Final answer:
The Fourier transform of x(t)=sinc(at)cos(at) is calculated by understanding that the sinc and cosine functions represent a modulated signal in time domain, converting to a combined effect in the frequency domain around the frequencies ±a.
Step-by-step explanation:
To compute the Fourier transform of the function x(t) = sinc(at)cos(at), where a is a real number greater than 0, we need to apply the properties and theorems related to Fourier transforms. The sinc function, which is sinc(at) = sin(at)/(at), represents a band-limited function that converts to a rectangular function in the frequency domain. When combined with the cos function, this operation can be viewed as a modulation of the sinc function. In the context of waves and oscillations, as seen in the provided snippets, this is akin to modifying the amplitude of a sinusoidal wave with a secondary harmonic function.
The Fourier transform of a product of two functions can usually be computed using convolution in the frequency domain, but in this case, since we know the transforms of both sinc and cosine, it might be more straightforward to use properties of transforms and modulation.
The Fourier Transform of a sinc function is a rectangular function with cutoff frequencies defined by the parameter a. The multiplication by cos(at) in the time domain corresponds to a frequency domain modulation that shifts and splits the spectrum. The Fourier transform of cos(at) is a pair of delta functions in the frequency domain at ±a. Hence, the overall Fourier transform of x(t) involves combining these effects. A precise calculation requires the use of integral properties, but without going into the details, we would expect the result to contain components centered around the frequencies ±a, modified by the effects of the sinc function.