---
title: "8-point DIF-FFT of {1,−1,3,2,1,1,3,−2}"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:difft-8pt-73c"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/difft-8pt-73c
site: BCT Engineering
---

# 8-point DIF-FFT of {1,−1,3,2,1,1,3,−2}

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

- Asked in: 2073 Chaitra, 6 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-73c
- 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, 3, 2, 1, 1, 3, −2} using Decimation in frequency Fast Fourier Transform (DIFFFT) algorithm.

## Short answer

Use radix-2 DIF FFT with outputs in bit-reversed order. X(k)={8, −4.2426−j1.4142, −4, 4.2426−j1.4142, 8, 4.2426+j1.4142, −4, −4.2426+j1.4142}.

## 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, 3, 2, 1, 1, 3, -2}

| Position | Input x(n) | After stage 1 | After stage 2 | After stage 3 |
|---|---|---|---|---|
| 0 | 1 | 2 | 8 | 8 |
| 1 | -1 | 0 | 0 | 8 |
| 2 | 3 | 6 | -4 | -4 |
| 3 | 2 | 0 | 0 | -4 |
| 4 | 1 | 0 | 0 | −4.2426 − j1.4142 |
| 5 | 1 | −1.4142 + j1.4142 | −4.2426 − j1.4142 | 4.2426 + j1.4142 |
| 6 | 3 | 0 | 0 | 4.2426 − j1.4142 |
| 7 | -2 | −2.8284 − j2.8284 | 4.2426 − j1.4142 | −4.2426 + j1.4142 |

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) = 8
X(1) = −4.2426 − j1.4142
X(2) = -4
X(3) = 4.2426 − j1.4142
X(4) = 8
X(5) = 4.2426 + j1.4142
X(6) = -4
X(7) = −4.2426 + j1.4142
```

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

**Answer: X(k) = {8, −4.2426 − j1.4142, -4, 4.2426 − j1.4142, 8, 4.2426 + j1.4142, -4, −4.2426 + j1.4142}.**

## Check yourself

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

1. Which pair gives the DIF butterfly outputs?
   A. A + WB and A - WB
   B. AB and A/B
   C. A + B and (A - B)W
   D. A - B and A + B with no twiddle
   Answer: C. The difference is multiplied by the twiddle. (Key point: DIF butterfly)

2. Stage 1 of an 8-point DIF pairs n with which sample?
   A. n + 1
   B. n + 3
   C. n + 4
   D. n + 2
   Answer: C. The first stage pairs samples N/2 = 4 apart. (Key point: Stage spacing)

3. How do you read X(k) from the final column?
   A. Positions 0 to 7 hold X(0), X(4), X(2), X(6), X(1), X(5), X(3), X(7)
   B. Reversed order X(7) to X(0)
   C. Only even k appear
   D. Positions are already X(0) to X(7)
   Answer: A. DIF output is bit-reversed so it must be rearranged. (Key point: Output order)

4. What is X(0) for {1,-1,3,2,1,1,3,-2}?
   A. 4
   B. 8, the sum
   C. 0
   D. 2
   Answer: B. 1 - 1 + 3 + 2 + 1 + 1 + 3 - 2 = 8. (Key point: Check)

## Close variants set in other exams

- [8-point DFT by FFT of {1,−1,2,2,1,1,2,2}](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Afft-8pt-73s) — Digital Signal Analysis and Processing, chapter 7, asked 4×

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