Question

Consider the function: \( x(t)=e^-a, a>0 \). Find its Fourier transform.

Answer 1

The Fourier transform of
`\( x(t) = e^(-a|t|) \), where \( a > 0 \),` is given by
`\( X(f) = (2a)/(a^2 + 4\pi^2f^2) \).`

Step-by-step explanation:

The given function is a decaying exponential function, and we want to find its Fourier transform. The Fourier transform
`\( X(f) \) of a function \( x(t) \)` is given by the integral:

`\[ X(f) = \int_(-\infty)^(\infty) x(t) e^(-j2\pi ft) \,dt \]`

For the function
`\( x(t) = e^(-a|t|) \), where \( a > 0 \)`, the integral can be solved using the properties of the Laplace transform. The result is:

`\[ X(f) = (2a)/(a^2 + (2\pi f)^2) \]`

This expression represents the amplitude spectrum of the function in the frequency domain. The denominator contains a term
`\( a^2 + (2\pi f)^2 \),` which is reminiscent of the Pythagorean identity. The numerator
`\( 2a \)`signifies the decay rate of the exponential function in the time domain.

The Fourier transform reveals that the frequency components of the original function are also decaying exponentially, with a rate determined by the parameter
`\( a \). The function \( e^(-a|t|) \)` has a symmetric spectrum, reflecting its even symmetry in the time domain. The result provides insight into how different frequencies contribute to the overall behavior of the original function in the time domain.