---
title: "Distortionless transmission conditions"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:distortionless-transmission"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/distortionless-transmission
site: BCT Engineering
---

# Distortionless transmission conditions

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2075 Baisakh, 4 marks
- Video: [Distortionless LTI System](https://www.youtube.com/watch?v=9VzN_CohuNs) — Neso Academy
- 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-transmission
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Derive the conditions for distortionless transmission for continuous-time LTI system.

## Short answer

Distortionless transmission requires y(t)=Kx(t−t0), so h(t)=Kδ(t−t0) and H(ω)=Ke^{-jωt0}. Conditions are constant magnitude |H|=K and linear phase θ=−ωt0 over the signal bandwidth.

## Answer

**Distortionless transmission** means the output is an exact copy of the input in shape; it may only be scaled in amplitude and delayed in time.

**Time-domain condition:**

$$
y(t) = K\,x(t - t_0)
$$

where $K$ is a constant gain and $t_0 \ge 0$ is a constant delay.

**Impulse response:** put $x(t) = \delta(t)$:

$$
h(t) = K\,\delta(t - t_0)
$$

**Frequency response:** take the Fourier transform of $y(t) = K x(t-t_0)$ using the time-shift property:

$$
\begin{aligned}
Y(\omega) &= K\,X(\omega)\,e^{-j\omega t_0} \\
H(\omega) &= \frac{Y(\omega)}{X(\omega)} = K\,e^{-j\omega t_0}
\end{aligned}
$$

**Conditions:**

1. **Constant magnitude:** $|H(\omega)| = K$ for all frequencies (at least over the band of the signal). Every frequency component is amplified equally, so there is no amplitude distortion.
2. **Linear phase:** $\theta(\omega) = \angle H(\omega) = -\omega t_0$ (a straight line through the origin with slope $-t_0$). Then every frequency component is delayed by the same time, so there is no phase distortion.

The delay is constant because the **group delay** is

$$
t_g = -\frac{d\theta(\omega)}{d\omega} = t_0
$$

```text
 |H(w)|                 angle H(w)
  K  +--------------      |\
     |                    |  \  slope = -t0
     |                    |    \
   --+-------------> w  --+------\----> w
```

In practice, it is enough for these conditions to hold over the bandwidth of the input signal.

## Check yourself

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

1. Distortionless transmission requires:
   A. y(t)=K x(t)^2
   B. y(t)=x(-t)
   C. y(t)=K x(t-t0)
   D. y(t)=dx/dt
   Answer: C. Scaling and delay only. (Key point: Time domain)

2. Corresponding h(t):
   A. K u(t-t0)
   B. Kδ(t)
   C. K e^(-t0 t)
   D. Kδ(t-t0)
   Answer: D. Put x=δ. (Key point: Impulse response)

3. Magnitude condition:
   A. |H(ω)|=K constant
   B. |H(ω)|=ω
   C. |H(ω)|=0
   D. |H(ω)|=1/ω
   Answer: A. No amplitude distortion. (Key point: Magnitude)

4. Phase condition:
   A. θ(ω)=ω^2
   B. θ(ω) = -ω t0, linear through origin
   C. θ(ω)=constant nonzero
   D. θ(ω)=0 for all ω needed always only for t0≠0
   Answer: B. Constant group delay -dθ/dω = t0. (Key point: Phase)

## Related in Signals and Systems

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- [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
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## Related in other subjects

- [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
- [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
- [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
- [Signum signal; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asignum-distortionless) — Communication Systems, chapter 1