Mathematical Representation of Signals

Introduction

A signal becomes useful in engineering when we can describe it mathematically. Instead of saying "the voltage rises, falls, and repeats," we write a function such as $$x(t)=A\cos(\omega t+\phi)$$ or a sequence such as $$x[n]$$.

This representation allows us to calculate energy, power, delay, frequency, sampling behavior, and system response. It is the bridge between physical waveforms and engineering design.

Why This Topic Matters

Prerequisites

• Understanding of signal as a time-varying quantity.
• Basic function notation and graph reading.
• Trigonometry for sinusoidal signals.
• Elementary algebra for shifting and scaling.
• Basic discrete indexing such as $$n=0,1,2,...$$.

Basic Intuition

A waveform on an oscilloscope is visual. A formula is precise. A table of samples is digital-friendly. A block diagram shows how it moves through a system. These are not separate ideas; they are different languages for the same signal.

Mathematical representation lets us stop guessing from the shape and start calculating exactly what the signal will do.

Core Theory Explanation

1. Continuous-time representation

Here time $$t$$ is continuous, so the signal has a value at every instant. Analog voltage, current, speech pressure, and RF carrier signals are commonly modeled this way.

2. Discrete-time representation

Here $$n$$ is an integer index. The signal is known only at sample instants. This is the natural language of DSP, digital audio, digital communication receivers, and sampled sensor data.

3. Standard signal building blocks

Step-by-Step Mathematical Derivation

1. Start from a physical waveform

Suppose an AC voltage has peak value $$A$$, frequency $$f$$, and phase $$\phi$$. A complete cycle corresponds to $$2\pi$$ radians.

2. Write the time-domain function

• $$A$$ tells maximum signal strength.
• $$\omega$$ tells how fast the waveform cycles.
• $$\phi$$ tells horizontal alignment.
• $$t$$ lets us compute the signal value at any instant.

3. Sample the same signal

If samples are taken every $$T_s$$ seconds, the sample time is $$t=nT_s$$.

This is the mathematical step from an analog waveform to a discrete-time sequence. The signal is no longer observed at every instant, only at sample instants.

4. Describe transformations

Working Principle

• Observe the real signal behavior on a graph or instrument.
• Choose the independent variable: continuous time $$t$$ or discrete index $$n$$.
• Identify amplitude, time location, period, frequency, and phase.
• Represent the waveform using standard signals and operations.
• Use the representation to compute energy, power, system output, or frequency behavior.

Diagram Explanation

Important Formulas

Real-World Applications

• Digital audio stores sound as discrete-time samples.
• Wireless systems represent carrier waves as sinusoids with amplitude, frequency, and phase.
• ECG and EEG signals are modeled for filtering and diagnosis.
• Image processing represents brightness as a two-dimensional signal.
• Radar and sonar use delayed and shifted signals to estimate distance and velocity.

Solved Examples

Beginner example: identify representation

An oscilloscope waveform of voltage versus time is naturally represented as $$v(t)$$ because time is continuous.

Intermediate numerical: sinusoid parameters

Given $$x(t)=4\cos(200\pi t+\pi/6)$$:

• Amplitude $$A=4$$.
• Angular frequency $$\omega=200\pi$$ rad/s.
• Frequency $$f=\omega/(2\pi)=100$$ Hz.
• Phase $$\phi=\pi/6$$ rad.

Advanced problem: sampled sinusoid

If $$x(t)=3\cos(100\pi t)$$ is sampled at $$T_s=0.001s$$:

The discrete sequence now describes the same waveform only at sample instants.

Common Mistakes

• Treating $$x(t)$$ and $$x[n]$$ as the same representation.
• Forgetting that $$n$$ is an integer index, not continuous time.
• Confusing time shifting with amplitude scaling.
• Writing $$x(t+t_0)$$ as a delay; it is actually an advance.
• Forgetting to substitute $$t=nT_s$$ while sampling.

Comparison Tables

Interview Questions

• What is the difference between $$x(t)$$ and $$x[n]$$?
• Why do we use standard signals like impulse and step?
• How does time shifting affect a waveform?
• What is the physical meaning of phase?
• How do you convert a continuous-time signal into a discrete-time sequence?
• Why are sinusoids important in ECE?

Exam-Oriented Notes

• For delay, use $$x(t-t_0)$$.
• For advance, use $$x(t+t_0)$$.
• For sampling, substitute $$t=nT_s$$.
• For time reversal, use $$x(-t)$$.
• Always identify whether the independent variable is $$t$$ or $$n$$ before solving.

Revision Summary

• Mathematical representation converts waveform behavior into exact analysis.
• $$x(t)$$ is continuous-time representation.
• $$x[n]$$ is discrete-time representation.
• Standard signals include impulse, step, ramp, exponential, and sinusoid.
• Signal operations include shifting, scaling, reversal, and sampling.

Practice Questions

Conceptual

• Explain why a formula is more precise than only a graph.
• Differentiate continuous-time and discrete-time signals.
• Explain the physical meaning of a time delay.

Numerical

• For $$x(t)=6\cos(50\pi t)$$, find amplitude and frequency.
• If $$x(t)$$ is sampled at $$T_s=0.01s$$, write $$x[n]$$.
• Sketch the effect of $$x(t-2)$$ and $$x(-t)$$.

MCQs

• The notation for discrete-time signal is: $$x(t)$$ / $$x[n]$$ / $$X(s)$$ / $$H(j\omega)$$.
• $$x(t-t_0)$$ represents: advance / delay / amplitude scaling / inversion.
• A ramp signal represents: constant value / sudden impulse / linear increase / oscillation.