Sampling Theorem, Pulse Amplitude Modulation (PAM), and Time-Division Multiplexing

Unit 6: Random processes, noise and communication channels

Learning objective: Model random voltage, PSD and receiver noise with consistent probabilistic and bandwidth conventions.

Physical perspective

Real signals may be stochastic in source content or channel disturbance. Expectation describes an ensemble and does not guarantee the outcome of a short observation. A wide-sense stationary model uses constant mean and lag-dependent autocorrelation; physical channels often violate that assumption over long intervals. The receiver must translate an SNR hypothesis into a specified detector performance under a stated noise model.

Extended engineering case: how to reason from requirements

Noise calculations are full of hidden reference choices. A density expressed in watts per hertz becomes a power only after integration over the filter response. Converting to decibels before integrating is generally wrong: integrate linear power first. A two-sided PSD over negative and positive frequencies differs numerically from a one-sided PSD defined only for nonnegative frequencies, but both predict the same total noise power when used consistently. A noise figure convention adds further assumptions about source temperature and a matched input. A simulation may omit amplifier gain compression, quantization and adjacent-channel interference; note these omissions in every result.

Matched filtering is a decision-oriented use of signal knowledge. If a known pulse occupies a finite interval, the receiver integrates its energy with a filter matched to its shape. This raises the sample SNR compared with observing one instantaneous noisy amplitude. When the interference spectrum is colored, weighting all frequencies equally is no longer optimal. A whitening filter followed by a matched filter can improve detection, but increases reliance on a correct noise model. Channel estimation uncertainty also matters: a receiver using the wrong pulse shape may suffer mismatch loss.

An apparently excellent simulated bit-error result deserves a statistical check. If only one thousand bits are sent and zero errors occur, it is not legitimate to report zero underlying error probability. The result supplies an upper statistical bound whose size depends on confidence. Approximately three expected-error opportunities for a zero-error trial gives the familiar rough 95% upper bound near 3/N for independent rare errors. Fading and burst interference violate independence, so interleaving or repeated independent trials may be needed to interpret data. Report trial count, random seed, confidence interval and Eb/N0 normalization alongside the headline BER.

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

• [6.1 Probability, autocorrelation and power spectra](../lessons/U06-L01-probability-autocorrelation-and-power-spectra.html) — model Rx(τ)=E[x(t)x*(t-τ)]; Sx(f)=F{Rx(τ)} for WSS processes
• [6.2 Thermal noise, matched filtering and equivalent noise bandwidth](../lessons/U06-L02-thermal-noise-matched-filtering-and-equivalent-noise-bandwidth.html) — model Nout=∫S_n(f)|H(f)|²df using compatible two-sided conventions
• [6.3 AWGN, Eb/N0 and probability of error](../lessons/U06-L03-awgn-eb-n0-and-probability-of-error.html) — model for coherent BPSK in AWGN, Pb=Q(sqrt(2Eb/N0))
• [6.4 Multipath, fading and channel coherence](../lessons/U06-L04-multipath-fading-and-channel-coherence.html) — model r(t)=∑k ak x(t-τk)+n(t) for a tapped delay-line channel

Unit assessment and project

• [Self-assessment](../assessments/unit-06.html)
• [Integrated project](../projects/unit-06.html)
• [Interactive model](../simulations/model-06.html)

Lesson 6.1 Probability, autocorrelation and power spectra

Unit: 6 — Random processes, noise and communication channels Core model: Rx(τ)=E[x(t)x*(t-τ)]; Sx(f)=F{Rx(τ)} for WSS processes Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

A random waveform is described by many possible outcomes, not by one deterministic plot. Autocorrelation describes similarity as a function of delay, while power spectral density describes the distribution of mean power over frequency. The Wiener–Khinchin relationship connects these descriptions under appropriate stationarity assumptions. White noise has ideal flat two-sided PSD and infinite total power if extended across all frequencies, which is a model abstraction rather than a realizable source. Noise observed after a filter has shaped PSD and potentially correlated samples.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for probability, autocorrelation and power spectra. The governing relationship in this lesson is Rx(τ)=E[x(t)x*(t-τ)]; Sx(f)=F{Rx(τ)} for WSS processes. 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 independent zero-mean samples have variance 0.04 V², their lag-zero correlation equals 0.04 V² and off-zero discrete-time correlation approaches zero in the ensemble model. Individual finite records still show accidental nonzero estimates.

Failure mode and model boundary

Never interpret a periodogram estimate as the exact underlying PSD. Window choice, averaging, record length and one-sided/two-sided scaling affect numerical values.

Investigation and design transfer

Simulate multiple short Gaussian-noise records; compare empirical variance and autocorrelation with theoretical targets, then average more trials.

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: Never interpret a periodogram estimate as the exact underlying PSD. Window choice, averaging, record length and one-sided/two-sided scaling affect numerical values. 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 multiple short Gaussian-noise records; compare empirical variance and autocorrelation with theoretical targets, then average more trials. 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 probability, autocorrelation and power spectra 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 6.2 Thermal noise, matched filtering and equivalent noise bandwidth

Unit: 6 — Random processes, noise and communication channels Core model: Nout=∫S_n(f)|H(f)|²df using compatible two-sided conventions Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

When noise passes through an LTI filter, its output variance is the integral of input PSD weighted by squared filter magnitude. Equivalent noise bandwidth compares the actual filter’s integrated noise passage to a rectangular reference filter with the same peak gain. For a fixed signal pulse in additive white Gaussian noise, a matched filter maximizes output SNR at the nominated sampling instant, assuming the noise is white and timing and waveform are known. Colored noise calls for whitening or an equivalent generalized detector.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for thermal noise, matched filtering and equivalent noise bandwidth. The governing relationship in this lesson is Nout=∫S_n(f)|H(f)|²df using compatible two-sided conventions. 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

An ideal one-sided thermal noise density of -174 dBm/Hz integrated over 1 MHz gives approximately -114 dBm before receiver noise figure. A 6 dB noise figure makes input-referred noise about -108 dBm under matched reference conditions.

Failure mode and model boundary

Filter bandwidth should be defined at the point in the system where noise and signal are being compared. A matched filter does not imply infinitely sharp stopband rejection or zero timing sensitivity.

Investigation and design transfer

Change pulse duration while keeping pulse energy fixed; compare peak matched-filter SNR with raw instantaneous sampling. Explain why matched integration is valuable.

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: Filter bandwidth should be defined at the point in the system where noise and signal are being compared. A matched filter does not imply infinitely sharp stopband rejection or zero timing sensitivity. 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: Change pulse duration while keeping pulse energy fixed; compare peak matched-filter SNR with raw instantaneous sampling. Explain why matched integration is valuable. 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 thermal noise, matched filtering and equivalent noise bandwidth 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 6.3 AWGN, Eb/N0 and probability of error

Unit: 6 — Random processes, noise and communication channels Core model: for coherent BPSK in AWGN, Pb=Q(sqrt(2Eb/N0)) Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

AWGN adds a random Gaussian voltage vector with a flat spectral model within the bandwidth of interest. It provides a tractable baseline for detector design and approximate link budgeting. Eb/N0 normalizes received energy per information bit against noise spectral density; it is not the same as broadband SNR unless the relevant bit rate and receiver bandwidth are specified. Uncoded coherent BPSK in AWGN has a simple Q-function bit-error expression with perfect synchronization. Applying that formula to fading, coding, differential detection or phase ambiguity without adjustments is incorrect.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for awgn, eb/n0 and probability of error. The governing relationship in this lesson is for coherent BPSK in AWGN, Pb=Q(sqrt(2Eb/N0)). 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/N0=6 dB, linear ratio is 3.981. The ideal BPSK bit-error probability is Q(√7.962)=Q(2.822)≈0.00239. This does not say a short burst will contain exactly 0.239% errors.

Failure mode and model boundary

Comparing BER curves requires the same coded versus uncoded bit-energy convention. Receiver estimates can also be biased if synchronization errors are ignored.

Investigation and design transfer

Generate 10,000 independent BPSK symbols with Gaussian noise at two Eb/N0 levels and compare measured BER with the ideal expression and binomial variability.

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: Comparing BER curves requires the same coded versus uncoded bit-energy convention. Receiver estimates can also be biased if synchronization errors are ignored. 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: Generate 10,000 independent BPSK symbols with Gaussian noise at two Eb/N0 levels and compare measured BER with the ideal expression and binomial variability. 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 awgn, eb/n0 and probability of error 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 6.4 Multipath, fading and channel coherence

Unit: 6 — Random processes, noise and communication channels Core model: r(t)=∑k ak x(t-τk)+n(t) for a tapped delay-line channel Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

A physical receiver often collects delayed copies of the same transmitted waveform. Their complex phasors may reinforce or cancel at different frequencies, producing frequency-selective fading. Channel coherence bandwidth is related to delay spread, while coherence time is related to Doppler spread; both relationships require a precise definition and statistical channel model. A flat-fading approximation is plausible only if the occupied signal bandwidth is small relative to channel variation in frequency. Equalization, diversity and adaptive coding address distinct risks.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for multipath, fading and channel coherence. The governing relationship in this lesson is r(t)=∑k ak x(t-τk)+n(t) for a tapped delay-line channel. 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

Two equal-amplitude paths with a 1 microsecond relative delay interfere constructively at some frequencies and destructively where phase difference equals odd multiples of π. The approximate notch spacing between equivalent destructive conditions is 1/τ=1 MHz.

Failure mode and model boundary

A deterministic two-tap example is not a Rayleigh or Rician statistical fading model unless a distribution for the path coefficients has been declared.

Investigation and design transfer

For 100 ns and 1 μs path separation, sketch magnitude versus frequency and predict which case creates more rapid frequency ripples.

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 deterministic two-tap example is not a Rayleigh or Rician statistical fading model unless a distribution for the path coefficients has been declared. 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: For 100 ns and 1 μs path separation, sketch magnitude versus frequency and predict which case creates more rapid frequency ripples. 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 multipath, fading and channel coherence 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 6 project — Random processes, noise and communication channels: an auditable synthetic design study

Problem statement. Model random voltage, PSD and receiver noise with consistent probabilistic and bandwidth conventions. The deliverable is a self-contained engineering analysis, not a claim that a device was measured or certified.

Scenario

Noise calculations are full of hidden reference choices. A density expressed in watts per hertz becomes a power only after integration over the filter response. Converting to decibels before integrating is generally wrong: integrate linear power first. A two-sided PSD over negative and positive frequencies differs numerically from a one-sided PSD defined only for nonnegative frequencies, but both predict the same total noise power when used consistently. A noise figure convention adds further assumptions about source temperature and a matched input. A simulation may omit amplifier gain compression, quantization and adjacent-channel interference; note these omissions in every result.

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, Probability, autocorrelation and power spectra: Simulate multiple short Gaussian-noise records; compare empirical variance and autocorrelation with theoretical targets, then average more trials. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 2, Thermal noise, matched filtering and equivalent noise bandwidth: Change pulse duration while keeping pulse energy fixed; compare peak matched-filter SNR with raw instantaneous sampling. Explain why matched integration is valuable. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 3, AWGN, Eb/N0 and probability of error: Generate 10,000 independent BPSK symbols with Gaussian noise at two Eb/N0 levels and compare measured BER with the ideal expression and binomial variability. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 4, Multipath, fading and channel coherence: For 100 ns and 1 μs path separation, sketch magnitude versus frequency and predict which case creates more rapid frequency ripples. 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 6 assessment — Random processes, noise and communication channels

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: Probability, autocorrelation and power spectra (20 points)

Problem. If independent zero-mean samples have variance 0.04 V², their lag-zero correlation equals 0.04 V² and off-zero discrete-time correlation approaches zero in the ensemble model. Individual finite records still show accidental nonzero estimates. 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 Rx(τ)=E[x(t)x*(t-τ)]; Sx(f)=F{Rx(τ)} for WSS processes. Never interpret a periodogram estimate as the exact underlying PSD. Window choice, averaging, record length and one-sided/two-sided scaling affect numerical values. 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: Thermal noise, matched filtering and equivalent noise bandwidth (20 points)

Problem. An ideal one-sided thermal noise density of -174 dBm/Hz integrated over 1 MHz gives approximately -114 dBm before receiver noise figure. A 6 dB noise figure makes input-referred noise about -108 dBm under matched reference conditions. 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 Nout=∫S_n(f)|H(f)|²df using compatible two-sided conventions. Filter bandwidth should be defined at the point in the system where noise and signal are being compared. A matched filter does not imply infinitely sharp stopband rejection or zero timing sensitivity. 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: AWGN, Eb/N0 and probability of error (20 points)

Problem. At Eb/N0=6 dB, linear ratio is 3.981. The ideal BPSK bit-error probability is Q(√7.962)=Q(2.822)≈0.00239. This does not say a short burst will contain exactly 0.239% errors. 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 for coherent BPSK in AWGN, Pb=Q(sqrt(2Eb/N0)). Comparing BER curves requires the same coded versus uncoded bit-energy convention. Receiver estimates can also be biased if synchronization errors are ignored. 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: Multipath, fading and channel coherence (20 points)

Problem. Two equal-amplitude paths with a 1 microsecond relative delay interfere constructively at some frequencies and destructively where phase difference equals odd multiples of π. The approximate notch spacing between equivalent destructive conditions is 1/τ=1 MHz. 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 r(t)=∑k ak x(t-τk)+n(t) for a tapped delay-line channel. A deterministic two-tap example is not a Rayleigh or Rician statistical fading model unless a distribution for the path coefficients has been declared. 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.