Discrete-Time Fourier Transform and Processing
Comprehensive university textbook unit covering fundamental principles, mathematical formulations, and practical engineering applications of Discrete-Time Fourier Transform and Processing.
§6.1 Z transform and region of convergence
Unit 6, Lesson 1: Z transform and region of convergence
Unit theme: Z transforms and discrete-time filters Level: Undergraduate engineering (idealized models) Learning objective: The bilateral Z transform sums x[n]z^(−n); its ROC distinguishes sequence sidedness.
1. Concept and mathematical language
The bilateral Z transform sums x[n]z^(−n); its ROC distinguishes sequence sidedness. The role of this principle in z transforms and discrete-time filters 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: (0.6)^n u[n] has X(z)=1/(1−0.6z^(−1)) and ROC |z|>0.6. 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: The expression X(z) without ROC does not uniquely identify a time sequence. 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: Derive the geometric sum and test convergence for z=0.5,0.8,1.2. 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 sensor 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 unit-circle pole location. 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: The expression X(z) without ROC does not uniquely identify a time sequence. 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. 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 transform and region of convergence 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 sensor 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.
§6.2 Discrete transfer functions and feedback recurrences
Unit 6, Lesson 2: Discrete transfer functions and feedback recurrences
Unit theme: Z transforms and discrete-time filters Level: Undergraduate engineering (idealized models) Learning objective: A recurrence yields H(z) only after treating initial states consistently; poles control the natural modes.
1. Concept and mathematical language
A recurrence yields H(z) only after treating initial states consistently; poles control the natural modes. The role of this principle in z transforms and discrete-time filters 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 y[n]=0.75 y[n−1]+0.25x[n], H(z)=0.25/(1−0.75z^(−1)) with DC gain 1. 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 direct-form implementation may overflow fixed-point arithmetic even when the mathematical model is stable. 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: Step the recursion with nonzero initial y and compare its transient to zero-state response. 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 sensor 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 unit-circle pole location. 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 direct-form implementation may overflow fixed-point arithmetic even when the mathematical model is stable. 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. 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 discrete transfer functions and feedback recurrences 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 sensor 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.
§6.3 FIR filter design and linear phase
Unit 6, Lesson 3: FIR filter design and linear phase
Unit theme: Z transforms and discrete-time filters Level: Undergraduate engineering (idealized models) Learning objective: Symmetric real FIR coefficients give a linear-phase frequency response except at phase discontinuities associated with zeros.
1. Concept and mathematical language
Symmetric real FIR coefficients give a linear-phase frequency response except at phase discontinuities associated with zeros. The role of this principle in z transforms and discrete-time filters 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: The 5-point moving average has h[k]=1/5 for k=0…4 and group delay 2 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 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: An averaging filter has spectral nulls and passes DC, but it is not an arbitrary sharp low-pass filter. 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: Graph impulse response and phase and verify the 2-sample delay of a low-frequency pulse. 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 sensor 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 unit-circle pole location. 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: An averaging filter has spectral nulls and passes DC, but it is not an arbitrary sharp low-pass filter. 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. 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 fir filter design and linear phase 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 sensor 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.
§6.4 IIR filters and numerical stability
Unit 6, Lesson 4: IIR filters and numerical stability
Unit theme: Z transforms and discrete-time filters Level: Undergraduate engineering (idealized models) Learning objective: IIR filters use feedback to realize poles; a causal rational stable filter has poles strictly inside the unit circle.
1. Concept and mathematical language
IIR filters use feedback to realize poles; a causal rational stable filter has poles strictly inside the unit circle. The role of this principle in z transforms and discrete-time filters 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: The one-pole filter y[n]=0.8 y[n−1]+0.2x[n] settles to 1 for a unit step. 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: Rounding coefficient 0.999 to 1 can change a decaying pole into a marginal pole. 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: Implement the same filter in floating point and low-precision fixed point and compare error. 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 sensor 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 unit-circle pole location. 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: Rounding coefficient 0.999 to 1 can change a decaying pole into a marginal pole. 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. 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 iir filters and numerical 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 sensor 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.
§6.5 Digital frequency response and group delay
Unit 6, Lesson 5: Digital frequency response and group delay
Unit theme: Z transforms and discrete-time filters Level: Undergraduate engineering (idealized models) Learning objective: The complex frequency response on z=e^(jω) reveals magnitude and phase, while group delay tracks phase slope.
1. Concept and mathematical language
The complex frequency response on z=e^(jω) reveals magnitude and phase, while group delay tracks phase slope. The role of this principle in z transforms and discrete-time filters 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 h=[1/2,1/2], H(e^(jω))=e^(−jω/2) cos(ω/2), with delay 0.5 sample away from zeros. 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: Group delay near frequency-response zeros is poorly conditioned and may not match intuitive pulse delay. 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 delay of a narrowband tone burst passing through the two-tap average. 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 sensor 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 unit-circle pole location. 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: Group delay near frequency-response zeros is poorly conditioned and may not match intuitive pulse delay. 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. 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 digital frequency response 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 embedded sensor 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.
Step-by-Step Signals and Systems Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Checkable numerical or symbolic example: (0.6)^n u[n] has X(z)=1/(1−0.6z^(−1)) and ROC |z|>0.6. 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.
Checkable numerical or symbolic example: (0.6)^n u[n] has X(z)=1/(1−0.6z^(−1)) and ROC |z|>0.6. 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.
Checkable numerical or symbolic example: For y[n]=0.75 y[n−1]+0.25x[n], H(z)=0.25/(1−0.75z^(−1)) with DC gain 1. 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.
Checkable numerical or symbolic example: For y[n]=0.75 y[n−1]+0.25x[n], H(z)=0.25/(1−0.75z^(−1)) with DC gain 1. 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.
Checkable numerical or symbolic example: The 5-point moving average has h[k]=1/5 for k=0…4 and group delay 2 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.
Checkable numerical or symbolic example: The 5-point moving average has h[k]=1/5 for k=0…4 and group delay 2 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.
Checkable numerical or symbolic example: The one-pole filter y[n]=0.8 y[n−1]+0.2x[n] settles to 1 for a unit step. 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.
Checkable numerical or symbolic example: The one-pole filter y[n]=0.8 y[n−1]+0.2x[n] settles to 1 for a unit step. 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.
Checkable numerical or symbolic example: For h=[1/2,1/2], H(e^(jω))=e^(−jω/2) cos(ω/2), with delay 0.5 sample away from zeros. 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.
Checkable numerical or symbolic example: For h=[1/2,1/2], H(e^(jω))=e^(−jω/2) cos(ω/2), with delay 0.5 sample away from zeros. 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.