---
title: "Short note: Hilbert transform"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:short-note-hilbert-transform"
source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-hilbert-transform
site: BCT Engineering
---

# Short note: Hilbert transform

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2081 Chaitra, 5 marks
- Video: [Hilber Transform](https://www.youtube.com/watch?v=wYrNwRuQFB4) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-hilbert-transform
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Write a short note on Hilbert transform.

## Short answer

Hilbert transform shifts all positive frequency components by -90° and negative ones by +90°. It is defined as x̂(t)=x(t)*1/(πt) and has transfer function -j sgn(f). Applications include generating SSB signals and forming the analytic signal.

## Answer

The **Hilbert transform** of a signal $x(t)$, written $\hat x(t)$, is the signal obtained by shifting the phase of all its positive-frequency components by $-90°$ and all negative-frequency components by $+90°$, while keeping amplitudes unchanged.

**Time domain:**

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

**Frequency domain:** the Hilbert transformer is a filter with

$$
H(f)=-j\,\text{sgn}(f),\qquad \hat X(f)=-j\,\text{sgn}(f)\,X(f)
$$

so $\lvert H(f)\rvert=1$ and $\angle H(f)=-90°$ for $f>0$, $+90°$ for $f<0$. It is an ideal wideband $90°$ phase shifter.

**Examples:** $\widehat{\cos\omega_0t}=\sin\omega_0t$, $\widehat{\sin\omega_0t}=-\cos\omega_0t$.

**Properties:**
1. $x(t)$ and $\hat x(t)$ have the same amplitude spectrum, energy and power.
2. $\hat{\hat x}(t)=-x(t)$.
3. $x(t)$ and $\hat x(t)$ are orthogonal: $\int x(t)\hat x(t)dt=0$.
4. For low-pass $m(t)$ and $f_c$ above its bandwidth, $\widehat{m(t)\cos\omega_ct}=m(t)\sin\omega_ct$.

**Applications:**
- **SSB generation:** $s(t)=m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct$ (phase-shift method).
- **Pre-envelope (analytic signal):** $x_+(t)=x(t)+j\hat x(t)$, whose spectrum is zero for $f<0$.
- **Band-pass signal representation** in terms of in-phase and quadrature components, and complex envelope analysis.

## Check yourself

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

1. The Hilbert transformer has frequency response:
   A. H(f) = j sgn(f)
   B. H(f) = -j sgn(f)
   C. H(f) = sgn(f)
   D. H(f) = 1/f
   Answer: B. It gives -90 degrees for f>0 and +90 for f<0 with unit magnitude. (Key point: Frequency response)

2. The Hilbert transform of cos w0 t is:
   A. -sin w0 t
   B. sin w0 t
   C. cos w0 t
   D. -cos w0 t
   Answer: B. A -90 degree shift turns cos w0 t into sin w0 t. (Key point: Examples)

3. Which statement is true for x(t) and its Hilbert transform?
   A. They are orthogonal and have the same amplitude spectrum
   B. They have different energies
   C. They are identical
   D. Their product integrates to the energy
   Answer: A. Only phase changes, so energy is equal, and integral of x x^ dt = 0. (Key point: Orthogonality)

## Related in Communication Systems

- [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 transform in frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-frequency-domain) — Communication Systems, chapter 2
- [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
- [Optimum detector impulse response; Hilbert transform](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response-hilbert) — Communication Systems, chapter 9
- [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×
- [Hilbert transform; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-ideal-lpf-noncausal) — Communication Systems, chapter 1