combined with dimming control in VLC. Against this background, we propose a general solution of joint dimming control and multiuser TPC design in VLC, for mitigating the multiuser interference while maintaining a specific indoor illumination level. The main contributions of the paper are summarized as follows:  
1. Instead of being restricted to the special case of zero-mean modulation, our scheme is capable of supporting a wide range of modulation schemes in VLC, such as pulse amplitude modulation (PAM) and PPM with a positive mean. 2. In contrast to RF communication, VLC is developed from the LED-based indoor lighting infrastructure. Therefore, several practical optical constraints including the LED non-linearity and illumination requirements are considered in our design for ensuring the normal operation of LEDs, as well as achieving the desired dimming level. 3. The optimal TPC subject to these practical constraints are developed with the objective of maximizing the minimum signal-to-interference-plus-noise ratio (SINR) among users. The proposed scheme exhibits a significant performance gain compared to the traditional pseudo-inverse-based ZF TPC. To facilitate the reader, we here summarize the following notations used throughout the paper. Boldface upper case and boldface lower case letters represent matrices and column vectors, respectively. Furthermore, \(\mathbf{I}_{N}\) is the identity matrix of size \(N\), and \(\mathbf{e}_{i}\) denotes the all-zero vector with a single one in the \(i^{\text{th}}\) element. \(||\cdot||_{2}\) denotes the \(l_{2}\) norm of a vector. The operators \(\text{vec}(\cdot)\) and \((\cdot)^{T}\) represent the vectorization and transpose of a matrix, respectively. The statistical expectation is denoted by \(\text{E}\{\cdot\}\). The operator \(\mathbf{A}\otimes\mathbf{B}\) is the Kronecker product of matrices \(\mathbf{A}\) and \(\mathbf{B}\).