Question

Compute the Fourier transform of x(t) = sin(a t) cos (a t)
where a is a real number > 0.

Answer 1

To compute the Fourier transform of x(t) = sin(at) cos(at), we first simplify the function using trigonometric identities and then apply the Fourier transform to get two impulses at frequencies ±2a.

Step-by-step explanation:

The question asks to compute the Fourier transform of x(t) = sin(at) cos(at), where a is a positive real number. First, we use a trigonometric identity to simplify the function:

x(t) = ½ sin(2at).

Now, taking the Fourier transform of the simplified function gives us:

Fourier transform: X(f) = ½[δ(f-2a) - δ(f+2a)],

where δ represents the Dirac delta function, and f is the frequency domain variable. This means that the Fourier transform of x(t) consists of two impulses at frequencies ±2a.