---
title: "FT of DC; meaning of ESD and PSD"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:ft-dc-esd-psd"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-dc-esd-psd
site: BCT Engineering
---

# FT of DC; meaning of ESD and PSD

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2078 Poush, 5+3 marks
- Video: [What is Power Spectral Density (PSD)?](https://www.youtube.com/watch?v=DoSLMEEo1Y0) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-dc-esd-psd
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Calculate Fourier transform of a constant DC function. What are the physical meanings of Energy spectral Density and Power spectral Density?

## Short answer

For constant x(t)=A, X(jω)=2πA δ(ω). ESD is |X(jω)|^2, showing how the finite energy of an energy signal is distributed over frequency (J/Hz). PSD is lim_{T→∞} |X_T(jω)|^2/T, giving the average power distribution of a power signal (W/Hz).

## Answer

#### Fourier transform of a constant (DC) signal

Let $x(t) = A$ for all $t$. This signal is not absolutely integrable, so the integral $\int A e^{-j\omega t}dt$ does not converge in the ordinary sense. We find the transform using the impulse function and duality.

**Step 1:** Consider $X(j\omega) = 2\pi A\,\delta(\omega)$ and take its inverse FT:

$$
\begin{aligned}
x(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} 2\pi A\,\delta(\omega)\,e^{j\omega t}\,d\omega \\
&= A\,e^{j0\cdot t} = A
\end{aligned}
$$

using the sifting property of $\delta(\omega)$.

**Step 2:** Since the inverse transform of $2\pi A\delta(\omega)$ is the constant $A$, the transform pair is

$$
A \;\longleftrightarrow\; 2\pi A\,\delta(\omega)
$$

(Equivalently, from duality: $\delta(t) \leftrightarrow 1$, so $1 \leftrightarrow 2\pi\delta(-\omega) = 2\pi\delta(\omega)$.) In terms of $f$ in Hz, $A \leftrightarrow A\,\delta(f)$.

**Meaning:** A DC signal has no variation, so all its content is at zero frequency. The spectrum is a single impulse at $\omega = 0$ with strength $2\pi A$.

```text
 x(t)                    X(jw)
  |                        ^ 2*pi*A
 A|-----------------       |
  |                        |
--+-----------> t   -------+-------> w
                           0
```

#### Physical meaning of ESD and PSD

- **Energy spectral density (ESD)**, $\Psi(\omega) = |X(j\omega)|^2$, applies to energy signals (finite energy, e.g. a pulse). It tells how the total energy of the signal is spread over frequency. The energy in a small band $d\omega$ around $\omega$ is $\frac{1}{2\pi}|X(j\omega)|^2d\omega$, and the total energy is $E = \frac{1}{2\pi}\int|X(j\omega)|^2d\omega$. Unit: joule per hertz.
- **Power spectral density (PSD)**, $S(\omega) = \lim_{T\to\infty}\frac{|X_T(j\omega)|^2}{T}$, applies to power signals (periodic or random signals with finite average power). It tells how the average power is spread over frequency; total power $P = \frac{1}{2\pi}\int S(\omega)\,d\omega$. Unit: watt per hertz.

In practice, the ESD/PSD shows the bandwidth a signal needs, which frequency bands carry most energy or power, and how much noise power passes through a filter. Both are real, non-negative and even functions of $\omega$ for real signals.

## Check yourself

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

1. The Fourier transform of x(t) = A (constant) is:
   A. A δ(ω)
   B. 2π A δ(ω)
   C. A/(jω)
   D. 2A
   Answer: B. A <-> 2πA δ(ω): a single impulse at ω = 0 of strength 2πA. (Key point: FT of DC)

2. Why can the DC transform not be found by ordinary integration?
   A. A is complex
   B. The signal is not absolutely integrable
   C. A is zero at t = 0
   D. The signal is periodic
   Answer: B. ∫A e^(-jωt)dt does not converge, so the impulse and inverse transform are used. (Key point: Why a special method)

3. Energy spectral density tells:
   A. How average power is spread over frequency, for power signals
   B. How total energy of an energy signal is distributed over frequency
   C. The phase at each frequency
   D. The sampling rate needed
   Answer: B. ESD = |X(jω)|^2 and integrates (with 1/2π) to the total energy. (Key point: ESD meaning)

4. Power spectral density is defined for:
   A. Finite-energy pulses
   B. Power signals, via the limit of |X_T|^2/T
   C. Only periodic signals with zero power
   D. Only discrete signals
   Answer: B. PSD = lim |X_T(jω)|^2/T describes how average power of periodic or random signals is distributed. (Key point: PSD definition)

## Related in Signals and Systems

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