---
title: "Hilbert vs Fourier transform; properties of HT"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:hilbert-vs-fourier-properties"
source: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-vs-fourier-properties
site: BCT Engineering
---

# Hilbert vs Fourier transform; properties of HT

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2073 Magh (CS I), 4+2+4 marks
- Video: [Hilbert Transform Explained: Basics, Definition, Function, and Properties](https://www.youtube.com/watch?v=UusIVHye4Jc) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-vs-fourier-properties
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Define Hilbert transform? How is it different from Fourier transform? State the properties of HT.

## Short answer

Hilbert transform shifts phase of positive frequencies by -90° and negative by +90°, keeping amplitudes; it maps time to time via convolution with 1/πt. Unlike Fourier transform (time to frequency), HT output is a real signal. Properties: same amplitude spectrum and energy, double HT yields -x(t), orthogonality with original, HT of even is odd and vice versa.

## Answer

#### Definition

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

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

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

Example: $\cos\omega_0t \to \sin\omega_0t$; $\sin\omega_0t \to -\cos\omega_0t$.

#### Hilbert transform vs Fourier transform

| Point | Hilbert transform | Fourier transform |
|---|---|---|
| Domain change | Time → time (same domain) | Time → frequency |
| Output | A real signal $\hat x(t)$ | Complex spectrum $X(f)$ |
| Nature | Convolution with $1/\pi t$ (an LTI filter) | Integral with kernel $e^{-j2\pi ft}$ |
| Effect | Only phase shift by ±90° | Gives amplitude and phase content |
| Applying twice | Gives $-x(t)$ | Gives $x(-t)$ |
| Use | SSB, analytic signal, envelope | Spectrum analysis, filtering, system response |

#### Properties of Hilbert transform

1. **Same amplitude spectrum:** $|\hat X(f)| = |X(f)|$, since $|-j\,\text{sgn}(f)| = 1$.
2. **Same energy / power:** $\int|\hat x(t)|^2dt = \int|x(t)|^2dt$; also same autocorrelation function.
3. **Double transform:** $\hat{\hat x}(t) = -x(t)$ (two $-90^\circ$ shifts give $-180^\circ$).
4. **Orthogonality:** $\int_{-\infty}^{\infty}x(t)\hat x(t)\,dt = 0$.
5. **Inverse:** $x(t) = -\dfrac{1}{\pi t} * \hat x(t)$.
6. **Symmetry:** HT of an even function is odd, and of an odd function is even.
7. **Linearity:** HT of $a x_1 + b x_2$ is $a\hat x_1 + b\hat x_2$.
8. **Modulation:** for a low-pass $m(t)$ band-limited below $f_c$, HT of $m(t)\cos\omega_ct$ is $m(t)\sin\omega_ct$.

These properties make HT the basis of the phase-shift SSB modulator, $s(t) = m(t)\cos\omega_ct \mp \hat m(t)\sin\omega_ct$.

## Check yourself

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

1. The Hilbert transform changes a signal by:
   A. Shifting positive frequencies by -90 degrees and negative by +90 degrees, amplitudes unchanged
   B. Doubling the amplitude of all components
   C. Converting it into the frequency domain
   D. Removing the negative frequencies
   Answer: A. HT is a -j sgn(f) filter: a pure phase shift of +/-90 degrees with unity magnitude. (Key point: Definition of HT)

2. Which statement correctly distinguishes HT from FT?
   A. HT gives a complex spectrum; FT gives a real time signal
   B. HT stays in the time domain (a convolution with 1/(pi t)); FT goes from time to frequency
   C. Both change the domain
   D. Applying FT twice gives -x(t)
   Answer: B. HT maps a real time signal to another real time signal; FT maps time to a complex spectrum. (Key point: HT versus FT)

3. Applying the Hilbert transform twice to x(t) gives:
   A. x(t)
   B. x(-t)
   C. -x(t)
   D. 0
   Answer: C. Two -90 degree shifts make -180 degrees, so the result is -x(t). (Key point: Double transform)

4. Which pair of HT properties is correct?
   A. x(t) and its HT have different energy; integral of x*xhat is 1
   B. |Xhat(f)| differs from |X(f)|; integral of x*xhat is nonzero
   C. |Xhat(f)| = |X(f)|, same energy, and integral of x(t) xhat(t) dt = 0
   D. HT of cos is -sin and HT of an even function is even
   Answer: C. Since |-j sgn f| = 1 amplitude and energy are preserved, and x and its HT are orthogonal. (Key point: Orthogonality and spectrum)

## 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 properties; energy and power of unit step](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-unit-step-energy) — Communication Systems, chapter 2
- [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1
- [Short note: Hilbert transform](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-hilbert-transform) — Communication Systems, chapter 2
- [Hilbert transform in frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-frequency-domain) — 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

## Related in other subjects

- [Properties of the Fourier transform](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-transform-properties) — Data Communication, chapter 3, asked 2×
- [Fourier transform vs Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-vs-series) — Applied Mathematics, chapter 4
- [Fourier series vs Fourier transform](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-vs-ft) — Data Communication, chapter 3
- [Fourier series vs transform; DTFT of aⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afs-vs-ft-dtft-geometric) — Signals and Systems, chapter 3