Engineering Signal Analysis: From Fourier to filtering: Theory

Engineering Signal Analysis: From Fourier to Filtering—Theory by Christian Tiberius and Max Mulder develops a connected account of signals in time and frequency. Eighteen chapters progress from signal models through Fourier methods, sampling, reconstruction, discrete transforms, spectral estimation, systems, measurement, and filters. The guide follows that sequence and emphasizes conventions, units, assumptions, and interpretation alongside mathematical operations. It is designed to accompany the theory volume; the separately published exercise book provides complementary practice and worked solutions.

Chapter-wise summaries and reading tasks

Chapter 1 Introduction

Signals represent quantities that vary, while systems transform or respond to those quantities. The opening chapter motivates time-domain and frequency-domain descriptions through engineering applications. These are complementary representations of one phenomenon rather than competing accounts. A useful signal model retains the features relevant to the question, and a system model describes the relationship between input and output. The chapter also establishes notation and the book’s structure. Readers should identify what is measured, its units, and the purpose of analysis before choosing a transform or algorithm; otherwise a technically correct calculation may answer the wrong practical question.

Study task. Describe one measurement as a signal and identify a system that acts on it. Explain what a time plot and a spectrum would each reveal.

Chapter 2 Signal preliminaries

Signal preliminaries distinguish deterministic and random, continuous and discrete, and periodic and aperiodic models. Symmetry can simplify analysis, while sinusoids and phasors provide building blocks for spectral descriptions. The chapter connects single- and double-sided conventions, signal transformations, and measures of energy and power. A shift or time scaling changes the mathematical argument and must be interpreted carefully. Energy and average power classify different kinds of behaviour, so neither is a universal substitute for signal amplitude. Establishing these definitions early makes later normalization and spectral calculations easier to check.

Study task. Classify several simple signals, decompose one into even and odd parts, and explain which energy or power measure is meaningful for each example.

Chapter 3 Real Fourier series

A real Fourier series represents a suitable periodic signal through harmonically related sine and cosine components. The chapter derives coefficients, examines conditions and symmetry, and connects the representation with a single-sided spectrum. The square-wave example illustrates how a complex waveform can be assembled from simpler components. A finite partial sum is an approximation whose behaviour should be distinguished from the full series, particularly near discontinuities. Parseval’s relationship links the time-domain power account with the coefficient representation. Keeping period, harmonic index, and frequency units explicit prevents confusion between the fundamental and its harmonics.

Study task. Predict which coefficients vanish from symmetry before integrating a periodic example. Compare a few partial sums and check their power interpretation against the source convention.

Chapter 4 Complex exponential Fourier series

The complex exponential series reorganizes the same periodic decomposition using complex coefficients. Positive and negative frequencies encode amplitude and phase relationships that can look unfamiliar but are mathematically systematic. For real signals, conjugate relationships connect the two sides of the spectrum. The chapter develops polar interpretation, symmetry, time-shift and differentiation properties, and Parseval’s relationship. Switching representations should preserve the underlying signal, so coefficient conversion provides a useful consistency check. A complex coefficient is not simply an amplitude to be read without regard to the chosen one- or two-sided convention.

Study task. Convert a simple trigonometric series into complex coefficients and reconstruct it. Explain how a time shift changes phase while preserving the relevant magnitude pattern.

Chapter 5 Fourier transform

The Fourier transform extends spectral analysis to aperiodic signals through a continuous frequency representation. The chapter develops the transform from the periodic framework, considers conditions, and interprets magnitude, phase, and symmetry. Limiting examples involving impulses, constants, and sinusoids connect ordinary signals with generalized representations. The transform’s units and normalization matter when moving between frequency-domain and time-domain quantities. Parseval’s relationship provides an energy check. Readers should distinguish a continuous spectrum from the discrete harmonic coefficients of a periodic series rather than treating every spectral picture as the same kind of object.

Study task. Compare the spectra of a finite pulse and a periodic repetition of that pulse. Explain why one uses a continuous transform and the other a harmonic series.

Chapter 6 Fourier transform theorems

Transform theorems make it possible to derive new results from known pairs without repeating every integral. The chapter connects linearity, shifts, scaling, reversal, duality, modulation, convolution, multiplication, differentiation, and integration. Each operation has a particular effect in the other domain, and combinations require tracking both amplitude factors and phase. A common source of error is applying a rule to a signal whose argument has been transformed differently from the assumed form. Writing intermediate steps preserves the structure of the operation and makes a final spectrum easier to interpret.

Study task. Derive a shifted and scaled pulse transform from a known pair. Annotate every amplitude and phase change and check the result using symmetry or a limiting case.

Chapter 7 Convolution

Convolution combines two functions by accumulating their overlap under a changing shift. The chapter connects its integral definition with properties, impulse behaviour, frequency-domain convolution, and examples. The operation differs from pointwise multiplication, even though transforms connect the two. A graphical construction helps identify where overlap begins and ends, while the algebra supplies the actual values. The result’s support and units offer independent checks. Understanding convolution here prepares the later interpretation of a linear time-invariant system’s response through its impulse response.

Study task. Sketch the support of a convolution before calculating it. Compare a graphical overlap argument with an analytical result and explain the effect of convolution with an impulse.

Chapter 8 Finite signal duration, leakage and windowing

A practical record is finite, so observing a signal means multiplying it by a time window. That operation changes the spectrum and can spread energy around a harmonic component. The chapter develops leakage, the difficulty of separating neighbouring frequencies, and the effect on energy and power interpretation. Windowing can alter sidelobes and main-lobe width, introducing trade-offs rather than eliminating every limitation. Record duration remains central to distinguishable frequency detail. Readers should separate the original signal’s properties from features introduced by the observation procedure.

Study task. Compare spectra from different record lengths and windows for the same sinusoid. Explain which changes reflect observation choices rather than new components in the underlying signal.

Chapter 9 Sampling

Sampling replaces a continuous-time signal with values at discrete instants. The chapter uses an impulse-train model to derive the spectral consequences and explain the sampling theorem and aliasing. Spectral replicas can overlap when the signal and sampling conditions are unsuitable, making different continuous signals produce indistinguishable samples. Merely increasing the number of stored points after acquisition cannot restore information lost this way. The relevant bandwidth and sample rate must be understood before interpretation. This connects an abstract transform result with practical acquisition design.

Study task. Draw spectral replicas for a band-limited example at two sample rates. Identify when overlap occurs and explain why later interpolation cannot undo aliasing.

Chapter 10 Signal reconstruction

Reconstruction asks under what conditions samples can determine the original continuous signal. The chapter contrasts ideal reconstruction with zero-order hold behaviour. Ideal results rely on assumptions about bandwidth, sampling, and the reconstruction operation; joining points in a convenient way is not automatically equivalent. A hold device produces a particular time-domain output and frequency response that should be analysed as part of the system. The chapter links acquisition with what happens when a discrete sequence is converted back to a continuous-time quantity.

Study task. Compare an ideal reconstruction model with a zero-order hold for one sampled signal. Identify the assumptions needed for exact recovery and a distortion introduced by the hold.

Chapter 11 Discrete-Time Fourier Transform

The discrete-time Fourier transform describes a sequence through a frequency function while retaining the effects of discrete sampling. The chapter develops forward and inverse forms, properties, and complex-exponential examples. A discrete-time sequence need not be finite, so this transform should not be conflated with a finite DFT. Periodicity in frequency follows from the sampling structure and affects how spectra are labelled and interpreted. Careful attention to sample interval and frequency units makes the connection with the continuous-time model transparent. Symmetry remains a useful check for real-valued sequences.

Study task. Explain how the DTFT differs from the transform of a continuous signal and from a finite DFT. Use a simple sequence to check spectral periodicity.

Chapter 12 Discrete Fourier Transform

The DFT provides a finite set of frequency-domain values for a finite sequence. The chapter connects its definition and inverse with frequency bins, symmetry, examples, leakage, and zero-padding. Bin spacing depends on the record and sampling choices, while zero-padding evaluates a denser representation without creating a longer observation. A real sinusoid’s appearance depends on its alignment with the analysis record and frequency grid. The DFT is therefore both a computational tool and a model with implicit structure. Scaling and one-sided conversion need to be justified before amplitudes are read from software output.

Study task. Compute a small DFT and its inverse, then zero-pad the sequence. Explain what changes in the plot and what does not change about the available frequency information.

Chapter 13 Energy and power spectral density

Energy and power spectral density describe how a signal’s energy or average power is distributed over frequency. The chapter treats continuous- and discrete-time deterministic signals and links integrated spectral quantities with time-domain measures. A spectral density has different units from a transform magnitude, so the distinction is more than a change of label. Periodic and finite-energy signals require appropriate treatment rather than one universal formula. The chapter creates the basis for later statistical estimation by establishing what quantity is being estimated and how its normalization can be checked.

Study task. Track units from a time-domain signal through its spectral-density representation. Verify that integration or the appropriate discrete sum recovers the intended energy or power.

Chapter 14 Spectral estimation

Spectral estimation addresses finite observations of signals with uncontrolled variability. The chapter states assumptions, introduces the periodogram, examines its statistical properties, and develops Welch averaging. A single estimate can vary substantially even when the underlying process is unchanged. Averaging can improve stability while choices about segment length and windows affect frequency detail. The practical task is to distinguish estimator variability from genuine structure and to report the method settings that shape the result. A smooth-looking spectrum is not sufficient evidence that an estimate is accurate or appropriately resolved.

Study task. Estimate a noisy signal’s spectrum with different segment lengths and averaging choices. Compare stability and frequency detail and explain the trade-off.

Chapter 15 Spectrogram

A spectrogram applies local spectral analysis across time to examine signals whose characteristics may change. Window length and overlap determine how time localization and frequency discrimination are balanced. The display depends on scaling, colour mapping, and the underlying estimation method, so a bright region requires interpretation rather than automatic identification as a physical event. The chapter connects the spectrogram with stationarity questions and time-varying examples. It helps readers recognize why a spectrum of the entire record can conceal the order in which different components occurred.

Study task. Compare a full-record spectrum with two spectrogram settings for a changing-frequency signal. Explain which temporal and frequency features each setting can distinguish.

Chapter 16 Linear time-invariant systems

Linear time-invariant systems have structural properties that allow their responses to be described through convolution and frequency response. The chapter examines system properties, differential-equation models, and the connection between time and frequency descriptions. Linearity concerns superposition, while time invariance concerns shifts; they are separate tests. A frequency-response function describes how sinusoidal components are modified under the stated conditions. Recognizing these assumptions is necessary before multiplying a spectrum by a system response. The chapter connects the earlier mathematical tools with a general framework for engineering analysis.

Study task. Test a simple input–output rule for linearity and time invariance separately. Explain when an impulse response is sufficient to characterize the system.

Chapter 17 Measuring signals

Measuring a signal involves a sensor and acquisition chain, not direct access to the ideal quantity of interest. The chapter connects sensor behaviour with the system concepts developed earlier. A device can change amplitude and phase, add limitations, or respond differently across frequencies. The measured output must therefore be interpreted through a model of the instrument and the conditions of use. A sensor’s static response does not necessarily describe its dynamic behaviour. Thinking through the full measurement chain helps distinguish properties of the physical input from properties introduced by the measuring system.

Study task. Map a measurement chain from physical input to recorded samples. Identify where sensor dynamics and sampling could alter the observed signal.

Chapter 18 Filters

Filters modify selected signal components according to a specified response. The chapter considers ideal and practical frequency-selective behaviour and relates filtering to the linear-system framework. Passbands, stopbands, transitions, gain, and phase all affect the result. A filter that suppresses unwanted components can also change useful information, so its purpose and limitations must be explicit. Time-domain effects follow from the same system description and should not be ignored when choosing by magnitude response alone. The final chapter brings transforms, convolution, system modelling, and practical interpretation together.

Study task. Specify a filtering objective with useful and unwanted components. Explain the required magnitude and phase considerations and how the output would be checked against the objective.

How to study this book

1. State the signal model, transform convention, sample interval, record duration, and units before calculating.
2. Use symmetry, support, dimensional analysis, and energy or power relationships as independent checks.
3. Compare an analytical result with a small numerical example, keeping normalization explicit.
4. Separate acquisition limitations from processing choices; do not interpret zero-padding as new information.
5. Read the corresponding exercise chapter after the theory and attempt problems before consulting worked solutions.

Questions for reflection

• How can a spectrum contain features introduced by measurement rather than by the source?
• Why do more plotted frequency points not necessarily mean better frequency resolution?
• Which assumptions justify describing a system through convolution?

Source edition and coverage

All eighteen numbered chapters of the Theory volume are covered. The mathematical appendices and companion Exercises volume remain available through the publisher. This original study guide explains the source in new wording and adds learning activities. Read the publisher’s book and complete exercises.