---
title: "4-point DFT of {0, 1, 2, 3}"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:dft-four-point-ramp"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dft-four-point-ramp
site: BCT Engineering
---

# 4-point DFT of {0, 1, 2, 3}

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2075 Baisakh, 6 marks
- Video: [Compute 4 point DFT using Radix-2 DIT FFT of the sequence x(n) = {0,1,2,3}](https://www.youtube.com/watch?v=kabwnw7a-gI) — VKY Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dft-four-point-ramp
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Compute four-point DFT of four-point sequence: x[n] = {0, 1, 2, 3}.

## Short answer

Using W4 = -j and the DFT matrix, X[0] = 6, X[1] = -2 + 2j, X[2] = -2, X[3] = -2 - 2j. Thus the four-point DFT is {6, -2+2j, -2, -2-2j}.

## Answer

**Formula:** For $N = 4$,

$$
X[k] = \sum_{n=0}^{3} x[n]\,W_4^{nk}, \qquad W_4 = e^{-j2\pi/4} = -j
$$

So $W_4^0 = 1$, $W_4^1 = -j$, $W_4^2 = -1$, $W_4^3 = j$.

**Matrix form:**

$$
\begin{bmatrix} X[0] \\ X[1] \\ X[2] \\ X[3] \end{bmatrix}
=
\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -j & -1 & j \\ 1 & -1 & 1 & -1 \\ 1 & j & -1 & -j \end{bmatrix}
\begin{bmatrix} 0 \\ 1 \\ 2 \\ 3 \end{bmatrix}
$$

**Each term:**

$$
\begin{aligned}
X[0] &= 0 + 1 + 2 + 3 = 6 \\
X[1] &= 0 + (1)(-j) + (2)(-1) + (3)(j) = -2 + 2j \\
X[2] &= 0 + (1)(-1) + (2)(1) + (3)(-1) = -2 \\
X[3] &= 0 + (1)(j) + (2)(-1) + (3)(-j) = -2 - 2j
\end{aligned}
$$

**Answer:** $X[k] = \{6,\; -2+2j,\; -2,\; -2-2j\}$

| $k$ | $X[k]$ | $\lvert X[k]\rvert$ | $\angle X[k]$ |
|---|---|---|---|
| 0 | $6$ | 6 | $0^\circ$ |
| 1 | $-2+2j$ | $2\sqrt2 = 2.828$ | $135^\circ$ |
| 2 | $-2$ | 2 | $180^\circ$ |
| 3 | $-2-2j$ | $2.828$ | $-135^\circ$ |

**Checks:**

- $X[0]$ equals the sum of samples (6).
- Since $x[n]$ is real, $X[3] = X^*[1]$, which holds.
- Parseval: $\sum|x[n]|^2 = 0+1+4+9 = 14$ and $\frac{1}{4}\sum|X[k]|^2 = \frac{1}{4}(36+8+4+8) = 14$.

## Check yourself

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

1. What is W_4 for a 4-point DFT?
   A. j
   B. -1
   C. 1
   D. -j
   Answer: D. W_4 = e^(-j2π/4) = -j. (Key point: Twiddle factor)

2. What is X[0] for {0,1,2,3}?
   A. 6
   B. 0
   C. 3
   D. 14
   Answer: A. X[0] is the sum of the samples: 0+1+2+3 = 6. (Key point: X[0])

3. What is X[1]?
   A. -2 - 2j
   B. -2 + 2j
   C. 2 - 2j
   D. -2
   Answer: B. X[1] = 0 + 1(-j) + 2(-1) + 3(j) = -2 + 2j. (Key point: X[1])

4. Which check confirms X[3]?
   A. X[3] must equal X[1]
   B. X[3] must be real
   C. Real x[n] gives X[3] = X*[1] = -2 - 2j
   D. X[3] must be zero for a ramp
   Answer: C. For a real sequence, X[N-k] = X*[k]. (Key point: Check)

## Related in Signals and Systems

- [4-point DFT of {3, −2, 1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adft-three-point-sequence) — Signals and Systems, chapter 3
- [4-point DFT of {2, 1+j, 1−j}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adft-complex-sequence) — Signals and Systems, chapter 3
- [4-point DFT by linear transformation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adft-linear-transformation) — Signals and Systems, chapter 3

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