---
title: "Why do we need DFT"
subject: "Digital Signal Analysis and Processing"
chapter: 7
key: "semester_7/digital_signal_analysis_and_processing:why-need-dft"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/why-need-dft
site: BCT Engineering
---

# Why do we need DFT

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

- Asked in: 2073 Shrawan, 2 marks; 2071 Chaitra, 2 marks; 2070 Asar, 2 marks; 2074 Chaitra, 3 marks; 2072 Kartik, 2 marks; 2070 Chaitra, 2 marks (asked 6 times)
- Video: [Digital Signal Processing 17: Why do we need Discrete Fourier Transform(DFT) ???](https://www.youtube.com/watch?v=00ExhN7boxo) — Study with Dr. Hisham أدرس مع د. هشام
- 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/why-need-dft
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-7

## Question

Why do we need the Discrete Fourier Transform (DFT) although we have the Discrete-time Fourier Transform (DTFT)?

## Answer

The DTFT X(e^jω) of a sequence is a **continuous** function of ω and needs an infinite sum over n, so it cannot be computed or stored on a computer. The DFT is needed because:

- **It is discrete in frequency.** It gives N samples of the DTFT at ω_k = 2πk/N, a finite set of numbers that a computer can store and process.
- **It is a finite sum.** X(k) = Σ (n = 0 to N−1) x(n) e^(−j2πkn/N) for a finite-length x(n); no infinite summation is required.
- **Fast algorithms exist (FFT)**, which cut the computation from N² to (N/2)log₂N operations.
- **Practical uses:** spectrum analysis, linear filtering by fast convolution (overlap-add/save), correlation, and filter design.

## Check yourself

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

1. Why can't the DTFT be computed on a computer?
   A. It is continuous in w and needs an infinite sum
   B. It is complex
   C. It is not invertible
   D. It is periodic only
   Answer: A. A computer needs finite, discrete numbers. (Key point: DTFT limit)

2. What does the N-point DFT give?
   A. N samples of x(n)
   B. N samples of the DTFT at w_k = 2 pi k/N
   C. The Z-transform at N points on the real axis
   D. An approximate DTFT with random frequencies
   Answer: B. It samples the DTFT at equally spaced frequencies. (Key point: DFT relation)

3. What does the FFT bring?
   A. Exact continuous spectrum
   B. Lower accuracy
   C. Reduces cost from N^2 to (N/2)log2 N
   D. Fewer samples
   Answer: C. Fast algorithms make the DFT practical. (Key point: FFT)

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