---
title: "8-point DIT-FFT of {1,1,0,1,0,1,2}"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:ditfft-8pt-72k"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/ditfft-8pt-72k
site: BCT Engineering
---

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

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

- Asked in: 2072 Kartik, 7 marks
- Video: [8-Point DFT Solved using Radix-2 DIT FFT Algorithm | Digital Signal Processing Problem Tutorial](https://www.youtube.com/watch?v=9BCirp2Ta6g) — 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/ditfft-8pt-72k
- 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, 0, 1, 0, 1, 2} using Decimation in Time Fast Fourier Transform (DITFFT) algorithm.

## Short answer

Zero-pad the seven given values with x(7)=0 and use radix-2 DIT FFT. X(k)={6, 0.2929+j1.2929, −1−j1, 1.7071−j2.7071, 0, 1.7071+j2.7071, −1+j1, 0.2929−j1.2929}.

## Answer

Only seven values are printed for an 8-point DFT, so I read the sequence as x(n) = {1, 1, 0, 1, 0, 1, 2} followed by x(7) = 0 (zero-padded to 8 points).

#### 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, and stage 3 uses W₈⁰ … W₈³.

x(n) = {1, 1, 0, 1, 0, 1, 2, 0}

| Position | Input (bit-reversed) | After stage 1 | After stage 2 | After stage 3 = X(k) |
|---|---|---|---|---|
| 0 | 1 | 1 | 3 | 6 |
| 1 | 0 | 1 | 1 + j2 | 0.2929 + j1.2929 |
| 2 | 0 | 2 | -1 | −1 − j1 |
| 3 | 2 | -2 | 1 − j2 | 1.7071 − j2.7071 |
| 4 | 1 | 2 | 3 | 0 |
| 5 | 1 | 0 | −j1 | 1.7071 + j2.7071 |
| 6 | 1 | 1 | 1 | −1 + j1 |
| 7 | 0 | 1 | j1 | 0.2929 − j1.2929 |


**Result:**

```text
X(0) = 6
X(1) = 0.2929 + j1.2929
X(2) = −1 − j1
X(3) = 1.7071 − j2.7071
X(4) = 0
X(5) = 1.7071 + j2.7071
X(6) = −1 + j1
X(7) = 0.2929 − j1.2929
```

Check: Σx(n) = 6 = X(0); the result equals the direct 8-point DFT.

**Answer: X(k) = {6, 0.2929 + j1.2929, −1 − j1, 1.7071 − j2.7071, 0, 1.7071 + j2.7071, −1 + j1, 0.2929 − j1.2929}.**

## Check yourself

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

1. Only seven values are given for an 8-point DFT. What is done?
   A. Drop one value
   B. Append x(7) = 0
   C. Repeat x(0)
   D. Use a 7-point DFT
   Answer: B. The sequence is zero-padded to 8 points. (Key point: Padding)

2. In what order is the input fed to the DIT-FFT?
   A. Bit-reversed
   B. Odd samples last only
   C. Descending
   D. Natural
   Answer: A. x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). (Key point: DIT input order)

3. Which butterfly is used?
   A. A' = A + B, B' = (A - B)W
   B. A' = A - B, B' = A + B
   C. A' = A·W, B' = B
   D. A' = A + W·B, B' = A - W·B
   Answer: D. DIT multiplies B by the twiddle first. (Key point: DIT butterfly)

4. Which value is X(0)?
   A. 8
   B. 0
   C. 1
   D. 6, the sum of the samples
   Answer: D. 1 + 1 + 0 + 1 + 0 + 1 + 2 + 0 = 6. (Key point: Check)

## Close variants set in other exams

- [8-point DIT-FFT of {1,1,0,0,1,1,2}](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aditfft-8pt-70c) — Digital Signal Analysis and Processing, chapter 7, asked 3×
- [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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