Structures for Discrete-Time Systems: Direct Form I/II, Cascade, and Parallel Realizations

Unit 4, Lesson 1: Z-plane poles and causal stability

Unit theme: Infinite impulse response filters and realization Level: Undergraduate engineering (idealized models) Learning objective: Poles of a causal rational digital filter must lie strictly within the unit circle for BIBO stability, assuming no unstable hidden modes.

1. Concept and mathematical language

Poles of a causal rational digital filter must lie strictly within the unit circle for BIBO stability, assuming no unstable hidden modes. The role of this principle in infinite impulse response filters and realization 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 mode proportional to 0.9^n decays; a mode 1.02^n grows with n. 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: Exact algebra can conceal cancellation of unstable internal modes, making numerical implementation behavior more complicated. 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: Plot impulse responses for poles at 0.8,0.98 and 1.01. 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 low-power smoothers, 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 stability and frequency-response measurements. 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: Exact algebra can conceal cancellation of unstable internal modes, making numerical implementation behavior more complicated. 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. 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 z-plane poles and causal stability 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 low-power smoothers, 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 4, Lesson 2: First-order recursive smoothing

Unit theme: Infinite impulse response filters and realization Level: Undergraduate engineering (idealized models) Learning objective: One-pole low-pass y[n]=a y[n−1]+(1−a)x[n] has unit DC gain for |a|<1 and a chosen zero initial state.

1. Concept and mathematical language

One-pole low-pass y[n]=a y[n−1]+(1−a)x[n] has unit DC gain for |a|<1 and a chosen zero initial state. The role of this principle in infinite impulse response filters and realization 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=0.75 and a unit-step input beginning at n=0, y[0]=0.25 and y[1]=0.4375. 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: Setting a close to 1 reduces bandwidth but lengthens transient and increases sensitivity to finite precision. 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: Sweep a from 0.2 to 0.98 and estimate 90% step settling sample. 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 low-power smoothers, 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 stability and frequency-response measurements. 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: Setting a close to 1 reduces bandwidth but lengthens transient and increases sensitivity to finite precision. 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. 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 first-order recursive smoothing 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 low-power smoothers, 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 4, Lesson 3: Second-order biquad resonators

Unit theme: Infinite impulse response filters and realization Level: Undergraduate engineering (idealized models) Learning objective: Complex-conjugate poles r exp(±jω0) create a resonant second-order response; pole radius controls decay.

1. Concept and mathematical language

Complex-conjugate poles r exp(±jω0) create a resonant second-order response; pole radius controls decay. The role of this principle in infinite impulse response filters and realization 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: At r=0.9, the free-oscillation envelope after 10 samples is r^10≈0.3487 of initial amplitude. 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: Biquad gain must be normalized to the chosen reference frequency and can become very large near the unit circle. 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: Simulate decaying resonances at radii 0.6,0.9,0.99. 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 low-power smoothers, 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 stability and frequency-response measurements. 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: Biquad gain must be normalized to the chosen reference frequency and can become very large near the unit circle. 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. 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 second-order biquad resonators 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 low-power smoothers, 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 4, Lesson 4: Analog prototype mapping and warping

Unit theme: Infinite impulse response filters and realization Level: Undergraduate engineering (idealized models) Learning objective: The bilinear transform s=(2/T)(1−z^(−1))/(1+z^(−1)) maps the stable analog half-plane into the digital unit disk.

1. Concept and mathematical language

The bilinear transform s=(2/T)(1−z^(−1))/(1+z^(−1)) maps the stable analog half-plane into the digital unit disk. The role of this principle in infinite impulse response filters and realization 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: At fs=1000 Hz and digital cutoff 100 Hz, prewarped analog frequency is 2fs tan(π·100/1000)≈649.84 rad/s. 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: Bilinear mapping is not a linear frequency conversion; prewarping is needed to exactly match a designated cutoff. 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 digital cutoff from naive analog and prewarped one-pole designs. 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 low-power smoothers, 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 stability and frequency-response measurements. 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: Bilinear mapping is not a linear frequency conversion; prewarping is needed to exactly match a designated cutoff. 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. 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 analog prototype mapping and warping 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 low-power smoothers, 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 4, Lesson 5: Second-order sections and numerical robustness

Unit theme: Infinite impulse response filters and realization Level: Undergraduate engineering (idealized models) Learning objective: Cascaded biquads limit the coefficient sensitivity and intermediate dynamic range problems of high-order direct forms.

1. Concept and mathematical language

Cascaded biquads limit the coefficient sensitivity and intermediate dynamic range problems of high-order direct forms. The role of this principle in infinite impulse response filters and realization 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 sixth-order IIR filter can be represented by three second-order sections when its roots can be paired into real sections. 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: Cascade order and scaling affect intermediate overflow even if the mathematical transfer function is unchanged. 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 float32 and 12-bit fixed-point cascading with alternate section orders. 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 low-power smoothers, 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 stability and frequency-response measurements. 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: Cascade order and scaling affect intermediate overflow even if the mathematical transfer function is unchanged. 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. 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 second-order sections and numerical robustness 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 low-power smoothers, 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 4 capstone — Low-power IIR architecture challenge

Engineering problem

Design a two-stage causal IIR smoother, show exact-real-number pole stability and compare 16-bit section ordering and saturation effects. The capstone intentionally uses invented signals, not field measurements. The unit provides the mathematical concepts: Z-plane poles and causal stability, First-order recursive smoothing, Second-order biquad resonators, Analog prototype mapping and warping, Second-order sections and numerical robustness. 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.