---
title: "Hilbert transform properties; energy and power of unit step"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:hilbert-properties-unit-step-energy"
source: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-properties-unit-step-energy
site: BCT Engineering
---

# Hilbert transform properties; energy and power of unit step

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2078 Chaitra, 2+5 marks
- Video: [Problem Based on Properties of Hilbert Transform Part-2 | Com Sys | R K Classes | Hindi | Lec 54 |](https://www.youtube.com/watch?v=Jnv9CCyV82w) — R K Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-properties-unit-step-energy
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Describe Hilbert Transformation and its properties. Compute the energy and power of unit step signal.

## Short answer

The Hilbert transform shifts all positive frequency components by −90° and negative by +90°, keeping magnitude unchanged. Properties: same amplitude spectrum, orthogonality to original, and double transform yields −x(t). The unit step signal has infinite energy and finite average power of 0.5 W (a power signal).

## Answer

#### Hilbert transform and its properties

The **Hilbert transform** of a signal $x(t)$ is the signal obtained by shifting the phase of every frequency component of $x(t)$ by $-90^\circ$ (positive frequencies) without changing amplitudes.

$$
\hat{x}(t) = x(t) * \frac{1}{\pi t} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau
$$

In the frequency domain:

$$
\hat{X}(f) = -j\,\text{sgn}(f)\,X(f), \quad H(f) = \begin{cases} -j, & f>0 \\ +j, & f<0 \end{cases}
$$

So the Hilbert transformer is an ideal $-90^\circ$ phase shifter with unit gain. Example: HT of $\cos\omega_0 t$ is $\sin\omega_0 t$, and HT of $\sin\omega_0 t$ is $-\cos\omega_0 t$.

**Properties**

1. $x(t)$ and $\hat{x}(t)$ have the **same amplitude spectrum** $|X(f)|$.
2. They have the **same energy (or power)** and the same autocorrelation.
3. **HT of HT** gives the negative: $\hat{\hat{x}}(t) = -x(t)$.
4. $x(t)$ and $\hat{x}(t)$ are **orthogonal**: $\int_{-\infty}^{\infty} x(t)\hat{x}(t)\,dt = 0$.
5. HT of an even function is odd and HT of an odd function is even.
6. Inverse HT: $x(t) = -\hat{x}(t) * \frac{1}{\pi t}$.

Use: generating SSB signals (phase-shift method), pre-envelope and analytic signal $x_+(t) = x(t) + j\hat{x}(t)$.

#### Energy and power of unit step

$u(t) = 1$ for $t \ge 0$, and $0$ for $t<0$.

**Energy:**

$$
E = \int_{-\infty}^{\infty}|u(t)|^2dt = \int_{0}^{\infty}1\,dt = \infty
$$

**Power:**

$$
\begin{aligned}
P &= \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|u(t)|^2dt \\
&= \lim_{T\to\infty}\frac{1}{2T}\int_{0}^{T}1\,dt = \lim_{T\to\infty}\frac{T}{2T} = \frac12
\end{aligned}
$$

**Answer:** $E = \infty$, $P = 0.5$ W (normalised to 1 Ω). Since energy is infinite but power is finite and non-zero, the unit step is a **power signal**.

## Check yourself

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

1. What does the Hilbert transform do to a signal's frequency components?
   A. Doubles the amplitude and keeps the phase
   B. Shifts the phase by -90° for positive frequencies without changing amplitude
   C. Shifts the phase by 180° and halves the amplitude
   D. Removes all negative frequency components
   Answer: B. H(f) = -j sgn(f) has unit magnitude and phase -90° for f > 0 (+90° for f < 0). (Key point: Hilbert transform action)

2. What results from applying the Hilbert transform twice to x(t)?
   A. x(t)
   B. 2x(t)
   C. 0
   D. -x(t)
   Answer: D. Two -90° shifts make a 180° shift, giving the negative of the signal. (Key point: Repeated HT)

3. What is the relation between x(t) and its Hilbert transform in terms of the integral of their product?
   A. The integral of x(t) x^(t) over all time is zero (orthogonal)
   B. The integral equals the energy of x(t)
   C. The integral equals twice the energy
   D. The integral is infinite
   Answer: A. x and its HT are orthogonal, e.g. cos and sin. (Key point: Orthogonality)

4. Why is the unit step a power signal with P = 0.5?
   A. Energy is finite and power is zero
   B. Energy and power are both infinite
   C. Energy is infinite but the average power over (-T, T) is T/2T = 1/2, finite and non-zero
   D. Energy is 0.5 and power is infinite
   Answer: C. The integral of 1 from 0 to infinity diverges, while the time-averaged power limit is 1/2. (Key point: Unit step type)

## Related in Communication Systems

- [Unit step in terms of signum; energy or power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aunit-step-signum-energy-power) — Communication Systems, chapter 1
- [Hilbert transformation; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-lti-properties) — Communication Systems, chapter 1
- [Hilbert transform properties; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-distortionless) — Communication Systems, chapter 2, asked 3×
- [Hilbert vs Fourier transform; properties of HT](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-vs-fourier-properties) — Communication Systems, chapter 2
- [SSB in terms of Hilbert transform; SSB generation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-hilbert-derivation-generation) — Communication Systems, chapter 2, asked 3×
- [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1

## Related in other subjects

- [Fourier transform of unit step](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unit-step) — Signals and Systems, chapter 3, asked 3×
- [Energy or power: unit step, impulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aenergy-power-check-signals) — Data Communication, chapter 2, asked 2×
- [Unit step signal and relation to impulse](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aunit-step-relation-impulse) — Digital Signal Analysis and Processing, chapter 1
- [Z-transform of k² e^(−ak) from unit step](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Az-transform-k2-e-ak) — Digital Control System, chapter 2