---
title: "Fourier transform of a DC signal"
subject: "Data Communication"
chapter: 3
key: "semester_5/data_communication:ft-dc-signal"
source: https://www.bctengineer.com/q/semester_5/data_communication/ft-dc-signal
site: BCT Engineering
---

# Fourier transform of a DC signal

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2079 Baisakh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/ft-dc-signal
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

Obtain the Fourier transform of a DC signal whose amplitude is unity at all time instants, and hence, plot its magnitude and phase spectrum.

## Short answer

X(ω) = 2π δ(ω). Magnitude spectrum is an impulse of area 2π at ω=0, zero elsewhere; phase spectrum is zero. Obtained via duality: δ(t)↔1 gives ∫1·e^{-jωt} dt = 2π δ(ω).

## Answer

x(t) = 1 for all t. This signal is not absolutely integrable, so the transform is found by the limit of a decaying signal or by duality.

#### Method (duality)
The impulse has the transform δ(t) ↔ 1. By the inverse transform, δ(t) = (1/2π) ∫ 1·e^(jωt) dω, or

∫ e^(jωt) dω = 2π δ(t)

Interchange the roles of t and ω: ∫ 1·e^(-jωt) dt = 2π δ(ω) (δ is even).

**X(ω) = 2π δ(ω)**

#### Alternative check (limit method)
x(t) = lim (a→0) e^(-a|t|) ↔ lim 2a/(a² + ω²). This has area ∫ 2a/(a²+ω²) dω = 2π for every a, and its width shrinks to zero as a → 0, which is exactly 2π δ(ω).

#### Spectra
- Magnitude: an impulse of area 2π at ω = 0, and zero everywhere else.
- Phase: 0 (the spectrum is real and positive).

```text
|X(w)|
      ^ (2*pi)
      |
      |
------+------> w        phase = 0 for all w
      0
```
All the power lies at zero frequency, which is natural: a DC signal does not vary in time.

## Check yourself

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

1. FT of x(t) = 1 is
   A. δ(ω)
   B. 1
   C. 2π
   D. 2π δ(ω)
   Answer: D. By duality. (Key point: Result)

2. Direct integration is not used because
   A. The signal is not absolutely integrable
   B. It is periodic
   C. It is complex
   D. It is zero
   Answer: A. Needs limit or duality. (Key point: Why not direct)

3. The spectrum is concentrated at
   A. ω = infinity
   B. All ω
   C. ω = 0
   D. ω = 2π
   Answer: C. DC has no variation. (Key point: Energy)

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