Propagation, Channels & Link Distributions

Before you begin

Bring Modules 06.1–06.4, especially realized gain and polarization and arrays and diversity, plus Path 01 fields and noise. The RF Systems lessons on requirements and sensitivity are recommended; the receiver screen needed here is supplied explicitly.

By the end, build an auditable link ledger; distinguish coherent geometry from statistical variation; calculate a received-power/SINR distribution; and state its population, percentile, model domain and missing evidence. Full terrain/ray tracing, channel-sounding execution, protocol receiver rules, detailed measurement and legal operation remain outside this lesson.

01 / 10

Failure: free-space path loss becomes a guaranteed range

The spreadsheet says 100 m. Which 100 m does it mean?

Our illustrative engineering case is the small 2.45 GHz telemetry node and its gateway. It works across an empty outdoor path. On a metal machine, the same separation sometimes fails; down an office corridor, turning the node changes the result. A nearby transmitter makes the failures more frequent. The original spreadsheet contained free-space loss and one receiver threshold.

The calculation may be correct for its inputs. The claim becomes wrong when one geometric reference is promoted into a product guarantee. A real link includes antennas in particular directions and mounting states, obstructed or reflected paths, interference, receiver behavior, and a population of places and times.

Think about itA median signal is above the receiver threshold. Must 95% of the links work?

Answer

No. The median describes the middle of the assumed population. In our office fixture, median SINR is 15.648006 dB against an 8 dB requirement, yet 6.305774% of the modeled received powers fall below the threshold. A favorable center can coexist with an unacceptable lower tail.

Repair the range claim before changing transmit power
Missing condition	Why it changes the decision
Antenna state	Frequency, direction, realized gain, polarization, body and mounting change spatial coupling.
Environment and geometry	LOS, ground, edges, doors, material layers and multipath determine whether the selected model applies.
Receiver and interferer	Same-band noise/interference and a named quality criterion determine the threshold.
Population and evidence	Distance alone names neither positions/times/orientations nor the fraction to be served, correlation, confidence or uncertainty.

Common misconceptionFree-space loss is a guaranteed product range.

It is a conditional propagation reference. Even a perfectly computed threshold crossing cannot validate an unobserved environment or antenna state. Retain the missing evidence as unknown.

The first repair is bookkeeping: write down exactly where the power is known and how it reaches the receiver.

02 / 10

Build the link ledger from transmit plane to receive decision

Where did each decibel enter the ledger?

Start with +10 dBm conducted at the transceiver’s R1-TX port. A matched 1 dB feed leaves +9 dBm incident at R2-TX. Candidate A’s idealized broadside realized-gain factor from 06.1 is 0.675, or −1.706962272 dBi. It already includes antenna mismatch and radiation loss. Multiplying by it produces directional EIRP at S0; it does not produce accepted power or TRP.

$\begin{aligned}P_{\mathrm{rx,R1}}&=P_{\mathrm{tx,R1}}-L_{\mathrm{tx}}+G_{\mathrm{real,tx}}-\mathrm{PL}\\&\quad+G_{\mathrm{real,rx}}-L_{\mathrm{rx}}+10\log_{10}(\mathrm{PLF})-L_{\mathrm{orientation}}\end{aligned}$ Power levels are dBm; feed, propagation, polarization and orientation terms are dB; both directional realized gains are dBi referenced to real 50 Ω. The propagation term is replaced, never stacked.

At receive, the separately assumed +2 dBi realized gain yields delivered power into the real 50 Ω R2-RX load. A matched 1 dB receive feed moves that result to R1-RX. Available receive power under conjugate match is a different quantity. Recovering it would require the receive gain before mismatch and the loading model; neither should be invented from the supplied realized gain.

Default office ledger · 100 m, state A, median channel · derived illustrative values
Step	Term	Result / plane
Transceiver output	Supplied +10 dBm	R1-TX: +10.000000 dBm
Transmit feed	−1 dB matched loss	R2-TX incident: +9.000000 dBm
Directional TX realized gain	10 log10(.675) = −1.706962272 dBi	S0 directional EIRP: +7.293037728 dBm
Office median channel	−82.231104909 dB	Isotropic-receiver reference: −74.938067181 dBm
Directional RX realized gain	+2 dBi, includes mismatch	R2-RX delivered: −72.938067181 dBm
Receive feed	−1 dB matched loss	R1-RX delivered: −73.938067181 dBm
State A	PLF = 1; pattern penalty = 0 dB	No additional change

R3 remains the detector/decision boundary. This local example refers its quality screen back to R1-RX using an assumed linear receiver. Supplied noise is −100 dBm and interference is −90 dBm, integrated over the same 1 MHz illustrative analysis bandwidth. This does not replace the portfolio waveform bandwidth or Path 05’s 20 kHz sensitivity fixture. Required SINR is a hypothetical 8 dB, not a PER specification.

$\begin{aligned}\mathrm{NI} &= 10 \log _{10}(10^{\frac{N}{10}} + 10^{\frac{I}{10}}) \\ \mathrm{SINR} &= P_{\mathrm{rx,R1}} - \mathrm{NI} \\ P_{\mathrm{threshold}} &= \mathrm{NI} + \mathrm{SINR}_{\mathrm{required}}\end{aligned}$ The exponentials are powers in mW. Independent additive same-band contributions are summed linearly; disabling I means 0 W, not 0 dBm.

NI = −89.586073148 dBm; threshold = −81.586073148 dBm at R1-RX. These equations approximate interference as additive in-band power. Structured blockers, receiver compression, timing and packet decoding can invalidate that screen.

Common misconceptionRealized gain still needs another mismatch and efficiency deduction.

That would charge the same effects twice. Likewise, a feed already included in a measured realized-gain reference cannot be deducted again. Name the included losses before combining data.

Go deeperR2-TX and R2-RX do not create new portfolio planes

They distinguish the two antenna feeds locally; both map to portfolio R2. S0 carries direction and polarization. With e^(+jωt), propagation carries negative phase along a path. The baseline TX direction is +x, θ=90°, φ=0° in the 06.1 specimen frame, with co eθ and cross eφ. The receiver looks back along the path using the aligned common transverse basis. A state-B pattern penalty is an assumed change, not a vector rotation computed from the earlier scalar pattern.

Now the propagation term has a defined job. Begin with the cleanest possible reference.

03 / 10

Friis and free-space basic transmission loss

How much of a spreading wave can a receiver collect?

In unobstructed far-field free space, the power flux from a fixed directional EIRP spreads over area proportional to r². A matched, aligned receiver collects flux through its effective aperture. Combining S = PₜGₜ/(4πr²) with Aₑ = Gᵣλ²/(4π) gives the familiar transmission equation. The gain and matching convention must stay consistent with the power planes.

$\begin{aligned}P_{r} &= P_{t} G_{t} G_{r} (\frac{\lambda }{4\pi r})^{2} \\ L_{\mathrm{FS}} &= 20 \log _{10}(\frac{4\pi r f}{c})\end{aligned}$ Linear watts and dimensionless gains; LOS, far field, polarization alignment and compatible matching. Use λ=c/f with exact c=299792458 m/s.

ITU-R P.525-5, Annex §2.3 defines free-space basic transmission loss between isotropic antennas. That loss is a ratio in dB; it is not the complete receive level. The ledger supplies conducted power, gains and the remaining losses separately.

Independent free-space anchors · 2.450 GHz · geometric baseline, not measured
Distance r	LFS · dB	Full default state-A R1-RX power
1 m	40.231104909	Algebra −31.938067181 dBm; suppressed by the default 2 m far-field screen
10 m	60.231104909	−51.938067181 dBm under LOS and aligned-gain assumptions
100 m	80.231104909	−71.938067181 dBm under the same assumptions

Think about itKeep both gains fixed. Multiply distance by ten. How much power remains?

Answer

The loss rises 20 dB; receive power becomes one hundredth. Doubling distance instead adds 20 log10(2)=6.020600 dB. Frequency is unchanged; this is geometric scaling under fixed antenna conditions.

Go deeperThe MHz/km constant and a conservative domain screen

With f in MHz and r in km, the exact-c constant is 20 log10(4π·10⁹/c) = 32.447783222 dB. P.525’s familiar rounded form uses 32.4. This workbench calculates from SI and rounds only for display.

For each ray, require r ≥ max(2Dtx²/λ, 2Drx²/λ, 10Dtx, 10Drx, 10λ). Dtx=0.10 m and Drx=0.20 m give a 2 m minimum here. This composite screen is a deliberately conservative lesson assumption, not a universal IEEE boundary or proof of quiet-zone accuracy. A failed screen suppresses receive-power and reliability conclusions; passing it does not establish clear LOS or a calibrated model.

Common misconceptionFree-space basic loss directly states the received power.

It omits the transmitter level, both antennas and their operating conditions. A loss of 80.231105 dB says nothing by itself about which receiver threshold will be met.

The λ² factor also raises a subtle question: what is being held fixed when frequency changes?

04 / 10

Frequency dependence is not one slogan

Does “higher frequency loses more” describe propagation, or the antennas you chose?

At fixed gain, Friis receive power scales as f⁻². But gain is a relationship to an isotropic reference, not a fixed collection area. For an electrically large aperture with constant physical area A and aperture efficiency ηₐ, G = 4πηₐA/λ². Raising frequency increases its ideal gain while narrowing its beam.

Double frequency, same distance and conducted power · ideal LOS comparison
Antenna constraint	Gain change	Receive-power change
Both supplied gains fixed	0 dB on both ends	−6.020600 dB; power × 1/4
One fixed physical aperture, other gain fixed	+6.020600 dB at the aperture	0 dB; ideal cancellation
Both physical apertures fixed	+6.020600 dB at each end	+6.020600 dB; power × 4

These comparisons assume electrically large efficient apertures, aligned pointing, compatible polarization and match, and far-field validity at both frequencies. The far-field distance itself can increase as wavelength falls. An arbitrary small printed antenna does not inherit this aperture scaling; its current distribution and efficiency may change instead.

Common misconceptionHigher frequency always gives less received power, regardless of the antenna constraint.

The three rows describe different engineering experiments. The workbench frequency control performs only the fixed-supplied-gain comparison; it never invents a larger gain or a new antenna response.

Real media add further frequency dependence. ITU-R P.676-13, Scope and Annex 1 §1 models dry-air and water-vapour attenuation using atmospheric conditions and spectral lines. Matching, polarization, material properties, path length and weather all matter. A frequency dial alone supplies none of those conditions.

Go deeperAbsorption is distinct from geometric spreading

P.676-13’s line-by-line method uses pressure, temperature and water vapour; its stated scope extends to 1000 GHz. The recommendation is a model route here, not an added scalar loss. The workbench implements no gas or rain correction. Absence of such a row means the effect is outside the modeled scenario, not that measured atmospheric attenuation is zero.

Before adding a generic “wall loss,” identify which physical mechanisms are present and which model could represent them.

05 / 10

Reflection, diffraction, scattering, penetration, and clutter

A steel cabinet blocks the direct path. Where can the received field come from?

Energy may reach the gateway by reflection from a wall, diffraction around an edge, scattering from rough or small objects, or transmission through surrounding material. These mechanisms alter phase, delay, angular structure and polarization as well as average power. An obstruction can create another useful path while also weakening the original one.

Mechanism → model route → evidence required · no generic wall coefficients
Mechanism	Model / source route	What must be known
Specular reflection	Coherent geometry and boundary model; two-ray is the bounded example below.	Surface geometry, incidence, complex permittivity, polarization and roughness relative to wavelength.
Diffraction around edges	ITU-R P.526-16 (11/2025)	Path profile, obstacle geometry and clearance; select the applicable diffraction case.
Scattering	Resolved paths or a justified statistical channel; Tse & Viswanath §2.1.7.	Size/roughness, angle/polarization distribution and unresolved-path population.
Penetration / building materials	Route through P.2040 / P.2109 in the P-series index.	Material and layers, moisture, thickness, incidence and whether loss means one wall or building entry.
Indoor path statistics	ITU-R P.1238-13 (09/2025)	Applicable building/path category, frequency, geometry, statistic and calibration population.
Vegetation / clutter	P.833 vegetation; P.2108-1 (09/2021) clutter.	Vegetation depth/state and water content; clutter geometry and model-specific terminal conditions.
Gas / rain	P.676-13; P.838-3 (03/2005).	Atmosphere and path integration; rain rate/polarization and a complete applicable prediction method.

The P-series catalogue was checked on 7 September 2026. Those entries establish routing and current identity. Only P.525’s point-to-point formula and P.676’s stated scope/introduction were read here as recommendation text; no unseen material table or indoor coefficient is imported.

Think about itCan you add a wall penalty to an empirical indoor model without further checking?

Answer

Only if the empirical model’s definition excludes that wall contribution. A model fitted across obstructed paths may already include it statistically. Adding it again can double count loss; treating it as independent can also invent a probability distribution.

Common misconceptionPath loss belongs to transmitter frequency alone.

Geometry, environment and the chosen loss definition are equally necessary. “At 2.45 GHz” does not identify a corridor, the terminal heights, a material stack or a population.

One coherent reflected path is enough to show why even the free-space baseline is not a pointwise lower-loss bound.

06 / 10

Two-ray interference creates distance-dependent nulls

Can moving farther from the node improve a link?

Two paths carry copies of the same narrowband signal. Their phases can reinforce or oppose. As horizontal range R changes, the direct and reflected lengths change by different amounts. A receiver can move out of a cancellation region even while geometric spreading increases.

$\begin{aligned}r_{d} &= \sqrt{R^{2} + (h_{t}-h_{r})^{2}} \\ r_{r} &= \sqrt{R^{2} + (h_{t}+h_{r})^{2}} \\ \Delta r &= \frac{4h_{t}h_{r}}{r_{r} + r_{d}}\end{aligned}$ Flat ground; positive R, ht and hr in metres. k=2π/λ. The stable Δr formula avoids subtracting nearly equal lengths.

$\begin{aligned}H &= (\frac{\lambda }{4\pi })[\frac{1}{r_{d}} + \frac{\Gamma _{\mathrm{ground}}e^{-jk\Delta r}}{r_{r}}] \\ \mathrm{PL}_{\mathrm{two-ray}} &= -10 \log _{10}|H|^{2}\end{aligned}$ e^(+jωt); the common direct-path phase has unit magnitude and is removed. Γground is a supplied complex field-reflection proxy, including the chosen polarization convention.

Replace the ledger’s entire path term with this result. Do not also subtract free-space loss. This simplified pair shares the same supplied antenna and polarization factors on both rays; angle-dependent vector patterns, roughness and multiple reflections are omitted. Each ray must pass the far-field screen.

Calculated at 2450 MHz; ht/hr = 1/2 m; Γ = 1 ∠ 180°. Both rays share supplied pattern/polarization factors. The sweep displays algebra over 2–500 m; points whose ray length is below 2.000000 m fail this lesson’s far-field screen. Selected R = 100 m: rd = 100.004999875 m, rr = 100.044989880 m, Δr = 0.039990005 m. No terrain, curvature or material solver.

Independent two-ray fixture · 2.450 GHz, ht=1 m, hr=2 m, Γ=−1
Horizontal R	Direct path rd	Coherent path loss	Direct-path Friis loss
10 m	10.049875621 m	59.389182377 dB	60.274318647 dB
20 m	20.024984395 m	60.965235911 dB	66.262548635 dB
50 m	50.009999000 m	69.245047965 dB	74.212241826 dB
100 m	100.004999875 m	75.567267219 dB	80.231539182 dB
500 m	500.000999999 m	101.999529643 dB	94.210522368 dB

At R=100 m, rd=100.004999875 m, rr=100.044989880 m and Δr=0.039990005 m. The coherent loss is 75.567267219 dB, less than the direct free-space value near 80.23 dB. Constructive interference raises power locally; it does not create energy or define a whole-space power balance.

Common misconceptionFree space is always the pointwise optimistic result.

Coherent multipath can produce enhancement or deep cancellation in a particular direction. A reflected field is not a separate uncorrelated power to add.

Go deeperWhen the inverse-fourth-power asymptote appears

For Γ≈−1, smooth flat ground, R much larger than both heights and kΔr≪1, Δr≈2hₜhᵣ/R and 1−e^(−jkΔr)≈jkΔr. Then |H|≈hₜhᵣ/R² and power scales approximately as R⁻⁴. These restrictions explain the asymptote; they do not replace the coherent curve at arbitrary distances. Tse & Viswanath, §2.1.5 and Exercise 2.5 develop the ground-plane mechanism.

Two exactly equal in-phase complex amplitudes give four times one-path power; equal opposite amplitudes give zero. Those are algebraic limiting cases. Positive-height geometry does not necessarily create an exact zero. With Γ=0 the model recovers Friis at rd, not horizontal R.

Geometry explains a particular point. A site with incomplete geometry often needs a different kind of statement: a distribution.

07 / 10

Path loss is often a distribution

What does “5 dB of variation” describe?

Imagine measuring local mean received powers across an agreed set of corridor positions. Their central trend may follow distance while obstructions create residual differences. A log-distance model expresses that trend in dB; Gaussian residuals in dB correspond to lognormal power. The coefficients must belong to that population and calibration procedure.

$\begin{aligned}\mathrm{PL}_{\mathrm{med}}(d) &= \mathrm{PL}_{0} + 10n \log _{10}(\frac{d}{d_{0}}) \\ \mathrm{PL}_{\mathrm{sample}} &= \mathrm{PL}_{\mathrm{med}} + \sigma Z \\ P_{\mathrm{rx,dBm}} &= \mu - \sigma Z\end{aligned}$ d0=10 m; PL0=LFS(d0)=60.231104909 dB is assumed, not fitted. n=2.2, σ=5 dB and Z~N(0,1) are chosen fixture values. Positive Z increases loss.

At 100 m, median loss is 82.231104909 dB and μ=−73.938067181 dBm at R1-RX for state A. In dB, this Gaussian’s mean and median coincide. In watts, its arithmetic mean is the median power multiplied by exp[½(ln(10)σ/10)²]. The distinction matters whenever averages cross between linear and logarithmic units.

$\begin{aligned}F_P(p)&=\Phi\!\left(\frac{p-\mu}{\sigma}\right)\\q_{0.05}&=\mu-1.6448536269514722\,\sigma\\\text{outage}&=\Pr(P<P_{\mathrm{threshold}})\end{aligned}$ For σ>0, Φ is the standard normal CDF. A 95% power-exceedance level is the lower 5th CDF percentile. A probability under an assumed model is not confidence in a field estimate.

Golden default · analytic distribution at R1-RX
Statistic	Value	Consequence
Median power / SINR	−73.938067181 dBm / 15.648005968 dB	Median exceeds the hypothetical 8 dB SINR criterion.
Lower 5th percentile	−82.162335316 dBm	Power exceeded by 95%; below the −81.586073148 dBm threshold.
Upper 95th percentile	−65.713799046 dBm	The favorable tail, not 95% coverage power.
Strict analytic outage	6.305773693%	Fails the conditional ≤5% outage target.
Primary 4096-draw outage	6.567382813%	Seeded finite-sample illustration; no field confidence interval.

Each curve is cumulative population mass, not a confidence band. Curves use a bounded display grid; tables retain exact analytic and weighted-sample summaries.

Common misconceptionAdding a fade margin guarantees a percentile.

A reserve becomes a percentile only through a validated distribution, population and dependency model. A scalar allowance by itself establishes none of them. Neighboring corridor samples may share the same obstruction; more closely spaced samples do not automatically provide more independent evidence.

Keep these quantities separate
Quantity	What it represents / evidence needed
Median / intercept / slope	Central scenario trend; fitted domain and holdout error needed.
Shadow variation	Large-scale variation across a named population; estimate residual distribution and correlation distance.
Fast fading / orientation	Distinct mechanisms; neither is a hidden additional draw in this workbench.
Model-parameter uncertainty	Limited knowledge of n, σ, intercept and model form; quantify from calibration/validation.
Measurement and sample uncertainty	Instrument/calibration effects and finite effective sample count; not the displayed shadow spread.
Production reserve / deliberate margin	Chosen design allowances; not independent random variables without evidence.

Go deeperHow this sample can be reproduced exactly

p06-normal-samples-v1 uses nonzero xorshift32 state 0x1A2B3C4D, or alternate 0x6D2B79F5. Each draw applies s ^= s<<13; s ^= s>>>17; s ^= s<<5, then unsigned 32-bit conversion. u=(s+0.5)/2³² is strictly inside (0,1). Successive u1/u2 pairs produce √(−2 ln u1)cos(2πu2) first and the sine result second.

The first primary states are 3388403996, 3984204854, 2523572680 and 2838152257. N=1024 is the prefix of N=4096. An equal-weight empirical percentile is x[ceil(pN)−1], with p=0 choosing the minimum. CDF accumulates weighted mass; outage counts only powers strictly below threshold. Mixtures evaluate both states at every same Z with weight 1/(2N).

Analytic mixed quantiles use a bounded monotone solve with 10⁻⁸ dBm interval and 10⁻¹⁰ probability stopping tolerances. Zero σ uses exact step quantiles. An ideal 90° single-path linear-polarization null retains exact zero watts rather than a plotting floor.

Power variation alone does not describe how the channel changes a waveform. Delay and Doppler supply the next two scales.

08 / 10

Delay, Doppler, and coherence connect channel to waveform

Two channels deliver equal average power. Must they affect symbols equally?

Copies arriving at different delays can overlap differently within a symbol. Copies arriving from different directions at a moving receiver can also change relative phase over time. A power-only distribution contains neither a power-delay profile nor a Doppler spectrum.

$\begin{aligned}\bar{\tau} &= \frac{\sum p_{i}\tau _{i}}{\sum p_{i} } \\ \tau _{\mathrm{rms}} &= \sqrt{\frac{\sum p_{i}(\tau _{i}-\bar{\tau})^{2}}{\sum p_{i}}}\end{aligned}$ τi are delays in seconds and pi are linear powers. Normalize by Σpi, not by the number of taps. The example uses delays 0/100 ns and normalized powers 0.8/0.2.

Independent channel-to-waveform scales · local illustrative examples
Quantity / convention	Derived value	What it does not establish
Mean excess delay	20 ns	Not RMS delay spread or a receiver timing offset estimate.
RMS delay spread	40 ns	Not the full 100 ns tap separation.
Reciprocal teaching scale 1/(5τrms)	5.000000 MHz	A heuristic scale; coherence needs a specified correlation measure and threshold.
Maximum Doppler magnitude v/λ at 3 m/s	24.516960997 Hz	Not automatically the Doppler spread; path angles and motion directions matter.
Chosen reciprocal time scale 1/(2fD,max)	20.394044762 ms	A convention-dependent comparison, not a guaranteed coherence time.
Portfolio symbol interval	100 μs	40 ns is 0.04% of this interval; the receiver’s tolerance still requires its waveform and processing.

Tse & Viswanath §§2.3.1–2.3.2 explain the reciprocal relationships and the dependence on definitions. Their significant-path delay extent is distinct from the RMS moment used above. Their time-scale convention uses Doppler spread and a different numerical factor; no universal equality is implied by our chosen 1/(2fD,max).

Goldsmith’s 2020 draft, §3.3.3, pp. 99–100, describes the 0.2/τrms approximation for a 0.5 correlation convention and explains why the factor depends on the profile and definition. Apply that qualification to our 5 MHz teaching scale.

Go deeperWhy 5 MHz is not an exact coherence boundary for this profile

Under uncorrelated taps, define normalized frequency correlation as R(Δf)=0.8+0.2e^(−j2πΔf·100 ns). Its magnitude at 5 MHz is 0.6, and 0.6 is its minimum. If “coherence bandwidth” means the first |R|=0.5 crossing, this two-tap profile has no such crossing. The often-used 1/(5τrms) estimate associated with a rough 50% correlation convention cannot be treated as that exact threshold here. The moment calculation is exact; the 5 MHz number is only a named reciprocal teaching scale.

For a single path, Doppler shift also depends on the cosine of its angle to velocity. If all significant paths have the same shift, a common frequency correction may remove it without fixing any multipath delay. Relative shifts drive time variation of the combined magnitude.

With many unresolved small paths and approximately independent uniform phases, a complex Gaussian channel can have a Rayleigh envelope. Adding a dominant specular component leads to a Rician alternative. For a separate unit-mean Rayleigh-power example, Pr(|h|²<0.1)=1−e⁻⁰·¹≈9.516258%. This is not the office shadowing distribution, and no such extra fading draw is applied by the workbench. Chapter 2 §2.4.2 states those mechanisms and assumptions.

The installed antenna can change which of those paths are strong. That makes orientation part of the scenario population, not a forgotten final margin.

09 / 10

Orientation, polarization, body, mounting, and diversity

Would the average of two orientations represent either one?

State A keeps the baseline broadside gain, aligned linear polarization and no pattern penalty. State B is a named local synthetic variant: 6 dB less directional pattern response plus ideal 45° linear misalignment. PLF=cos²45°=0.5 adds 3.010299957 dB loss. The total state change is 9.010299957 dB, but its probability weight must remain explicit.

$F_{\mathrm{mixture}}(p) = 0.5F_{\mathrm{A}}(p) + 0.5F_{\mathrm{B}}(p)$ An equally weighted use population contains two complete state distributions. Apply each state’s own gain/polarization configuration before mixing probability.

Averaging component dB quantiles does not generally produce mixture quantiles. For our equal-weight, equal-σ Gaussian components the symmetry happens to put the mixture median halfway between the two component medians; the model still solves the CDF. That symmetry is not an algorithm for other weights, variances or tails.

Think about itSet σ to zero while retaining the 50/50 A/B mixture. Does every result become identical?

Answer

No. Each state collapses to one power, but the mixture retains two different point masses. Equality to the threshold is not outage. A custom 90° single-state polarization has exactly 0 W in this ideal model, with its full probability mass in outage for any positive signal threshold.

Common misconceptionOne antenna orientation represents the product.

Body proximity, mounting metal, cable attachment and rotation can change the pattern and polarization. A null in a required state should stay visible in the test matrix rather than being buried inside one arbitrary average margin.

The earlier 06.3 scalar integration model does not supply a complex channel or an installed vector pattern for this lesson. Nor do 06.4’s eight synthetic diversity samples calibrate our shadowing distribution. Carry the evidence identities forward, but do not silently promote one model’s illustrative outputs into another model’s field data.

Go deeperDiversity needs a joint distribution

Two antennas may offer different orientations or polarization responses, but a shared body shadow or blockage can affect both. Request joint signal, interference and noise behavior before computing selection or coherent-combining reliability. Marginal outage numbers alone cannot determine the benefit; neither can treating correlated margins as independent dB allowances.

We can now compare three complete conditional statements with the same ledger and receiver assumptions.

10 / 10

Build three node link distributions

Which model result would you defend, and which measurement would you request first?

Freeze the source and receiver defaults while changing only the declared scenario. All three are calculated illustrations at 100 m and 2.450 GHz, not “typical” site measurements. The office and hall intercepts are assumed free-space values at 10 m; no field-derived coefficient is supplied.

Read the default office median, lower 5th percentile and strict outage. Explain why the 95% target fails.
Change distance only and Apply. Check the ledger and lower tail together.
Change frequency with fixed gains. Explain separately what one or two fixed physical apertures would mean.
Restore the office, change only σ, and Apply. The same seeded normal sequence changes scale.
Select A/B 50/50 and Apply; inspect both complete component states and the mixed CDF.
Turn the nearby interferer off, then on. The received signal stays fixed; the threshold moves.
Choose two-ray, audit ht/hr, Γ and both ray lengths, then test a distance that fails the field-region screen.
Restore a named scenario and write its conditional validation request, including the still-unknown field population and confidence.

Calculate, compare, qualify

Link Distribution Workbench

Can the lower tail meet the proposed target? Change one assumption, apply it, and explain which population the result describes. These are synthetic scenarios, not surveyed environments.

Scenario preset · restores all inputs

1 · Choose the model and population

Propagation model

Product orientation population

Frequency (MHz)100…100000 MHz; step 1; default 2450.

Distance / horizontal R (m)0.1…100000 m; step 0.1; default 100.

Log-distance exponent n (dimensionless)1.2…6 dimensionless; step 0.1; default 2.2.

Shadow standard deviation σ (dB)0…15 dB; step 0.5; default 5.

Frequency edits keep the supplied gains fixed. n and σ are chosen illustrations; their application domain is 10–1000 m at 2450 MHz, with an assumed free-space reference at 10 m. They are not universal office or factory coefficients.

2 · Define the receiver screen and synthetic sample

Nearby interferer

Requested power exceedance

Supplied noise · R1-RX / 1 MHz (dBm)-150…-20 dBm; step 1; default -100.

Supplied interference · R1-RX / 1 MHz (dBm)-150…-20 dBm; step 1; default -90.

Required SINR · R3 screen (dB)-10…30 dB; step 0.5; default 8.

Standard-normal draws N

Reproducible seed

Antenna, feed and polarization assumptions

3 · Audit each port-to-space term

Conducted transmit power · R1-TX (dBm)-30…30 dBm; step 1; default 10.

Matched transmit feed loss · R1 → R2 (dB)0…10 dB; step 0.1; default 1.

Matched receive feed loss · R2 → R1 (dB)0…10 dB; step 0.1; default 1.

Directional realized TX gain (dBi)-20…30 dBi; step 0.1; default 10 log10(0.675), exact off-step value.

Directional realized RX gain (dBi)-20…30 dBi; step 0.1; default 2.

Largest TX dimension Dtx (m)0.01…2 m; step 0.01; default 0.1.

Largest RX dimension Drx (m)0.01…2 m; step 0.01; default 0.2.

Single-state polarizationThe fixed 50/50 mixture cannot be edited. A custom angle changes only the selected single state.

Custom linear misalignment ψ (°)Enable a custom single-state polarization to edit.

Two-ray geometry and complex reflection proxy

4 · Geometry is used only by two-ray

Transmit height ht (m)Only the two-ray model uses ground geometry.

Receive height hr (m)Only the two-ray model uses ground geometry.

Ground field-reflection magnitude (linear)Only the two-ray model uses ground geometry.

Ground reflection phase (°)Only the two-ray model uses ground geometry.

Flat ground, narrowband complex fields, e^(+jωt). Γ is an assumed field-reflection proxy with a declared phase; it is not a material coefficient generated from the selected frequency.

Illustrative engineering case · committed result

link-distribution-workbench/2.0 · p06-m05-link-v1. 2450 MHz; 100 m path distance; log-distance; orientation A. Real 50 Ω sources/loads at R1/R2. Fixed supplied directional realized gains include antenna mismatch once.

Conditional model result · field coverage unknown

Dtx/Drx = 0.1/0.2 m. Required ray length ≥ 2.000000 m under the conservative lesson screen. Geometric screen met; it is not proof of model accuracy.

Complete term and reference-plane ledger · no added mismatch or fading row
Term / plane	Committed term	Meaning and included effects
Conducted source · R1-TX	10.000000 dBm	Power supplied to the matched feed
Matched feed · R1-TX → R2-TX	−1.0 dB → 9.000000 dBm	Incident power at R2-TX; accepted/radiated totals are not independently supplied
Realized TX gain · R2-TX → S0	-1.706962272 dBi	Direction +x broadside for baseline A; EIRP in that direction 7.293038 dBm. Includes mismatch and dissipation; not TRP.
Selected channel term · S0	82.231104909 dB loss	Median PL0 = 60.231104909 dB at assumed d0 = 10 m; n = 2.2; σ = 5 dB.
Realized RX gain · S0 → R2-RX	2.000000 dBi	Separately supplied gain into real 50 Ω includes receive mismatch. Available receive power needs gain before that mismatch; not supplied separately.
Matched receive feed · R2-RX → R1-RX	−1.0 dB	Deduct once from delivered R2-RX power
Polarization / pattern states · S0	A: 0° / 0 dB pattern penalty	Add 10 log10(PLF) and subtract the separate pattern penalty; no average-dB mixture.
Noise + interference · R1-RX	-100.000000 dBm + -90.000000 dBm → -89.586073 dBm	Add watts over the same supplied 1 MHz analysis bandwidth. Receiver-added noise already included in supplied noise.
Receiver decision screen · R3	-81.586073 dBm required at R1-RX	8 dB SINR required in an assumed linear same-band receiver. R3 is the decision boundary; powers are referred back to R1-RX.

Analytic median receive power · R1-RX: -73.938067 dBm
Analytic median same-band SINR · R3 screen: 15.648006 dB
Lower 5th percentile · 95% exceedance level: -82.162335 dBm
Strict model outage · P < threshold: 6.305774%

Conditional 95% target not met. The allowed outage is 5%; this distribution gives 6.305774%. This is a model probability, not confidence in product coverage.

Orientation populations · deterministic state, conditional shadow distribution
State and weight	PLF / pattern loss	R1-RX median	Outage for this state
A · 100.000000%	1.000000000 linear at 0° / 0 dB	-73.938067 dBm	6.305774%

Each curve is cumulative population mass, not a confidence band. Curves use a bounded display grid; tables retain exact analytic and weighted-sample summaries.

Analytic versus sampled quantiles · received power at R1-RX
CDF percentile	Power exceeded by	Analytic · dBm	Finite sample · dBm
1%	99% (continuous case)	-85.569807 dBm	-85.808059 dBm
5%	95% (continuous case)	-82.162335 dBm	-82.251264 dBm
10%	90% (continuous case)	-80.345825 dBm	-80.327373 dBm
50%	50% (continuous case)	-73.938067 dBm	-73.937531 dBm
90%	10% (continuous case)	-67.530309 dBm	-67.733330 dBm
95%	5% (continuous case)	-65.713799 dBm	-65.646055 dBm
99%	1% (continuous case)	-62.306328 dBm	-62.615662 dBm

CDF / outage / sample audit · no field confidence interval
Quantity	Analytic	Seeded finite sample
CDF at threshold: P ≤ threshold	6.305774%	6.567383%
Strict outage: P < threshold	6.305774%	6.567383%
Population	1 orientation state(s); shadow σ = 5 dB	4096 Z draws; 4096 weighted rows; each row mass 1/4096
Method	Gaussian marginal / exact point masses / weighted mixture	p06-normal-samples-v1; xorshift32 → Box–Muller z0 then z1; primary seed 0x1A2B3C4D
Mean in linear power	7.834592609e-11 W at R1-RX	The arithmetic mean W generally differs from the W value at the dB median.

For mixtures, both states use every same Z with weight 1/(2N). Pairing supports controlled comparisons; 2N rows do not mean 2N independent shadow observations. Seed/count changes affect sampled results, never analytic probability. Zero σ removes shadow spread only; orientation states may still differ. A custom 90° linear mismatch is exactly 0 W, an ideal polarization null; no finite dBm floor replaces it.

One change at a time

Each row starts from the committed configuration and reuses the same normal draws. These effects interact; do not add the row changes as independent margins.

Controlled sensitivity · analytic R1-RX lower-tail power and strict outage
Only change	Requested lower percentile	Outage	Decision domain
Distance × 2	-88.784995 dBm	41.875903%	conditional
Fixed-gain frequency × 2	-88.182935 dBm	37.240845%	extrapolation
Shadow σ + 1 dB	-83.807189 dBm	10.121345%	conditional
A/B 50/50	-89.384806 dBm	33.521155%	conditional
Interferer off	-82.162335 dBm	0.015170%	conditional

Compare a CDF · view only

Each curve is cumulative population mass, not a confidence band. Distance × 2: only the named input changes; the same Z sequence is reused. Curves use a bounded display grid; tables retain exact analytic and weighted-sample summaries.

Frequency comparison: for free space with both gains fixed, doubling f changes receive power by −6.020600 dB. One fixed efficient electrically large aperture offsets this with +6.020600 dB gain; two such apertures give a net +6.020600 dB. Those are alternative antenna constraints; the gain controls above never apply aperture scaling automatically.

Write the conditional validation request

Record this committed 2450 MHz, 100 m log-distance case; TX/RX reference planes and gain convention; orientation A; the 95% exceedance target; and the supplied 1 MHz noise/interference screen. Then name the actual site and installation population, correlation model, independent holdout locations/times, measurement uncertainty and the evidence that could refute the assumed tail.

Model result: 6.305774% strict outage. Field result, confidence, parameter calibration and receiver packet criterion: unknown. Detailed measurement execution belongs to later material.

Three frozen scenario records

These records remain canonical when the workbench is edited. Tables show all seven analytic and primary-sample quantiles; 95% exceedance means the lower 5th percentile. The deterministic outdoor record is a point mass.

Illustrative · p06-m05-link-v1 · office

Office corridor

log-distance; n=2.2, shadow σ=5 dB; orientation A; +10 dBm R1-TX, 1/1 dB feeds, −1.706962272/+2 dBi realized gains. N/I=−100/−90 dBm over 1 MHz at R1-RX; 8 dB hypothetical R3 SINR requirement.

Office corridor · frozen model and empirical decision
Statistic	Analytic model	Primary sample
R1-RX median	-73.938067 dBm	-73.937531 dBm
Component medians	A: -73.938067 dBm	Both states share each Z in a mixture
Strict outage	6.305774%	6.567383%
95% exceedance target	Not met	4096 rows; 4096 synthetic Z draws; no field confidence

Office corridor · frozen analytic and empirical quantiles · received power at R1-RX
CDF percentile	Power exceeded by	Analytic · dBm	Finite sample · dBm
1%	99% (continuous case)	-85.569807 dBm	-85.808059 dBm
5%	95% (continuous case)	-82.162335 dBm	-82.251264 dBm
10%	90% (continuous case)	-80.345825 dBm	-80.327373 dBm
50%	50% (continuous case)	-73.938067 dBm	-73.937531 dBm
90%	10% (continuous case)	-67.530309 dBm	-67.733330 dBm
95%	5% (continuous case)	-65.713799 dBm	-65.646055 dBm
99%	1% (continuous case)	-62.306328 dBm	-62.615662 dBm

Dominant uncertainty / next evidence: Fit the intercept, slope, spatial correlation and residual tails on held-out corridors. Field uncertainty and confidence are unknown. The selected n and σ are teaching choices, restricted to 10–1000 m at 2.450 GHz; validate the environmental model before assigning a product probability.

Illustrative · p06-m05-link-v1 · outdoor

Outdoor LOS

free-space; n=not used, shadow σ=0 dB; orientation A; +10 dBm R1-TX, 1/1 dB feeds, −1.706962272/+2 dBi realized gains. N/I=−100/−90 dBm over 1 MHz at R1-RX; 8 dB hypothetical R3 SINR requirement.

Outdoor LOS · frozen model and empirical decision
Statistic	Analytic model	Primary sample
R1-RX median	-71.938067 dBm	-71.938067 dBm
Component medians	A: -71.938067 dBm	Both states share each Z in a mixture
Strict outage	0.000000%	0.000000%
95% exceedance target	Conditionally met	4096 rows; 4096 synthetic Z draws; no field confidence

Outdoor LOS · frozen analytic and empirical quantiles · received power at R1-RX
CDF percentile	Power exceeded by	Analytic · dBm	Finite sample · dBm
1%	99% (continuous case)	-71.938067 dBm	-71.938067 dBm
5%	95% (continuous case)	-71.938067 dBm	-71.938067 dBm
10%	90% (continuous case)	-71.938067 dBm	-71.938067 dBm
50%	50% (continuous case)	-71.938067 dBm	-71.938067 dBm
90%	10% (continuous case)	-71.938067 dBm	-71.938067 dBm
95%	5% (continuous case)	-71.938067 dBm	-71.938067 dBm
99%	1% (continuous case)	-71.938067 dBm	-71.938067 dBm

Dominant uncertainty / next evidence: Verify clear LOS, field region, pointing, ground geometry and the actual interferer. Field uncertainty and confidence are unknown. Verify the assumed direct path before claiming outdoor performance.

Illustrative · p06-m05-link-v1 · hall

Machine hall / on metal

log-distance; n=3.2, shadow σ=7 dB; orientation mixture; +10 dBm R1-TX, 1/1 dB feeds, −1.706962272/+2 dBi realized gains. N/I=−100/−90 dBm over 1 MHz at R1-RX; 8 dB hypothetical R3 SINR requirement.

Machine hall / on metal · frozen model and empirical decision
Statistic	Analytic model	Primary sample
R1-RX median	-88.443217 dBm	-88.370428 dBm
Component medians	A: -83.938067 dBm; B: -92.948367 dBm	Both states share each Z in a mixture
Strict outage	78.964465%	79.382324%
95% exceedance target	Not met	8192 rows; 4096 synthetic Z draws; no field confidence

Machine hall / on metal · frozen analytic and empirical quantiles · received power at R1-RX
CDF percentile	Power exceeded by	Analytic · dBm	Finite sample · dBm
1%	99% (continuous case)	-107.383668 dBm	-107.892403 dBm
5%	95% (continuous case)	-102.110619 dBm	-102.150279 dBm
10%	90% (continuous case)	-99.212020 dBm	-99.022366 dBm
50%	50% (continuous case)	-88.443217 dBm	-88.370428 dBm
90%	10% (continuous case)	-77.674414 dBm	-77.895799 dBm
95%	5% (continuous case)	-74.775815 dBm	-75.102495 dBm
99%	1% (continuous case)	-69.502766 dBm	-69.564649 dBm

Dominant uncertainty / next evidence: Measure installed vector patterns and joint orientation, blockage and interference populations. Field uncertainty and confidence are unknown. The selected n and σ are teaching choices, restricted to 10–1000 m at 2.450 GHz; validate the environmental model before assigning a product probability.

Turn the result into a validation campaign

Propose five named node prototypes and five gateways, with crossed A/B orientations in machine hall, office corridor and outdoor LOS. Freeze assembly, mounting metal, feed/cable routes, coordinate frames and interferer operating states. Survey distances and heights; retain timestamps and positions with every synchronized signal/interference record.

Estimate local means and correlation from pilot data, then choose independent site/time blocks and a sample size appropriate to the requested tail. Keep calibration data separate from held-out validation. Report per-state distributions before use-weighted mixtures; include production spread, instrument calibration, receiver mode and confidence method. Thousands of adjacent RSSI readings or a few successful packets cannot prove coverage.

p06-evidence-map-v1 · complete local snapshot; missing evidence remains unknown
Stable ID / owner / question	Specimen, state and plane	Quantity / statistic / evidence	Decision and next evidence
M01-A · 06.1 Does good match prove a good antenna?	p06-m01-power-pattern-v1 2.450 GHz only Analytic free-space stand-in, O1 specimen frame. No physical build or installation prediction. Right-handed specimen x/y/z; θ from +z, φ from +x toward +y. Total scalar power; vector co/cross/AR unknown. R2: real 50 Ω, matched source; S0: radiation. No upstream feed loss. Realized gain includes mismatch and antenna dissipation once.	S11 −10 dB; ηrad 50%; ηtot 45%; Prad 0.45 mW; Greal(+x) 0.675 (−1.706962 dBi) Deterministic single fixture, full-sphere total and named +x direction; no unit population. Illustrative · port-pattern-ledger/2.0; independent anchors in antenna-golden.json 0 dBm incident; dipole-like scalar shape. Model arithmetic is exact to stated numerical tolerance; physical uncertainty unknown.	Port acceptance is known within the illustration. Product adequacy is unknown. Next: R2 match plus independent radiation-efficiency and vector-pattern evidence for actual N1 states.
M01-INSTALL-UNKNOWN · 06.1 Will the node work in its intended states?	N1 product requirement; no specimen evidence supplied 2.400–2.500 GHz requirement band Free space, metal-machine plate, plastic enclosure, hand phantom; O1 +z upright, O2 rotated 90° about +y. Drawings/phantom specification still required. Right-handed specimen x/y/z; θ from +z, φ from +x toward +y. Total scalar power; vector co/cross/AR unknown. R2: real 50 Ω, matched source; S0: radiation. No upstream feed loss. Realized gain includes mismatch and antenna dissipation once.	Match, efficiency, gain, vector pattern, and channel/link performance: unknown Five proposed prototypes; no observed population or uncertainty yet. Illustrative · Illustrative engineering case / proposed evidence plan Gateway may use two antennas; no diversity or channel benefit assigned. Missing evidence is unknown.	No installed antenna, channel, or product compliance decision. Next: Freeze mechanics and coordinates; family selection in 06.2, installation in 06.3, channel in 06.5, measurement execution in 06.6.
M05-LEDGER · 06.5 Can each loss be assigned once?	p06-m05-link-v1 2.450 GHz; single-frequency local fixture Local synthetic node/gateway channel case at 100 m; no supplied measured antenna installation or site geometry. S0 baseline TX +x broadside, θ=90°, φ=0° from 06.1; co eθ / cross eφ. RX looks back along the path in the common transverse linear basis. B is an assumed 6 dB pattern penalty plus 45° mismatch, not an imported 3D rotation. R1-TX +10 dBm → 1 dB matched feed → R2-TX; Greal,TX = 10log10(.675) dBi → S0 channel/PLF/pattern → Greal,RX = +2 dBi into real 50 Ω R2-RX → 1 dB feed → R1-RX. R3 decision uses input-referred N/I over 1 MHz. Both gains include mismatch once.	Greal,TX=−1.706962271689751 dBi; baseline directional EIRP=7.293037728 dBm; threshold=−81.586073148 dBm at R1-RX. Analytic power CDF and strict P<threshold outage; p06-normal-samples-v1, primary seed 0x1A2B3C4D, N=4096; paired A/B uses 8192 rows but 4096 shadow realizations. No field confidence. Illustrative · link-distribution-workbench/2.0; p06-m05-link-v1; independent Python oracle Fictional engineering case; supplied N=−100 and I=−90 dBm, same 1 MHz at R1-RX; hypothetical required SINR 8 dB at R3. No standards PER, fast-fading draw, design margin or measured uncertainty.	Use Candidate A’s broadside gain factor without repeating mismatch or efficiency. Accepted/radiated totals for this link remain unmeasured. Next: Matched conducted powers and feed loss, calibrated installed directional realized gains and compatible receiver noise/interference planes.
M05-OUTDOOR · 06.5 What does an aligned free-space baseline establish?	p06-m05-link-v1 2.450 GHz; single-frequency local fixture Outdoor-LOS local variant; clear far-field LOS at 100 m, state A only. No ground or obstruction terms included. S0 baseline TX +x broadside, θ=90°, φ=0° from 06.1; co eθ / cross eφ. RX looks back along the path in the common transverse linear basis. B is an assumed 6 dB pattern penalty plus 45° mismatch, not an imported 3D rotation. R1-TX +10 dBm → 1 dB matched feed → R2-TX; Greal,TX = 10log10(.675) dBi → S0 channel/PLF/pattern → Greal,RX = +2 dBi into real 50 Ω R2-RX → 1 dB feed → R1-RX. R3 decision uses input-referred N/I over 1 MHz. Both gains include mismatch once.	PL=80.231104909 dB; R1-RX P=−71.938067181 dBm; σ=0; analytic and sample outage=0. Analytic power CDF and strict P<threshold outage; p06-normal-samples-v1, primary seed 0x1A2B3C4D, N=4096; paired A/B uses 8192 rows but 4096 shadow realizations. No field confidence. Illustrative · link-distribution-workbench/2.0; p06-m05-link-v1; independent Python oracle Fictional engineering case; supplied N=−100 and I=−90 dBm, same 1 MHz at R1-RX; hypothetical required SINR 8 dB at R3. No standards PER, fast-fading draw, design margin or measured uncertainty.	Threshold clears in this deterministic model; actual outdoor coverage unknown. Next: Survey LOS/ground geometry, installed antenna states and nearby interferer; compare power versus distance to the conditional baseline.
M05-OFFICE · 06.5 Does a positive median margin meet 95% exceedance?	p06-m05-link-v1 2.450 GHz; single-frequency local fixture Office corridor local variant; A only, 100 m; assumed free-space PL0 at d0=10 m, applicability 10–1000 m at 2.450 GHz. S0 baseline TX +x broadside, θ=90°, φ=0° from 06.1; co eθ / cross eφ. RX looks back along the path in the common transverse linear basis. B is an assumed 6 dB pattern penalty plus 45° mismatch, not an imported 3D rotation. R1-TX +10 dBm → 1 dB matched feed → R2-TX; Greal,TX = 10log10(.675) dBi → S0 channel/PLF/pattern → Greal,RX = +2 dBi into real 50 Ω R2-RX → 1 dB feed → R1-RX. R3 decision uses input-referred N/I over 1 MHz. Both gains include mismatch once.	Assumed n=2.2, σ=5 dB; μ=−73.938067181 dBm; q05=−82.162335316 dBm; analytic outage=6.305773693%; primary empirical=6.567382813%. Analytic power CDF and strict P<threshold outage; p06-normal-samples-v1, primary seed 0x1A2B3C4D, N=4096; paired A/B uses 8192 rows but 4096 shadow realizations. No field confidence. Illustrative · link-distribution-workbench/2.0; p06-m05-link-v1; independent Python oracle Fictional engineering case; supplied N=−100 and I=−90 dBm, same 1 MHz at R1-RX; hypothetical required SINR 8 dB at R3. No standards PER, fast-fading draw, design margin or measured uncertainty.	Conditional 95% model target not met. Synthetic draw count is not field confidence. Next: Fit intercept/exponent/residual distribution and correlation to an agreed corridor population; validate on independent holdout corridors/time blocks.
M05-HALL · 06.5 What does an orientation mixture reveal?	p06-m05-link-v1 2.450 GHz; single-frequency local fixture Machine-hall/on-metal local variant; A/B 50/50; same 100 m and receiver settings. 6 dB B penalty is chosen, not a metal model or 06.3 perturbation output. S0 baseline TX +x broadside, θ=90°, φ=0° from 06.1; co eθ / cross eφ. RX looks back along the path in the common transverse linear basis. B is an assumed 6 dB pattern penalty plus 45° mismatch, not an imported 3D rotation. R1-TX +10 dBm → 1 dB matched feed → R2-TX; Greal,TX = 10log10(.675) dBi → S0 channel/PLF/pattern → Greal,RX = +2 dBi into real 50 Ω R2-RX → 1 dB feed → R1-RX. R3 decision uses input-referred N/I over 1 MHz. Both gains include mismatch once.	Assumed n=3.2, σ=7 dB; component medians −83.938067181 / −92.948367138 dBm; solved mixture median −88.443217159 dBm; analytic outage=78.964464910%. Analytic power CDF and strict P<threshold outage; p06-normal-samples-v1, primary seed 0x1A2B3C4D, N=4096; paired A/B uses 8192 rows but 4096 shadow realizations. No field confidence. Illustrative · link-distribution-workbench/2.0; p06-m05-link-v1; independent Python oracle Fictional engineering case; supplied N=−100 and I=−90 dBm, same 1 MHz at R1-RX; hypothetical required SINR 8 dB at R3. No standards PER, fast-fading draw, design margin or measured uncertainty.	Neither a mean orientation penalty nor this uncalibrated mixture supports a universal range claim. Next: Installed pattern/polarization states and their use weights; synchronized joint signal/interference records with blockage, machine operation and temporal/spatial registration.
M05-TWO-RAY · 06.5 Can free space be beaten at a point?	p06-m05-link-v1 2.450 GHz; single-frequency local fixture Separate two-ray local variant, state A; Dtx/Drx=0.10/0.20 m; conservative far-field screen met. S0 baseline TX +x broadside, θ=90°, φ=0° from 06.1; co eθ / cross eφ. RX looks back along the path in the common transverse linear basis. B is an assumed 6 dB pattern penalty plus 45° mismatch, not an imported 3D rotation. R1-TX +10 dBm → 1 dB matched feed → R2-TX; Greal,TX = 10log10(.675) dBi → S0 channel/PLF/pattern → Greal,RX = +2 dBi into real 50 Ω R2-RX → 1 dB feed → R1-RX. R3 decision uses input-referred N/I over 1 MHz. Both gains include mismatch once.	ht=1 m, hr=2 m, R=100 m; Γ=1∠180°; rd=100.004999875 m, rr=100.044989880 m; coherent path loss=75.567267219 dB. One deterministic narrowband flat-ground geometry. No probability assigned. Illustrative · link-distribution-workbench/2.0; p06-m05-link-v1; independent Python oracle Fictional engineering case; supplied N=−100 and I=−90 dBm, same 1 MHz at R1-RX; hypothetical required SINR 8 dB at R3. No standards PER, fast-fading draw, design margin or measured uncertainty.	Constructive local interference can increase received power. This does not create power or establish a whole-space energy budget. Next: Geometry, ground polarization/reflection, roughness, angle-dependent vector patterns and additional paths.
M05-VALIDATION · 06.5 What evidence would make a conditional link statement useful?	Proposed N1/gateway field campaign; unit and drawing revisions pending 2.450 GHz; single-frequency local fixture Freeze node/gateway mechanics, on-metal separation, orientation registrations, nearby interferer modes and receiver operating mode. Actual coordinates/sites still unknown. S0 baseline TX +x broadside, θ=90°, φ=0° from 06.1; co eθ / cross eφ. RX looks back along the path in the common transverse linear basis. B is an assumed 6 dB pattern penalty plus 45° mismatch, not an imported 3D rotation. R1-TX +10 dBm → 1 dB matched feed → R2-TX; Greal,TX = 10log10(.675) dBi → S0 channel/PLF/pattern → Greal,RX = +2 dBi into real 50 Ω R2-RX → 1 dB feed → R1-RX. R3 decision uses input-referred N/I over 1 MHz. Both gains include mismatch once.	Proposed ≥95% R1-RX power exceedance above the stated threshold, for a pre-agreed population; field result and confidence unknown. Five proposed nodes × five gateways; crossed A/B and three environments. Independent site/time blocks, sample size and confidence method must be agreed from measured dependence, not 4096 synthetic draws. Requirement · link-distribution-workbench/2.0; p06-m05-link-v1; independent Python oracle Requirement ≠ result. Spatial/temporal correlation, model error, measurement uncertainty and deliberate reserves remain separate.	Collect evidence before claiming range. A confidence-bound acceptance rule requires an agreed population and justified effective sample size. Next: Calibrated R1/R2 powers/gains, synchronized RSSI/SINR and packet records, timestamps/positions, uncertainty budget, held-out sites and a documented criterion appropriate to the eventual waveform.

A complete conditional statement

For the fictional 2.450 GHz office corridor at 100 m, state A, the declared real 50 Ω R1/R2 ledger and assumed n=2.2/σ=5 dB distribution predict 93.694226307% of received powers at or above −81.586073148 dBm. That threshold represents an 8 dB R3 SINR screen with the supplied 1 MHz noise/interference powers. It misses the proposed 95% target. Field coverage, confidence and eventual packet performance are unknown.

No universal range specification follows from these three examples. They tell us which assumptions to test and which evidence the next measurement must supply.

Ungraded review

Check your understanding

Answer each question in your own words, then reveal the model answer.

01At fixed antenna gains and 2.450 GHz, what changes between 10 m and 100 m in free space?
Model answer
Free-space basic transmission loss rises exactly 20 dB, from 60.231104909 to 80.231104909 dB. With every other ledger term fixed, R1-RX power drops 20 dB, a factor of 100. Both points still need LOS, polarization and field-region validity; this scaling is not a range guarantee.
02A ledger subtracts mismatch and radiation efficiency after using directional realized gains. Repair it.
Model answer
Remove those extra deductions: the supplied realized gains already include antenna mismatch and dissipation at their real 50 Ω references. Keep the R1-to-R2 matched feed losses, the selected channel term, polarization factor and any separate declared orientation-pattern change once. R3 remains the decision boundary; noise/interference here are referred back to R1-RX.
03Can the office n=2.2 and σ=5 dB be used for a 60 GHz path through wet vegetation?
Model answer
Only as labelled algebraic extrapolation; calibration is unknown. The supplied local fixture applies over 10–1000 m at 2.450 GHz and has no field-derived coefficients. Vegetation, material, weather, polarization and antenna constraints require their own applicable evidence/models. An ITU catalogue scope is a routing aid, not a coefficient table or validation.
04Why can a farther two-ray point be stronger than a nearer one?
Model answer
The direct and reflected complex fields change relative phase with geometry. A farther point can leave a destructive-interference null. Sum fields before taking squared magnitude and replace the path term with the result. Γ=0 recovers Friis at the slant direct length rd. A nonmonotone curve cannot support an unqualified one-number range inversion.
05The office median clears the threshold by 7.648006 dB. Does it meet 95% exceedance?
Model answer
No. At R1-RX the lower 5th CDF percentile is −82.162335316 dBm, below the −81.586073148 dBm threshold. Strict analytic outage is 6.305773693%, above the allowed 5%. The seeded 6.567382813% sample outage is a finite illustrative estimate, not a confidence interval. With σ=0, equality meets the threshold; orientation mixtures can still contain multiple point masses.
06Write a complete conditional 100 m statement and the evidence needed before publishing a range specification.
Model answer
For the fictional 2.450 GHz office corridor at 100 m, state A, the stated real 50 Ω R1/R2 ledger, assumed PL0 at 10 m with n=2.2/σ=5 dB, and N/I=−100/−90 dBm over the same 1 MHz at R1-RX, the model predicts 93.694226307% of received powers at or above the threshold for an 8 dB R3 SINR screen. This is not 95% coverage and has no field confidence. Specify sites, mechanics, orientation weights, interferer modes, correlation/effective sample size, calibration/measurement uncertainty, holdout data and the eventual waveform criterion before assessing a product requirement.

References and further reading

Principles, models and access

Access/status checked 7 September 2026. Model link-distribution-workbench/2.0; fixture p06-m05-link-v1; sampler p06-normal-samples-v1; evidence snapshot p06-evidence-map-v1. All scenario coefficients, orientation penalties, receiver powers and criteria are deliberate teaching inputs. The Python oracle independently checks the fixture; it does not validate a physical product.

ITU-R P.525-5, Calculation of free-space attenuation (11/2024). In force. Annex §§2.2–2.3, equations (3)–(6) actually read for flux, aperture and basic free-space loss. The implementation derives the MHz/km constant from exact c.
ITU-R P.676-13, Attenuation by atmospheric gases and related effects (08/2022). In force. Scope and Annex 1 §1 read for atmospheric inputs and line-by-line model routing. No gas attenuation algorithm or coefficient table is implemented.
Current ITU-R P-series catalogue. P.526-16 (11/2025), P.1238-13 (09/2025), P.2108-1 (09/2021) and P.838-3 (03/2005) official status pages checked. Catalogue scopes route diffraction, indoor, clutter and rain questions; their full normative models were not read or implemented.
David Tse and Pramod Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. Author-hosted Chapter 2, §§2.1.5–2.1.7, 2.3.1–2.3.2 and 2.4.2 actually read for ground reflection, shadowing, delay/Doppler and Rayleigh/Rician distinctions. Their order-of-magnitude conventions are distinguished from our RMS moments and local reciprocal scales.
Andrea Goldsmith, Wireless Communications, draft of second edition, Chapters 1–7 (8 February 2020). Author-hosted draft, §3.3.3, pp. 99–100 read on 8 September 2026 for the profile-dependent 0.2/τrms correlation-bandwidth approximation. This draft identity is not a claim about the latest published edition.
C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016. Official edition/contents verified. Friis, gain, aperture and polarization foundations are further reading; the full text was not accessed. No unseen book passage or clause is asserted as evidence.
IEEE 145-2025, IEEE Standard for Definitions of Terms for Antennas. Official active status and public scope consulted; published 31 March 2026. The full normative text was not accessed; the local power and coordinate conventions are stated explicitly.