---
title: "Frequency response of a CT LTI system"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:ct-frequency-response"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-frequency-response
site: BCT Engineering
---

# Frequency response of a CT LTI system

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2082 Kartik, 5 marks; 2083 Bhadra (new course), 3 marks; 2083 Baisakh (new course), 4 marks (asked 3 times)
- Video: [Frequency Response Descriptions for LTI Systems](https://www.youtube.com/watch?v=E3QfYXscsCc) — Barry Van Veen
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-frequency-response
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Describe the frequency response (transfer function) of a continuous time LTI system.

## Short answer

The frequency response H(jω) is the Fourier transform of the impulse response, giving the system's gain and phase shift at each frequency. For an input e^{jωt}, output is H(jω)e^{jωt}. Y(jω) = X(jω)H(jω). For real h(t), |H| is even and phase odd.

## Answer

The **frequency response** $H(j\omega)$ of a continuous-time LTI system is the Fourier transform of its impulse response. It tells how the system changes the amplitude and phase of each sinusoidal (complex exponential) component of the input.
$$
H(j\omega) = \int_{-\infty}^{\infty} h(t)e^{-j\omega t}\,dt
$$

#### Eigenfunction property
If $x(t) = e^{j\omega t}$, the output is
$$
\begin{aligned}
y(t) &= \int_{-\infty}^{\infty} h(\tau)e^{j\omega(t-\tau)}\,d\tau \\
&= e^{j\omega t}\int_{-\infty}^{\infty} h(\tau)e^{-j\omega\tau}\,d\tau = H(j\omega)\,e^{j\omega t}
\end{aligned}
$$
Complex exponentials pass through unchanged in form, only multiplied by the complex number $H(j\omega)$.

#### Transfer function
From the convolution property, $y(t) = x(t)*h(t) \Rightarrow Y(j\omega) = X(j\omega)H(j\omega)$, so
$$
H(j\omega) = \frac{Y(j\omega)}{X(j\omega)}
$$
For a system described by $\sum_k a_k \frac{d^k y}{dt^k} = \sum_k b_k \frac{d^k x}{dt^k}$:
$$
H(j\omega) = \frac{\sum_k b_k (j\omega)^k}{\sum_k a_k (j\omega)^k}
$$

#### Magnitude and phase
$H(j\omega) = |H(j\omega)|\,e^{j\angle H(j\omega)}$
- $|H(j\omega)|$: **gain** (magnitude response), often plotted in dB on a Bode plot.
- $\angle H(j\omega)$: **phase shift**; $-d\angle H/d\omega$ is the group delay.
- For real $h(t)$: $|H|$ is even and $\angle H$ is odd in $\omega$.
- A sinusoid $\cos\omega_0 t$ gives output $|H(j\omega_0)|\cos(\omega_0 t + \angle H(j\omega_0))$.

**Example:** RC low-pass filter, $H(j\omega) = \dfrac{1}{1 + j\omega RC}$: $|H| = 1/\sqrt{1 + (\omega RC)^2}$, $\angle H = -\tan^{-1}\omega RC$. Low frequencies pass, high frequencies are attenuated.

## Check yourself

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

1. H(jω) of a CT LTI system is:
   A. The Fourier transform of h(t)
   B. The derivative of h(t)
   C. The Laplace transform of x(t)
   D. h(t) evaluated at t = ω
   Answer: A. H(jω) = ∫h(t)e^{-jωt}dt. (Key point: Definition)

2. If x(t) = e^{jωt}, the output is:
   A. h(t) e^{jωt}
   B. H(jω) e^{jωt}
   C. e^{jωt}/H(jω)
   D. H(jω) alone
   Answer: B. Complex exponentials are eigenfunctions of LTI systems. (Key point: Eigenfunction)

3. The transfer function in terms of spectra is:
   A. X(jω)/Y(jω)
   B. Y(jω) X(jω)
   C. H(jω) = Y(jω)/X(jω)
   D. Y(jω) - X(jω)
   Answer: C. Convolution becomes multiplication so Y = XH. (Key point: Transfer function)

4. For input cos ω0t, the output is:
   A. cos(ω0 t) only
   B. |H(jω0)| cos(ω0 t - 90°)
   C. H(0) cos(ω0 t)
   D. |H(jω0)| cos(ω0 t + ∠H(jω0))
   Answer: D. The system scales amplitude by |H| and shifts phase by ∠H. (Key point: Sinusoid response)

## Related in Signals and Systems

- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5
- [Frequency response; ideal LPF impulse response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-cutoff-impulse) — Signals and Systems, chapter 5, asked 2×
- [Commutative property of CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acommutative-property-ct) — Signals and Systems, chapter 5
- [Cascade of two CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acascade-impulse-response) — Signals and Systems, chapter 5
- [DT transfer function; drawing a Bode plot](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-transfer-function-bode) — Signals and Systems, chapter 6
- [Transfer function of y[n] = 0.4y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adifference-equation-0-4) — Signals and Systems, chapter 6, asked 9×

## Related in other subjects

- [Impulse response, transfer function and distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adistortionless-transfer-function) — Communication Systems, chapter 1
- [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2×
- [Impulse response and transfer function significance](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-transfer-significance) — Communication Systems, chapter 1
- [Transfer function from overshoot and settling time](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-from-mp25-ts6) — Control Systems, chapter 3