IDFT Problems with Solutions.
Problem 1:
Find the inverse discrete Fourier transform (IDFT) of the following sequence:
X[k] = {1, 2, 3, 4}
Solution:
To find the IDFT, we use the formula:
x[n] = (1/N) * Σ(k=0 to N-1) X[k] * e^(j2πnk/N)
where N is the length of the sequence.
In this case, N = 4, so the formula becomes:
x[n] = (1/4) * Σ(k=0 to 3) X[k] * e^(j2πnk/4)
x[0] = (1/4) * (1e^0 + 2e^0 + 3e^0 + 4e^0) = 2.5
x[1] = (1/4) * (1e^0 + 2e^(jπ/2) + 3e^(jπ) + 4e^(3jπ/2)) = -0.5 + j
x[2] = (1/4) * (1e^0 + 2e^(jπ) + 3e^(2jπ) + 4e^(3jπ)) = -2.5
x[3] = (1/4) * (1e^0 + 2e^(3jπ/2) + 3e^(jπ) + 4e^(jπ/2)) = -0.5 – j
Therefore, the IDFT of X[k] is x[n] = {2.5, -0.5 + j, -2.5, -0.5 – j}.
Problem 2:
Find the IDFT of the following sequence:
X[k] = {1, -j, -1, j}
Solution:
Using the same formula as in Problem 1, we get:
x[n] = (1/4) * Σ(k=0 to 3) X[k] * e^(j2πnk/4)
x[0] = (1/4) * (1e^0 – je^0 – 1e^0 + je^0) = 0
x[1] = (1/4) * (1e^0 – je^(jπ/2) – 1e^(jπ) + je^(3jπ/2)) = 1
x[2] = (1/4) * (1e^0 + je^(jπ) – 1e^(2jπ) – je^(3jπ)) = 0
x[3] = (1/4) * (1e^0 + je^(3jπ/2) – 1e^(jπ) – je^(jπ/2)) = 1
Therefore, the IDFT of X[k] is x[n] = {0, 1, 0, 1}.