---
title: "8-point DIF-FFT of {1,1,2,2,1,1,2,1}"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:difft-8pt-69c"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/difft-8pt-69c
site: BCT Engineering
---

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

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

- Asked in: 2069 Chaitra, 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/difft-8pt-69c
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

Find 8-point DFT of sequence x[n] = {1, 1, 2, 2, 1, 1, 2, 1} using Decimation in frequency FFT (DIFFFT) algorithm.

## Short answer

Use radix-2 DIF FFT with outputs in bit-reversed order. X(k)={11, −0.7071−j0.7071, −2+j1, 0.7071−j0.7071, 1, 0.7071+j0.7071, −2−j1, −0.7071+j0.7071}.

## 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) = {1, 1, 2, 2, 1, 1, 2, 1}

| Position | Input x(n) | After stage 1 | After stage 2 | After stage 3 |
|---|---|---|---|---|
| 0 | 1 | 2 | 6 | 11 |
| 1 | 1 | 2 | 5 | 1 |
| 2 | 2 | 4 | -2 | −2 + j1 |
| 3 | 2 | 3 | j1 | −2 − j1 |
| 4 | 1 | 0 | 0 | −0.7071 − j0.7071 |
| 5 | 1 | 0 | −0.7071 − j0.7071 | 0.7071 + j0.7071 |
| 6 | 2 | 0 | 0 | 0.7071 − j0.7071 |
| 7 | 1 | −0.7071 − j0.7071 | 0.7071 − j0.7071 | −0.7071 + j0.7071 |

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) = 11
X(1) = −0.7071 − j0.7071
X(2) = −2 + j1
X(3) = 0.7071 − j0.7071
X(4) = 1
X(5) = 0.7071 + j0.7071
X(6) = −2 − j1
X(7) = −0.7071 + j0.7071
```

Check: Σx(n) = 11 = X(0); the result equals the direct 8-point DFT.

**Answer: X(k) = {11, −0.7071 − j0.7071, −2 + j1, 0.7071 − j0.7071, 1, 0.7071 + j0.7071, −2 − j1, −0.7071 + j0.7071}.**

## Check yourself

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

1. The DIF butterfly gives?
   A. A + B and (A - B)W
   B. A + WB and A - WB
   C. A·B
   D. A + W
   Answer: A. Sum straight, difference times twiddle. (Key point: DIF butterfly)

2. Stage 2 pairs n with?
   A. n + 2
   B. n + 3
   C. n + 4
   D. n + 1
   Answer: A. Stage 2 uses spacing N/4 = 2 with W4^n. (Key point: Stage spacing)

3. Which X(k) is at position 4?
   A. X(7)
   B. X(2)
   C. X(1)
   D. X(4)
   Answer: C. Bit reversal of 100 is 001, so X(1). (Key point: Output order)

4. X(0) is?
   A. 1
   B. 0
   C. 11, the sum
   D. 6
   Answer: C. 1+1+2+2+1+1+2+1 = 11. (Key point: Check)

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