Lambertian Channel Model

Lambertian Channel Model for Optical Wireless Communication

The code you provided uses a Lambertian channel model for optical wireless communication, specifically for modeling the received power in a system with a light-emitting diode (LED) and a photodetector (PD).

Key Components of the Channel Model

1. Lambertian Emission

• The code uses the Lambertian model to describe the light emission from the LED. This model assumes that the LED emits light according to a Lambertian distribution, which is typical for LED lights.
• The Lambertian order (m) is calculated as:

  m = -log(2) / log(cos(θ))

  where (θ) is the semi-angle at half-power.
• The emission pattern follows a cosine law, meaning the intensity decreases as the angle from the LED’s optical axis increases.

2. Geometrical Path Loss

• The distance between the source (LED) and the receiver (PD) is computed using the Euclidean distance formula:

  D1 = sqrt((XR - XT(1,1))^2 + (YR - YT(1,1))^2 + h^2)

  where (XR, YR) are receiver coordinates, (XT, YT) is the LED position, and (h) is the height.

3. Received Power Calculation

• The received power is calculated using:

  H_A1 = (m+1) * Adet * cos(φ)^(m+1) / (2 * π * D1^2)

• H_A1 is the channel gain.
• Adet is the photodetector area.
• φ is the angle between receiver and LED.
• D1 is the distance between source and receiver.

4. Optical Concentrator Gain

• The optical concentrator gain is computed as:

  G_Con = (n^2) / sin(FOV)

  where (n) is the refractive index and (FOV) is the field of view.

5. Channel DC Gain

• The channel DC gain (H_A1) models how much transmitted optical power is captured by the photodetector.

Channel Model Type

• Free-Space Optical Communication: The model is based on free-space optical propagation, combining path loss, Lambertian emission, and concentrator gain. It is commonly used in indoor optical wireless communication (OWC) and Li-Fi systems.

Summary of the Channel Model Used

The code uses the Lambertian model combined with geometrical path loss and concentrator gain to calculate received power. This approach is widely used in indoor optical wireless communication systems.

Semi-Angle at Half Power

The semi-angle at half power, denoted as θ_HP, is the angle at which emitted power falls to 50% of its maximum value.

It is widely used to describe the beam width or spread of radiation patterns.

Lambertian Emission

The intensity follows a cosine law:

I(θ) = Imax · cos(θ)^m

The semi-angle at half power satisfies:

I(θHP) = Imax / 2

The relationship between Lambertian order and semi-angle is:

m = log(2) / log(cos(θHP))

Example

• An LED with a semi-angle at half power of 70° has half its maximum intensity at 70° from the optical axis.