Method of linear transformation

Watson is working on a task involving the analysis of signals. He needs to process signals using mathematical techniques. He applies a linear transformation method to the signal. The shape of the signal can be described by the formula:

,

where \Psi(x') is a function that describes the position and intensity of particular components of the signal—a Gaussian function describing the shape and width of particular spectral lines. The coefficients $\mu$ and $\sigma$ are provided as input to Watson, and the coefficient $A$ is calculated using the following formula:

The function definition for $\Psi (x)$ can be given in this way: , where the Fourier transform of the function $F(x)$ is calculated as: .

To perform precise computations in signal analysis, Watson needs to know the numerical value of the coefficient $A$.