# Understanding Maximum likelihood detector expression?

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.

You may look at the solution I gave at Why Isn't the ML Estimator (MLE) in MIMO Spatial Multiplexing Obtained by the Least Squares Solution.

You have 2 options:

1. Brute Force: Try each signal in your dictionary an see which one minimizes the LS objective.
2. Solve the LS objective without the constraint, then search for the most similar signal out of the dictionary.
• Thank u so much sir for your detailed answer.... Commented Apr 9, 2023 at 2:08
• How do we +1.... Commented Apr 9, 2023 at 7:56
• @Heretolearn, On the left you have up arrow. Above the V you marked.
– Royi
Commented Apr 9, 2023 at 8:08
• Could u pls show it with some pic or snap.... Commented Apr 9, 2023 at 11:46
• @Heretolearn, therre you go: i.imgur.com/f4G9RzU.png
– Royi
Commented Apr 9, 2023 at 12:00