---
title: "FFT of {1,1,2,4,3,1,2,1} by DIT-FFT"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:dit-fft-72c"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/dit-fft-72c
site: BCT Engineering
---

# FFT of {1,1,2,4,3,1,2,1} by DIT-FFT

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

- Asked in: 2072 Chaitra, 8 marks
- Video: [Using DIT-FFT algorithm compute the DFT of a sequence x[n]=[1,1,1,1,0,0,0,0]](https://www.youtube.com/watch?v=RHXvrzH0XvA) — Engg-Course-Made-Easy
- 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/dit-fft-72c
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

Find the FFT of the signal x[n] = {1, 1, 2, 4, 3, 1, 2, 1} using DIT-FFT algorithm.

## Short answer

Use radix-2 DIT FFT with bit-reversed input order. X(k)={15, −4.1213−j2.1213, j3, 0.1213−j2.1213, 1, 0.1213+j2.1213, −j3, −4.1213+j2.1213}.

## Answer

#### Method: radix-2 decimation in time (N = 8, 3 stages)
Input in bit-reversed order: x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). Each butterfly: A' = A + W·B, B' = A − W·B, with W₈⁰ = 1, W₈¹ = 0.7071 − j0.7071, W₈² = −j, W₈³ = −0.7071 − j0.7071. Stage 1 uses W₂ (= ±1), stage 2 uses W₄⁰ = 1 and W₄¹ = −j, stage 3 uses W₈⁰ … W₈³.

x(n) = {1, 1, 2, 4, 3, 1, 2, 1}

| Position | Input (bit-reversed) | After stage 1 | After stage 2 | After stage 3 = X(k) |
|---|---|---|---|---|
| 0 | 1 | 4 | 8 | 15 |
| 1 | 3 | -2 | -2 | −4.1213 − j2.1213 |
| 2 | 2 | 4 | 0 | j3 |
| 3 | 2 | 0 | -2 | 0.1213 − j2.1213 |
| 4 | 1 | 2 | 7 | 1 |
| 5 | 1 | 0 | −j3 | 0.1213 + j2.1213 |
| 6 | 4 | 5 | -3 | −j3 |
| 7 | 1 | 3 | j3 | −4.1213 + j2.1213 |


**Result:**

```text
X(0) = 15
X(1) = −4.1213 − j2.1213
X(2) = j3
X(3) = 0.1213 − j2.1213
X(4) = 1
X(5) = 0.1213 + j2.1213
X(6) = −j3
X(7) = −4.1213 + j2.1213
```

Check: Σx(n) = 15 = X(0); the result equals the direct 8-point DFT.

**Answer: X(k) = {15, −4.1213 − j2.1213, j3, 0.1213 − j2.1213, 1, 0.1213 + j2.1213, −j3, −4.1213 + j2.1213}.**

## Check yourself

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

1. How many stages does the 8-point radix-2 DIT-FFT have?
   A. 8
   B. 2
   C. 4
   D. 3
   Answer: D. log2 8 = 3. (Key point: Stages)

2. What twiddle does stage 1 use?
   A. -j
   B. W2, equal to +1 and -1, so just add and subtract
   C. W8^1
   D. 0.7071
   Answer: B. Stage 1 pairs only need sum and difference. (Key point: Stage 1 twiddle)

3. Which twiddles does stage 2 use?
   A. j and -1
   B. W8^1 and W8^3
   C. W4^0 = 1 and W4^1 = -j
   D. Only 1
   Answer: C. The 4-point sub-DFTs use 1 and -j. (Key point: Stage 2 twiddles)

4. X(0) for {1,1,2,4,3,1,2,1} is?
   A. 1
   B. 8
   C. 15, the sum
   D. 14
   Answer: C. 1+1+2+4+3+1+2+1 = 15. (Key point: Check)

## Related in Digital Signal Analysis and Processing

- [DIT-FFT of {1,1,2,0,1,2,0,1} and spectrum](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adit-fft-spectrum-71s) — Digital Signal Analysis and Processing, chapter 7
- [How fast is FFT; X(7) by DIT-FFT](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Afft-speed-x7) — Digital Signal Analysis and Processing, chapter 7, asked 2×
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- [8-point DIF-FFT of {1/2,1/2,1/2,1/2,0,0,0,0}](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adif-fft-half-ones) — Digital Signal Analysis and Processing, chapter 7
- [8-point DIF-FFT of {1,1,2,2,1,1,2,1}](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adifft-8pt-69c) — Digital Signal Analysis and Processing, chapter 7

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