Finite Register Length Effects, Coefficient Quantization, and Roundoff Noise

Unit 7, Lesson 1: Periodogram normalization and units

Unit theme: Spectral estimation and feature extraction Level: Undergraduate engineering (idealized models) Learning objective: A periodogram converts a finite windowed DFT into a spectral power estimate; density units depend on sample rate and window energy.

1. Concept and mathematical language

A periodogram converts a finite windowed DFT into a spectral power estimate; density units depend on sample rate and window energy. The role of this principle in spectral estimation and feature extraction is to connect the expression on paper with an observable prediction. Before manipulating symbols, define the independent variable, the range of allowed inputs, the output quantity, and the reference condition. In particular, identify whether the argument concerns a discrete-time signal. The same algebraic shape may describe completely different physical situations when time is measured in seconds in one case and by integer samples in another. A meaningful graph must identify its axes and units, the origin of the record, and whether the plotted values are exact or illustrative.

2. A quantitative example

Checkable numerical or symbolic example: For N independent zero-mean samples with variance 1, average time-domain sample power is expected to be 1 in dimensionless squared units. Explain what the chosen values mean, list any units, and reproduce the result without replacing the governing mathematical relation with a memorized rule. A numerical example answers a narrowly framed question; it does not by itself certify an entire signal-processing chain. Describe the source or test signal, observation interval, reference time, and the output that would be compared with the calculated answer. If the expression uses a delta or step, say whether it represents an ideal mathematical distribution or a realizable finite-width experiment.

3. Derivation strategy and model boundaries

Start from the smallest version of the problem that keeps its essential structure. Represent a pulse with known support; annotate the independent coordinate and the nonzero support or spectral components. Derive the relevant expression step by step, naming the rule used to move from one line to the next. When integrating or summing, show the limits; when using a transform, write its kernel and normalization convention. If the quantity is complex, separate magnitude and phase before interpreting it as a real measurement. Avoid silently switching between angular frequency ω and cyclic frequency f.

A model boundary is a statement about where the prediction can fail. For this lesson, pay particular attention to the following: A raw |FFT|² plot is not automatically a PSD in V²/Hz; missing normalization causes incorrect integrated power. A useful report says whether the discrepancy originates in a violated hypothesis, finite measurement resolution, numerical rounding, a different Fourier or transform convention, or an unmodeled physical effect. Fitting a parameter after observing a failure does not retrospectively validate the original prediction. State any fitted parameters separately from independently measured ones.

4. Engineering investigation

Task: Verify Parseval for a tone and for a seeded noise record. Begin by writing a one-paragraph specification of the input, expected output, and permitted error. Choose an observable output rather than a qualitative claim such as “the plot looks correct.” If the application is rotating machine condition monitoring, explain what sensor or computation would provide the input and how the result could inform a decision. List relevant data units, an example operating range, and one adverse condition. Do not treat a synthetic input as a measurement of a real device.

Connect the investigation to time-domain energy versus spectral integration. Specify what a genuinely independent verification would require: a different derivation, a second calculation package, or a controlled instrument measurement with calibrated reference. When measurements are possible, identify bandwidth, resolution and uncertainty before interpreting a small residual as meaningful. When physical testing is unavailable, label all results “analytical” or “synthetic simulation.” A design choice should remain provisional until its assumptions have survived checks under conditions resembling the intended application.

5. Interpretation and common mistakes

Pitfall to diagnose: A raw |FFT|² plot is not automatically a PSD in V²/Hz; missing normalization causes incorrect integrated power. If an equation is linear, verify linearity on two different input functions instead of one. A second frequent mistake is confusing a transformation's algebraic existence with its physical realizability. An ideal bandlimited spectrum, an impulse with zero duration, infinite observation time and perfect reconstruction are mathematical conveniences. Each may be useful, but laboratory claims must include finite time, bandwidth, quantization and uncertainty when relevant. This distinction becomes essential when a simulation is used to select acquisition hardware or establish safety margins.

6. Guided exercises and assessment

1. Reconstruction: Starting from the definitions, derive the relation used in the quantitative example. State every assumption, not only the final substitution. Reproduce a dimensional or normalization check.

2. Boundary test: Recompute the lesson's core prediction for a zero input and a doubled input parameter where meaningful. Explain whether the expected change follows from linearity, dimensional reasoning, or a different property.

3. Counterexample: Design a case in which periodogram normalization and units would be incorrectly applied. Name the violated assumption and the observable symptom.

4. Independent result: Design a calculation or simulation that would check the mathematical expression without copying its exact implementation. Provide numerical values and expected units.

5. Application decision: For rotating machine condition monitoring, propose a measurement or controlled experiment that would distinguish the simplified model from one plausible alternative. Include a measurable acceptance criterion.

Self-check and marking guidance

An adequate answer identifies signal type and variables; states the relevant model equation; shows intermediate steps; reports units and reference convention; handles one limit or counterexample; and distinguishes analytic prediction from experimental verification. Work that quotes a final value without hypotheses is incomplete. A passing simulation alone does not establish the physical correctness of its underlying model. Assess whether the calculation would still be valid under a different sample rate, initial state or signal class, whichever is relevant here.

Editorial note: These examples are for instruction. Their idealized component values and invented datasets are not design specifications for safety-critical, mains-connected, medical or certified systems.

Unit 7, Lesson 2: Welch averaging and estimator variance

Unit theme: Spectral estimation and feature extraction Level: Undergraduate engineering (idealized models) Learning objective: Welch divides a record into windowed blocks, computes spectra and averages them to reduce random variation.

1. Concept and mathematical language

Welch divides a record into windowed blocks, computes spectra and averages them to reduce random variation. The role of this principle in spectral estimation and feature extraction is to connect the expression on paper with an observable prediction. Before manipulating symbols, define the independent variable, the range of allowed inputs, the output quantity, and the reference condition. In particular, identify whether the argument concerns a discrete-time signal. The same algebraic shape may describe completely different physical situations when time is measured in seconds in one case and by integer samples in another. A meaningful graph must identify its axes and units, the origin of the record, and whether the plotted values are exact or illustrative.

2. A quantitative example

Checkable numerical or symbolic example: A 4096-sample record with nonoverlapping 512-sample segments contains eight segments for averaging. Explain what the chosen values mean, list any units, and reproduce the result without replacing the governing mathematical relation with a memorized rule. A numerical example answers a narrowly framed question; it does not by itself certify an entire signal-processing chain. Describe the source or test signal, observation interval, reference time, and the output that would be compared with the calculated answer. If the expression uses a delta or step, say whether it represents an ideal mathematical distribution or a realizable finite-width experiment.

3. Derivation strategy and model boundaries

Start from the smallest version of the problem that keeps its essential structure. Represent a sinusoid with a calibrated phase origin; annotate the independent coordinate and the nonzero support or spectral components. Derive the relevant expression step by step, naming the rule used to move from one line to the next. When integrating or summing, show the limits; when using a transform, write its kernel and normalization convention. If the quantity is complex, separate magnitude and phase before interpreting it as a real measurement. Avoid silently switching between angular frequency ω and cyclic frequency f.

A model boundary is a statement about where the prediction can fail. For this lesson, pay particular attention to the following: Overlapping segments are correlated, so 50% overlap does not double the number of independent estimates. A useful report says whether the discrepancy originates in a violated hypothesis, finite measurement resolution, numerical rounding, a different Fourier or transform convention, or an unmodeled physical effect. Fitting a parameter after observing a failure does not retrospectively validate the original prediction. State any fitted parameters separately from independently measured ones.

4. Engineering investigation

Task: Compare mean and spread across repeated synthetic records for 1,4,8 segment averages. Begin by writing a one-paragraph specification of the input, expected output, and permitted error. Choose an observable output rather than a qualitative claim such as “the plot looks correct.” If the application is rotating machine condition monitoring, explain what sensor or computation would provide the input and how the result could inform a decision. List relevant data units, an example operating range, and one adverse condition. Do not treat a synthetic input as a measurement of a real device.

Connect the investigation to time-domain energy versus spectral integration. Specify what a genuinely independent verification would require: a different derivation, a second calculation package, or a controlled instrument measurement with calibrated reference. When measurements are possible, identify bandwidth, resolution and uncertainty before interpreting a small residual as meaningful. When physical testing is unavailable, label all results “analytical” or “synthetic simulation.” A design choice should remain provisional until its assumptions have survived checks under conditions resembling the intended application.

5. Interpretation and common mistakes

Pitfall to diagnose: Overlapping segments are correlated, so 50% overlap does not double the number of independent estimates. If a spectrum is complex, report both phase convention and amplitude normalization. A second frequent mistake is confusing a transformation's algebraic existence with its physical realizability. An ideal bandlimited spectrum, an impulse with zero duration, infinite observation time and perfect reconstruction are mathematical conveniences. Each may be useful, but laboratory claims must include finite time, bandwidth, quantization and uncertainty when relevant. This distinction becomes essential when a simulation is used to select acquisition hardware or establish safety margins.

6. Guided exercises and assessment

1. Reconstruction: Starting from the definitions, derive the relation used in the quantitative example. State every assumption, not only the final substitution. Calculate a boundary value without a calculator.

2. Boundary test: Recompute the lesson's core prediction for a zero input and a doubled input parameter where meaningful. Explain whether the expected change follows from linearity, dimensional reasoning, or a different property.

3. Counterexample: Design a case in which welch averaging and estimator variance would be incorrectly applied. Name the violated assumption and the observable symptom.

4. Independent result: Design a calculation or simulation that would check the mathematical expression without copying its exact implementation. Provide numerical values and expected units.

5. Application decision: For rotating machine condition monitoring, propose a measurement or controlled experiment that would distinguish the simplified model from one plausible alternative. Include a measurable acceptance criterion.

Self-check and marking guidance

An adequate answer identifies signal type and variables; states the relevant model equation; shows intermediate steps; reports units and reference convention; handles one limit or counterexample; and distinguishes analytic prediction from experimental verification. Work that quotes a final value without hypotheses is incomplete. A passing simulation alone does not establish the physical correctness of its underlying model. Assess whether the calculation would still be valid under a different sample rate, initial state or signal class, whichever is relevant here.

Editorial note: These examples are for instruction. Their idealized component values and invented datasets are not design specifications for safety-critical, mains-connected, medical or certified systems.

Unit 7, Lesson 3: Harmonics and spectral peak discrimination

Unit theme: Spectral estimation and feature extraction Level: Undergraduate engineering (idealized models) Learning objective: Peak detection uses a calibrated noise floor, minimum spacing, and knowledge of harmonic structure.

1. Concept and mathematical language

Peak detection uses a calibrated noise floor, minimum spacing, and knowledge of harmonic structure. The role of this principle in spectral estimation and feature extraction is to connect the expression on paper with an observable prediction. Before manipulating symbols, define the independent variable, the range of allowed inputs, the output quantity, and the reference condition. In particular, identify whether the argument concerns a discrete-time signal. The same algebraic shape may describe completely different physical situations when time is measured in seconds in one case and by integer samples in another. A meaningful graph must identify its axes and units, the origin of the record, and whether the plotted values are exact or illustrative.

2. A quantitative example

Checkable numerical or symbolic example: A 125 Hz fundamental with harmonics at 250 and 375 Hz produces a 3rd harmonic at 375 Hz. Explain what the chosen values mean, list any units, and reproduce the result without replacing the governing mathematical relation with a memorized rule. A numerical example answers a narrowly framed question; it does not by itself certify an entire signal-processing chain. Describe the source or test signal, observation interval, reference time, and the output that would be compared with the calculated answer. If the expression uses a delta or step, say whether it represents an ideal mathematical distribution or a realizable finite-width experiment.

3. Derivation strategy and model boundaries

Start from the smallest version of the problem that keeps its essential structure. Represent a step that starts at the declared reference instant; annotate the independent coordinate and the nonzero support or spectral components. Derive the relevant expression step by step, naming the rule used to move from one line to the next. When integrating or summing, show the limits; when using a transform, write its kernel and normalization convention. If the quantity is complex, separate magnitude and phase before interpreting it as a real measurement. Avoid silently switching between angular frequency ω and cyclic frequency f.

A model boundary is a statement about where the prediction can fail. For this lesson, pay particular attention to the following: A weak aliased interferer can appear as a harmonic, and leakage can create local maxima without distinct tones. A useful report says whether the discrepancy originates in a violated hypothesis, finite measurement resolution, numerical rounding, a different Fourier or transform convention, or an unmodeled physical effect. Fitting a parameter after observing a failure does not retrospectively validate the original prediction. State any fitted parameters separately from independently measured ones.

4. Engineering investigation

Task: Design peak tests for a tone mixed with impulsive and broadband interference. Begin by writing a one-paragraph specification of the input, expected output, and permitted error. Choose an observable output rather than a qualitative claim such as “the plot looks correct.” If the application is rotating machine condition monitoring, explain what sensor or computation would provide the input and how the result could inform a decision. List relevant data units, an example operating range, and one adverse condition. Do not treat a synthetic input as a measurement of a real device.

Connect the investigation to time-domain energy versus spectral integration. Specify what a genuinely independent verification would require: a different derivation, a second calculation package, or a controlled instrument measurement with calibrated reference. When measurements are possible, identify bandwidth, resolution and uncertainty before interpreting a small residual as meaningful. When physical testing is unavailable, label all results “analytical” or “synthetic simulation.” A design choice should remain provisional until its assumptions have survived checks under conditions resembling the intended application.

5. Interpretation and common mistakes

Pitfall to diagnose: A weak aliased interferer can appear as a harmonic, and leakage can create local maxima without distinct tones. If a pole or time constant appears, check the expected long-time behavior. A second frequent mistake is confusing a transformation's algebraic existence with its physical realizability. An ideal bandlimited spectrum, an impulse with zero duration, infinite observation time and perfect reconstruction are mathematical conveniences. Each may be useful, but laboratory claims must include finite time, bandwidth, quantization and uncertainty when relevant. This distinction becomes essential when a simulation is used to select acquisition hardware or establish safety margins.

6. Guided exercises and assessment

1. Reconstruction: Starting from the definitions, derive the relation used in the quantitative example. State every assumption, not only the final substitution. Sketch the expected waveform or spectral trend.

2. Boundary test: Recompute the lesson's core prediction for a zero input and a doubled input parameter where meaningful. Explain whether the expected change follows from linearity, dimensional reasoning, or a different property.

3. Counterexample: Design a case in which harmonics and spectral peak discrimination would be incorrectly applied. Name the violated assumption and the observable symptom.

4. Independent result: Design a calculation or simulation that would check the mathematical expression without copying its exact implementation. Provide numerical values and expected units.

5. Application decision: For rotating machine condition monitoring, propose a measurement or controlled experiment that would distinguish the simplified model from one plausible alternative. Include a measurable acceptance criterion.

Self-check and marking guidance

An adequate answer identifies signal type and variables; states the relevant model equation; shows intermediate steps; reports units and reference convention; handles one limit or counterexample; and distinguishes analytic prediction from experimental verification. Work that quotes a final value without hypotheses is incomplete. A passing simulation alone does not establish the physical correctness of its underlying model. Assess whether the calculation would still be valid under a different sample rate, initial state or signal class, whichever is relevant here.

Editorial note: These examples are for instruction. Their idealized component values and invented datasets are not design specifications for safety-critical, mains-connected, medical or certified systems.

Unit 7, Lesson 4: Envelope, cepstrum and modulation features

Unit theme: Spectral estimation and feature extraction Level: Undergraduate engineering (idealized models) Learning objective: Demodulation and cepstral methods expose periodic modulation structures that may be masked in a raw amplitude spectrum.

1. Concept and mathematical language

Demodulation and cepstral methods expose periodic modulation structures that may be masked in a raw amplitude spectrum. The role of this principle in spectral estimation and feature extraction is to connect the expression on paper with an observable prediction. Before manipulating symbols, define the independent variable, the range of allowed inputs, the output quantity, and the reference condition. In particular, identify whether the argument concerns a discrete-time signal. The same algebraic shape may describe completely different physical situations when time is measured in seconds in one case and by integer samples in another. A meaningful graph must identify its axes and units, the origin of the record, and whether the plotted values are exact or illustrative.

2. A quantitative example

Checkable numerical or symbolic example: For a 200 Hz carrier amplitude modulated at 10 Hz, ideal sidebands occur at 190 Hz and 210 Hz. Explain what the chosen values mean, list any units, and reproduce the result without replacing the governing mathematical relation with a memorized rule. A numerical example answers a narrowly framed question; it does not by itself certify an entire signal-processing chain. Describe the source or test signal, observation interval, reference time, and the output that would be compared with the calculated answer. If the expression uses a delta or step, say whether it represents an ideal mathematical distribution or a realizable finite-width experiment.

3. Derivation strategy and model boundaries

Start from the smallest version of the problem that keeps its essential structure. Represent a bandlimited synthetic record; annotate the independent coordinate and the nonzero support or spectral components. Derive the relevant expression step by step, naming the rule used to move from one line to the next. When integrating or summing, show the limits; when using a transform, write its kernel and normalization convention. If the quantity is complex, separate magnitude and phase before interpreting it as a real measurement. Avoid silently switching between angular frequency ω and cyclic frequency f.

A model boundary is a statement about where the prediction can fail. For this lesson, pay particular attention to the following: Envelope detection by rectification or analytic-signal magnitude can fail with multiple comparable carriers. A useful report says whether the discrepancy originates in a violated hypothesis, finite measurement resolution, numerical rounding, a different Fourier or transform convention, or an unmodeled physical effect. Fitting a parameter after observing a failure does not retrospectively validate the original prediction. State any fitted parameters separately from independently measured ones.

4. Engineering investigation

Task: Create a synthetic amplitude-modulated trace and recover its 10 Hz envelope rate. Begin by writing a one-paragraph specification of the input, expected output, and permitted error. Choose an observable output rather than a qualitative claim such as “the plot looks correct.” If the application is rotating machine condition monitoring, explain what sensor or computation would provide the input and how the result could inform a decision. List relevant data units, an example operating range, and one adverse condition. Do not treat a synthetic input as a measurement of a real device.

Connect the investigation to time-domain energy versus spectral integration. Specify what a genuinely independent verification would require: a different derivation, a second calculation package, or a controlled instrument measurement with calibrated reference. When measurements are possible, identify bandwidth, resolution and uncertainty before interpreting a small residual as meaningful. When physical testing is unavailable, label all results “analytical” or “synthetic simulation.” A design choice should remain provisional until its assumptions have survived checks under conditions resembling the intended application.

5. Interpretation and common mistakes

Pitfall to diagnose: Envelope detection by rectification or analytic-signal magnitude can fail with multiple comparable carriers. If data are sampled, state the sample rate and anti-aliasing assumption. A second frequent mistake is confusing a transformation's algebraic existence with its physical realizability. An ideal bandlimited spectrum, an impulse with zero duration, infinite observation time and perfect reconstruction are mathematical conveniences. Each may be useful, but laboratory claims must include finite time, bandwidth, quantization and uncertainty when relevant. This distinction becomes essential when a simulation is used to select acquisition hardware or establish safety margins.

6. Guided exercises and assessment

1. Reconstruction: Starting from the definitions, derive the relation used in the quantitative example. State every assumption, not only the final substitution. Identify a counterexample to an unstated assumption.

2. Boundary test: Recompute the lesson's core prediction for a zero input and a doubled input parameter where meaningful. Explain whether the expected change follows from linearity, dimensional reasoning, or a different property.

3. Counterexample: Design a case in which envelope, cepstrum and modulation features would be incorrectly applied. Name the violated assumption and the observable symptom.

4. Independent result: Design a calculation or simulation that would check the mathematical expression without copying its exact implementation. Provide numerical values and expected units.

5. Application decision: For rotating machine condition monitoring, propose a measurement or controlled experiment that would distinguish the simplified model from one plausible alternative. Include a measurable acceptance criterion.

Self-check and marking guidance

An adequate answer identifies signal type and variables; states the relevant model equation; shows intermediate steps; reports units and reference convention; handles one limit or counterexample; and distinguishes analytic prediction from experimental verification. Work that quotes a final value without hypotheses is incomplete. A passing simulation alone does not establish the physical correctness of its underlying model. Assess whether the calculation would still be valid under a different sample rate, initial state or signal class, whichever is relevant here.

Editorial note: These examples are for instruction. Their idealized component values and invented datasets are not design specifications for safety-critical, mains-connected, medical or certified systems.

Unit 7, Lesson 5: Time-frequency localization and feature reliability

Unit theme: Spectral estimation and feature extraction Level: Undergraduate engineering (idealized models) Learning objective: Spectral centroid, band power and short-time energy respond differently to shifts in window size, amplitude and noise.

1. Concept and mathematical language

Spectral centroid, band power and short-time energy respond differently to shifts in window size, amplitude and noise. The role of this principle in spectral estimation and feature extraction is to connect the expression on paper with an observable prediction. Before manipulating symbols, define the independent variable, the range of allowed inputs, the output quantity, and the reference condition. In particular, identify whether the argument concerns a discrete-time signal. The same algebraic shape may describe completely different physical situations when time is measured in seconds in one case and by integer samples in another. A meaningful graph must identify its axes and units, the origin of the record, and whether the plotted values are exact or illustrative.

2. A quantitative example

Checkable numerical or symbolic example: For two equal-power spectral lines at 100 and 300 Hz, power-weighted centroid is (100+300)/2=200 Hz. Explain what the chosen values mean, list any units, and reproduce the result without replacing the governing mathematical relation with a memorized rule. A numerical example answers a narrowly framed question; it does not by itself certify an entire signal-processing chain. Describe the source or test signal, observation interval, reference time, and the output that would be compared with the calculated answer. If the expression uses a delta or step, say whether it represents an ideal mathematical distribution or a realizable finite-width experiment.

3. Derivation strategy and model boundaries

Start from the smallest version of the problem that keeps its essential structure. Represent a two-component signal with separated scales; annotate the independent coordinate and the nonzero support or spectral components. Derive the relevant expression step by step, naming the rule used to move from one line to the next. When integrating or summing, show the limits; when using a transform, write its kernel and normalization convention. If the quantity is complex, separate magnitude and phase before interpreting it as a real measurement. Avoid silently switching between angular frequency ω and cyclic frequency f.

A model boundary is a statement about where the prediction can fail. For this lesson, pay particular attention to the following: Centroid and kurtosis are not universal condition indicators and can be unstable under interference or clipping. A useful report says whether the discrepancy originates in a violated hypothesis, finite measurement resolution, numerical rounding, a different Fourier or transform convention, or an unmodeled physical effect. Fitting a parameter after observing a failure does not retrospectively validate the original prediction. State any fitted parameters separately from independently measured ones.

4. Engineering investigation

Task: Measure feature changes with window length and added 300 Hz noise. Begin by writing a one-paragraph specification of the input, expected output, and permitted error. Choose an observable output rather than a qualitative claim such as “the plot looks correct.” If the application is rotating machine condition monitoring, explain what sensor or computation would provide the input and how the result could inform a decision. List relevant data units, an example operating range, and one adverse condition. Do not treat a synthetic input as a measurement of a real device.

Connect the investigation to time-domain energy versus spectral integration. Specify what a genuinely independent verification would require: a different derivation, a second calculation package, or a controlled instrument measurement with calibrated reference. When measurements are possible, identify bandwidth, resolution and uncertainty before interpreting a small residual as meaningful. When physical testing is unavailable, label all results “analytical” or “synthetic simulation.” A design choice should remain provisional until its assumptions have survived checks under conditions resembling the intended application.

5. Interpretation and common mistakes

Pitfall to diagnose: Centroid and kurtosis are not universal condition indicators and can be unstable under interference or clipping. If the argument treats a finite record as infinitely long, write down the window explicitly. A second frequent mistake is confusing a transformation's algebraic existence with its physical realizability. An ideal bandlimited spectrum, an impulse with zero duration, infinite observation time and perfect reconstruction are mathematical conveniences. Each may be useful, but laboratory claims must include finite time, bandwidth, quantization and uncertainty when relevant. This distinction becomes essential when a simulation is used to select acquisition hardware or establish safety margins.

6. Guided exercises and assessment

1. Reconstruction: Starting from the definitions, derive the relation used in the quantitative example. State every assumption, not only the final substitution. State an acceptance tolerance before simulating.

2. Boundary test: Recompute the lesson's core prediction for a zero input and a doubled input parameter where meaningful. Explain whether the expected change follows from linearity, dimensional reasoning, or a different property.

3. Counterexample: Design a case in which time-frequency localization and feature reliability would be incorrectly applied. Name the violated assumption and the observable symptom.

4. Independent result: Design a calculation or simulation that would check the mathematical expression without copying its exact implementation. Provide numerical values and expected units.

5. Application decision: For rotating machine condition monitoring, propose a measurement or controlled experiment that would distinguish the simplified model from one plausible alternative. Include a measurable acceptance criterion.

Self-check and marking guidance

An adequate answer identifies signal type and variables; states the relevant model equation; shows intermediate steps; reports units and reference convention; handles one limit or counterexample; and distinguishes analytic prediction from experimental verification. Work that quotes a final value without hypotheses is incomplete. A passing simulation alone does not establish the physical correctness of its underlying model. Assess whether the calculation would still be valid under a different sample rate, initial state or signal class, whichever is relevant here.

Editorial note: These examples are for instruction. Their idealized component values and invented datasets are not design specifications for safety-critical, mains-connected, medical or certified systems.

Unit 7 capstone — Feature extraction under uncertainty

Engineering problem

Generate weak periodic tones plus seeded broadband noise, compare raw and Welch PSD estimates, and evaluate features against false-alarm and sensitivity requirements. The capstone intentionally uses invented signals, not field measurements. The unit provides the mathematical concepts: Periodogram normalization and units, Welch averaging and estimator variance, Harmonics and spectral peak discrimination, Envelope, cepstrum and modulation features, Time-frequency localization and feature reliability. Write a concise problem statement and choose consistent notation for time, sample index, physical units and the transform convention. Keep raw input parameters in a table separate from derived outputs.

Required submissions

1. A mathematical model listing all assumptions, independent variables and initial conditions or finite record lengths.
2. A hand-verifiable result with the steps from governing equation to value or functional expression.
3. One independent computation, using a different method from the hand calculation, with an explicit discrepancy tolerance.
4. A plot that labels units, sample rate, window and phase/frequency conventions when relevant.
5. Two stress tests: one boundary case and one deliberate model violation with the expected failure symptoms.
6. A short engineering report distinguishing verification of code from validation against physical data.

Evaluation rubric (100 points)

• Mathematical and physical specification: 20.
• Correct derivations and units: 25.
• Independently reproducible numerical result: 20.
• Boundary and violation tests: 20.
• Clear interpretation and data provenance: 15.

Discussion prompts

Explain why your assumptions are necessary; demonstrate whether changing the record length or excitation changes the result; discuss which error is numerical and which is modeling error; and propose a minimal real-world test to determine whether the simplified model is appropriate. A visually convincing trace without a numeric reference earns no numerical verification credit. Refer to the corresponding lessons for the domain-specific limitations.