- 2074 Asoj · 7 marks
8-point DIF-FFT of {1/2,1/2,1/2,1/2,0,0,0,0}
Short answer
Use an 8-point radix-2 DIF-FFT with three stages and bit-reversed output. The result is X(k) = {2, 0.5-j1.2071, 0, 0.5-j0.2071, 0, 0.5+j0.2071, 0, 0.5+j1.2071}.
Compute the 8-point DFT of the sequence x[n] = {1/2, 1/2, 1/2, 1/2, 0, 0, 0, 0} using Decimation in Frequency Fast Fourier Transform (DIF-FFT) algorithm.
Method: radix-2 decimation in frequency (N = 8, 3 stages)
Each butterfly: A' = A + B, B' = (A − B)·W. Stage 1 pairs n and n + 4 (twiddle W₈ⁿ on the difference), stage 2 pairs n and n + 2 (W₄ⁿ), stage 3 pairs adjacent samples (W₂⁰ = 1). The outputs come out in bit-reversed order.
x(n) = {0.5, 0.5, 0.5, 0.5, 0, 0, 0, 0}
| Position | Input x(n) | After stage 1 | After stage 2 | After stage 3 |
|---|---|---|---|---|
| 0 | 0.5 | 0.5 | 1 | 2 |
| 1 | 0.5 | 0.5 | 1 | 0 |
| 2 | 0.5 | 0.5 | 0 | 0 |
| 3 | 0.5 | 0.5 | 0 | 0 |
| 4 | 0 | 0.5 | 0.5 − j0.5 | 0.5 − j1.2071 |
| 5 | 0 | 0.3536 − j0.3536 | −j0.7071 | 0.5 + j0.2071 |
| 6 | 0 | −j0.5 | 0.5 + j0.5 | 0.5 − j0.2071 |
| 7 | 0 | −0.3536 − j0.3536 | −j0.7071 | 0.5 + j1.2071 |
Output positions 0, 1, 2, 3, 4, 5, 6, 7 hold X(0), X(4), X(2), X(6), X(1), X(5), X(3), X(7) (bit-reversed). Rearranging:
Result:
X(0) = 2
X(1) = 0.5 − j1.2071
X(2) = 0
X(3) = 0.5 − j0.2071
X(4) = 0
X(5) = 0.5 + j0.2071
X(6) = 0
X(7) = 0.5 + j1.2071
Check: Σx(n) = 2 = X(0); the result equals the direct 8-point DFT.
Answer: X(k) = {2, 0.5 − j1.2071, 0, 0.5 − j0.2071, 0, 0.5 + j0.2071, 0, 0.5 + j1.2071}.
✓ Answer checked by Claude Opus 5.5 (high effort)
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