Formula sheet
Wireless and RF Communication Systems Formula Sheet
Wireless and RF formulas for EIRP, path loss, Fresnel clearance, receiver sensitivity, noise figure, SINR, fade, Doppler, OFDM, linearity, and validation.
This formula sheet collects first-pass equations for wireless and RF communication systems. It is intended for link screening, receiver-margin review, site-survey preparation, commissioning evidence, and failure diagnosis. The equations are useful only when the engineering boundary is explicit: antenna port, receiver input, demodulator input, packet-service boundary, or field measurement point.
Use these formulas with the companion wireless topic and exercises. This page is not a propagation-standard replacement and not a regulatory rulebook. It gives a disciplined calculation framework for engineers who need to connect RF power, antennas, propagation, bandwidth, receiver noise, interference, waveform timing, and validation evidence.
How to Use This Formula Sheet
Use this sheet as a system-level RF review tool. Start with the service decision, not the equation: coverage, throughput, telemetry reliability, command link, interference mitigation, field commissioning, failure diagnosis or regulatory headroom. The required evidence changes when the boundary moves from antenna port to receiver input, demodulator input, packet-service interface or installed field measurement point.
Then build the calculation in layers. First check power units, EIRP and regulatory headroom. Next estimate path loss, Fresnel clearance, antenna pointing and received power. Then calculate receiver noise, required SNR, waveform margin, fade allowance, interference, blocker linearity and timing effects. Finally compare predictions with field measurements and service metrics.
Use the wireless formulas here for RF-system decisions. Use the separate telecommunications link-design sheet for general link-budget accounting, the digital modulation sheet for waveform and coding efficiency, and the applied electromagnetics sheet for field, antenna and wave-propagation physics outside a communication-service boundary.
Symbols and Conventions
| Symbol | Meaning | Typical unit |
|---|---|---|
| P_t | transmitter output power at radio output | dBm or dBW |
| P_r | received power at receiver input | dBm |
| G_t, G_r | transmit and receive antenna gain | dBi or linear |
| L_t, L_r | transmitter and receiver feeder or connector loss | dB |
| L_{path} | propagation path loss | dB |
| L_{misc} | miscellaneous implementation, polarization, pointing, body, radome, or installation loss | dB |
| B | receiver noise bandwidth | Hz |
| NF | receiver noise figure | dB |
| SNR | signal-to-noise ratio at the stated boundary | linear or dB |
| SINR | signal-to-interference-plus-noise ratio | linear or dB |
| M | margin against a stated requirement | dB |
| f | carrier frequency | Hz, MHz, or GHz |
| \lambda | wavelength | m |
| d | path distance | m or km |
| v | relative speed | m/s |
| \tau | excess delay or propagation delay | s |
Use dB for ratios, dBm or dBW for absolute power, dBi for antenna gain, and dB-Hz for carrier-to-noise density. Do not add a dBm value to another dBm value. Add gains and losses in dB to an absolute power value only when the references are compatible.
Basis and Validity Limits
The formulas in this sheet are first-pass wireless and RF engineering relationships. They assume that the reference plane, bandwidth, antenna state, polarization, channel condition and service requirement are explicit. A link that closes on received power can still fail through obstruction, interference, receiver overload, timing spread, mobility, protocol behavior, installation drift or field uncertainty.
Free-space path loss is valid for clear far-field paths. It is not enough for indoor, urban, industrial, maritime, underground, body-area, vehicular or obstructed links unless the extra losses and variability are measured or modelled. Fresnel clearance depends on survey accuracy, terrain, vegetation growth, antenna height, earth curvature and installation tolerance. Antenna gain depends on orientation, polarization, mounting, nearby structures and cable state.
Receiver calculations depend on the stated noise bandwidth, waveform, demodulator point and implementation loss. Sensitivity, SINR, Eb/N0, EVM, packet loss and service availability are related but not interchangeable. Linearity formulas are screens unless blocker levels, reference planes, filtering and front-end configuration match the installed receiver. Field acceptance requires calibrated measurements and service evidence, not only a spreadsheet margin.
Power Units and Decibel Accounting
Power ratio:
Absolute power in dBm:
Absolute power in dBW:
Conversion:
Linear power from dBm:
Engineering use: keep a link budget as a dB ledger. Start with a known absolute power, then add gains and subtract losses. Convert to linear units only when ratios must be multiplied, when noise factors are cascaded, or when independent noise and interference powers must be combined.
Frequency, Wavelength, and Propagation Time
Wavelength:
where c\approx3.0\times10^8\ \text{m/s} in free space.
One-way propagation delay:
where v_p is propagation velocity. For free-space radio paths, v_p is approximately c. In cables or waveguides, use the actual velocity factor or group velocity.
Worked screen: a 5.8\ \text{GHz} carrier has:
An 8.0\ \text{km} microwave path has free-space propagation delay:
The delay is usually small for ordinary telemetry, but it can matter in time synchronization, ranging, radar, control loops, and packet-latency budgets.
EIRP and Regulatory Headroom
Effective isotropic radiated power:
where P_t is conducted transmitter power, L_t includes feeder, connector, filter, switch, duplexer, and lightning-protection losses before the antenna, and G_t is antenna gain relative to isotropic gain.
Regulatory or coordination headroom:
Equivalent radiated power in a specified polarization or antenna reference must follow the applicable rule or coordination basis. Do not assume that a legal EIRP makes the link technically robust, and do not assume that a technically useful antenna setting is legal.
Worked screen: if P_t=18\ \text{dBm}, feeder loss is 1.4\ \text{dB}, antenna gain is 9\ \text{dBi}, and the reviewed EIRP limit is 30\ \text{dBm}:
The configuration has regulatory headroom, but extra transmit power may worsen coexistence or receiver overload elsewhere. Increasing power should be a design decision, not the default fix for poor field performance.
Free-Space Path Loss and Received Power
Free-space path loss:
Using distance in kilometers and frequency in MHz:
Received power at the receiver input:
where L_{path} may be free-space loss plus obstruction, atmospheric, rain, vegetation, building-penetration, body-blocking, diffraction, polarization, or fading allowance, depending on the scenario.
Validity limits:
- Free-space loss assumes a clear unobstructed path and far-field operation.
- Indoor, urban, industrial, underground, maritime, and mobile channels usually need measured or statistical corrections.
- Link closure against sensitivity is not the same as service acceptance.
Worked screen: a 5.8\ \text{GHz} link over 2.4\ \text{km} uses P_t=20\ \text{dBm}, L_t=1.0\ \text{dB}, G_t=16\ \text{dBi}, G_r=16\ \text{dBi}, L_r=1.2\ \text{dB}, and L_{misc}=4.0\ \text{dB}.
This is a credible first-pass received level. It still needs comparison with receiver noise, required waveform SNR, interference, fade margin, and field measurements.
Fresnel-Zone Clearance
First Fresnel-zone radius at a path point:
With d_1 and d_2 in kilometers, frequency in GHz, and r_1 in metres:
Clearance screen:
where C_F is the required clearance fraction. A common screening target is C_F=0.6, but project requirements, terrain, antenna height uncertainty, vegetation growth, and survey uncertainty may justify more.
Worked screen: for d_1=0.7\ \text{km}, d_2=1.1\ \text{km}, and f=5.8\ \text{GHz}:
Required 60\% clearance:
If the surveyed clearance is 2.4\ \text{m}, the screen fails by:
A visual line of sight is therefore not enough. Raise antennas, change the path, reduce obstruction, or justify the loss with measurement and service evidence.
Antenna Aperture, Beamwidth, and Far Field
Antenna effective aperture:
where G is linear gain, not dBi.
Approximate far-field distance:
where D is the largest antenna dimension.
Approximate half-power beamwidth for a circular aperture:
where k depends on illumination and aperture efficiency. Values near 1.0 to 1.2 radians are common for screening.
Engineering use: higher antenna gain improves link budget in the pointed direction but narrows the beam, increases alignment sensitivity, and may reduce tolerance to mast movement, vibration, installation error, or mobile geometry. Antenna gain is not generated power; it is spatial redistribution.
Receiver Noise Floor and Sensitivity
Thermal-noise power:
At approximately 290\ \text{K}:
Receiver noise floor including noise figure:
Required received signal:
where SNR_{req} is the detector, demodulator, or modem SNR requirement for the stated waveform, coding, error target, and measurement point. M_{impl} covers implementation loss, calibration allowance, demodulator degradation, or project-specific guard.
Operating margin against receiver requirement:
Worked screen: for B=1.0\ \text{MHz}, NF=6.0\ \text{dB}, SNR_{req}=14\ \text{dB}, and M_{impl}=2.0\ \text{dB}:
If the received power estimate is -69.5\ \text{dBm}:
This is strong thermal-noise margin, but it is not a complete acceptance decision. Fade, interference, receiver linearity, service load, and field uncertainty still have to be covered.
Cascaded Noise Figure
Noise factor:
Noise figure:
Friis cascade equation:
Use linear gains and linear noise factors. A passive loss before the first low-noise amplifier has:
where L is the linear loss factor.
Worked screen: a 2.0\ \text{dB} pre-LNA cable loss is followed by an LNA with NF=1.5\ \text{dB} and gain 18\ \text{dB}, then a receiver stage with NF=7.0\ \text{dB}.
Convert:
Cascade:
The front-end cable loss dominates the result. Moving the LNA closer to the antenna or reducing pre-LNA loss can improve receiver sensitivity more effectively than changing a later receiver stage.
Link Margin Decomposition
Total available receiver margin:
Guarded design margin:
where:
- M_{fade} covers fading, rain, shadowing, or multipath allowance;
- M_I covers interference and coexistence uncertainty;
- M_{install} covers antenna alignment, cable tolerance, connector loss, and mounting variation;
- M_{aging} covers corrosion, water ingress, antenna drift, repairs, or equipment drift;
- M_u covers measurement and model uncertainty.
Pass condition:
Engineering use: avoid hiding all allowances inside one unexplained margin. Separate thermal-noise margin from fade, interference, installation, aging, and uncertainty. Otherwise a link may appear conservative while one real-world effect consumes the entire reserve.
Carrier-to-Noise Density and Energy per Bit
Carrier-to-noise density:
For satellite, telemetry, or long-range links using G/T:
with:
Energy per bit to noise density:
Margin:
Use this form when the design basis is bit energy, coding, symbol timing, or link availability rather than only received power.
Interference, SINR, and Coexistence
Linear signal-to-interference-plus-noise ratio:
If SNR=C/N and C/I are known:
Carrier-to-interference margin:
Noise-rise from interference:
Engineering use: a receiver may pass a thermal-sensitivity test and fail near strong emitters. Coexistence review should include in-channel interference, adjacent-channel leakage, blocking, intermodulation, receiver compression, pulsed transmitters, switching equipment, and intermittent site activity.
Worked screen: suppose thermal SNR=25\ \text{dB} and carrier-to-interference ratio is 18\ \text{dB}.
Interference dominates the usable margin. Improving receiver sensitivity would not solve the main problem unless it also improves coexistence or filtering.
Fading, Rain, and Availability Screening
Received power during an additional fade:
Fade margin against required received signal:
Rain attenuation screen for a microwave path:
where \gamma_R is specific attenuation in dB/km and L_{eff} is effective path length in km for the rain condition used by the project.
Adaptive-capacity check:
when the link falls back to a lower modulation and coding mode during fade.
Engineering use: a radio carrier may remain locked while service capacity fails. Rain fade, multipath, foliage, body blocking, vehicle shadowing, antenna movement, or indoor obstruction should be checked against the service requirement, not only carrier availability.
Doppler Shift and Coherence Time
Maximum Doppler shift for relative radial speed:
Approximate coherence time:
This is a screening approximation, not a guarantee. Real channels depend on scattering geometry, motion direction, antenna pattern, carrier frequency, and estimator design.
Worked screen: a mobile terminal travels at 20\ \text{m/s} at 2.4\ \text{GHz}.
If channel estimation is updated every 20\ \text{ms}, the update is too slow for this screen. The system may need more frequent pilots, diversity, lower-order modulation, stronger coding, or a mobility-specific validation test.
Delay Spread and OFDM Guard Interval
Guard-interval pass condition:
where T_{cp} is cyclic prefix or guard interval, \tau_{max} is the maximum relevant excess delay, and M_\tau is timing margin for measurement uncertainty and channel variation.
OFDM payload-time efficiency:
where T_u is useful symbol time.
Coherence bandwidth screening:
where \sigma_\tau is RMS delay spread. This approximation is used only for early screening; actual modem performance depends on equalization, coding, pilot density, interleaving, and channel estimation.
Worked screen: T_u=64\ \mu\text{s}, T_{cp}=3.2\ \mu\text{s}, and measured excess delay is 4.0\ \mu\text{s}. The guard interval fails by:
Extended prefix T_{cp}=6.4\ \mu\text{s} gives:
The extended prefix improves robustness but costs about 4.3 percentage points of payload-time efficiency. The release decision should compare EVM, packet loss, capacity, and field delay-spread evidence, not only the guard interval formula.
Receiver Linearity and Intermodulation
Third-order intermodulation screen for two blockers:
for the product at 2f_1-f_2 or the corresponding alternate product. Use powers and intercept at the same receiver reference plane.
Carrier-to-intermodulation ratio:
Pass condition:
Compression guard:
Worked screen: a desired signal is P_C=-83\ \text{dBm}. Two blockers at the receiver input are P_1=-35\ \text{dBm} and P_2=-38\ \text{dBm}. Receiver IIP3=-6\ \text{dBm}.
If the required ratio is 18\ \text{dB}, the receiver fails by:
A preselector, antenna null, channel move, more linear front end, lower blocker exposure, or shielding change may solve this. Increasing desired transmit power may improve C/IM3 at this receiver but can create new coexistence problems elsewhere.
Sampling, FFT, and Spectrum Measurements
Ideal sampling condition:
FFT bin spacing:
Observation time:
Resolution bandwidth, window function, detector mode, averaging, span, antenna factor, preamplifier state, and calibration all affect spectrum evidence. A spectrum trace without settings is weak engineering evidence.
Packet or service metrics:
Use RF metrics and service metrics together. Good received power does not guarantee packet-loss, latency, jitter, handover, or application update-rate performance.
Guarded Field Acceptance
Measured margin above required received signal:
Guarded margin with combined uncertainty:
where u_c is combined standard uncertainty in dB and k is the coverage factor used by the project.
Prediction error:
Release condition:
Worked screen: predicted received power is -69.5\ \text{dBm}, measured received power is -72.0\ \text{dBm}, required received power is -92.0\ \text{dBm}, u_c=1.5\ \text{dB}, k=2, and release requires 15\ \text{dB} guarded margin.
The guarded margin passes by:
The prediction is 2.5\ \text{dB} optimistic, so the acceptance record should still investigate or explain the difference. A passed field margin is stronger when measured power, spectrum occupancy, packet loss, latency, jitter, antenna state, weather, and configuration evidence all agree.
Validation Evidence Package
Before accepting a wireless or RF calculation, assemble evidence that connects the formula result to the installed service. The package should let another engineer identify the power reference plane, antenna configuration, path condition, bandwidth, service requirement, field method and margin policy without reconstructing the work from memory.
Include calibrated or traceable records for conducted power, feeder loss, antenna gain basis, antenna alignment, path survey, received power, spectrum occupancy, noise floor, interference, blocker exposure, packet loss, latency, jitter, timing error and configuration state where those quantities affect acceptance. Record spectrum-analyzer settings, detector mode, RBW/VBW or equivalent noise bandwidth, averaging, antenna factor, cable loss, preamplifier state and measurement uncertainty.
For service release, include the guarded margin, prediction error, weather or obstruction state, traffic load, adaptive modulation state, fallback capacity, alarm state, firmware or modem configuration, retest trigger and responsible reviewer. A wireless site should not be accepted from a single received-power reading when coexistence, availability or packet-service metrics are part of the requirement.
Calculation Workflow Checklist
For a wireless or RF design review:
- Define the service requirement: data rate, coverage, availability, latency, jitter, packet loss, update interval, or measurement quality.
- State the RF boundary for every power value: transmitter output, antenna port, free-space path, receiver input, demodulator input, or packet-service interface.
- Calculate EIRP and regulatory headroom.
- Estimate path loss, received power, Fresnel clearance, and antenna-alignment sensitivity.
- Calculate receiver noise floor, required signal, and thermal-noise margin.
- Separate fade, interference, installation, aging, and uncertainty margins.
- Check blockers, intermodulation, compression, dynamic range, and coexistence when strong signals are credible.
- Check Doppler, delay spread, sampling, clocking, and guard interval when mobility or multipath is credible.
- Validate with calibrated field measurements and service tests.
- Record the assumptions so future engineers can tell whether a site change invalidates the calculation.
Common Formula Mistakes
Common mistakes include mixing dB and linear units, using transmitter power instead of EIRP, counting antenna gain without feeder loss, applying free-space loss to an obstructed path without allowance, treating visual line of sight as Fresnel clearance, and assuming receiver sensitivity is independent of bandwidth and waveform.
Other frequent mistakes are accepting a link from received power alone, ignoring interference after a clean bench sensitivity test, hiding all risk in one unlabelled margin, measuring spectrum without settings, accepting adaptive fallback without checking service capacity, and failing to preserve field evidence for later maintenance.
The strongest RF calculation is not the one with the largest nominal margin. It is the one whose assumptions, units, boundaries, margins, and validation evidence remain understandable when the spectrum changes, the antenna is moved, the cable ages, or the service requirement becomes stricter.