Digital Baseband Transmission, Inter-Symbol Interference (ISI), and Nyquist Criterion

Unit 8: Coherent digital modulation and detection foundations

Learning objective: Introduce coherent ASK/BPSK/QPSK, constellation decisions, pulse shaping and timing recovery.

Physical perspective

A digital receiver decides which symbol most likely generated a noisy observation. A constellation plots a selected pair of sufficient-statistic coordinates, not the raw traveling radio wave. The distance between candidates and the noise covariance control detection reliability. Receiver timing, pulse shape and carrier phase errors can be as important as noise magnitude.

Extended engineering case: how to reason from requirements

A constellation is not a picture of electromagnetic radiation traveling through space. It is the receiver's collection of noisy coordinates after filtering, frequency translation and timing selection. The ideal points are deliberately separated so the detector can use geometry. In AWGN with equal noise variance on orthogonal axes, nearest-point detection is maximum likelihood for equal priors. A carrier-phase error rotates the point cloud; an I/Q gain imbalance stretches it, a DC offset shifts it and timing errors smear the clusters. Correcting only SNR cannot repair those systematic effects.

BPSK and QPSK offer a useful comparison. At equal energy per bit and coherent detection in ideal AWGN, Gray-coded QPSK does not inherently have a worse BER than BPSK. Its benefit is higher bits per symbol rather than a free reduction in needed Eb/N0. Increasing constellation order without increasing power places points closer together and ordinarily demands better SNR at a given error target. Pulse shaping further complicates the link: the transmitted waveform is a sum of overlapping pulses, and a matched receiver uses the expected pulse shape to recover clean sampling instants. Choosing a smaller raised-cosine rolloff reduces theoretical spectral occupancy but increases filter length and timing sensitivity.

An engineering review should show at least four displays: time waveform, frequency spectrum, constellation and eye diagram. Each reveals a different failure mode. A clean-looking constellation at correct timing says little about adjacent-channel emissions; a narrow spectrum does not guarantee accurate clock recovery. The eight-unit course culminates in a disciplined link assessment that states known assumptions, separates synthetic models from hardware results and reserves space for uncertain channel conditions.

Connections and study order

The next four lessons move from the simplest tractable model to a broader engineering treatment of this unit. Solve their worked cases separately first, then assemble the assumptions in a single coherent link design. For every step, distinguish data representing the actual message from a signal generated by the transceiver and from noise introduced by the environment or receiver. Use the model's limiting cases to decide whether parameters can be safely neglected. Finally, specify how any simulation could be validated through an independent analytic calculation and, if qualified laboratory access exists, through a supervised physical measurement.

Unit lessons

• [8.1 Binary ASK, OOK and energy detection](../lessons/U08-L01-binary-ask-ook-and-energy-detection.html) — model OOK symbols s0(t)=0 and s1(t)=A p(t) under the chosen mapping
• [8.2 BPSK and matched-filter decisions](../lessons/U08-L02-bpsk-and-matched-filter-decisions.html) — model BPSK symbols ±sqrt(Eb) in normalized signal space
• [8.3 QPSK, quadrature carrier mixing and symbol mapping](../lessons/U08-L03-qpsk-quadrature-carrier-mixing-and-symbol-mapping.html) — model QPSK symbols (±1±j)/sqrt(2); Es=2Eb for uncoded equal-probability mapping
• [8.4 Pulse shaping, Nyquist ISI criterion and eye diagrams](../lessons/U08-L04-pulse-shaping-nyquist-isi-criterion-and-eye-diagrams.html) — model ideal raised-cosine passband width ≈Rs(1+α) with rolloff α in [0,1]

Unit assessment and project

• [Self-assessment](../assessments/unit-08.html)
• [Integrated project](../projects/unit-08.html)
• [Interactive model](../simulations/model-08.html)

Lesson 8.1 Binary ASK, OOK and energy detection

Unit: 8 — Coherent digital modulation and detection foundations Core model: OOK symbols s0(t)=0 and s1(t)=A p(t) under the chosen mapping Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

On–off keying maps one bit to transmitted energy and the other to nominal silence. Coherent and noncoherent receivers use different decision variables and have different performance under phase uncertainty. An optical direct-detection system has additional shot-noise and intensity constraints beyond a simple bipolar AWGN voltage model. Threshold selection must reflect prior probabilities, noise variance and relative costs of false alarms versus misses. A receiver built only for equal symbol probabilities may be unsuitable for a rare-event telemetry link.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for binary ask, ook and energy detection. The governing relationship in this lesson is OOK symbols s0(t)=0 and s1(t)=A p(t) under the chosen mapping. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

If the zero-symbol decision statistic is distributed N(0,σ²) and one-symbol statistic is N(A,σ²) with equal priors and costs, the optimal midpoint threshold is A/2. Unequal priors shift the likelihood-ratio threshold.

Failure mode and model boundary

Electrical OOK transmit amplitude is not the same as optical field amplitude. Do not mix squared amplitude conventions when computing received optical power.

Investigation and design transfer

Plot false-alarm and missed-detection probabilities as threshold moves across two Gaussian distributions. Find a threshold for equal priors and discuss the effect of asymmetric costs.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: Electrical OOK transmit amplitude is not the same as optical field amplitude. Do not mix squared amplitude conventions when computing received optical power. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Plot false-alarm and missed-detection probabilities as threshold moves across two Gaussian distributions. Find a threshold for equal priors and discuss the effect of asymmetric costs. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from binary ask, ook and energy detection inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Lesson 8.2 BPSK and matched-filter decisions

Unit: 8 — Coherent digital modulation and detection foundations Core model: BPSK symbols ±sqrt(Eb) in normalized signal space Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

BPSK uses opposite phases of the same carrier or equivalent baseband pulse. With perfect coherent phase reference and symmetric AWGN, the maximum-likelihood decision is simply the sign of the matched-filter projection. Normalizing signal-space coordinates makes geometric distance explicit: the points lie 2√Eb apart. An unknown 180-degree phase ambiguity swaps every bit, which motivates pilots, differential encoding or a code structure capable of detecting inversion. The BER function assumes the receiver samples at the intended pulse time.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for bpsk and matched-filter decisions. The governing relationship in this lesson is BPSK symbols ±sqrt(Eb) in normalized signal space. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

At Eb=1 normalized unit and N0=0.5, Eb/N0=2 (3.01 dB). Ideal coherent BPSK bit-error rate is Q(2)≈0.02275. Raising energy by 3 dB improves performance nonlinearly, not by a fixed additive BER amount.

Failure mode and model boundary

A hard-decision detector discards reliability information that a soft-input decoder could exploit. The simple sign rule also fails with nonzero decision bias or correlated interference.

Investigation and design transfer

Simulate a scatter of noisy symbols along the real axis. Record how often samples fall on the wrong side of zero at several Eb/N0 levels.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: A hard-decision detector discards reliability information that a soft-input decoder could exploit. The simple sign rule also fails with nonzero decision bias or correlated interference. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Simulate a scatter of noisy symbols along the real axis. Record how often samples fall on the wrong side of zero at several Eb/N0 levels. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from bpsk and matched-filter decisions inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Lesson 8.3 QPSK, quadrature carrier mixing and symbol mapping

Unit: 8 — Coherent digital modulation and detection foundations Core model: QPSK symbols (±1±j)/sqrt(2); Es=2Eb for uncoded equal-probability mapping Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

QPSK conveys two coded bits per symbol using orthogonal I and Q components. With a Gray mapping, nearest-neighbor symbol errors usually change one bit, improving BER relative to arbitrary bit assignments at moderate noise. Under equal energy per bit, ideal coherent Gray-coded QPSK in AWGN has the same bit-error probability as BPSK, despite transmitting twice as many bits per symbol. Spectral occupation also depends on pulse shaping, not only the number of constellation points.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for qpsk, quadrature carrier mixing and symbol mapping. The governing relationship in this lesson is QPSK symbols (±1±j)/sqrt(2); Es=2Eb for uncoded equal-probability mapping. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

To transmit 100 kb/s uncoded with QPSK, nominal symbol rate is 50 ksym/s. With raised-cosine roll-off α=0.25, the idealized complex-baseband occupied bandwidth scale is Rs(1+α)=62.5 kHz; be explicit about passband or complex-baseband width convention.

Failure mode and model boundary

A rotated constellation can look like a different mapping. Clock recovery and carrier recovery must be evaluated before blaming random noise for systematic symbol errors.

Investigation and design transfer

Rotate four points by 22.5 degrees and add Gaussian noise. Compare a fixed-axis slicer with a receiver that estimates and removes the known rotation.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: A rotated constellation can look like a different mapping. Clock recovery and carrier recovery must be evaluated before blaming random noise for systematic symbol errors. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Rotate four points by 22.5 degrees and add Gaussian noise. Compare a fixed-axis slicer with a receiver that estimates and removes the known rotation. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from qpsk, quadrature carrier mixing and symbol mapping inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Lesson 8.4 Pulse shaping, Nyquist ISI criterion and eye diagrams

Unit: 8 — Coherent digital modulation and detection foundations Core model: ideal raised-cosine passband width ≈Rs(1+α) with rolloff α in [0,1] Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

A zero-ISI pulse has the property that its sampled values at symbol-spaced offsets vanish except at the origin, under ideal synchronization. Raised-cosine pulses are common examples, with α trading spectral compactness against sharper time-domain sidelobes. In many transmitters a root-raised-cosine filter is used, and a matching receiver filter produces the overall raised-cosine response. An eye diagram superimposes symbol intervals to reveal timing margin and interference. The eye is a diagnostic, not a direct absolute BER certificate.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for pulse shaping, nyquist isi criterion and eye diagrams. The governing relationship in this lesson is ideal raised-cosine passband width ≈Rs(1+α) with rolloff α in [0,1]. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

For Rs=200 ksym/s and α=0.35, the usual ideal raised-cosine null-to-null passband bandwidth estimate is 270 kHz. Actual emissions extend beyond finite filters and must be compared against spectral-mask measurements.

Failure mode and model boundary

Lower α does not always improve a real link: narrow pulse transitions are more sensitive to timing errors and demand longer filter impulse responses.

Investigation and design transfer

Sweep rolloff factor and timing error in a synthetic pulse-shaping display. Identify when the eye is widest at the receiver sampling instant.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: Lower α does not always improve a real link: narrow pulse transitions are more sensitive to timing errors and demand longer filter impulse responses. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Sweep rolloff factor and timing error in a synthetic pulse-shaping display. Identify when the eye is widest at the receiver sampling instant. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from pulse shaping, nyquist isi criterion and eye diagrams inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Unit 8 project — Coherent digital modulation and detection foundations: an auditable synthetic design study

Problem statement. Introduce coherent ASK/BPSK/QPSK, constellation decisions, pulse shaping and timing recovery. The deliverable is a self-contained engineering analysis, not a claim that a device was measured or certified.

Scenario

A constellation is not a picture of electromagnetic radiation traveling through space. It is the receiver's collection of noisy coordinates after filtering, frequency translation and timing selection. The ideal points are deliberately separated so the detector can use geometry. In AWGN with equal noise variance on orthogonal axes, nearest-point detection is maximum likelihood for equal priors. A carrier-phase error rotates the point cloud; an I/Q gain imbalance stretches it, a DC offset shifts it and timing errors smear the clusters. Correcting only SNR cannot repair those systematic effects.

Required work products

1. Requirements document. State the message type, sample or symbol rate, occupied-bandwidth convention, reference impedances or powers and a numerical reliability criterion. Differentiate functional requirements (what users need) from implementation choices (how blocks operate).

2. Equation notebook. For each of the unit's four governing relationships, derive the formula assumptions, provide an annotated worked calculation and evaluate a limiting case. Keep all intermediate calculations in linear units until a logarithmic representation is actually necessary.

3. Simulation experiment. Open the associated synthetic model. Take screenshots or download CSV for three input settings: nominal, stressed, and nearly ideal. Record why each setting was chosen. Compare at least two numerical points with a direct formula calculation independent of the browser code.

4. Failure analysis. Write one paragraph each on systematic bias, random variation and model mismatch. State the measurement instrument or independent experiment needed to distinguish the alternatives in a supervised hardware implementation.

5. Decision memo. Recommend a configuration for the stated application. Include a tradeoff table of bandwidth, power, complexity, delay, expected errors and uncertainty. Explicitly identify the two most consequential unverified assumptions.

Suggested numerical investigation

• Experiment 1, Binary ASK, OOK and energy detection: Plot false-alarm and missed-detection probabilities as threshold moves across two Gaussian distributions. Find a threshold for equal priors and discuss the effect of asymmetric costs. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 2, BPSK and matched-filter decisions: Simulate a scatter of noisy symbols along the real axis. Record how often samples fall on the wrong side of zero at several Eb/N0 levels. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 3, QPSK, quadrature carrier mixing and symbol mapping: Rotate four points by 22.5 degrees and add Gaussian noise. Compare a fixed-axis slicer with a receiver that estimates and removes the known rotation. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 4, Pulse shaping, Nyquist ISI criterion and eye diagrams: Sweep rolloff factor and timing error in a synthetic pulse-shaping display. Identify when the eye is widest at the receiver sampling instant. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.

Evaluation rubric (100 points)

• Model derivations and units: 25 points. A formula with unexplained symbols does not earn full marks.
• Independent arithmetic and numerical consistency: 20 points. Reproduce at least one graph point without reusing the graph's implementation.
• Simulation experiment and transparent provenance: 20 points. Identify every result as synthetic.
• Failure analysis and sensitivity: 20 points. Include meaningful nonideal tests, not only cosmetic parameter changes.
• Clear engineering recommendation: 15 points. Tie the recommendation to the defined objective and state where further evidence is required.

Safety: do not build or operate radio transmitters, powered antennas or high-voltage test apparatus as part of this offline project.

Unit 8 assessment — Coherent digital modulation and detection foundations

Instructions: show the reference planes, numerical steps and units. These are explanatory solution outlines, not a substitute for deriving the equations. Allocate 100 points: 20 each for Questions 1–4 and 20 for the integration problem.

Question 1: Binary ASK, OOK and energy detection (20 points)

Problem. If the zero-symbol decision statistic is distributed N(0,σ²) and one-symbol statistic is N(A,σ²) with equal priors and costs, the optimal midpoint threshold is A/2. Unequal priors shift the likelihood-ratio threshold. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with OOK symbols s0(t)=0 and s1(t)=A p(t) under the chosen mapping. Electrical OOK transmit amplitude is not the same as optical field amplitude. Do not mix squared amplitude conventions when computing received optical power. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 2: BPSK and matched-filter decisions (20 points)

Problem. At Eb=1 normalized unit and N0=0.5, Eb/N0=2 (3.01 dB). Ideal coherent BPSK bit-error rate is Q(2)≈0.02275. Raising energy by 3 dB improves performance nonlinearly, not by a fixed additive BER amount. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with BPSK symbols ±sqrt(Eb) in normalized signal space. A hard-decision detector discards reliability information that a soft-input decoder could exploit. The simple sign rule also fails with nonzero decision bias or correlated interference. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 3: QPSK, quadrature carrier mixing and symbol mapping (20 points)

Problem. To transmit 100 kb/s uncoded with QPSK, nominal symbol rate is 50 ksym/s. With raised-cosine roll-off α=0.25, the idealized complex-baseband occupied bandwidth scale is Rs(1+α)=62.5 kHz; be explicit about passband or complex-baseband width convention. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with QPSK symbols (±1±j)/sqrt(2); Es=2Eb for uncoded equal-probability mapping. A rotated constellation can look like a different mapping. Clock recovery and carrier recovery must be evaluated before blaming random noise for systematic symbol errors. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 4: Pulse shaping, Nyquist ISI criterion and eye diagrams (20 points)

Problem. For Rs=200 ksym/s and α=0.35, the usual ideal raised-cosine null-to-null passband bandwidth estimate is 270 kHz. Actual emissions extend beyond finite filters and must be compared against spectral-mask measurements. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with ideal raised-cosine passband width ≈Rs(1+α) with rolloff α in [0,1]. Lower α does not always improve a real link: narrow pulse transitions are more sensitive to timing errors and demand longer filter impulse responses. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 5: integrate the unit into a receiver chain (20 points)

Problem. Propose a complete signal chain for the unit's principal application. Show where source content, oscillator/reference information, channel distortion, random noise and decision logic enter. For each of the four lesson mechanisms, label an observable that a student could calculate or simulate. Choose one validation check independent of the simulation itself.

Solution outline. A strong answer includes a correctly ordered block diagram, a table of input/output units, a resource budget (power, spectrum or sampling), and at least one numerical limiting case. A student's independent calculation must use the same model conventions as the interactive graph. Discuss which missing impairment could overturn the proposed design. The answer should explain why a locally accurate mechanism does not guarantee successful end-to-end information delivery.

Marking: 5 points for the architecture, 5 for model consistency, 5 for independent validation, 5 for credible limitations.