Mathematical Foundations and Signal Representation

Introduction

Signals and systems provide the mathematical language for describing information, physical phenomena, and computational processes. In electronics and communication engineering, a signal may represent speech, temperature, image intensity, sensor output, or a modulated radio waveform. A system transforms an input signal into an output signal. Before analyzing filtering, modulation, sampling, or communication channels, students must be able to represent signals precisely and classify their fundamental properties.

This chapter develops the mathematical foundations required for Signals and Systems. It introduces complex numbers and Euler’s relation, functions and sequences, calculus, difference equations, and basic linear algebra. These tools are then applied to classify continuous-time and discrete-time signals, periodic and aperiodic signals, energy and power signals, even and odd signals, deterministic and random signals, and analog and digital signals.

Learning Objectives

After studying this chapter, you should be able to:

1. Use complex numbers, polar notation, and Euler’s relation in signal analysis.
2. Distinguish functions, continuous-time signals, and discrete-time sequences.
3. Apply differentiation, integration, summation, and finite differences to signals.
4. Formulate and interpret first-order difference equations.
5. Represent vectors, matrices, and linear combinations relevant to signal models.
6. Classify signals using standard engineering criteria.
7. Select suitable signal representations for practical communication and instrumentation problems.

1. Signals as Mathematical Functions

A signal is a function that conveys information about a physical quantity or an abstract data sequence. A continuous-time signal is written as x(t), where t is time and normally has units of seconds. A discrete-time signal is written as x[n], where n is an integer index.

The notation is important. In x(t), time varies continuously; in x[n], values exist only at integer-indexed instants. A discrete-time signal may result from sampling a continuous-time signal, but a discrete-time sequence can also originate directly from a digital algorithm.

A system is a rule or physical device that maps an input to an output:

Here, T denotes the system, x(t) is the input, and y(t) is the output. For discrete time, the corresponding expression is y[n] = T\{x[n]\}.

Common elementary signals

The unit step is defined, for continuous time, by

The value at t = 0 may be assigned differently without affecting most ordinary signal calculations. The discrete-time unit step is u[n] = 1 for n ≥ 0 and zero otherwise.

The unit impulse, or Dirac delta, is not an ordinary finite-valued function. It is characterized by

and by the sifting property

provided x(t) is sufficiently well behaved at t₀. The discrete impulse is defined by δ[n] = 1 when n = 0 and zero otherwise. Thus, for a sequence,

These signals are fundamental because complicated waveforms can often be described using shifted, scaled, and combined elementary signals.

2. Complex Numbers and Euler’s Relation

A complex number has the form

where a is the real part, b is the imaginary part, and j = √(-1). Electrical engineering uses j rather than i, because i commonly denotes current.

The complex conjugate is z* = a - jb. The magnitude and phase are

The two-argument function atan2 is preferred because it identifies the correct quadrant. A nonzero complex number can therefore be written in polar form as

where θ = ∠z is measured in radians.

Euler’s relation is

It follows that

and

These identities allow sinusoidal signals to be treated as combinations of complex exponentials. This is especially useful because differentiation and integration preserve the form of a complex exponential:

where s may be real or complex and has units of inverse seconds when t is in seconds.

Worked Example 1: Rectangular and polar forms

Let z = 3 + j4. Then

|z| = √(3² + 4²) = 5.

Its phase is atan2(4,3) ≈ 0.927 radians, or approximately 53.13°. Therefore,

z = 5e^j0.927.

Conversely, 5e^j0.927 = 5(cos 0.927 + j sin 0.927) ≈ 3 + j4.

In phasor analysis, multiplication in polar form is particularly simple: magnitudes multiply and phases add. This makes complex notation valuable for analyzing sinusoidal steady-state circuits and communication carriers.

3. Functions, Transformations, and Sequences

A function assigns at most one output to each permitted input. Signal operations frequently involve amplitude scaling, time shifting, time reversal, and time scaling.

For a continuous-time signal x(t):

• Amplitude scaling: Ax(t).
• Time shift: x(t - t₀) delays the signal by t₀ when t₀ > 0.
• Time reversal: x(-t).
• Time scaling: x(at). For |a| > 1, the waveform is compressed; for 0 < |a| < 1, it is expanded.

For discrete-time sequences, x[n-n₀] is a delay by n₀ samples when n₀ is positive. Unlike continuous time, arbitrary time scaling such as x[0.5n] is not generally meaningful because the index must remain an integer. Expressions such as x[2n] and x[n/2] require careful interpretation and may involve compression, expansion, or interpolation conventions.

A sequence may be represented explicitly, recursively, graphically, or as a sum of shifted impulses. For example,

This identity is exact for any sequence for which the expression is defined. It is the discrete-time counterpart of decomposing a continuous signal using impulses.

4. Calculus and Difference Operations

Calculus describes both the local variation and accumulated content of a signal. The derivative of x(t) is

and represents instantaneous rate of change. The integral over an interval is

where the result has units equal to the units of x(t) multiplied by seconds.

For discrete-time signals, the first forward difference is

and a backward difference is

A running accumulation can be written as

provided the sum converges or is interpreted over a specified finite interval. Difference operations are central to digital filters, numerical approximations, and discrete-time models.

Difference equations

A difference equation relates present and past values of a sequence. A first-order linear equation may be written as

where a and b are constants, x[n] is the input, and y[n] is the output. A complete solution requires an initial condition, such as y[-1]. Without an initial condition, the output is not uniquely determined.

For the zero-input case, where x[n] = 0,

for indices beginning at n₀. The magnitude of a determines the behavior: values decay when |a| < 1, remain constant in magnitude when |a| = 1, and grow when |a| > 1. This statement concerns the homogeneous response and assumes the recurrence is evaluated forward in increasing integer time.

Worked Example 2: A first-order recurrence

Consider y[n] = 0.8y[n-1] + x[n] with y[-1] = 0 and input x[0] = 2, x[1] = 1, and x[2] = 0. Then

y[0] = 0.8(0)+2 = 2,

y[1] = 0.8(2)+1 = 2.6,

y[2] = 0.8(2.6)+0 = 2.08.

The factor 0.8 causes memory of previous samples to decay, although continuing inputs would also contribute to later outputs.

5. Basic Linear Algebra for Signal Representation

Many signals are represented by vectors. A finite sequence can be written as

where T denotes transpose and N is the number of samples. A matrix maps an input vector to an output vector:

Here, A is a matrix whose dimensions must be compatible with x. If A has M rows and N columns, then x has N components and y has M components.

A linear combination of basis signals is

where φ_k are basis vectors or basis functions and c_k are coefficients. Orthogonal basis functions satisfy

For finite real vectors, the inner product is

For complex vectors, the appropriate inner product is ⟨x,y⟩ = xᴴy, where H denotes conjugate transpose. The Euclidean norm is

and measures signal magnitude in vector space. These ideas support coordinate transformations, least-squares estimation, Fourier representations, and digital communication models.

6. Classification of Signals

Classification is not merely descriptive. It determines which mathematical tools and engineering models are appropriate.

Continuous-time and discrete-time

A continuous-time signal is defined for every real time in an interval. A discrete-time signal is defined only at indexed instants. If a continuous signal is sampled at rate f_s samples per second, the sampling interval is

and ideal uniform sampling produces x[n] = x_c(nT_s), where x_c(t) is the original continuous-time signal. Sampling does not automatically make amplitudes digital; amplitude quantization is a separate operation.

Analog and digital

An analog signal generally has a continuous amplitude range. A digital signal uses a finite or countable set of amplitude levels, often represented by binary symbols. Thus, “continuous-time/discrete-time” concerns the time axis, while “analog/digital” primarily concerns amplitude representation. A signal can be continuous-time and analog, discrete-time and analog-valued, or discrete-time and digital.

Periodic and aperiodic

A continuous-time signal is periodic if there exists a positive T₀ such that

The smallest positive such value is the fundamental period. For a discrete-time signal, periodicity requires a positive integer N₀ satisfying

A discrete-time sinusoid x[n] = cos(ω₀n + φ) is periodic if and only if ω₀/(2π) is rational. If ω₀/(2π) = p/q in lowest terms, the fundamental period is N₀ = q when the corresponding cosine does not introduce a smaller period through symmetry. For a continuous sinusoid cos(ω₀t + φ), the period is T₀ = 2π/|ω₀| for nonzero angular frequency ω₀ in radians per second.

Energy and power signals

The energy of a continuous-time signal is

and the energy of a discrete-time signal is

The average power is, respectively,

and

when these limits exist. An energy signal has finite positive energy and zero average power. A power signal has finite positive average power and usually infinite energy. A nonzero signal cannot be both an energy signal and a power signal under these standard definitions. The zero signal is a special case with both energy and power equal to zero.

For a sinusoid x(t)=A cos(ω₀t+φ), the average power is A²/2 when power is normalized to a one-ohm load or when signal power is defined as mean-square amplitude. Physical electrical power additionally depends on impedance and voltage/current units.

Even and odd signals

A continuous or discrete signal is even if

and odd if

Every signal can be decomposed into even and odd parts. For continuous time,

so that x(t)=x_e(t)+x_o(t). The same formulas apply with t replaced by n.

Deterministic and random signals

A deterministic signal is completely specified by a mathematical rule or by known data. A random, or stochastic, signal cannot be predicted exactly at every time using available information; it is described statistically. Noise in a receiver is commonly modeled as random, whereas a known carrier waveform is deterministic.

Random-signal analysis uses quantities such as mean and autocorrelation. For a random process X(t), the mean is

where E[·] denotes expected value. The autocorrelation is

for a complex-valued process. These are statistical descriptions and do not imply that an individual realization equals its mean or correlation function.

7. Worked Classification Example

Consider x(t)=3cos(20πt). It is continuous-time, deterministic, real-valued, and analog in the usual physical interpretation. Its angular frequency is ω₀=20π rad/s, so

T₀=2π/(20π)=0.1 s.

It is periodic. It is even because cos(-20πt)=cos(20πt). Its mean-square value is 3²/2=4.5, so it is a power signal under the normalized definition. Its total energy over infinite time is not finite, because a nonzero periodic sinusoid persists indefinitely.

Now consider x[n]=(0.5)ⁿ u[n]. It is discrete-time, deterministic, and aperiodic. Its energy is

because the geometric-series formula ∑ₙ₌₀^∞rⁿ=1/(1-r) applies when |r|<1. Its average power is zero, so it is an energy signal rather than a nonzero power signal.

8. Applied Case: Sensor Acquisition for a Motor Controller

Background

An industrial motor controller monitors vibration using an analog accelerometer. The sensor output is a continuous-time voltage containing a slowly varying mechanical component, high-frequency interference, and random electronic noise.

Problem

The controller is digital and can process only samples. Engineers must choose a sampling rate and represent the signal without confusing time discretization with amplitude digitization.

Analysis

Suppose the highest significant vibration frequency is 4 kHz. Sampling at f_s=10 kHz gives T_s=100 μs. The sampled sequence is x[n]=x_c(nT_s). Since the sampling rate exceeds twice the highest frequency, ideal band-limited reconstruction is theoretically possible under the Nyquist sampling condition. In practice, an anti-aliasing low-pass filter is required because real signals are not perfectly band-limited and filters do not have infinitely sharp transitions.

The converter then quantizes each sample to a finite number of voltage levels. This introduces quantization error, which is distinct from sampling error. Random noise may be modeled statistically, while a calibration waveform used during testing is deterministic.

Approach and outcome

The engineering design uses an analog anti-aliasing filter, a sampling rate selected with transition-band margin, and an analog-to-digital converter with sufficient resolution. The digital controller applies a difference equation to smooth the measured sequence. This workflow illustrates the chain:

physical phenomenon → analog sensor signal → filtering → sampling → quantization → digital processing.

Learning points

1. Time and amplitude classifications are independent.
2. Sampling is not quantization.
3. Periodicity and energy properties depend on the mathematical model, not merely on the physical source.
4. Noise should be characterized statistically rather than treated as a known deterministic waveform.
5. Engineering specifications must include units, bandwidth, sampling rate, resolution, and assumptions.

9. Limitations, Best Practices, and Contemporary Relevance

Signal classifications can depend on idealizations. An infinite-duration sinusoid is a power signal mathematically, whereas a finite measurement of it has finite energy. Similarly, an impulse is an ideal model used to simplify analysis; physical systems approximate it with short, high-amplitude pulses.

Good practice includes defining the time variable and units, stating initial conditions for recurrences, distinguishing angular frequency from ordinary frequency, checking whether a period is a positive real number or a positive integer, and separating statistical models from individual observations. Engineers should also document sampling assumptions and avoid claiming that digital processing can recover information removed by inadequate sampling.

Modern signal-processing systems increasingly use embedded processors, machine learning, and neural networks. Nevertheless, the underlying data remain signals represented by sequences, vectors, and matrices. Complex arithmetic supports wireless communication, while difference equations and linear algebra remain essential for real-time filters and feature extraction. AI-based systems do not eliminate the need for sampling theory, unit consistency, reproducibility, and responsible treatment of measurement uncertainty.

Chapter Summary

Signals are mathematical functions or sequences used to represent information and physical measurements. Complex numbers provide magnitude and phase descriptions, and Euler’s relation converts between complex exponentials and trigonometric functions. Functions can be shifted, reversed, and scaled, while calculus describes continuous-time change and accumulation. Difference equations describe discrete-time systems and require appropriate initial conditions.

Vectors, matrices, inner products, and norms provide a foundation for finite-dimensional signal representation and computational models. Signals may be classified as continuous-time or discrete-time, analog or digital, periodic or aperiodic, energy or power, even or odd, and deterministic or random. These classifications are logically distinct and guide the choice of analytical tools. Practical acquisition systems demonstrate why sampling, quantization, filtering, and noise modeling must be treated as separate concepts.

Key Terms

Signal: A function or sequence that conveys information.

System: A rule or device that maps an input signal to an output signal.

Continuous-time signal: A signal defined for every real value of time.

Discrete-time signal: A sequence defined at integer-indexed instants.

Complex number: A number of the form a+jb.

Euler’s relation: The identity e^jθ=cosθ+jsinθ.

Magnitude: The nonnegative size of a complex number or vector.

Phase: The angular position of a complex number.

Unit step: A signal that changes from zero to one at a reference point.

Impulse: An ideal signal characterized by unit area or a single nonzero sequence value.

Difference equation: An equation relating samples at different indices.

Vector: An ordered collection of numerical components.

Matrix: A rectangular array used to represent linear mappings.

Periodic signal: A signal that repeats after a positive period.

Energy signal: A signal with finite, nonzero total squared magnitude.

Power signal: A signal with finite, nonzero long-term average squared magnitude.

Even signal: A signal symmetric about the origin.

Odd signal: A signal antisymmetric about the origin.

Deterministic signal: A signal completely specified by a rule or known data.

Random signal: A signal described using probability or statistical properties.

Review Questions

Short Answer Questions

1. State the difference between a continuous-time signal and a discrete-time signal.
2. Write Euler’s relation and explain the meaning of its variables.
3. What initial information is required to solve a first-order difference equation uniquely?
4. Define an energy signal and a power signal.
5. Explain why sampling and quantization are different operations.

Long Answer Questions

1. Explain rectangular and polar representations of complex numbers, including magnitude, phase, and conjugation.
2. Derive the even and odd components of a signal and prove that their sum equals the original signal.
3. Discuss periodicity in continuous-time and discrete-time sinusoids, emphasizing the integer-period condition in discrete time.
4. Explain how vectors, matrices, inner products, and norms support signal representation.
5. Describe the role of calculus and difference equations in continuous-time and discrete-time signal processing.

Analytical / Application Questions

1. Classify x(t)=2sin(40πt+π/4) according to time type, determinism, periodicity, symmetry, and energy/power category. Justify each answer.
2. For y[n]=0.6y[n-1]+x[n], with y[-1]=1, x[0]=2, and x[1]=−1, calculate y[0] and y[1], and interpret the role of the coefficient 0.6.
3. Design a conceptual acquisition chain for a sensor whose useful bandwidth is 3 kHz. State a suitable minimum ideal sampling-rate condition and explain why an anti-aliasing filter and quantizer are required.

Multiple Choice Questions with Answers

Question 1

Which expression is Euler’s relation?

A. e^jθ=cosθ+jsinθ

B. e^jθ=sinθ+jcosθ

C. e^jθ=cosθ−jsinθ

D. e^jθ=θ+jθ

Correct Answer: A. e^jθ=cosθ+jsinθ

Question 2

What is the magnitude of z=3+j4?

A. 3

B. 4

C. 5

D. 7

Correct Answer: C. 5

Question 3

A discrete-time sinusoid is periodic only when which condition holds?

A. Its amplitude is greater than one

B. ω₀/(2π) is rational

C. Its phase is zero

D. Its frequency is negative

Correct Answer: B. ω₀/(2π) is rational

Question 4

For x[n]=(0.5)ⁿ u[n], what is the total energy?

A. 1/2

B. 1

C. 4/3

D. 2

Correct Answer: C. 4/3

Question 5

Which operation primarily changes the amplitude while leaving the time axis unchanged?

A. x(t−t₀)

B. x(−t)

C. Ax(t)

D. x(at)

Correct Answer: C. Ax(t)

Question 6

Which statement correctly distinguishes sampling from quantization?

A. Sampling discretizes amplitude, while quantization discretizes time

B. Sampling discretizes time, while quantization discretizes amplitude

C. Both operations only change phase

D. They are identical operations

Correct Answer: B. Sampling discretizes time, while quantization discretizes amplitude

Question 7

If |a|<1 in the homogeneous equation y[n]=ay[n−1], the response generally:

A. Grows without bound

B. Decays toward zero

C. Becomes periodic with every period

D. Becomes undefined

Correct Answer: B. Decays toward zero

Question 8

Which formula gives the even part of a continuous-time signal?

A. [x(t)−x(−t)]/2

B. [x(t)+x(−t)]/2

C. x(t)x(−t)

D. x(t)/x(−t)

Correct Answer: B. [x(t)+x(−t)]/2

Question 9

A nonzero periodic sinusoid extending for all time is normally classified as a:

A. Finite-energy signal

B. Power signal

C. Zero signal

D. Finite-duration impulse

Correct Answer: B. Power signal

Question 10

For a sensor with highest useful frequency 4 kHz, the ideal Nyquist condition requires a sampling frequency:

A. Less than 2 kHz

B. Exactly 2 kHz in every practical system

C. Greater than 8 kHz

D. Equal to 4 kHz

Correct Answer: C. Greater than 8 kHz