Engineering Signal Analysis: From Fourier to filtering: Exercises

Engineering Signal Analysis: From Fourier to Filtering—Exercises accompanies the theory volume with problems, computer activities, and worked solutions. Its eighteen chapters mirror the progression from signal descriptions to Fourier analysis, sampling, spectral methods, systems, measurement, and filters. This guide explains what each problem group is intended to develop and how to check reasoning. It does not reproduce the publisher’s questions or answer key; use the original exercises to practise and the solutions to diagnose errors after an independent attempt.

Chapter-wise summaries and reading tasks

Chapter 1 Introduction

The opening exercises establish the connection between an engineering signal and its time- and frequency-domain descriptions. The goal is to identify meaningful features before relying on numerical output. Computer activities provide an opportunity to relate a mathematical expression to a plotted observation, but settings and units must be recorded. A correct graph can still answer the wrong question if the time range or frequency axis is misunderstood. This chapter is best used to build a routine of predicting, calculating, and interpreting rather than comparing only the final appearance with a solution.

Study task. Before running an introductory computer activity, predict its key time and spectral features. Record the settings and explain one difference between the prediction and output.

Chapter 2 Signal preliminaries

Signal-preliminary problems practise classification, symmetry, sinusoidal descriptions, operations, and energy or power. They require careful reading of the signal argument and of distinctions such as periodic versus aperiodic. Quantization examples also connect mathematical range with a finite representation. Working by sketch first can reveal the consequences of a shift, reversal, or scaling before algebra becomes involved. A useful correction process identifies which definition was misapplied, not merely which sign differs from the solution. The chapter develops habits that will be needed in every later transform calculation.

Study task. Choose one operation problem and solve it by both sketch and formula. Explain how you checked its support, symmetry, and relevant energy or power classification.

Chapter 3 Real Fourier series

Real Fourier-series exercises connect harmonic coefficients with recognizable signal structure. Some results can be obtained through trigonometric identities or symmetry without starting from an integral. Other problems require integrating over a consistent period and interpreting a one-sided representation. Computer work helps compare partial reconstructions with the target waveform. The aim is to understand why coefficients appear or vanish and how they contribute to the signal. Matching a coefficient list alone is insufficient if the fundamental frequency or normalization has been misunderstood.

Study task. Predict the nonzero coefficient types for a periodic example, then calculate them. Reconstruct several partial sums and explain the visible approximation behaviour.

Chapter 4 Complex exponential Fourier series

The complex-series chapter practises conversion between real and complex forms, polar interpretation, symmetry, examples, shift and differentiation relationships, and power checks. Positive and negative harmonic coefficients must be handled consistently, particularly for real signals. A result that differs by a conjugate or factor may reveal a convention error rather than a completely failed method. Writing the reconstruction equation makes that distinction visible. The worked solutions are most useful when compared line by line with an independently reasoned derivation, including the interpretation of phase.

Study task. Convert a simple real series to complex coefficients and back. Use reconstruction and a power check to locate any missing factors or incorrect conjugate relationships.

Chapter 5 Fourier transform

Fourier-transform problems develop magnitude and phase interpretation, symmetry, transform pairs, limiting forms, and energy checks. A real signal’s conjugate symmetry provides a strong diagnostic tool, while evenness or oddness can simplify calculations. Computer exercises offer visual comparisons but need careful frequency and amplitude scaling. The chapter encourages students to connect the transform with the original signal rather than treat it as an unrelated expression. When a result appears plausible, an inverse or limiting-case check can reveal a mistake that a picture alone would miss.

Study task. Solve a transform problem and verify its symmetry and units. Explain how one feature of the original signal is represented in the resulting magnitude or phase.

Chapter 6 Fourier transform theorems

The theorem exercises ask students to construct transform results through structured operations such as shifts, scaling, reversal, modulation, convolution, and differentiation. Recognizing a signal as a combination of known building blocks is often more efficient than direct integration. The challenge is to preserve operation order and all associated factors. A short derivation with named rules is easier to audit than one compressed leap to a final expression. Computer comparisons can then test whether predicted support, phase, and symmetry agree. The method builds transferable reasoning rather than memorization of isolated pairs.

Study task. Decompose a target signal into known operations and state the transform rule at each step. Check the final expression using an independent property.

Chapter 7 Convolution

Convolution practice develops the relationship between graphical overlap, integration limits, and the resulting function. Reversing and shifting one signal changes where overlap occurs, so sketches should guide the piecewise calculation. The exercises also examine properties such as commutativity and connect them with alternate ways to perform the same computation. A numerical convolution is useful as a check only when the sampling and scaling conventions match the continuous or discrete problem. The aim is to understand how the result is built rather than obtain an unexplained software array.

Study task. Determine convolution support and interval boundaries before integrating. Compare two equivalent constructions and explain why they produce the same result.

Chapter 8 Finite signal duration, leakage and windowing

The finite-duration chapter uses computer experiments to expose leakage and window effects. Changing the observation interval alters the spectrum even when the underlying sinusoid remains the same. Comparing windows reveals trade-offs between main-lobe width and sidelobes, which influence the visibility of nearby components. Students should vary one setting at a time and retain a common scale so differences can be interpreted. The exercise is not to select the prettiest plot but to explain how the observation procedure creates the pattern and what limitations remain.

Study task. Repeat a spectral experiment with different durations and windows. Keep the source signal unchanged and explain each observed difference through the analysis settings.

Chapter 9 Sampling

Sampling exercises focus on spectral replicas, aliasing, and the relationship between continuous frequencies and recorded samples. A sketch of the shifted spectra can make an ambiguity clear before any detailed calculation. The important question is whether different source components become indistinguishable under the chosen sampling process. A later plot may look well behaved despite aliasing, so visual smoothness is not a sufficient check. The problems develop the ability to justify a sample rate and interpret recorded frequencies under explicit assumptions about the original signal.

Study task. For one sampling scenario, draw the relevant replicas and identify a possible alias. Explain what prior bandwidth information is needed to interpret the samples uniquely.

Chapter 10 Signal reconstruction

Reconstruction problems reverse the sampling perspective by asking which continuous signal can be recovered through a specified filter or hold process. The analysis must account for the sampled spectrum and the reconstruction response together. Ideal recovery is conditional on the original assumptions, while practical reconstruction can introduce its own frequency-dependent effects. Students should separate information already lost at acquisition from distortion introduced during reconstruction. The solutions are most instructive when the student can explain why a particular frequency component is retained, removed, or changed.

Study task. Trace a sampled spectrum through a reconstruction filter. Distinguish aliasing that has already occurred from an effect caused by the reconstruction stage.

Chapter 11 Discrete-Time Fourier Transform

DTFT practice develops finite-sum manipulations, symmetry proofs, and the interpretation of frequency periodicity for discrete-time sequences. Even when a sequence has a finite nonzero segment, its transform remains a function of frequency rather than only the finite DFT grid. A geometric-series derivation can connect an algebraic expression with the duration and phase of that segment. Students should record which frequency variable and sample-interval normalization are in use. This prevents later confusion when moving between a DTFT expression and numerical DFT values.

Study task. Derive the DTFT of a short finite sequence and check its periodicity and real-signal symmetry. Explain how its values relate to a selected finite frequency grid.

Chapter 12 Discrete Fourier Transform

DFT exercises combine derivations, explicit small transforms, sinusoidal examples, and computer checks. The chapter requires careful distinctions between sequence length, frequency-bin index, and physical frequency. Leakage and zero-padding revisit earlier observation limits in a finite computational setting. An inverse transform provides an immediate consistency check, while a power or energy relationship tests scaling. Students should explain how a real sequence’s symmetry constrains the output and why a denser plotted grid does not by itself create more measured information.

Study task. Compute a small DFT by hand and with software, then reconstruct the sequence. Explain any scaling difference and distinguish zero-padding from collecting a longer record.

Chapter 13 Energy and power spectral density

Spectral-density exercises connect component descriptions with energy or power in specified frequency regions. The key is to identify the correct quantity and convention before summing or integrating. A one-sided and a two-sided representation can describe the same physical total while assigning values differently. The problems therefore reward dimensional and total-power checks rather than reliance on the height of a plotted peak. Students should also distinguish a deterministic spectral representation from the estimates developed in the following chapter.

Study task. Calculate power in a stated frequency interval for a simple multicomponent signal. Verify the total against a time-domain account and explain the role of the spectral convention.

Chapter 14 Spectral estimation

Spectral-estimation computer work introduces variability through noisy simulated measurements. Changing record and segment settings shows how estimates fluctuate and how averaging alters their appearance. A single realization cannot establish the estimator’s general behaviour, so repeated trials are valuable. Students should compare the measured estimate with the known simulated signal while recognizing uncertainty rather than expect perfect agreement. The aim is to explain a trade-off between stability and frequency detail and to document the settings needed for another reader to reproduce the analysis.

Study task. Repeat a noisy spectral experiment across several realizations. Compare raw and averaged estimates and explain which differences reflect random variability rather than a changed source.

Chapter 15 Spectrogram

Spectrogram exercises investigate signals whose frequency content changes over time. The display makes it possible to compare a local time-frequency account with an overall spectrum, but window length and overlap affect what can be seen. A changing-frequency example is useful because the intended behaviour is known before computation. Students should examine both the time and frequency axes and avoid reading colour intensity without its scale. Comparing settings develops an understanding of resolution limits rather than the assumption that one plot is a complete view of the signal.

Study task. Generate a known changing-frequency signal and compare spectrograms using two window lengths. Explain the loss and gain in localization in each representation.

Chapter 16 Linear time-invariant systems

System exercises practise identifying order, linearity, and time invariance from mathematical descriptions. These properties must be tested separately; a differential equation’s presence does not automatically guarantee the desired assumptions. Later calculations connect input, output, and system response using the theory volume’s framework. The challenge is to justify why a method applies before applying it. A counterexample can disprove a claimed property, while a general argument is needed to establish one. Keeping this distinction clear prevents a few successful numerical trials from being treated as a proof.

Study task. Test a proposed system for linearity and time invariance using definitions. Provide a counterexample where a property fails and explain the consequences for response analysis.

Chapter 17 Measuring signals

Measurement exercises use a sensor example to connect physical dynamics with the recorded signal. The task is to identify what the device senses, what it outputs, and how its model modifies the relationship across frequency. A measured displacement or voltage may represent another physical quantity only through that calibrated relationship. Students should track units and distinguish the input process from instrument response. This chapter brings theoretical system analysis into practical interpretation and shows why measurement cannot be treated as a perfectly transparent step.

Study task. Draw the input–output model of a sensor and label the physical quantities and units. Explain one situation where ignoring its dynamic response would distort interpretation.

Chapter 18 Filters

Filter problems combine spectral interpretation, system response, and checks on output energy or power. A desired output can constrain the filter response at the frequencies actually present without uniquely determining its behaviour everywhere. That distinction helps students see why several mathematical responses may satisfy one example but differ for other inputs. The chapter also connects idealized filters with practical reasoning about retained and rejected components. A complete solution explains feasibility, conventions, and the information available, rather than presenting one curve as the only possible answer without justification.

Study task. Determine which parts of a filter response are constrained by a chosen input–output pair. Explain whether the available information determines the response uniquely.

How to study this book

1. Attempt each problem independently before reading its worked solution, and retain the unsuccessful reasoning for comparison.
2. State conventions, units, assumptions, and expected qualitative behaviour at the start of each solution.
3. Use independent checks such as symmetry, support, inverse transformation, and energy or power consistency.
4. For computer exercises, save parameters and vary one setting at a time so differences remain interpretable.
5. After checking a solution, rework a related problem without the answer visible and explain the method in your own words.

Questions for reflection

• What can an independent check reveal that matching a plotted shape cannot?
• When can a numerical experiment illustrate a property without proving it?
• How should an error log distinguish convention mistakes from conceptual misunderstandings?

Source edition and coverage

All eighteen exercise chapters are covered. This is an original practice guide, not a reproduction of the questions or worked solutions in the publisher’s exercise book. This original study guide explains the source in new wording and adds learning activities. Read the publisher’s book and complete exercises.