--- title: "Hilbert transformation; properties of LTI system" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:hilbert-define-lti-properties" source: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-define-lti-properties site: BCT Engineering --- # Hilbert transformation; properties of LTI system Communication Systems · Chapter 1: Introduction - Asked in: 2071 Magh (CS I), 2+4 marks - Video: [18EC61 : L1 : Hilbert Transform](https://www.youtube.com/watch?v=RBt0N-hc-sw) — Few Minutes Learning - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-define-lti-properties - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question Define Hilbert transformation. State and explain the properties of LTI system. ## Short answer The Hilbert transform shifts positive-frequency phase by −90° and negative by +90° without altering amplitudes, used for SSB and analytic signals. LTI systems obey superposition and time invariance; key properties include causality (h(t)=0 for t<0), stability (∫|h|dt finite), and eigenfunction property where sinusoids only change amplitude and phase, no new frequencies. ## Answer #### Hilbert transformation The **Hilbert transform** $\hat{x}(t)$ of a signal $x(t)$ shifts the phase of all its positive-frequency components by $-90^\circ$ and negative-frequency components by $+90^\circ$, leaving amplitudes unchanged: $$ \hat{x}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau = x(t) * \frac{1}{\pi t}, \qquad \hat{X}(f) = -j\,\text{sgn}(f)\,X(f) $$ Example: $\cos 2\pi f_ct \to \sin 2\pi f_ct$. It is used for SSB generation and for the analytic signal $x_+(t) = x(t) + j\hat x(t)$. #### Properties of LTI systems An LTI system is completely described by its impulse response $h(t)$ (or transfer function $H(f)$), and $y(t) = x(t) * h(t)$. 1. **Linearity (superposition):** $a x_1 + b x_2 \to a y_1 + b y_2$. Allows analysing a complex input as a sum of simple ones (e.g. sinusoids). 2. **Time invariance:** $x(t - t_0) \to y(t - t_0)$; the system behaves the same at all times. 3. **Commutative:** $x * h = h * x$. 4. **Associative (cascade):** $(x * h_1) * h_2 = x * (h_1 * h_2)$; two systems in cascade have $h = h_1 * h_2$ and $H = H_1H_2$. 5. **Distributive (parallel):** $x * (h_1 + h_2) = x * h_1 + x * h_2$; parallel systems add. 6. **Causality:** causal if $h(t) = 0$ for $t < 0$. 7. **Stability (BIBO):** stable if $\int|h(t)|dt < \infty$. 8. **Eigenfunction property:** an input $e^{j2\pi f_0t}$ gives $H(f_0)e^{j2\pi f_0t}$; sinusoids pass through with only gain and phase change, no new frequencies. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the Hilbert transform do? A. Shifts positive-frequency components by -90 degrees and negative ones by +90 degrees B. Doubles the amplitude C. Shifts all components by +90 degrees D. Removes negative frequencies Answer: A. X_hat(f) = -j sgn(f) X(f), amplitudes unchanged. (Key point: Hilbert transform) 2. The Hilbert transform of cos(2 pi fc t) is: A. -cos(2 pi fc t) B. sin(2 pi fc t) C. -sin(2 pi fc t) D. cos(4 pi fc t) Answer: B. A -90 degree shift on cosine gives sine. (Key point: Example) 3. Two LTI systems in cascade have overall response: A. h1 + h2 B. h1 / h2 C. h1 * h2, with H = H1 H2 D. H = H1 + H2 Answer: C. Associativity: cascade gives convolution in time and product in frequency. (Key point: Cascade) 4. What happens to e^(j2 pi f0 t) passing through an LTI system? A. It becomes a different frequency B. It is scaled and phase shifted by H(f0), with no new frequencies C. It is unchanged exactly D. It gets squared Answer: B. Complex exponentials are eigenfunctions: output is H(f0)e^(j2 pi f0 t). (Key point: Eigenfunction) ## Related in Communication Systems - [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 - [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 - [Band pass and band limited signals; LTI properties](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Abandpass-bandlimited-lti-properties) — Communication Systems, chapter 1 - [Parseval's theorem; output PSD of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-power-lti-psd) — Communication Systems, chapter 1 - [LTI system output in time domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-output-time-domain) — Communication Systems, chapter 1 ## Related in other subjects - [Describe LTI system properties](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-describe) — Signals and Systems, chapter 5, asked 3× - [Plot sequence; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aplot-sequence-lti-properties) — Digital Signal Analysis and Processing, chapter 1 - [Properties of DT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-lti-properties) — Signals and Systems, chapter 6, asked 3× - [Define LTI; properties of DT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adefine-lti-dt-properties) — Signals and Systems, chapter 6, asked 3×