---
title: "8-point DIF-FFT of {1/2,1/2,1/2,1/2,0,0,0,0}"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:dif-fft-half-ones"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dif-fft-half-ones
site: BCT Engineering
---

# 8-point DIF-FFT of {1/2,1/2,1/2,1/2,0,0,0,0}

Digital Signal Analysis and Processing · Chapter 7: Discrete Fourier Transform

- Asked in: 2074 Asoj, 7 marks
- Video: [Problem on 8 point DFT using DIF FFT in digital signal processing || EC Academy](https://www.youtube.com/watch?v=nMfr5dcV860) — EC Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dif-fft-half-ones
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

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.

## 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}.

## Answer

#### 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:**

```text
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}.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In a radix-2 DIF butterfly, what are the outputs for inputs A and B?
   A. A·B and A/B
   B. A + B and A - B with no twiddle ever
   C. A + B and (A - B)W
   D. (A + W B) and (A - W B)
   Answer: C. In DIF the sum goes straight through and the difference is multiplied by the twiddle factor. (Key point: DIF butterfly)

2. In stage 1 for N = 8, which samples are paired?
   A. n and n + 4
   B. n and n + 8
   C. n and n + 1
   D. n and n + 2
   Answer: A. The first stage pairs samples N/2 apart, with twiddle W8^n on the difference. (Key point: Stage 1 pairing)

3. In what order do the DIF outputs appear?
   A. Natural order
   B. Bit-reversed order
   C. Reverse natural order
   D. Random order
   Answer: B. Positions 0 to 7 hold X(0), X(4), X(2), X(6), X(1), X(5), X(3), X(7). (Key point: Output order)

4. Which value of X(0) is expected and why?
   A. 2, the sum of the samples
   B. 0.5, the first sample
   C. 4, twice the sum
   D. 0, as half the samples are zero
   Answer: A. X(0) = sum x(n) = 4 x 0.5 = 2. (Key point: Result check)

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