Consider the MIMO system which has $N_t$ antennas at transmitter and $N_r$ antennas at receiver and uses Generalized space shift keying (GSSK) modulation. The received signal is given by:
$$Y = H X + N_w\tag 1$$
where, $H$ is the channel of dimension $N_r \times N_t$, $N_w$ is AWGN of dimension $N_r \times L$, $X = [\mathbf{x_1},\mathbf{x_2},...,\mathbf{x_L}]$ is transmitted signal sequence of dimension $N_t \times L$ and $L$ denotes total time slots.
Each transmitted signal is drawn from the transmit symbol set $\mathbb{X}_{\text{GSSK}}=\{\mathbf{x[1],\mathbf{x}[2],...,\mathbf{x}[N]}\}$, where $\mathbf{x}[n]$ is $n^{\text{th}}$ GSSK modulated symbol of dimension $N_t \times 1$ and $\mathbf{x}[n] \in \mathbb{X}_{\text{GSSK}}$.
Then according to the paper titled "Supervised Learning-Based Semi-Blind Detection for Generalized Space Shift Keying MIMO Systems", it is found that the Maximum Likelihood detection for the system in equation (1) is given as:
$$(\hat{H}, \hat{X}) = \underset{H \in \mathbb{C}^{N_r \times N_t}\\X \in \mathbb{X}_{\text{GSSK}}^L}{\arg\,\min}\, \| Y- HX\|_{\text{F}}^2 \tag2 $$
such that $\mathbb{X}_{\text{GSSK}}^L$ is $L$ dimensional $\mathbb{X}_{\text{GSSK}}$ and $|\mathbb{X}_{\text{GSSK}}^L| = N^L$, where $|\cdot|$ denotes the cardinality of set, $\|\cdot\|_{\text{F}}$ denotes Frobenius norm.
My query is that what does the equation (2) indicates and how to proceed for its implementation in MATLAB.
