MATLAB Guide: Modeling UVLC Capacity Lower Bounds

UVLC Capacity

Calculating the point-to-point capacity of an underwater optical link requires modeling Gaussian noise and water attenuation (Beer-Lambert Law).

1. The Capacity Lower Bound Formula

C_LB = h(X) + ln(Hc) - f(rho) - h(Y|X)

Breaking Down the Symbols

This equation is essentially a "Data Budget." You start with a total amount of information and subtract everything that ruins it along the way.

h(X)

The Volume of your Shout (Input Entropy)
This is the total information you are trying to send. A "louder" signal with more variety allows for more data.

ln(H_c)

The Fog Tax (Channel Loss)
Water isn't clear. It absorbs light. This term is a "tax" you pay to the ocean. The murkier the water, the more data you lose here.

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f(ρ)

The LED Penalty (Constraints)
Unlike a radio, an LED cannot have "negative light." It's either on or off. This mathematical penalty accounts for the physical limits of using light bulbs to send data.

h(Y|X)

The Background Splashing (Noise)
This is the "static" in the water—sunlight, heat, or electronic interference. If the splashing is too loud, the message gets garbled.

Why Does This Matter?

Engineers use this formula to build Autonomous Underwater Vehicles (AUVs). By solving this equation, they can predict exactly how far a robot can swim away from its base before the "Underwater Wi-Fi" cuts out.

If the result (C_LB) is high, you can stream 4K video from the ocean floor. If it's low, you might only be able to send a simple text message.

2. Implementation Snippet

Use the following MATLAB function to solve for the Lagrange multiplier (λ*) in Case I distributions:

function lambda = solve_lambda()
    % Solves for input optimization
    fun = @(x) 1 - (1./x - exp(-x)./(1-exp(-x)));
    lambda = fzero(fun, 1);
end

3. Visualizing Results

To plot the OSNR (Optical Signal-to-Noise Ratio) against capacity in nats/Hz, you must account for different extinction coefficients (c):

• Clear Ocean: c = 0.151
• Coastal Water: c = 0.305
• Turbid Harbor: c = 2.17

This code allows engineers to predict link performance before deploying expensive underwater AUVs.