【Advanced】Signal Multiple Antenna Technology and Beamforming in MATLAB

# 2.1 Multiple Antenna Array Model

The multiple antenna array model is the foundation of multi-antenna signal technology, describing the geometric shape of the antenna array and the interactions among the antennas.

### 2.1.1 Array Factor

The array factor characterizes the gain pattern of the antenna array in a specific direction, resulting from the coherent superposition of the radiated signals from each antenna in the array. The array factor expression is:

AF(θ, φ) = ∑_{n=1}^{N} a_n e^(-j2π(x_n sin(θ) cos(φ) + y_n sin(θ) sin(φ) + z_n cos(θ)))

Where:

- `θ` and `φ` are the direction and azimuth angles
- `N` is the number of antennas in the array
- `a_n` is the complex amplitude of the nth antenna
- `x_n`, `y_n`, and `z_n` are the coordinates of the nth antenna

# 2. Signal Multi-Antenna Technology Implementation in MATLAB

### 2.1 Multiple Antenna Array Model

#### 2.1.1 Array Factor

The array factor describes the radiation pattern formed by the superposition of phase and amplitude of signals from each antenna in a multi-antenna array. For a uniform linear array, the array factor is:

AF(θ) = ∑_{n=0}^{N-1} w_n * e^(-j2πn * d * sin(θ) / λ)

Where:

- `θ` is the direction of arrival
- `N` is the number of antennas
- `w_n` is the weight of the nth antenna
- `d` is the antenna spacing
- `λ` is the signal wavelength

#### 2.1.2 Beamforming

Beamforming utilizes the array factor to control the radiation direction and gain of the antenna array. By adjusting the antenna weights, the radiated energy of the antenna array can be concentrated in a specific direction, forming a beam.

### 2.2 Multi-Antenna Channel Estimation

Channel estimation is the process of estimating the signal transmission characteristics in a multi-antenna channel, including path loss, time delay, ***mon methods for channel estimation include:

#### 2.2.1 Minimum Mean Square Error Estimation

The Minimum Mean Square Error estimation (MMSE) estimates the channel by minimizing the mean square value of the channel estimation error. The estimation formula is:

h_MMSE = R_hh * H^H * (H * R_hh * H^H + R_nn)^-1 * y

Where:

- `h_MMSE` is the estimated channel
- `R_hh` is the channel autocorrelation matrix
- `H` is the channel matrix
- `R_nn` is the noise covariance matrix
- `y` is the received signal

#### 2.2.2 Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) estimates the channel by maximizing the likelihood function of the received signal. The estimation formula is:

h_MLE = arg max_h P(y | h)

Where:

- `h_MLE` is the estimated channel
- `P(y | h)` is the likelihood function of the received signal given the channel condition

### 2.3 Multi-Antenna Detection and Decoding

Multi-antenna ***mon methods include:

#### 2.3.1 Maximum Likelihood Detection

Maximum Likelihood Detection (MLD) detects the transmitted signal by maximizing the likelihood function of the received signal. The detection formula is:

s_MLD = arg max_s P(y | s)

Where:

- `s_MLD` is the estimated transmitted signal
- `P(y | s)` is the likelihood function of the received signal given the transmitted signal

#### 2.3.2 Minimum Mean Square Error Detection

Minimum Mean Square Error Detection (MMSE) detects the transmitted signal by minimizing the mean square value of the detection error. The detection formula is:

s_MMSE = E[s | y]

Where:

- `s_MMSE` is the estimated transmitted signal
- `E[s | y]` is the expected value of the transmitted signal given the received signal

# 3.1 Mean Square Error Beamforming

#### 3.1.1 Fundamental Principles

Mean Square Error (MSE) beamforming is an adaptive beamforming algorithm aimed at optimizing the beam direction by minimizing the mean square error between the received signal and the desired signal. Its fundamental principles are as follows:

***Array Response Vector:** Define the array response vector as:

a(θ) = [a1(θ), a2(θ), ..., aN(θ)]^T

where `aN(θ)` represents the response of the nth antenna, and `θ` is the signal arrival angle (AOA).

***Desired Signal:** The desired signal is a signal from a specific direction, with its array response vector represented as:

s(θ) = [s1(θ), s2(θ), ..., sN(θ)]^T

***Mean Square Error:** MSE is defined as the mean square error between the received signal `y` and the desired signal `s`:

MSE = E[||y - s||^2]

#### 3.1.2 Algorithm Implementation

The steps for implementing the MSE beamforming algorithm are as follows:

1. **Calculate the Array Response Vector:** Calculate the array response vector for each antenna based on the antenna positions and the signal arrival angle.
2. **Calculate the Covariance Matrix:** Calculate the covariance matr