Wireless and RF Communication Systems Exercises

These exercises practise wireless and RF communication engineering as field-oriented design work. The calculations cover EIRP, antenna and feeder losses, path and Fresnel clearance, receiver sensitivity, C/N0, Eb/N0, bandwidth, Doppler shift, OFDM guard interval, rain fade, fading outage, adaptive modulation, intermodulation, reciprocal mixing, antenna pointing, polarization mismatch, measurement uncertainty, and release evidence.

Assume simplified screening models unless an exercise states otherwise. Real wireless systems also require regulatory approval, antenna-pattern data, terrain and clutter review, calibrated measurements, spectrum occupancy evidence, equipment configuration records, cybersecurity and network policy review, and site-specific acceptance criteria.

Release Evidence Notes

Use these exercises as screening evidence for wireless/RF field acceptance, not as proof that a link or coverage zone is ready because one margin calculation passes. A credible release should connect each result to the installed RF boundary, spectrum condition, antenna geometry, receiver state, traffic requirement, environmental exposure and operational fallback.

The minimum evidence set is:

• installation evidence for transmit power reference, EIRP limit, feeder and connector loss, antenna model, pattern, height, azimuth, downtilt, polarization, Fresnel clearance, obstruction survey and grounding or enclosure state;
• receiver evidence for sensitivity basis, bandwidth, noise figure, G/T, implementation loss, front-end compression, adjacent-channel rejection, intermodulation risk, reciprocal mixing and gain-state behavior;
• channel evidence for spectrum occupancy, co-channel and adjacent-channel interferers, SINR, fading statistics, rain region, delay spread, Doppler range, cyclic-prefix margin, multipath state and diversity independence;
• service evidence for traffic load, MCS policy, retry behavior, latency, packet loss, outage definition, availability target, degraded-mode capacity, alarm thresholds and rollback or channel-change action;
• validation evidence for calibrated field measurements, capture duration, measurement uncertainty, firmware configuration, antenna alignment record, before/after survey, regulatory record, commissioning log and release authority.

Treat a numerical pass as provisional when it depends on conducted power instead of EIRP, clear-channel sensitivity, visual line of sight without Fresnel clearance, average SINR, a quiet one-time spectrum scan, independent diversity that has not been proven or outage minutes counted without service capacity. Wireless release decisions should support operation only when the evidence covers the same installed site, traffic state, weather basis and interference environment.

How to Use These Exercises

For each problem:

1. define the RF boundary where power, loss, bandwidth, or delay is measured;
2. keep absolute units such as dBm separate from ratios such as dB;
3. state whether the decision is coverage, compliance, capacity, coexistence, availability, or release;
4. separate thermal-noise margin from fade, interference, installation, and uncertainty margins;
5. identify which field measurement would confirm the calculation.

The common mistake is treating a wireless link as a single received-power number. A credible RF review connects power, antennas, propagation, receiver noise, linearity, waveform timing, interference, traffic, and validation evidence.

Engineering Boundary Notes

Wireless evidence must state the RF reference plane. Conducted transmitter power, antenna-port power, EIRP, received antenna power, receiver input power, demodulator SINR and service goodput are different boundaries. Losses, gains, implementation reserves and regulatory limits must be assigned to the correct one.

The radio channel is not a fixed loss term. Fading, rain, obstruction, Fresnel clearance, polarization, antenna pointing, clutter, Doppler, delay spread and co-channel occupancy can change the release decision even when the nominal link budget passes. Exercises that produce one margin should be read as screening checks unless the field condition is measured or bounded.

Receiver linearity is a release boundary, not an afterthought. Blockers, reciprocal mixing, adjacent-channel leakage, intermodulation, desensitization and spectrum occupancy can consume margin without lowering the wanted-signal budget in an obvious way. A field release should therefore pair link-budget evidence with spectrum and receiver-state evidence.

Common Release Mistakes

• using conducted power as if it were EIRP after feeder loss, antenna gain and regulatory limits;
• approving a path from visual line of sight while ignoring Fresnel clearance and obstruction growth;
• treating receiver sensitivity as valid under blockers, interference, compression or gain-state changes;
• accepting average SINR while fading, rain or traffic load controls outage probability;
• assuming diversity independence without spacing, polarization, route or fading-correlation evidence;
• validating spectrum occupancy from a short quiet scan instead of the operating traffic window;
• releasing a power-control change without checking coexistence, EIRP compliance and fallback capacity.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Link-budget closure	1, 2, 3, 17, 18	EIRP, path loss, Fresnel clearance, sensitivity, bandwidth penalty, Eb/N0 data-rate margin and transmit-power clamp	Decide whether the path is legal, clear and robust enough for the selected mode.
Time-varying channel behavior	4, 5, 6, 15	Doppler, coherence time, cyclic prefix, delay spread, rain fade and outage probability	Select receiver tracking, guard interval, diversity and adaptive modulation behavior.
Interference and linearity	7, 9, 12, 13, 16, 18	Intermodulation, co-channel SINR, adjacent-channel leakage, reciprocal mixing, spectrum occupancy, power control and uncertainty	Decide whether filtering, channel change, coordination, oscillator quality, rejection or power reduction is required.
Antenna and receive-system design	10, 11, 14	Downtilt footprint, half-power coverage, pointing loss, polarization mismatch, system noise temperature and G/T	Check whether the installed RF geometry and receive chain match the service area.
Field acceptance and release	8, 12, 14, 15	Guarded measured margin, installation-loss evidence, occupancy evidence, outage model and release criteria	Accept, reject or conditionally release the link based on reproducible evidence.

Validation Package Checklist

Before treating a wireless calculation as release evidence, collect:

1. RF boundary, transmit power reference, feeder losses, antenna model and EIRP limit;
2. antenna height, azimuth, downtilt, polarization, pointing record and Fresnel clearance;
3. receiver bandwidth, sensitivity basis, noise figure, gain state and linearity limits;
4. spectrum occupancy, blockers, adjacent-channel leakage and interference coordination;
5. fading, rain, Doppler, delay spread, diversity and outage assumptions;
6. traffic load, MCS policy, retry behavior, latency and degraded-mode capacity;
7. calibrated field measurements, uncertainty allowance and capture duration;
8. release decision, channel change, power clamp, restriction or retest requirement.

Exercise 1: EIRP, Link Budget, and Margin

A point-to-point industrial wireless link operates at 2.4\ \text{GHz} over 3.2\ \text{km}. The transmitter output is 23\ \text{dBm}, transmitter feeder loss is 1.8\ \text{dB}, transmitter antenna gain is 14\ \text{dBi}, receiver antenna gain is 10\ \text{dBi}, receiver feeder loss is 1.2\ \text{dB}, and miscellaneous implementation loss is 3.0\ \text{dB}.

The applicable EIRP limit is 36\ \text{dBm}. Receiver sensitivity for the selected mode is -82\ \text{dBm}, and the design requires 15\ \text{dB} fade and implementation margin above sensitivity.

Find EIRP, received power, and design margin.

Solution

EIRP is:

Regulatory headroom:

Free-space path loss with distance in kilometers and frequency in MHz:

Received power:

Margin above sensitivity:

The link misses the required margin by:

Engineering Comment

The link is legal under the EIRP limit and closes against receiver sensitivity, but it does not meet the required design margin. Because EIRP headroom is only 0.8\ \text{dB}, simply increasing transmitter power is probably not the right fix. Better options include lower feeder loss, improved antenna alignment, higher receive antenna gain if permitted, a more robust modulation mode, or a better path.

Plausibility Check

A 2.4\ \text{GHz}, 3.2\ \text{km} free-space loss near 110\ \text{dB} is plausible. The result should still be field-checked because vegetation, mounting height, polarization error, and local interference can easily consume several dB.

Exercise 2: Fresnel-Zone Clearance

A 5.8\ \text{GHz} link has a total path length of 2.0\ \text{km}. At a rooftop obstruction, the distances to the two antennas are:

The straight line between antennas clears the obstruction by 2.4\ \text{m}. Use the first Fresnel-zone radius approximation:

where distances are in kilometers and r_1 is in metres. Check whether 60\% first-Fresnel clearance is met.

Solution

First Fresnel-zone radius:

Required 60\% clearance:

Actual clearance:

Clearance shortfall:

Engineering Comment

The path does not meet the 60\% first-Fresnel clearance screen at the obstruction. Raising one or both antennas by at least 0.6\ \text{m} may satisfy the simple geometry, but a practical design should add survey tolerance, mast movement, rooftop equipment growth, vegetation, and installation error.

Plausibility Check

At higher frequency the Fresnel radius is smaller than at lower frequency, but it is not zero. A path can have visual line of sight and still lose margin because the Fresnel zone is partially blocked.

Exercise 3: Receiver Sensitivity and Bandwidth Penalty

A narrowband receiver has noise bandwidth:

receiver noise figure:

required detector SNR:

and implementation allowance:

The desired received signal is -91\ \text{dBm}. Find the receiver sensitivity and operating margin. Then estimate the new margin if a wider mode increases the noise bandwidth to 1.0\ \text{MHz} with all other values unchanged.

Solution

Noise floor:

For 200\ \text{kHz}:

Required signal:

Operating margin:

For 1.0\ \text{MHz}:

New margin:

Engineering Comment

The wider mode consumes about 7\ \text{dB} of sensitivity margin because the receiver admits more thermal noise. A wider channel may increase throughput, but it can reduce range or availability unless antenna gain, transmit power, coding, or fade margin compensates.

Plausibility Check

Increasing bandwidth from 200\ \text{kHz} to 1.0\ \text{MHz} is a factor of 5. The noise increase is:

which matches the margin reduction.

Exercise 4: Doppler Shift and Coherence Time

A mobile telemetry receiver operates at 915\ \text{MHz}. The mobile unit travels at:

Estimate the maximum Doppler shift using:

and:

Then estimate coherence time using:

Compare this with a channel-estimation update interval of 20\ \text{ms}.

Solution

Wavelength:

Doppler shift:

Coherence time:

The channel-estimation update interval is:

Engineering Comment

The update interval is too slow for this simple coherence-time screen. The receiver may still work if the channel is benign or the waveform is robust, but a mobile fading channel should update channel estimates, pilots, equalizers, or diversity decisions faster than the channel changes.

Plausibility Check

At sub-GHz frequency and highway-like speed, Doppler shifts of tens of hertz are reasonable. The issue is not the absolute hertz value alone; it is whether the receiver algorithms track channel variation quickly enough.

Exercise 5: OFDM Cyclic Prefix and Multipath Delay

An OFDM system has useful symbol time:

and normal cyclic prefix:

A reflected path arrives:

after the main path. Its power is 12\ \text{dB} below the main path. Estimate whether the normal cyclic prefix covers the reflection. Then estimate throughput efficiency for normal cyclic prefix and an extended cyclic prefix of 6.4\ \mu\text{s}.

Solution

Delay beyond the normal cyclic prefix:

The normal cyclic prefix does not fully cover the delayed path.

Convert reflected-path power ratio:

A rough excess-delay interference screen can scale the reflected energy by the fraction beyond the delayed path:

In dB relative to the main path:

Normal cyclic-prefix efficiency:

Extended cyclic-prefix efficiency:

Throughput efficiency penalty:

Engineering Comment

The normal cyclic prefix is too short for the significant delayed path. The extended prefix covers the reflection with:

of timing margin, but it reduces payload efficiency. The engineering decision is a tradeoff between robustness and throughput. Field validation should confirm delay spread, EVM, packet loss, and whether antenna changes can reduce the reflection without sacrificing efficiency.

Plausibility Check

A few microseconds of excess delay corresponds to hundreds of metres of extra path length. In industrial, urban, or mountainous environments, that can occur through large reflectors even when received power is strong.

Exercise 6: Rain Fade and Adaptive Capacity

A microwave backhaul link has clear-weather received power:

and receiver noise floor:

The high-capacity modulation mode requires 27\ \text{dB} detector SNR plus 3\ \text{dB} implementation margin. During heavy rain, attenuation adds 16\ \text{dB} loss. A fallback mode requires 10\ \text{dB} detector SNR plus the same 3\ \text{dB} implementation margin, but its net capacity is 55\ \text{Mbit/s}. The service requires 80\ \text{Mbit/s} during the reviewed condition.

Determine whether the high mode survives rain and whether the fallback mode satisfies the service.

Solution

Clear-weather SNR:

High-mode requirement:

Clear high-mode margin:

Rain reduces received power by 16\ \text{dB}:

Rain SNR:

High-mode rain margin:

The high mode fails during the heavy-rain condition.

Fallback requirement:

Fallback rain margin:

The fallback RF mode is robust enough, but its capacity is:

so it does not satisfy the service requirement.

Engineering Comment

Adaptive modulation preserves carrier availability but not necessarily service availability. The link remains up in fallback mode, but the committed capacity is not met. Corrective options include more fade margin, larger antennas, diversity, shorter path, lower required committed capacity, traffic shaping, or a backup route.

Plausibility Check

The result explains a common field symptom: the radio does not fully drop, but packet queues, latency, and service alarms appear during rain because adaptive fallback reduces capacity below demand.

Exercise 7: Third-Order Intermodulation and Filter Tradeoff

An RF receiver has desired signal:

Two nearby blockers at the receiver input are:

and:

The receiver input third-order intercept is:

For a two-tone screen, estimate:

The required carrier-to-intermodulation ratio is 15\ \text{dB}. Then evaluate a preselector that attenuates each blocker by 12\ \text{dB} before the nonlinear stage but adds 1\ \text{dB} loss to the desired signal.

Solution

Before the filter:

Carrier-to-intermodulation ratio:

This fails the 15\ \text{dB} requirement by:

After filtering, blocker levels at the nonlinear stage become:

Desired signal becomes:

New intermodulation level:

New carrier-to-intermodulation ratio:

Engineering Comment

The preselector costs 1\ \text{dB} of desired-signal margin, but it improves the intermodulation condition decisively. This is often the correct tradeoff when field failures are caused by receiver linearity rather than weak-signal sensitivity.

Plausibility Check

Third-order products fall rapidly when blocker levels are reduced. A 12\ \text{dB} reduction in both blockers gives much more than 12\ \text{dB} improvement in the third-order product estimate.

Exercise 8: Field Measurement Guard Margin

A commissioning calculation predicted received power:

During field acceptance, the measured received power is:

Receiver sensitivity is:

The required guarded operating margin is 8.0\ \text{dB}. Measurement and installation uncertainty are combined as:

Use coverage factor:

Check whether the guarded field margin meets the release criterion.

Solution

Difference between predicted and measured received power:

The field level is 2.9\ \text{dB} weaker than predicted.

Nominal margin above sensitivity:

Guarded margin:

Compare with the release criterion:

The guarded margin misses the criterion by:

Engineering Comment

The nominal link appears to pass, but the guarded field margin does not. A 0.4\ \text{dB} miss is small, yet it is evidence that the release decision is thin. Before accepting the link, the team should check antenna alignment, feeder loss, connector condition, polarization, spectrum occupancy, measurement calibration, and whether the required margin already includes separate fade and interference allowances.

Plausibility Check

The measured value being nearly 3\ \text{dB} below prediction is not unusual in field RF work. That is large enough to matter because 3\ \text{dB} is a factor of two in power.

Exercise 9: Co-Channel Interference and SINR Capacity Screen

A wireless channel has desired received carrier power:

The receiver noise floor across the occupied channel is:

A co-channel interferer is measured at:

The channel bandwidth is 10\ \text{MHz}. The selected modulation and coding mode requires at least 14\ \text{dB} SINR after implementation allowances. Estimate SINR with interference and compare the Shannon capacity screen with the no-interference case.

Solution

Combine noise and interference in linear power. With powers expressed in dBm, the combined interference-plus-noise level can be written as:

The interferer is 6\ \text{dB} above the noise floor:

Therefore:

SINR:

The selected mode misses the requirement by:

Convert SINR to a linear ratio:

Shannon capacity screen:

Without the interferer, SNR would be:

Engineering Comment

The carrier is not especially weak; the problem is that the interference raises the effective impairment floor. In field work, the corrective action may be channel coordination, antenna discrimination, polarization change, tighter filtering, power control, scheduling, or moving the service to a cleaner channel. Capacity estimates should be treated as screens, not guaranteed payload throughput.

Plausibility Check

The interferer is only 12\ \text{dB} below the desired carrier but 6\ \text{dB} above the noise floor, so it should dominate the impairment budget. A drop from roughly 60\ \text{Mbit/s} to roughly 38\ \text{Mbit/s} is directionally consistent with that impairment.

Exercise 10: Antenna Downtilt and Half-Power Footprint

A sector antenna is mounted at 35\ \text{m} above ground. The target receiver height is 1.5\ \text{m}. The combined electrical and mechanical downtilt is 6^\circ below horizontal. The vertical half-power beamwidth is 8^\circ.

Estimate the ground distance to the beam centerline and the approximate near and far half-power footprint distances. Check whether a service zone from 250\ \text{m} to 800\ \text{m} is inside the half-power footprint.

Solution

Height difference:

Beam centerline distance:

The half-power angular bounds are:

Near half-power distance:

Far half-power distance:

The requested service zone from 250\ \text{m} to 800\ \text{m} lies inside the approximate half-power footprint.

Engineering Comment

This geometric screen is useful for first-pass planning, but it is not an antenna-pattern simulation. Real coverage depends on azimuth pattern, sidelobes, clutter, rooftop diffraction, local elevation, user distribution, mounting tolerance, polarization, cable loss, transmit power, interference and receiver sensitivity.

Plausibility Check

A small downtilt angle produces a long footprint because the tangent is small. The far half-power distance should therefore be much larger than the centerline distance when the lower beam edge is only 2^\circ below horizontal.

Exercise 11: Receive-System Noise Temperature and G/T

A receive station has antenna gain:

The feeder loss before the low-noise amplifier is:

Antenna noise temperature is:

Receiver noise temperature after the feeder is:

Assume feeder physical temperature is 290\ \text{K}. Estimate system noise temperature referred to the antenna input and the receive G/T. The minimum required G/T is 8.0\ \text{dB/K}.

Solution

Convert feeder loss to a linear loss factor:

Equivalent system noise temperature referred to the antenna input:

Effective receive gain after feeder loss:

Noise temperature in dBK:

Receive figure of merit:

Margin above the requirement:

Engineering Comment

Loss ahead of the low-noise amplifier hurts twice: it reduces gain and adds thermal noise before the receiver can amplify the signal. For receive-limited systems, reducing feeder loss or moving the low-noise amplifier closer to the antenna can be more valuable than increasing downstream gain.

Plausibility Check

A system noise temperature around 350\ \text{K} is plausible for a warm feeder and moderate receiver noise temperature. Because 10\log_{10}(350) is about 25\ \text{dB}, a mid-30s dBi effective gain should produce a single-digit positive G/T in dB/K.

Exercise 12: Spectrum-Occupancy Guard for Field Release

A commissioning team wants to release a wireless channel. The planned received carrier is:

The measured in-channel noise floor is:

An intermittent co-channel signal is observed at:

with measured duty cycle of 18\% during the observation window. The release criterion requires:

• nominal C/(I+N)\ge16\ \text{dB};
• guarded C/(I+N)\ge16\ \text{dB} after applying k=2 and analyzer amplitude uncertainty u=1.5\ \text{dB} adversely to both carrier and impairment;
• occupancy below 10\% for the reviewed channel.

Check whether the channel can be released.

Solution

Combine noise and interference:

Nominal carrier-to-impairment ratio:

The nominal ratio passes the 16\ \text{dB} threshold.

The adverse guard lowers carrier by:

and raises the impairment estimate by the same amount. Therefore:

The guarded ratio fails the threshold:

The occupancy rule also fails:

The channel should not be released under the stated criteria.

Engineering Comment

The nominal RF margin alone would approve the channel, but the guarded margin and occupancy rule reject it. This is exactly why release decisions need measurement uncertainty, time occupancy and service requirements, not just a single spectrum-analyzer screenshot.

Plausibility Check

The interferer is 19\ \text{dB} below the carrier, so a nominal pass is plausible. The uncertainty guard removes 6\ \text{dB} from the ratio and the occupancy exceeds the allowed duty cycle, so the release decision reasonably changes from pass to fail.

Exercise 13: Adjacent-Channel Rejection and Blocker Leakage

A wireless receiver has desired carrier power:

The in-channel noise floor is:

The selected mode requires:

An adjacent-channel blocker is measured at the receiver input:

The current receiver filter provides adjacent-channel rejection:

An improved preselector provides:

but adds desired-signal insertion loss:

Estimate the blocker leakage after the current filter, the combined impairment level, current SINR, the ACR required after allowing the preselector insertion loss, and the release margin with the improved preselector.

Solution

Blocker leakage after the current adjacent-channel rejection:

Combine leakage with noise in linear power. A convenient dBm form is:

The leakage is:

above the noise floor, so:

Current SINR:

Current release margin:

The current filter fails the mode badly.

With the improved preselector, desired carrier becomes:

The maximum allowed combined impairment level is:

Convert that combined allowance into a maximum allowed leakage level above the fixed noise floor:

Required adjacent-channel rejection after allowing insertion loss:

The improved preselector provides:

of rejection margin.

New blocker leakage:

New combined impairment:

New SINR:

Release margin:

The improved preselector makes the mode releasable under this screen, but the margin is still modest.

Engineering Comment

Adjacent-channel rejection is not just a compliance number. A strong nearby signal can leak through the receiver selectivity and dominate the impairment floor even when it is outside the nominal channel. The filter tradeoff must include both the blocker reduction and the desired-signal insertion loss.

Plausibility Check

The current blocker is 48\ \text{dB} stronger than the desired carrier, so 54\ \text{dB} of rejection leaves leakage only 6\ \text{dB} below the carrier. That is not enough for a 12\ \text{dB} SINR mode. Raising rejection to 68\ \text{dB} pushes leakage below the thermal noise floor, so the final SINR is mainly limited by the original noise and the 1.2\ \text{dB} filter loss.

Exercise 14: Antenna Pointing and Polarization Release Margin

A point-to-point RF link had a commissioned fade and implementation margin of:

after receiver sensitivity and required fade reserve. A field audit finds installation defects:

• transmit antenna azimuth error: 3.0^\circ with 10^\circ half-power beamwidth;
• receive antenna azimuth error: 4.0^\circ with 8^\circ half-power beamwidth;
• polarization mismatch angle: 15^\circ;
• water ingress in one feeder adds 1.4\ \text{dB} loss.

Use the main-lobe pointing-loss screen:

and the polarization mismatch loss:

The release rule requires at least:

of remaining margin. After corrective work, the transmit error is reduced to 1.0^\circ, the receive error to 1.5^\circ, polarization mismatch to 5^\circ, and feeder ingress loss to 0\ \text{dB}. Estimate margin before and after correction and decide whether the link can be released.

Solution

Transmit pointing loss before correction:

Receive pointing loss before correction:

Polarization mismatch before correction:

Total installation loss before correction:

Remaining margin before correction:

The link fails the release rule before correction:

After correction, transmit pointing loss is:

Receive pointing loss is:

Polarization mismatch after correction:

Total corrected installation loss:

Remaining margin after correction:

Release margin above the rule:

The corrected installation passes the release rule.

Engineering Comment

Pointing and polarization errors are installation losses, not abstract antenna details. They should be counted separately from rain fade, aging, receiver sensitivity and interference margin. The parabolic pointing approximation is only a main-lobe screen; final acceptance should use antenna-pattern data, measured received level, mechanical alignment records, polarization checks, feeder sweep or return-loss evidence and a weatherproofing inspection.

Plausibility Check

The receive antenna error is half of its half-power beamwidth, so a 3.0\ \text{dB} pointing loss is physically plausible. Before correction, installation defects consume more than half of the 11.0\ \text{dB} margin and the link fails. After correction, residual installation loss is below 1\ \text{dB}, leaving 10.43\ \text{dB} of margin and a credible release surplus.

Exercise 15: Rayleigh Fading Outage and Diversity Release

A mobile industrial telemetry link has enough average received power in a clear drive test, but the service requirement is based on outage probability. The selected waveform needs usable SNR of at least:

The measured average SNR on one antenna branch is:

The release rule requires at least:

availability against short Rayleigh flat-fading outages. Use the screening model:

where SNR values are linear ratios, not dB values. Then estimate the average SNR required for 1.0\% outage. Finally check a proposed two-branch selection-diversity receiver with independent branch fading and:

on each branch.

Solution

Convert the required SNR to a linear ratio:

Convert the one-branch average SNR:

One-branch outage probability is:

As a percentage:

The corresponding availability is:

The one-branch link fails the 99.0\% availability rule.

For a target outage probability:

solve the Rayleigh screen for required average SNR:

Convert back to dB:

Additional average SNR needed without diversity is:

For the proposed diversity receiver, each branch has:

Single-branch outage at the improved branch SNR is:

For ideal independent selection diversity with two equal branches:

As a percentage:

Diversity-case availability is:

Availability surplus above the rule is:

The diversity proposal passes the simplified availability screen if the independence assumption is credible.

Engineering Comment

Average SNR is not the same as availability. A link with strong average margin can still violate a service rule if fading regularly drives the instantaneous SNR below the demodulation threshold. The outage model must match the channel: Rayleigh flat fading is a screen for severe non-line-of-sight multipath, not a universal propagation law.

Selection diversity helps only when branch fades are sufficiently independent. The release package should show antenna spacing, polarization or pattern diversity, correlation evidence, branch SNR logs, packet-loss or BER counters, mobility conditions, interference state and the exact rule for counting an outage. If both branches fade together behind the same obstruction, the squared-outage estimate is optimistic.

Plausibility Check

The average branch SNR is 12\ \text{dB} above the required threshold, but Rayleigh fading has a deep lower tail, so a 6.11\% outage estimate is plausible. Requiring 1\% outage pushes the needed average SNR to about 32\ \text{dB}, or 8\ \text{dB} more than the measured branch. Two independent improved branches reduce the outage by multiplication, from 2.48\% per branch to about 0.0615\% together.

Exercise 16: Reciprocal Mixing Noise-Floor Rise

A receiver passes a clean sensitivity test, but packet errors appear when a nearby transmitter is active. The wanted carrier at the receiver input is:

The quiet in-channel noise floor over the receiver noise bandwidth is:

The receiver noise bandwidth is:

The selected mode requires:

A nearby off-channel blocker is measured at the receiver input:

At the blocker offset, the local oscillator single-sideband phase noise is:

The existing pre-mixer filter attenuates the blocker by:

The receiver input one-dB compression point is:

Use the reciprocal-mixing screen:

where N_{RM,dens} is in \text{dBm/Hz}. Integrated reciprocal-mixing noise is:

Check compression headroom, reciprocal-mixing noise, effective noise floor, SINR and release margin. Then check an improved preselector that adds 18\ \text{dB} more blocker attenuation but adds 1.0\ \text{dB} insertion loss to the wanted carrier.

Solution

Blocker power at the mixer input with the existing filter:

Compression headroom is:

The receiver is not failing the simple compression screen.

Reciprocal-mixing noise density:

Receiver noise bandwidth contribution:

Integrated reciprocal-mixing noise:

Combine quiet noise and reciprocal-mixing noise:

Noise-floor rise:

SINR with the blocker active:

Release margin:

The current receiver state fails the release screen even though the blocker is below the one-dB compression limit.

With the improved preselector, blocker power at the mixer is:

New reciprocal-mixing noise density:

New integrated reciprocal-mixing noise:

New combined noise floor:

The wanted carrier after the preselector insertion loss is:

New SINR:

New release margin:

The improved preselector makes the mode releasable under this simplified reciprocal-mixing screen.

Engineering Comment

Reciprocal mixing is why a receiver can have compression headroom and still lose packets near a strong off-channel transmitter. The blocker is mixed by an oscillator whose phase-noise skirt is not zero at the blocker offset, so part of the blocker energy appears as an in-channel noise rise.

A release package should state the blocker level at the mixer reference plane, blocker offset, pre-mixer filtering, LO phase-noise curve at that offset, receiver bandwidth, quiet and loaded noise-floor measurements, gain state, packet-error or EVM evidence and uncertainty allowance. Adjacent-channel rejection alone does not prove reciprocal mixing is acceptable.

Plausibility Check

The blocker reaching the mixer is 59\ \text{dB} above the quiet noise floor, so even a phase-noise density of -94\ \text{dBc/Hz} can matter after integration over 100\ \text{kHz}. Adding 18\ \text{dB} of blocker attenuation drops the reciprocal-mixing noise below the quiet floor, leaving only about 1.8\ \text{dB} of noise rise while the wanted signal loses 1.0\ \text{dB}. That trade is plausible for a preselector fix.

Exercise 17: C/N0 to Eb/N0 Data-Rate Gate

A low-power telemetry link has measured received carrier power at the receiver reference plane:

The receiver noise figure is:

Use thermal noise density:

The selected coding and demodulator require:

after accounting for target error rate. Implementation loss is:

The release rule requires at least:

of margin above the required (E_b/N_0). Check the current payload bit rate:

and a proposed high-rate mode:

assuming received carrier power and receiver noise figure do not change.

Solution

Receiver input noise density including noise figure is:

Carrier-to-noise-density ratio is:

Bit-rate term for the current mode:

Available (E_b/N_0) after implementation loss is:

Margin above the demodulator requirement is:

Release surplus after the required margin is:

The current (250\ \text{kbit/s}) mode passes the release screen.

For the proposed (1.0\ \text{Mbit/s}) mode:

Available (E_b/N_0) is:

Margin above the demodulator requirement is:

Release surplus is:

The proposed high-rate mode fails the release screen unless received carrier power, coding gain, antenna gain, noise figure, bandwidth plan or required margin changes.

Engineering Comment

(C/N_0) is useful because it separates the received carrier and receiver noise density from the chosen bit rate. Raising bit rate without raising carrier power reduces energy per bit. A link can therefore pass at a low telemetry rate and fail at a higher payload rate even when RSSI, EIRP and antenna alignment are unchanged. The release package should state the receiver reference plane, noise-figure basis, coding mode, implementation loss, target error rate and payload bit rate.

Plausibility Check

Increasing bit rate from (250\ \text{kbit/s}) to (1.0\ \text{Mbit/s}) is a factor of four, which costs (10\log_{10}(4)=6.0\ \text{dB}) in (E_b/N_0). The current mode has exactly (6.0\ \text{dB}) above the demodulator requirement, so the proposed high-rate mode should land exactly at the requirement with no release margin. That matches the calculated change from (15.0\ \text{dB}) to (9.0\ \text{dB}).

Exercise 18: Transmit Power Control Clamp and Coexistence Gate

A private wireless uplink uses transmit power control to reach a base-station receiver target. The terminal conducted transmit power is P_{tx}. The terminal feeder loss is:

and the main antenna gain toward the serving base station is:

The serving path loss is:

The base-station receive antenna gain is:

and receive feeder loss is:

The high-rate mode requires received power:

The legal EIRP limit is:

A nearby co-channel receiver sees the terminal through a side-lobe direction. The neighbor path loss is:

The terminal side-lobe gain toward the neighbor is initially:

The neighbor receive antenna gain is:

and its receive feeder loss is:

The neighbor interference limit is:

Check the conducted transmit power required for the high-rate target, the EIRP clamp, the serving received power at the legal EIRP limit and the neighbor interference level. Then evaluate a corrected antenna orientation with:

and a fallback mode with received-power target:

The fallback mode carries:

while protected traffic requires:

Solution

Serving received power is:

Solve for the conducted transmit power needed for the high-rate target:

The corresponding EIRP would be:

The high-rate target would require 2\ \text{dB} more EIRP than the legal limit allows.

Maximum legal conducted transmit power is:

Serving received power at that legal clamp is:

High-rate received-power margin is:

The high-rate mode fails at the legal EIRP clamp.

Neighbor interference at the legal clamp and original side-lobe direction is:

Neighbor interference margin is:

The original antenna state also fails the neighbor coexistence gate.

With corrected side-lobe gain:

Corrected neighbor margin:

The corrected antenna state passes the neighbor interference limit at the legal EIRP clamp.

Check fallback received-power margin at the same legal clamp:

Fallback capacity margin:

The high-rate mode cannot be released, but the corrected antenna state can support the fallback mode for protected traffic with 4\ \text{dB} received-power margin, 3\ \text{dB} neighbor-interference margin and 3\ \text{Mbit/s} protected-capacity margin.

Engineering Comment

Transmit power control is a constrained release problem, not a command to raise power until RSSI looks good. The serving receiver target, legal EIRP limit, antenna pattern, neighbor interference limit, amplifier linearity, battery or thermal budget and selected MCS must all be true at the same time.

The release package should state conducted power calibration, antenna gain, feeder loss, EIRP basis, serving receiver target, MCS fallback rule, neighbor measurement reference plane, channel occupancy, side-lobe or downtilt evidence, power-control clamp, alarm behavior and protected-traffic policy. If the high-rate mode cannot meet the target inside the EIRP and coexistence limits, the correct answer may be fallback mode, antenna correction, channel coordination or service rejection, not hidden power increase.

Plausibility Check

The required high-rate EIRP is 32\ \text{dBm}, only 2\ \text{dB} above the limit, so it is plausible that the received-power shortfall at the legal clamp is also 2\ \text{dB}. The original side-lobe sends too much energy toward the neighbor, while a 6\ \text{dB} side-lobe reduction changes a 3\ \text{dB} interference failure into a 3\ \text{dB} pass. That symmetry is expected from dB arithmetic.

Review Checklist

After solving a wireless or RF exercise, ask:

1. Are dB, dBm, dBW, dBi, and linear ratios kept separate?
2. Is EIRP below the applicable limit with antenna and feeder losses included?
3. Is the path budget separated from fade, interference, installation, and aging margins?
4. Is receiver sensitivity tied to bandwidth, noise figure, waveform, coding, and target error rate?
5. Is C/N0 converted to Eb/N0 using the actual bit rate, implementation loss, coding mode and target error rate?
6. Are blockers, intermodulation, compression, reciprocal mixing and adjacent-channel effects checked when strong signals are credible?
7. Is adjacent-channel leakage combined with thermal noise before judging SINR?
8. Does the waveform timing tolerate the measured delay spread and Doppler condition?
9. Does fallback modulation preserve the service requirement, not just carrier lock?
10. Is co-channel interference combined with noise in linear power before calculating SINR?
11. Does antenna downtilt place the half-power footprint over the real service area?
12. Are pointing loss, polarization mismatch and feeder defects counted as installation losses?
13. Are feeder loss, antenna noise temperature and receiver noise temperature included in receive-system checks?
14. Are occupancy, uncertainty and intermittent interferers included before release?
15. Does field evidence include calibrated power, spectrum, packet loss, latency, jitter, configuration, and environmental context?
16. Do outage and availability claims state the fading model, service threshold, observation rule and diversity independence evidence?
17. Does transmit power control respect EIRP, receiver target, neighbor interference, antenna pattern and fallback capacity at the same time?

The strongest wireless calculation is not the one with the largest margin. It is the one whose assumptions can be found again when the site changes, the spectrum fills, the weather shifts, or the installation ages.

Common Mistakes

• Mixing dB ratios with dBm, dBW or dBi absolute references when calculating EIRP, received power or margins.
• Treating EIRP compliance as a link-quality pass while ignoring receiver sensitivity, SINR, interference and fading outage.
• Using a receiver sensitivity value from a different bandwidth, waveform, coding rate or target error rate.
• Treating RSSI or received carrier power as a data-rate pass without converting C/N0 to Eb/N0.
• Counting antenna gain but omitting feeder loss, connector loss, radome loss, installation loss or polarization mismatch.
• Declaring a Fresnel path clear from a map without checking actual antenna height, terrain, buildings, vegetation growth and clearance standard.
• Checking rain fade or average SNR while ignoring co-channel interference, adjacent-channel blockers, intermodulation and receiver compression.
• Passing a blocker check from adjacent-channel rejection or compression headroom while ignoring LO phase noise and reciprocal mixing.
• Treating fallback modulation as acceptable when the required payload rate, latency or availability is no longer met.
• Assuming diversity gains without proving branch independence, antenna correlation, polarization separation or blockage diversity.
• Using Doppler or OFDM timing calculations without checking oscillator error, mobility profile, delay spread and receiver tracking limits.
• Releasing a field measurement without calibration, uncertainty guard, spectrum context, packet counters, configuration record and environmental conditions.
• Raising transmit power to close a serving link while violating EIRP, worsening co-channel interference or hiding that the selected MCS is not releasable.