Muhammad Abdullah

To directly implement the Swerling 0 characteristic function (CF) in MATLAB without using Fourier integral pairs, you can follow these steps:

1. Define the Swerling 0 CF equation in MATLAB:

```matlab

function cf = swerling0_cf(x, N, s)

cf = exp(-N * x * (s / (s + 1))) / (s + 1)^N;

end

```

Here, `x` is the signal-to-noise ratio (SNR), `N` is the number of pulses, and `s` is the complex variable representing the Laplace transform.

2. To calculate the probability of detection (`Pd`) directly from the Swerling 0 CF, you can use the following steps:

```matlab

function Pd = swerling0_Pd(x, N, threshold)

% Define the integration range for the real and imaginary parts of s

s_real = linspace(-100, 100, 1001);

s_imag = linspace(-100, 100, 1001);

% Initialize the probability of detection

Pd = 0;

% Iterate through the real and imaginary parts of s

for i = 1:length(s_real)

for j = 1:length(s_imag)

s = s_real(i) + 1i * s_imag(j);

cf = swerling0_cf(x, N, s);

Pd = Pd + exp(-threshold) * (threshold)^(N - 1) / (s + 1)^N * real(cf) * (s_real(2) - s_real(1)) * (s_imag(2) - s_imag(1));

end

end

end

```

In this code, we define the integration range for the real and imaginary parts of `s`, and then iterate through these values to numerically integrate the Swerling 0 CF. The `real()` function is used to extract the real part of the CF, as the probability of detection is based on the real part of the Laplace transform.

To obtain the Confluent Hypergeometric function from the Swerling 0 CF, you can use the following MATLAB function:

```matlab

function M = confluent_hypergeometric(a, b, z)

M = hypergeom([a], [b], z);

end

```

Here, `a` and `b` are the parameters of the Confluent Hypergeometric function, and `z` is the argument.

To use these functions, you would call them with the appropriate parameters, like this:

```matlab

x = 10; % SNR

N = 1; % Number of pulses

threshold = 10; % Detection threshold

Pd = swerling0_Pd(x, N, threshold);

M = confluent_hypergeometric(N, 1, -N * x / (1 + x));

```

This will calculate the probability of detection (`Pd`) for the Swerling 0 case and the Confluent Hypergeometric function based on the provided parameters.

Please note that the numerical integration approach may require some fine-tuning to ensure accurate results, especially for large values of `N` or extreme values of `x` and `threshold`.

Good luck; partial credit ai