Final answer:
The Fourier transform of x(t) = sinc(t)(sinc(t) + 2e^(j2t)sinc(t)) involves finding the transform of the sinc function and then applying the modulation property to incorporate the e^(j2t) term, resulting in a superposition of shifted frequency spectra.
Step-by-step explanation:
The question concerns computing the Fourier transform of a given wave function, specifically x(t) = sinc(t)(sinc(t) + 2e^(j2t)sinc(t)). The process involves transforming this time domain signal into its frequency domain representation. The sinc function, often represented as sinc(t) = sin(πt)/(πt), is a common function encountered in signal processing. The Fourier transform of the sinc function is well-known and appears as a rectangular pulse in the frequency domain. Given the additional exponential term e^(j2t), which represents a frequency shift, the Fourier transform will entail the convolution of the transform of sinc(t) with that of e^(j2t). This results in a shifted frequency spectrum due to the modulation.
The approach to computing the Fourier transform would be to first find the transform of sinc(t), denoted as F[sinc(t)], and then apply the modulation property of Fourier transforms to account for the e^(j2t) term. The final result would be the superposition of a rectangular pulse and its shifted version in the frequency domain due to the exponential component.