---
title: "Distortionless transmission; ideal LPF step response"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:distortionless-ideal-lpf-step"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/distortionless-ideal-lpf-step
site: BCT Engineering
---

# Distortionless transmission; ideal LPF step response

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2070 Bhadra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/distortionless-ideal-lpf-step
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

What is distortionless transmission? Derive the expression for unit step response of ideal low pass filter.

## Short answer

Distortionless transmission means y(t)=K x(t-t0), requiring constant magnitude and linear phase. For an ideal LPF with cutoff ω_c, the unit step response is s(t)=1/2 + (1/π) Si[ω_c(t-t0)], with rise time inversely proportional to bandwidth and about 9% overshoot.

## Answer

#### Distortionless transmission

Transmission is **distortionless** when the output has exactly the same shape as the input, only scaled by a constant $K$ and delayed by a constant $t_0$:

$$
y(t) = K\,x(t-t_0) \;\Rightarrow\; H(\omega) = K\,e^{-j\omega t_0}
$$

So the system needs (1) constant magnitude $|H(\omega)| = K$ and (2) linear phase $\angle H(\omega) = -\omega t_0$ over the signal bandwidth.

#### Unit step response of the ideal LPF

Ideal LPF (unit gain, delay $t_0$, cutoff $\omega_c$):

$$
H(\omega) = e^{-j\omega t_0}\ \text{for } |\omega| < \omega_c,\quad 0 \text{ otherwise}
$$

Its impulse response (inverse Fourier transform) is

$$
h(t) = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega(t-t_0)}d\omega = \frac{\sin\omega_c(t-t_0)}{\pi(t-t_0)}
$$

The step response is the integral of $h(t)$:

$$
s(t) = \int_{-\infty}^{t} \frac{\sin\omega_c(\lambda-t_0)}{\pi(\lambda-t_0)}\,d\lambda
$$

Put $x = \omega_c(\lambda - t_0)$:

$$
\begin{aligned}
s(t) &= \frac{1}{\pi}\int_{-\infty}^{\omega_c(t-t_0)}\frac{\sin x}{x}dx \\
&= \frac{1}{\pi}\left[\frac{\pi}{2} + \mathrm{Si}\left(\omega_c(t-t_0)\right)\right] \\
&= \frac{1}{2} + \frac{1}{\pi}\mathrm{Si}\left[\omega_c(t-t_0)\right]
\end{aligned}
$$

where $\mathrm{Si}(y) = \int_0^{y}\frac{\sin x}{x}dx$ is the sine integral.

- $s(t_0) = 0.5$, $s(\infty) = 1$, $s(-\infty) = 0$.
- Rise time $\approx \pi/\omega_c$, inversely proportional to bandwidth.
- About 9% overshoot with ringing (Gibbs phenomenon), and response before $t = 0$ (non-causal).

## Check yourself

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

1. Transmission is distortionless when:
   A. y(t) = K x(t - t0), a scaled and delayed copy of the input
   B. y(t) = x(t) exactly with no delay and no scaling
   C. y(t) = K x(t) with random delay
   D. The output contains only the low frequencies
   Answer: A. The shape is unchanged; only a constant gain K and constant delay t0 are allowed. (Key point: Distortionless definition)

2. Which pair of conditions on H(w) is needed for distortionless transmission over the signal bandwidth?
   A. Constant magnitude and linear phase
   B. Constant phase and linear magnitude
   C. Linear magnitude and linear phase
   D. Zero magnitude outside and constant phase
   Answer: A. H(w) = K e^(-jw t0) means |H| = K and phase = -w t0. (Key point: Conditions)

3. How is the step response of the ideal LPF obtained from the impulse response?
   A. By differentiating h(t)
   B. By integrating h(t) from -inf to t
   C. By multiplying h(t) by u(t)
   D. By squaring h(t)
   Answer: B. The step is the integral of the impulse, so s(t) is the running integral of h. (Key point: Step from h to s)

4. Which statement about the ideal LPF step response is correct?
   A. s(t0) = 1, no overshoot
   B. s(t0) = 0.5 with about 9% overshoot and ringing (Gibbs phenomenon)
   C. s(t0) = 0 and it is causal
   D. It rises exponentially to 1 with no ringing
   Answer: B. s(t) = 1/2 + Si[wc(t - t0)]/pi, giving 0.5 at t0, overshoot of about 9% and response before t = 0. (Key point: Step response features)

## Related in Signals and Systems

- [Impulse response of ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-response) — Signals and Systems, chapter 5, asked 5×
- [Impulse response of DT ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-discrete-time) — Signals and Systems, chapter 6
- [Ideal LPF impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-step) — Signals and Systems, chapter 5, asked 5×
- [Ideal LPF realizability; step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-realizability-step) — Signals and Systems, chapter 5
- [Ideal low pass and high pass filters](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afrequency-response-lp-hp-filters) — 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×

## 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
- [Impulse response; conditions for distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-distortionless-conditions) — Communication Systems, chapter 1
- [RC low pass filter operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Arc-low-pass-filter) — Basic Electronics Engineering, chapter 1, asked 5×
- [Matched filter approximation using ideal LPF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-ideal-lpf) — Communication Systems, chapter 9, asked 2×