Answer:
Therefore, the 4-point DFT of the signal [3, 2, -5, 4] is [4, 5 - 9i, 0, -1 + i].
Explanation:
To find the 4-point Discrete Fourier Transform (DFT) of the given signal [3, 2, -5, 4], we can use the formula for DFT:
X(k) = Σ[n=0 to N-1] x(n) * exp(-j * 2π * k * n / N)
where X(k) is the DFT coefficient at frequency index k, x(n) is the input signal, N is the length of the signal, and j is the imaginary unit.
For the given signal [3, 2, -5, 4], with N = 4, the 4-point DFT coefficients can be calculated as follows:
For k = 0:
X(0) = (3 * exp(-j * 2π * 0 * 0 / 4)) + (2 * exp(-j * 2π * 0 * 1 / 4)) + (-5 * exp(-j * 2π * 0 * 2 / 4)) + (4 * exp(-j * 2π * 0 * 3 / 4))
= 3 + 2 - 5 + 4
= 4
For k = 1:
X(1) = (3 * exp(-j * 2π * 1 * 0 / 4)) + (2 * exp(-j * 2π * 1 * 1 / 4)) + (-5 * exp(-j * 2π * 1 * 2 / 4)) + (4 * exp(-j * 2π * 1 * 3 / 4))
= 3 + 2 - 5i - 4i
= 5 - 9i
For k = 2:
X(2) = (3 * exp(-j * 2π * 2 * 0 / 4)) + (2 * exp(-j * 2π * 2 * 1 / 4)) + (-5 * exp(-j * 2π * 2 * 2 / 4)) + (4 * exp(-j * 2π * 2 * 3 / 4))
= 3 - 2 - 5 + 4
= 0
For k = 3:
X(3) = (3 * exp(-j * 2π * 3 * 0 / 4)) + (2 * exp(-j * 2π * 3 * 1 / 4)) + (-5 * exp(-j * 2π * 3 * 2 / 4)) + (4 * exp(-j * 2π * 3 * 3 / 4))
= 3 - 2 + 5i - 4i
= -1 + i
Therefore, the 4-point DFT of the signal [3, 2, -5, 4] is [4, 5 - 9i, 0, -1 + i].