Question

The Fourier transform of sinc³(t) is given as follows:

F [sinc³(t)] {(a) : f ≤ -3/2
{(b) : -3/2 < f ≤ -1/2
{(c) : -1/2 < f ≤ 1/2
{(d) : 1/2 < f ≤ 3/2
{(e) : 3/2 < f
Determine the corresponding values or the corresponding expressions in (a), (b), (c), (d), and (e).

Answer 1

The corresponding values or expressions for the Fourier transform of sinc³(t) in different frequency intervals are determined through the evaluation of integrals.

Step-by-step explanation:

The Fourier transform of sinc³(t) can be calculated as follows:

F [sinc³(t)] = ∫ sinc³(t) e-2πift dt

where f is the frequency.

To determine the corresponding values or expressions in each interval, we can evaluate the integral for each interval:

• For f ≤ -3/2, the Fourier transform is 0.
• For -3/2 < f ≤ -1/2, the Fourier transform expression is 1/16(3 + 2πif) + 9/16(πif)sin(2πif) - 27/16(1/(2πif))cos(2πif).
• For -1/2 < f ≤ 1/2, the Fourier transform expression is 1/16(3 + 2πif) - 9/16(πif)sin(2πif) - 27/16(1/(2πif))cos(2πif).
• For 1/2 < f ≤ 3/2, the Fourier transform expression is 1/16(3 - 2πif) - 9/16(πif)sin(2πif) - 27/16(1/(2πif))cos(2πif).
• For f > 3/2, the Fourier transform is 0.