Fast Fourier Transform (FFT) Decimation-in-Time and Decimation-in-Frequency Algorithms

Unit 3, Lesson 1: FIR filtering and group delay

Unit theme: Finite impulse response filter design Level: Undergraduate engineering (idealized models) Learning objective: A length-M real symmetric FIR h[k]=h[M−1−k] has nominal linear-phase group delay (M−1)/2 samples away from zero crossings.

1. Concept and mathematical language

A length-M real symmetric FIR h[k]=h[M−1−k] has nominal linear-phase group delay (M−1)/2 samples away from zero crossings. The role of this principle in finite impulse response filter design 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 nine-tap symmetric FIR introduces nominal delay (9−1)/2=4 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 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: Symmetry alone says nothing about attenuation quality; an FIR can be linear-phase while failing the desired magnitude mask. 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 the center of a filtered pulse against the expected group delay. 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 sensor anti-interference filtering, 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 dense-grid magnitude/phase checks. 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: Symmetry alone says nothing about attenuation quality; an FIR can be linear-phase while failing the desired magnitude mask. 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 fir filtering and group delay 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 sensor anti-interference filtering, 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 3, Lesson 2: Moving average and comb-shaped nulls

Unit theme: Finite impulse response filter design Level: Undergraduate engineering (idealized models) Learning objective: A boxcar FIR averages M consecutive samples and creates regularly spaced spectral nulls.

1. Concept and mathematical language

A boxcar FIR averages M consecutive samples and creates regularly spaced spectral nulls. The role of this principle in finite impulse response filter design 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 M=8 at fs=800 Hz, nulls occur at nonzero integer multiples of fs/M=100 Hz below Nyquist. 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: A moving average is not an ideal brick-wall low-pass; sidelobes can pass unwanted signals. 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: Filter a 10 Hz signal mixed with 100 Hz interference; test M=4,8,16. 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 sensor anti-interference filtering, 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 dense-grid magnitude/phase checks. 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 moving average is not an ideal brick-wall low-pass; sidelobes can pass unwanted signals. 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 moving average and comb-shaped nulls 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 sensor anti-interference filtering, 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 3, Lesson 3: Windowed-sinc low-pass construction

Unit theme: Finite impulse response filter design Level: Undergraduate engineering (idealized models) Learning objective: Truncating the ideal sinc low-pass with a finite symmetric window yields implementable FIR coefficients with a transition band.

1. Concept and mathematical language

Truncating the ideal sinc low-pass with a finite symmetric window yields implementable FIR coefficients with a transition band. The role of this principle in finite impulse response filter design 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 normalized cutoff fc/fs=0.1, ideal center coefficient h[0]=2fc/fs=0.2 before windowing/normalization. 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: Cutoff is usually understood at a specified response point; finite-length design can shift the effective −3 dB frequency. 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: Generate 21- and 81-tap Hann-windowed kernels and measure actual attenuation. 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 sensor anti-interference filtering, 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 dense-grid magnitude/phase checks. 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: Cutoff is usually understood at a specified response point; finite-length design can shift the effective −3 dB frequency. 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 windowed-sinc low-pass construction 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 sensor anti-interference filtering, 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 3, Lesson 4: Spectral inversion and high-pass FIR

Unit theme: Finite impulse response filter design Level: Undergraduate engineering (idealized models) Learning objective: A high-pass impulse response can be formed by subtracting a low-pass response from a centered unit impulse.

1. Concept and mathematical language

A high-pass impulse response can be formed by subtracting a low-pass response from a centered unit impulse. The role of this principle in finite impulse response filter design 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 odd-length FIR with low-pass h[center]=0.2, spectral inversion gives high-pass center coefficient 1−0.2=0.8. 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: For even-length designs the delay center is fractional and a direct integer impulse subtraction needs reformulation. 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 response sums Hlow+Hhigh against a pure delay. 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 sensor anti-interference filtering, 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 dense-grid magnitude/phase checks. 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: For even-length designs the delay center is fractional and a direct integer impulse subtraction needs reformulation. 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 spectral inversion and high-pass fir 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 sensor anti-interference filtering, 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 3, Lesson 5: FIR ripple, order and coefficient effects

Unit theme: Finite impulse response filter design Level: Undergraduate engineering (idealized models) Learning objective: Sharper transition bands and stricter attenuation generally demand a longer FIR; quantized coefficients perturb response.

1. Concept and mathematical language

Sharper transition bands and stricter attenuation generally demand a longer FIR; quantized coefficients perturb response. The role of this principle in finite impulse response filter design 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=2000 Hz, passband edge 350 Hz and stopband edge 450 Hz imply transition width 100 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 Kaiser or equiripple design order estimate is not a guarantee; verify the realized frequency response on a dense grid. 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: Quantize a 31-tap low-pass to 8,12,16 fractional bits and compare stopband leakage. 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 sensor anti-interference filtering, 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 dense-grid magnitude/phase checks. 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 Kaiser or equiripple design order estimate is not a guarantee; verify the realized frequency response on a dense grid. 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 fir ripple, order and coefficient effects 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 sensor anti-interference filtering, 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 3 capstone — Linear-phase FIR anti-interference design

Engineering problem

Specify ripple, rejection, transition width, group delay and coefficient precision for a 4 kHz digitized signal with a 1.2 kHz interfering tone. The capstone intentionally uses invented signals, not field measurements. The unit provides the mathematical concepts: FIR filtering and group delay, Moving average and comb-shaped nulls, Windowed-sinc low-pass construction, Spectral inversion and high-pass FIR, FIR ripple, order and coefficient effects. 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.