Amplitude Modulation: DSB-FC, DSB-SC, SSB, and VSB Schemes

Unit 2: Signals, spectra, filters and equivalent baseband

Learning objective: Understand waveform descriptions in both time and frequency, and define a consistent one-sided or two-sided spectral convention.

Physical perspective

For every signal, distinguish instantaneous amplitude, average power and energy. Sinusoidal examples may be power signals with infinite total energy, while finite pulses are finite-energy waveforms. The Fourier transform is a coordinate change that exposes filtering and channel occupancy. A filter that changes waveform shape also changes interference and noise, so there is no universal bandwidth independent of the measurement and the desired error rate.

Extended engineering case: how to reason from requirements

The foundation of spectral reasoning is to ask what a display actually computes. A laboratory spectrum analyzer can report power in a resolution bandwidth, a digital FFT routine can report a complex DFT coefficient, and a communications simulation may report a continuous density estimate. Those three views are not interchangeable until window energy, reference impedance, sample rate and calibration are specified. A sinusoid between FFT bins leaks across nearby estimates even if the underlying signal contains only one frequency. Increasing FFT size by zero-padding interpolates the same finite-record spectrum; it does not create additional observed information or improve fundamental resolution without longer data. Window tapering lowers distant sidelobes while changing the main-lobe width and noise bandwidth.

The LTI viewpoint unifies several devices: a copper cable, an acoustic room, a front-end anti-alias filter and a matched receiver filter. Each can be described locally as a convolution if its parameters do not change during the measurement. Apply a known test pulse and compare the observed shape with the impulse-response prediction. Then apply two tones with different phase. If a high-frequency tone is attenuated, the receiver may boost it but cannot undo noise that has already been added after attenuation. If two delayed components cancel exactly at a frequency, inverting the channel is ill-conditioned. A robust design often accepts some residual distortion to avoid extreme noise enhancement. Complex baseband is most useful when translating this reasoning from a 100 MHz carrier to a slowly varying envelope, but the sign convention and units must remain consistent.

As a quality audit, write a three-column notebook with time-domain prediction, spectral prediction and simulation observation. A correct simulation should satisfy each independent description within numerical tolerance. For example, a time delay of 2 ms appears as a linearly increasing negative phase with frequency, but preserves magnitude for an ideal delay-only channel. An amplitude notch implies a cancellation mechanism and cannot be inferred from phase slope alone.

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

• [2.1 Fourier transforms and spectral lines](../lessons/U02-L01-fourier-transforms-and-spectral-lines.html) — model x(t)=A cos(2 pi f0 t) ↔ (A/2)[δ(f-f0)+δ(f+f0)]
• [2.2 Sampling, aliasing and reconstruction](../lessons/U02-L02-sampling-aliasing-and-reconstruction.html) — model fs>2B for a strictly bandlimited baseband signal with highest positive frequency B
• [2.3 Linear time-invariant channel and group delay](../lessons/U02-L03-linear-time-invariant-channel-and-group-delay.html) — model y(t)=h(t)*x(t); H(f)=∫h(t)e^(-j2πft)dt
• [2.4 Complex envelopes and I/Q representation](../lessons/U02-L04-complex-envelopes-and-i-q-representation.html) — model s_RF(t)=Re{u(t)e^(j2πfct)} with complex envelope u(t)

Unit assessment and project

• [Self-assessment](../assessments/unit-02.html)
• [Integrated project](../projects/unit-02.html)
• [Interactive model](../simulations/model-02.html)

Lesson 2.1 Fourier transforms and spectral lines

Unit: 2 — Signals, spectra, filters and equivalent baseband Core model: x(t)=A cos(2 pi f0 t) ↔ (A/2)[δ(f-f0)+δ(f+f0)] Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

A single real sinusoid has conjugate-symmetric positive and negative Fourier components in the mathematical two-sided spectrum. Multiplying by a cosine produces shifted spectra; this is the mechanism behind upconversion and downconversion. Engineers sometimes show only a positive-frequency spectrum for a real radio signal, but the axis convention must be declared. For finite observation windows, tones appear as broadened peaks because multiplication by a time window convolves their ideal line spectrum with the window transform.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for fourier transforms and spectral lines. The governing relationship in this lesson is x(t)=A cos(2 pi f0 t) ↔ (A/2)[δ(f-f0)+δ(f+f0)]. 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 a 1 kHz sine observed over 10 ms, exactly ten cycles fit in the window. With a coherent sampled record, an FFT bin may align with the tone. Moving to 1.05 kHz while keeping a 10 ms record creates spectral leakage across bins.

Failure mode and model boundary

FFT-bin amplitude is not a calibrated power spectral density unless normalization, sample rate and window equivalent noise bandwidth are handled.

Investigation and design transfer

Generate tones whose frequencies do and do not fit an integer number of cycles into a record. Compare rectangular and Hann window frequency responses.

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: FFT-bin amplitude is not a calibrated power spectral density unless normalization, sample rate and window equivalent noise bandwidth are handled. 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 tones whose frequencies do and do not fit an integer number of cycles into a record. Compare rectangular and Hann window frequency responses. 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 fourier transforms and spectral lines 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 2.2 Sampling, aliasing and reconstruction

Unit: 2 — Signals, spectra, filters and equivalent baseband Core model: fs>2B for a strictly bandlimited baseband signal with highest positive frequency B Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

Sampling repeats a continuous-time spectrum every sampling-frequency interval. If the replicas overlap, distinct analog inputs produce identical sample sequences and cannot be uniquely reconstructed. In practice, analog anti-alias filters limit the energy before the ADC, while finite aperture and clock jitter cause other impairments. The Nyquist statement is a sufficient sampling condition for a precisely bandlimited baseband waveform; bandpass sampling has a different set of feasibility constraints.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for sampling, aliasing and reconstruction. The governing relationship in this lesson is fs>2B for a strictly bandlimited baseband signal with highest positive frequency B. 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

A 7 kHz tone sampled at 10 ksample/s produces the same discrete-time samples as a suitably phased 3 kHz alias. The observed folded frequency is |7-10|=3 kHz. Merely interpolating samples cannot reveal which original analog tone was present.

Failure mode and model boundary

Do not confuse the Nyquist frequency fs/2 with the filter transition width. Real filters need a guard band, and high-amplitude out-of-band signals may overload the front end.

Investigation and design transfer

Design a sampling plan for 0–20 kHz audio-like content. Specify a sample rate, anti-alias transition region and amplitude range.

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: Do not confuse the Nyquist frequency fs/2 with the filter transition width. Real filters need a guard band, and high-amplitude out-of-band signals may overload the front end. 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: Design a sampling plan for 0–20 kHz audio-like content. Specify a sample rate, anti-alias transition region and amplitude range. 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 sampling, aliasing and reconstruction 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 2.3 Linear time-invariant channel and group delay

Unit: 2 — Signals, spectra, filters and equivalent baseband Core model: y(t)=h(t)*x(t); H(f)=∫h(t)e^(-j2πft)dt Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

An LTI channel can attenuate, delay and disperse the constituent frequencies of an input. Its impulse response contains the same input–output information as its transfer function, provided the model assumptions hold. Constant delay corresponds to a phase slope linear in frequency. If phase is nonlinear, spectral components travel with different group delays and the waveform can spread, increasing intersymbol interference even when the amplitude response seems flat. A causal system cannot depend on future inputs, and a physically stable impulse response should be compatible with the assumed energy or input–output stability definition.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for linear time-invariant channel and group delay. The governing relationship in this lesson is y(t)=h(t)*x(t); H(f)=∫h(t)e^(-j2πft)dt. 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 tones at 1 and 4 kHz pass through a channel with gains of 1 and 0.4 respectively. Even with zero added noise the output shape changes because high-frequency content is weaker. A subsequent equalizer can restore amplitude approximately while amplifying noise in the attenuated region.

Failure mode and model boundary

Zero-forcing equalization is not always optimal: a spectral null makes inversion arbitrarily noise-sensitive. A perfectly rectangular channel frequency response also implies a noncausal or infinite-duration idealization.

Investigation and design transfer

Draw magnitude and phase responses for a low-pass channel; propose a transmitted pulse and estimate how dispersion changes adjacent symbols.

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: Zero-forcing equalization is not always optimal: a spectral null makes inversion arbitrarily noise-sensitive. A perfectly rectangular channel frequency response also implies a noncausal or infinite-duration idealization. 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: Draw magnitude and phase responses for a low-pass channel; propose a transmitted pulse and estimate how dispersion changes adjacent symbols. 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 linear time-invariant channel and group delay 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 2.4 Complex envelopes and I/Q representation

Unit: 2 — Signals, spectra, filters and equivalent baseband Core model: s_RF(t)=Re{u(t)e^(j2πfct)} with complex envelope u(t) Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

A real passband radio signal can be represented through a complex envelope containing in-phase and quadrature parts. The complex representation is a mathematical convenience for analyzing amplitude and phase without numerically resolving every carrier cycle. If u=I+jQ, multiplying by a complex exponential rotates phase; time variation in phase creates frequency offsets. Practical I/Q modulators can have gain imbalance, quadrature error and DC offsets. Those impairments create image components, carrier leakage and biased constellation measurements.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for complex envelopes and i/q representation. The governing relationship in this lesson is s_RF(t)=Re{u(t)e^(j2πfct)} with complex envelope u(t). 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 u=0.6+j0.8, the magnitude is one and the envelope phase is atan2(0.8,0.6)≈53.13 degrees. A 90 degree rotation gives -0.8+j0.6. The physical passband waveform remains real after taking the real part of the modulated complex exponential.

Failure mode and model boundary

The sign convention for Q depends on whether an upconverter uses cos minus Q sin or cos plus Q sin. Never merge formulas with inconsistent conventions.

Investigation and design transfer

Sketch four QPSK baseband points, rotate them by a fixed carrier phase error and describe what a constellation display should show.

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: The sign convention for Q depends on whether an upconverter uses cos minus Q sin or cos plus Q sin. Never merge formulas with inconsistent conventions. 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: Sketch four QPSK baseband points, rotate them by a fixed carrier phase error and describe what a constellation display should show. 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 complex envelopes and i/q representation 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 2 project — Signals, spectra, filters and equivalent baseband: an auditable synthetic design study

Problem statement. Understand waveform descriptions in both time and frequency, and define a consistent one-sided or two-sided spectral convention. The deliverable is a self-contained engineering analysis, not a claim that a device was measured or certified.

Scenario

The foundation of spectral reasoning is to ask what a display actually computes. A laboratory spectrum analyzer can report power in a resolution bandwidth, a digital FFT routine can report a complex DFT coefficient, and a communications simulation may report a continuous density estimate. Those three views are not interchangeable until window energy, reference impedance, sample rate and calibration are specified. A sinusoid between FFT bins leaks across nearby estimates even if the underlying signal contains only one frequency. Increasing FFT size by zero-padding interpolates the same finite-record spectrum; it does not create additional observed information or improve fundamental resolution without longer data. Window tapering lowers distant sidelobes while changing the main-lobe width and noise bandwidth.

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, Fourier transforms and spectral lines: Generate tones whose frequencies do and do not fit an integer number of cycles into a record. Compare rectangular and Hann window frequency responses. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 2, Sampling, aliasing and reconstruction: Design a sampling plan for 0–20 kHz audio-like content. Specify a sample rate, anti-alias transition region and amplitude range. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 3, Linear time-invariant channel and group delay: Draw magnitude and phase responses for a low-pass channel; propose a transmitted pulse and estimate how dispersion changes adjacent symbols. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 4, Complex envelopes and I/Q representation: Sketch four QPSK baseband points, rotate them by a fixed carrier phase error and describe what a constellation display should show. 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 2 assessment — Signals, spectra, filters and equivalent baseband

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: Fourier transforms and spectral lines (20 points)

Problem. For a 1 kHz sine observed over 10 ms, exactly ten cycles fit in the window. With a coherent sampled record, an FFT bin may align with the tone. Moving to 1.05 kHz while keeping a 10 ms record creates spectral leakage across bins. 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 x(t)=A cos(2 pi f0 t) ↔ (A/2)[δ(f-f0)+δ(f+f0)]. FFT-bin amplitude is not a calibrated power spectral density unless normalization, sample rate and window equivalent noise bandwidth are handled. 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: Sampling, aliasing and reconstruction (20 points)

Problem. A 7 kHz tone sampled at 10 ksample/s produces the same discrete-time samples as a suitably phased 3 kHz alias. The observed folded frequency is |7-10|=3 kHz. Merely interpolating samples cannot reveal which original analog tone was present. 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 fs>2B for a strictly bandlimited baseband signal with highest positive frequency B. Do not confuse the Nyquist frequency fs/2 with the filter transition width. Real filters need a guard band, and high-amplitude out-of-band signals may overload the front end. 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: Linear time-invariant channel and group delay (20 points)

Problem. Two tones at 1 and 4 kHz pass through a channel with gains of 1 and 0.4 respectively. Even with zero added noise the output shape changes because high-frequency content is weaker. A subsequent equalizer can restore amplitude approximately while amplifying noise in the attenuated region. 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 y(t)=h(t)*x(t); H(f)=∫h(t)e^(-j2πft)dt. Zero-forcing equalization is not always optimal: a spectral null makes inversion arbitrarily noise-sensitive. A perfectly rectangular channel frequency response also implies a noncausal or infinite-duration idealization. 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: Complex envelopes and I/Q representation (20 points)

Problem. If u=0.6+j0.8, the magnitude is one and the envelope phase is atan2(0.8,0.6)≈53.13 degrees. A 90 degree rotation gives -0.8+j0.6. The physical passband waveform remains real after taking the real part of the modulated complex exponential. 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 s_RF(t)=Re{u(t)e^(j2πfct)} with complex envelope u(t). The sign convention for Q depends on whether an upconverter uses cos minus Q sin or cos plus Q sin. Never merge formulas with inconsistent conventions. 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.