Multirate Digital Signal Processing: Decimation, Interpolation, and Filter Banks

Unit 8, Lesson 1: Fixed-point representation and saturation

Unit theme: Real-time DSP implementation and verification Level: Undergraduate engineering (idealized models) Learning objective: Fixed-point DSP represents samples with scale factors and limited integer range; intermediate sums may overflow before final clipping.

1. Concept and mathematical language

Fixed-point DSP represents samples with scale factors and limited integer range; intermediate sums may overflow before final clipping. The role of this principle in real-time dsp implementation and verification 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 embedded and measured-data 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: Signed Q1.15 uses step 2^(−15) and representable positive maximum 1−2^(−15). 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: Wraparound and saturation are different failure modes; test arithmetic paths at maximum amplitude. 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 16-bit saturating and wrapping addition for two 0.75-scale operands. 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 embedded real-time controller telemetry, 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 golden-reference test signals and execution-time budgets. 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: Wraparound and saturation are different failure modes; test arithmetic paths at maximum amplitude. 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. 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 fixed-point representation and saturation 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 embedded real-time controller telemetry, 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 8, Lesson 2: Block convolution and overlap-save

Unit theme: Real-time DSP implementation and verification Level: Undergraduate engineering (idealized models) Learning objective: Overlap-save uses FFT-based circular convolution plus discarded contaminated samples to implement exact linear convolution in blocks.

1. Concept and mathematical language

Overlap-save uses FFT-based circular convolution plus discarded contaminated samples to implement exact linear convolution in blocks. The role of this principle in real-time dsp implementation and verification 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 embedded and measured-data 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 an M=65 tap FIR and N=256 FFT, each block yields N−M+1=192 usable output samples. 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: Discarding too few contaminated samples creates periodic wraparound artifacts; overlap-save still incurs buffer latency. 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 direct convolution to overlap-save on a known impulse and random sequence. 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 embedded real-time controller telemetry, 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 golden-reference test signals and execution-time budgets. 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: Discarding too few contaminated samples creates periodic wraparound artifacts; overlap-save still incurs buffer latency. 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. 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 block convolution and overlap-save 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 embedded real-time controller telemetry, 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 8, Lesson 3: Real-time deadlines and memory budgets

Unit theme: Real-time DSP implementation and verification Level: Undergraduate engineering (idealized models) Learning objective: A streaming pipeline must finish each frame before the next input frame arrives, including copying, conversion and worst-case interrupt delays.

1. Concept and mathematical language

A streaming pipeline must finish each frame before the next input frame arrives, including copying, conversion and worst-case interrupt delays. The role of this principle in real-time dsp implementation and verification 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 embedded and measured-data 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: With frames of B=256 samples at fs=48000 Hz, frame period is 256/48000≈5.333 ms. 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: Average processing time below the deadline is insufficient if occasional worst-case processing exceeds it. 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: Build a CPU budget with filter computation, data transfer and an explicit timing margin. 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 embedded real-time controller telemetry, 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 golden-reference test signals and execution-time budgets. 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: Average processing time below the deadline is insufficient if occasional worst-case processing exceeds it. 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. 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 real-time deadlines and memory budgets 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 embedded real-time controller telemetry, 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 8, Lesson 4: Independent regression and measurement traceability

Unit theme: Real-time DSP implementation and verification Level: Undergraduate engineering (idealized models) Learning objective: DSP output should be compared against an independent reference at reference inputs, extremes and adversarial settings.

1. Concept and mathematical language

DSP output should be compared against an independent reference at reference inputs, extremes and adversarial settings. The role of this principle in real-time dsp implementation and verification 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 embedded and measured-data 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 arrays y_ref and y_test, RMSE=sqrt(Σ(y_test−y_ref)²/N); report full-scale and absolute error too. 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: Two implementations sharing the same bug may agree; physical validity requires independent evidence and calibrated instruments. 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 regression suite with DC, impulse, Nyquist-edge tone and seeded 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 embedded real-time controller telemetry, 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 golden-reference test signals and execution-time budgets. 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: Two implementations sharing the same bug may agree; physical validity requires independent evidence and calibrated instruments. 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. 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 independent regression and measurement traceability 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 embedded real-time controller telemetry, 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 8, Lesson 5: Integrated condition-monitoring DSP capstone

Unit theme: Real-time DSP implementation and verification Level: Undergraduate engineering (idealized models) Learning objective: An auditable end-to-end DSP system connects sampling, windowing, filtering, feature extraction, alarms and uncertainty.

1. Concept and mathematical language

An auditable end-to-end DSP system connects sampling, windowing, filtering, feature extraction, alarms and uncertainty. The role of this principle in real-time dsp implementation and verification 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 embedded and measured-data 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: At fs=4000 and N=2000, a full block spans 0.5 s, while raw Fourier-bin spacing is 2 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: A benchmark on one invented bearing trace cannot justify real-world failure detection or a maintenance decision. 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 a reproducible spectral-monitoring chain with measurable passband and latency targets. 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 embedded real-time controller telemetry, 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 golden-reference test signals and execution-time budgets. 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 benchmark on one invented bearing trace cannot justify real-world failure detection or a maintenance decision. 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. 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 integrated condition-monitoring dsp capstone 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 embedded real-time controller telemetry, 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 8 capstone — Integrated real-time signal processor

Engineering problem

Implement a block-based pipeline with filtering and spectral reporting, document worst-case frame deadline, RAM requirements, test vectors and simulated failure injection. The capstone intentionally uses invented signals, not field measurements. The unit provides the mathematical concepts: Fixed-point representation and saturation, Block convolution and overlap-save, Real-time deadlines and memory budgets, Independent regression and measurement traceability, Integrated condition-monitoring DSP capstone. 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.