---
title: "Impulse response, transfer function and distortionless transmission"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:distortionless-transfer-function"
source: https://www.bctengineer.com/q/semester_6/communication_systems/distortionless-transfer-function
site: BCT Engineering
---

# Impulse response, transfer function and distortionless transmission

Communication Systems · Chapter 1: Introduction

- Asked in: 2077 Chaitra (CS I), 2+6 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_6/communication_systems/distortionless-transfer-function
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define impulse response and transfer function of a system. Derive the transfer function for a system that provides distortionless transmission and draw its amplitude response as well.

## Short answer

The impulse response h(t) is the output for a unit impulse δ(t); the transfer function H(f) is the Fourier transform of h(t), giving amplitude and phase. For distortionless transmission, H(f) = K e^{-j2π f t_d}, so |H(f)| must be constant K and phase θ(f) linear, θ(f) = -2π f t_d. The amplitude response is a flat line.

## Answer

#### Impulse response and transfer function
- **Impulse response** $h(t)$: the output of an LTI system when the input is a unit impulse $\delta(t)$, with zero initial conditions. For any input, $y(t) = x(t) * h(t) = \int x(\lambda)h(t-\lambda)\,d\lambda$.
- **Transfer function** $H(f)$: the Fourier transform of the impulse response, $H(f) = \int h(t)e^{-j2\pi ft}\,dt = Y(f)/X(f)$. It gives the gain $|H(f)|$ and phase shift $\theta(f)$ the system applies to each frequency: $H(f) = |H(f)|e^{j\theta(f)}$.

#### Transfer function for distortionless transmission
Transmission is distortionless when the output is an exact replica of the input except for a constant gain $K$ and a constant delay $t_d$:

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

Take the Fourier transform of both sides, using the time-shift property $x(t - t_d) \leftrightarrow X(f)e^{-j2\pi f t_d}$:

$$
Y(f) = K\,X(f)\,e^{-j2\pi f t_d}
$$

Therefore

$$
H(f) = \frac{Y(f)}{X(f)} = K\,e^{-j2\pi f t_d}
$$

and the impulse response is $h(t) = K\,\delta(t - t_d)$.

Comparing with $H(f) = |H(f)|e^{j\theta(f)}$, the two conditions are:
1. **Amplitude response constant:** $|H(f)| = K$ for all frequencies in the signal band.
2. **Phase response linear in frequency:** $\theta(f) = -2\pi f t_d$, i.e. constant time delay $t_d = -\frac{1}{2\pi}\frac{d\theta}{df}$ for every component. (A phase of $-2\pi f t_d \pm n\pi$ is also acceptable; $\pm\pi$ only inverts the sign.)

#### Amplitude and phase response

```text
 |H(f)|
    |
  K +-------------------------
    |
    +-------------------------> f
   -B            0           B

 theta(f)
    |\
    | \   slope = -2*pi*td
 ---+--\----------------------> f
    |   \
    |    \
```

The amplitude response is a horizontal line at height $K$ and the phase response is a straight line through the origin with negative slope $-2\pi t_d$. In practice these need to hold only over the bandwidth $-B \le f \le B$ of the signal. If $|H(f)|$ varies, **amplitude distortion** results; if $\theta(f)$ is not linear, **phase (delay) distortion** results.

## Check yourself

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

1. What is the impulse response h(t) of an LTI system?
   A. The Fourier transform of the output
   B. The output for a unit step input
   C. The output when the input is a unit impulse, with zero initial conditions
   D. The input needed to make the output zero
   Answer: C. h(t) is the response to delta(t) with the system initially at rest; y(t)=x(t)*h(t). (Key point: Impulse response definition)

2. Distortionless transmission requires the output to be:
   A. y(t) = x(t)^2
   B. Any waveform with the same energy as the input
   C. y(t) = K x(t) with delay allowed to vary with frequency
   D. y(t) = K x(t - td), a constant gain and constant delay
   Answer: D. The output must be an exact replica apart from constant gain K and constant delay td. (Key point: Distortionless condition)

3. Using the time-shift property, the transfer function for distortionless transmission is:
   A. H(f) = K delta(f - td)
   B. H(f) = K / (1 + j2 pi f td)
   C. H(f) = K e^{+j2 pi f td^2}
   D. H(f) = K e^{-j2 pi f td}
   Answer: D. Fourier transform of Kx(t-td) is K X(f) e^{-j2 pi f td}, so H = Y/X = K e^{-j2 pi f td}. (Key point: Derived transfer function)

4. What does the amplitude response of a distortionless system look like over the signal band?
   A. A triangle peaking at f = 0
   B. A straight line through the origin with slope -2 pi td
   C. A decaying exponential
   D. A horizontal line at height K
   Answer: D. |H(f)| = K, constant, across the signal band; the sloped straight line is the phase response. (Key point: Amplitude response)

## Related in Communication Systems

- [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 and transfer function significance](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-transfer-significance) — Communication Systems, chapter 1
- [Analog communication components; distortionless transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-components-distortionless-tf) — Communication Systems, chapter 1
- [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×
- [Signum signal; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asignum-distortionless) — Communication Systems, chapter 1
- [Short note: Distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-distortionless-transmission) — Communication Systems, chapter 1, asked 2×

## Related in other subjects

- [Distortionless transmission; ideal LPF step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adistortionless-ideal-lpf-step) — Signals and Systems, chapter 5
- [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5
- [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×
- [Distortionless transmission conditions](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adistortionless-transmission) — Signals and Systems, chapter 5