---
title: "Signum signal; distortionless transmission"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:signum-distortionless"
source: https://www.bctengineer.com/q/semester_6/communication_systems/signum-distortionless
site: BCT Engineering
---

# Signum signal; distortionless transmission

Communication Systems · Chapter 1: Introduction

- Asked in: 2072 Magh (CS I), 2+5 marks
- Video: [Distortion Less Transmission of the System](https://www.youtube.com/watch?v=PYZ84ZpMGAI) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/signum-distortionless
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define signum signal. Explain the significance of distortionless transmission in communication system with its necessary derivation.

## Short answer

The signum function sgn(t) is +1 for t>0, 0 at t=0, -1 for t<0. Distortionless transmission means y(t)=K x(t-t_d), requiring H(f)=K e^{-j2π f t_d}: constant amplitude |H(f)|=K and linear phase θ(f)= -2π f t_d. This ensures no amplitude or phase distortion, preserving pulse shapes and signal quality.

## Answer

#### Signum signal
The **signum** (sign) function gives the sign of its argument:

$$
\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}
$$

It is an odd function, related to the unit step by $\text{sgn}(t) = 2u(t) - 1$, and its Fourier transform is $\frac{1}{j\pi f}$. It is used in defining the Hilbert transform: $\hat X(f) = -j\,\text{sgn}(f)X(f)$.

#### Distortionless transmission: significance and derivation
In a communication system the receiver must get the **same waveform** that was sent; otherwise pulses overlap (ISI), speech/music lose quality and video gets smeared. Transmission is distortionless when the output is only a scaled, delayed copy of the input:

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

Taking the Fourier transform (time-shift property):

$$
\begin{aligned}
Y(f) &= K\,X(f)\,e^{-j2\pi ft_d} \\
H(f) &= \frac{Y(f)}{X(f)} = K\,e^{-j2\pi ft_d}
\end{aligned}
$$

So $|H(f)| = K$ and $\theta(f) = -2\pi ft_d$, and $h(t) = K\delta(t - t_d)$. The conditions are:
1. **Constant amplitude response** over the signal band (no amplitude distortion).
2. **Linear phase response**, i.e. constant delay $t_d$ for all components (no phase distortion).

```text
 |H(f)|               theta(f)
 K +-----------          |\
   |                     | \ slope -2*pi*td
 --+-----------> f     --+--\-------> f
                         |   \
```

**Significance:**
- Gives the requirement for designing channels, amplifiers and filters: flat gain and linear phase over the signal bandwidth.
- Explains **amplitude distortion** (non-flat gain) and **phase/delay distortion** (non-linear phase); the ear is not very sensitive to phase distortion, but video and digital data are.
- Shows why **equalizers** are used in telephone lines, modems and TV to correct the channel.
- Since an exact distortionless system is ideal, the conditions only need to hold over the bandwidth of the transmitted signal.

## Check yourself

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

1. Which relation holds for the signum function?
   A. sgn(t) = 2u(t) - 1
   B. sgn(t) = u(t) + 1
   C. sgn(t) = u(t) - 2
   D. sgn(t) = 1 - 2u(t)
   Answer: A. sgn(t) is +1 for t>0 and -1 for t<0, so sgn(t) = 2u(t) - 1. (Key point: Signum function)

2. What is the time-domain condition for distortionless transmission?
   A. y(t) = K x(t) + t_d
   B. y(t) = x(t)^2
   C. y(t) = K x(t - t_d)
   D. y(t) = x(t) * x(t)
   Answer: C. Output must be a scaled, delayed copy of the input, K x(t - t_d). (Key point: Distortionless output)

3. Taking the Fourier transform of y(t)=K x(t-t_d), what is H(f)?
   A. K e^(-j2 pi f t_d)
   B. K e^(j2 pi f t_d)^2
   C. K / (j2 pi f t_d)
   D. K sinc(f t_d)
   Answer: A. The time-shift property gives Y = K X e^(-j2 pi f t_d), so H = K e^(-j2 pi f t_d). (Key point: Transfer function)

4. What phase response is needed for no phase distortion?
   A. Constant phase independent of f
   B. Linear phase theta = -2 pi f t_d giving constant delay
   C. Phase that increases as f^2
   D. Zero phase only at the carrier
   Answer: B. Linear phase means every component is delayed by the same t_d. (Key point: Phase condition)

## 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, transfer function and distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adistortionless-transfer-function) — 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×
- [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×
- [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
- [Unit step in terms of signum; energy or power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aunit-step-signum-energy-power) — Communication Systems, chapter 1

## Related in other subjects

- [Distortionless transmission conditions](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adistortionless-transmission) — Signals and Systems, chapter 5
- [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
- [Significance of transmission voltage level, 132 vs 400 kV](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Avoltage-level-132kv-vs-400kv) — Power System Analysis I, chapter 1
- [Bypass capacitor significance; CE small signal analysis](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Ace-bypass-capacitor-small-signal) — Electronics Devices and Circuit, chapter 2, asked 2×