Modulation Quality & Waveform Tradeoffs

One EVM, two failures

If two transmitters both report about 5% RMS EVM, should they receive the same fix?

No. The two deterministic teaching cases below are intentionally close in RMS EVM. One spends the error randomly across directions; the other rotates every symbol by nearly the same angle. The scalar result is similar, while the mechanisms, conditional views, and responsible subsystem are different.

Evidence Both panels are generated by impairment-trade/1.0 from the same PRBS-9 source, QPSK mapping, RRC pair, selected-symbol rule, and plot scale. Only the stated impairment/reference settings differ.

Define EVM and reference processing

What exactly is the percentage measuring?

At selected matched-filter decision times, let the known reference symbol be s[k] and the observed symbol after the declared processing be ŝ[k]. The error vector is e[k]=ŝ[k]−s[k]. This lesson uses population RMS EVM normalized by RMS reference energy:

Under an ideal unit-energy, symbol-plane AWGN convention, EVM≈1/√SNR. At 25 dB this predicts 5.6234%. The live finite-burst result need not equal it exactly because the model includes finite pulse filters, a fixed noise realization, and—at the default state—a lightly driven memoryless PA.

Evidence The definition, normalization warning, per-symbol basis, and need to compare like references follow Rohde & Schwarz's EVM guidance; the exact population and processing contract is local and versioned.

EVM summary and conditional views

Which view turns a failed score into a testable hypothesis?

Start with RMS, peak, and percentile EVM, then condition the same symbol errors on time, intended amplitude, frequency, and decision timing. A constellation compresses time; a spectrum discards symbol correspondence; an eye overlays sequences. Diagnosis comes from agreement across views, not from recognizing one attractive shape.

Evidence R&S documents EVM versus time/symbol, frequency, power, and joint power/frequency as complementary diagnostic views. This lesson adds eye, CCDF, and counterfactual linkage to the frozen Path 02 waveform.

Noise and interference signatures

When does a cloud stop behaving like noise?

Circular complex AWGN spreads repeated references in many directions with no stable symbol-index pattern. Its EVM follows the declared SNR scale approximately and improves monotonically in the ideal branch. A coherent in-band interferer instead produces a structured or periodic error vector; viewed against symbol index or error frequency, its repeatability can remain visible even when the power spectrum barely separates it.

• Noise hypothesis: diffuse direction, weak time correlation, EVM falls predictably as SNR increases.
• Spur/interference hypothesis: periodic error or a localized EVM-frequency feature, often tied to clock, LO, supply, or another carrier.
• Observation-path check: vary analyzer reference level, averaging, input attenuation, and instrument floor before assigning the error to the DUT.

Evidence R&S identifies noise as constellation spreading and notes that EVM versus frequency can reveal narrowband spurious signals that are difficult to see on a conventional magnitude trace.

Frequency, phase, I/Q, and DC signatures

Which errors move every symbol together, and which create an image?

Common phase multiplies the envelope by one fixed phasor. Constant frequency offset adds phase linearly with normalized time. I/Q gain or quadrature mismatch is different: it creates both desired and conjugated-envelope terms. DC offset adds a fixed vector and, in a zero-IF transmitter, corresponds to energy at the LO centre.

The pinned +1 dB/+3° case gives 23.988 dB analytic image rejection. Common phase/gain correction cannot generally remove the b·u* term; calibration must address the I/Q transform itself. A rotating constellation points to frequency offset; a rigid rotation points to phase; a translated centre points to DC.

Evidence The I/Q desired/image equation is derived from the frozen complex-envelope convention. Analog Devices documents gain imbalance, quadrature cross-coupling, and DC offset as distinct analytical I/Q modulator/demodulator impairments.

Compression and regrowth

Why can EVM and adjacent energy worsen together as backoff is spent?

The lesson uses a synthetic memoryless Rapp model. Its AM-AM law approaches unity gain for small envelopes and asymptotically limits magnitude near A_sat. Input backoff is referenced to mean D3 complex-envelope power. A smooth deterministic AM-PM law is parameterized by phase at |u|=A_sat.

Compression creates error correlated with intended magnitude: outer 16QAM points pull inward first, while phase conversion bends their direction. The nonlinearity also mixes spectral components, raising energy in the visible adjacent teaching bands. That joint movement is stronger evidence than either RMS EVM or a spectrum alone.

Evidence Rapp's 1991 HPA work connects nonlinear amplification, backoff, and spectral degradation. Analog Devices distinguishes static nonlinear behavior from short- and long-term memory effects; this lab intentionally models only the former.

Timing and ISI fingerprints

What does the constellation hide that the eye retains?

The fixed QPSK baseline uses α=0.35, span 8 symbols, 8 samples/symbol, and separate energy-normalized 65-tap RRC transmit and receive filters. Each filter delays 32 samples or four symbols; the pair delays 0.8 ms at 80 ksample/s. With ideal sampling and the impairment branches bypassed, finite truncation leaves the checked residual below.

A fractional timing error moves all decisions away from the eye opening. A delayed echo adds sequence-dependent contributions, so repeated symbols split into correlated clusters. The lab evaluates both echo delay and decision timing with a 16-point Hann-windowed sinc interpolator, not a nearest-sample shortcut. The eye is drawn at R3 before reference correction.

• 01Symbols
• 02RRC TX
• 03I/Q + DC
• 04Carrier errors
• 05Rapp PA
• 06Two-path channel
• 07AWGN
• 08RRC RX
• 09Reference processing
• 10Decision

Evidence Tap count, delay, and residual are recomputed by the production function from the frozen Path 02.6 convention. The echo is a synthetic two-path teaching channel, not a fitted propagation measurement.

PAPR and CCDF population

How much confidence belongs in one reported peak?

PAPR is the largest sampled instantaneous power divided by average power over a declared population. A CCDF reports how often power exceeds the average by a threshold. Both depend on pulse shape, mapping, oversampling, record length, seed/data, transient handling, filtering, and reference plane. Change the population and the tail can change without any transmitter hardware change.

• Use CCDF to connect waveform peaks to headroom and backoff probability, not to promise a universal crest factor.
• Keep the sample grid and reconstruction assumption visible; intersample peaks can exceed sampled peaks.
• Compare waveforms with identical population rules, seed policy, pulse filters, and reference plane.

Evidence The live CCDF uses one fixed 256-symbol PRBS-9 burst, explicit transient exclusions, D3 reference plane, and the selected sample grid. It is a repeatable teaching population, not a long-run tail guarantee.

Counterfactual diagnosis

Which mechanism must disappear before the evidence improves?

Use the same symbols and noise direction, remove one enabled branch, and rerun the complete pipeline. A mechanism gains credibility when its removal restores the expected conditional views—not merely when total EVM drops. Then challenge the hypothesis with a second change: frequency reference, backoff, timing, equalization, or observation setup.

Accessible fixed-case fallback

Each case is generated on the server by the same production function as the live lab.

Evidence All live and fallback numerics use impairment-trade/1.0, seed 0x02A5C0DE, and identical reference/population rules. Plots and tables are simulation outputs, not decorative traces or measurements.

Final waveform decision record

What evidence makes a waveform recommendation reviewable?

Finish the path by turning the 20.0 kbit/s condition-monitoring case into a bounded decision. The baseline is unit-energy QPSK at 10 ksymbol/s with α=0.35 RRC pulse shaping, span 8, 8 samples/symbol, and the frozen I/Q sign convention. The record must show when 16QAM's lower symbol rate is worth its tighter error margin and amplitude sensitivity.

Decision

Choose a mapping and operating point; do not write “lowest EVM wins” without spectral, peak, and implementation constraints.

Reference contract

Freeze I/Q sign, symbol normalization, planes D3/R0/R3, rate ledger, filtering, synchronization, and equalization before comparing.

Evidence

Use the same configuration across constellation/error, eye, time/power, spectrum, CCDF, and leave-one-out views.

Boundaries

Label derived and simulated results; reserve measured claims for calibrated hardware evidence with uncertainty.

Mitigation

Connect each proposed action to a signature it should restore, then name the follow-up test that could falsify it.

Handoff

Pass PA architecture and hardware linearity to Path 04; pass propagation, receiver, measurement, and compliance questions to their owning paths.

90-minute capstone · waveform decision record v7

Submit one concise engineering record with these nine deliverables:

1. A one-paragraph decision statement naming the link requirement, candidate mapping, and explicit pass/fail criteria.
2. The full I/Q and real-RF sign convention: u=I+jQ and sRF=I cos(2πfct)−Q sin(2πfct).
3. An uncoded 20.0 kbit/s rate ledger covering bits/symbol, symbol rate, samples/symbol, and complex sample rate.
4. The RRC definition, α, span, tap count, filter delays, transient exclusions, and finite residual-ISI fixture.
5. An EVM contract naming reference, plane, processing, normalization, statistic, and population.
6. Linked constellation/error, eye, time/power, normalized-spectrum, and CCDF evidence from the same configuration.
7. A bandwidth and adjacent-band statement with drawn integration edges and an explicit non-ACLR boundary.
8. A counterfactual diagnosis table plus a signature-to-mitigation argument that does not add leave-one-out deltas.
9. A provenance and limitations note separating derived, simulated, and measured claims and handing PA hardware depth to Path 04.

Source ledger and scope

Primary measurement guidance

Rohde & Schwarz, Understanding error vector magnitude and its Version 01.00 white paper: EVM definition, normalization, conditional views, measurement setup, and constellation signatures.

Normative terminology boundary

ITU-R SM.328-12 (09/2025), in force: spectra and bandwidth-of-emissions terminology. The lesson does not claim its local adjacent-band integration is a standards measurement.

PA model provenance

C. Rapp, 1991, HPA nonlinearity study: nonlinearity, backoff, and spectral degradation context. The repository metadata notes that full text is not hosted there; the implemented equation is explicitly frozen in the local model contract.

System-model boundary

Analog Devices, Modeling and Simulation of RF and Microwave Systems: I/Q gain/phase/DC models and the distinction between static PA nonlinearity and memory effects.

Local curriculum evidence

Path 02 portfolio conventions and specs P02-S1 through P02-S7 define the PRBS-9, transform sign, I/Q convention, QPSK normalization, RRC parameters, sample plan, reference planes, fixture values, and artifact handoff.