---
title: "4-point DFT by linear transformation"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:dft-linear-transformation"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dft-linear-transformation
site: BCT Engineering
---

# 4-point DFT by linear transformation

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2079 Asoj, 5 marks
- Video: [Problem to find 4-point DFT using matrix method or Linear Transformation method || EC Academy](https://www.youtube.com/watch?v=wkJAmZLqPEY) — EC 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-linear-transformation
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Determine the 4-point DFT of the signal x[n] = u[n] + u[n−1] − u[n−3] − u[n−4] using linear transformation method.

## Short answer

The 4-point sequence is x[n] = {1, 2, 2, 1} for n=0..3. Using the DFT matrix with W₄ = −j: X[0]=6, X[1]=−1−j, X[2]=0, X[3]=−1+j. DFT = {6, −1−j, 0, −1+j}.

## Answer

#### Find the sequence
$x[n] = u[n] + u[n-1] - u[n-3] - u[n-4]$. Evaluate it term by term:

| n | u[n] | u[n−1] | −u[n−3] | −u[n−4] | x[n] |
|---|---|---|---|---|---|
| 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 0 | 2 |
| 2 | 1 | 1 | 0 | 0 | 2 |
| 3 | 1 | 1 | −1 | 0 | 1 |
| ≥4 | 1 | 1 | −1 | −1 | 0 |

So $x[n] = \{1, 2, 2, 1\}$ for $n = 0, 1, 2, 3$.

#### Linear transformation (matrix) method
$X_N = W_N x_N$, where $W_N^{kn} = e^{-j2\pi kn/N}$. For $N = 4$, $W_4 = e^{-j\pi/2} = -j$, with $W^0 = 1$, $W^1 = -j$, $W^2 = -1$ and $W^3 = j$:
$$
W_4 = \begin{bmatrix}1 & 1 & 1 & 1\\ 1 & -j & -1 & j\\ 1 & -1 & 1 & -1\\ 1 & j & -1 & -j\end{bmatrix}
$$
$$
\begin{bmatrix}X[0]\\X[1]\\X[2]\\X[3]\end{bmatrix} = \begin{bmatrix}1 & 1 & 1 & 1\\ 1 & -j & -1 & j\\ 1 & -1 & 1 & -1\\ 1 & j & -1 & -j\end{bmatrix}\begin{bmatrix}1\\2\\2\\1\end{bmatrix}
$$
Row by row:
$$
\begin{aligned}
X[0] &= 1 + 2 + 2 + 1 = 6 \\
X[1] &= 1 - 2j - 2 + j = -1 - j \\
X[2] &= 1 - 2 + 2 - 1 = 0 \\
X[3] &= 1 + 2j - 2 - j = -1 + j
\end{aligned}
$$

**Answer:** $X[k] = \{6,\ -1 - j,\ 0,\ -1 + j\}$.

**Check:** $X[3] = X^*[1]$, as expected for a real sequence. By Parseval, $\sum|x[n]|^2 = 1 + 4 + 4 + 1 = 10$, and $\frac{1}{4}\sum|X[k]|^2 = \frac{1}{4}(36 + 2 + 0 + 2) = 10$.

## Check yourself

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

1. x[n] = u[n] + u[n-1] - u[n-3] - u[n-4] equals which sequence for n = 0..3?
   A. {1, 1, 1, 1}
   B. {1, 2, 2, 1}
   C. {1, 2, 1, 0}
   D. {2, 2, 1, 1}
   Answer: B. Evaluating term by term gives 1, 2, 2, 1 and 0 for n >= 4. (Key point: Finding the sequence)

2. For N = 4, the twiddle factor W4 = e^(-j2π/4) equals:
   A. j
   B. -j
   C. -1
   D. 1
   Answer: B. e^(-jπ/2) = -j, so W^0=1, W^1=-j, W^2=-1, W^3=j. (Key point: Twiddle factor)

3. In the linear transformation method, the DFT is computed as:
   A. X = W x, with W[k][n] = e^(-j2πkn/N)
   B. X = x / W
   C. X = W^T W x
   D. X = x convolved with W
   Answer: A. The DFT vector is the N x N matrix of twiddle factors multiplying the sample vector. (Key point: Matrix method)

4. What is the 4-point DFT of {1, 2, 2, 1}?
   A. {6, -1 - j, 0, -1 + j}
   B. {6, 1 - j, 0, 1 + j}
   C. {6, 0, -1 - j, -1 + j}
   D. {6, -1 + j, 0, -1 - j}
   Answer: A. X[0]=6, X[1] = 1 - 2j - 2 + j = -1 - j, X[2] = 0, X[3] = conj(X[1]). (Key point: Result and check)

## Related in Signals and Systems

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