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Wireless Signal Point Processes with a Reflecting Wall

H. Paul Keeler, George C. Alexandropoulos

arXiv:2610.34101v1cs.ITmath.IT

TL;DR

A reflecting wall pairs each wireless transmitter with a mirror image, raising the question of what signal point processes result when paired powers are kept separate or recombined. The paper derives mean measures from threshold regions and compares the two treatments, finding a large-threshold ratio independent of density and wall distance but generally dependent on paired fading. Under common fading, the limit depends only on the reflection coefficient and path-loss exponent; Poisson-transmitter results also characterize paired counts and the strongest path.

  • Problem

    A fixed reflecting wall creates paired direct and reflected paths, but the resulting signal point processes under separate and recombined power treatments need characterization.

    Source: §I · p. 1, §V · p. 12

  • Method

    The paper derives threshold-region mean measures using Campbell’s theorem, then analyzes separate and recombined signals and Poisson-transmitter count and strongest-path properties.

    Source: A. Random transmitter field · p. 3, A. Signals kept separate · p. 5, B. Signals recombined · p. 6

  • Results

    At large thresholds, the recombined-to-superposed mean-measure ratio is independent of transmitter density and wall distance but generally depends on joint fading, while common fading leaves a limit determined only by reflection coefficient and path-loss exponent.

    Source: C. Bounds and asymptotics · p. 7, §V · p. 12

  • Takeaways & Limitations

    The fixed-wall model isolates mirror-geometry dependence and provides a reference case for reflected wireless signals.

    Source: §V · p. 12

    The fixed specular wall does not model programmable surfaces.

    Source: §V · p. 12

Abstract

from arXiv · show

A long reflecting wall gives every wireless transmitter a mirror image, pairing its direct and reflected paths. Motivated by intelligent surfaces, we study the resulting signal point processes under two treatments: keeping the powers apart and superposing their point processes, or adding the powers transmitter by transmitter to form a recombined process. A threshold on inverse received power selects a region of transmitter locations, so Campbell's theorem reduces the mean measures to area calculations without assuming Poisson transmitters. For power-law path loss and paired fading marks, we derive the ratio of the recombined and superposed mean measures at large thresholds. The limit is independent of transmitter density and wall distance, but generally depends on the joint fading distribution. Under common fading, that dependence disappears and the limit depends only on the reflection coefficient and path-loss exponent. For Poisson transmitters, we derive paired count fluctuations and convergence of the count ratio. We also calculate the probability that the strongest individual path is direct. When the wall passes through the receiver, this probability has a closed form and the strongest received power has an exact Fr\'echet distribution. These results provide a reference model for wireless networks with reflecting surfaces.

I. INTRODUCTION

The paper models how a fixed reflecting wall pairs each wireless transmitter with a mirror image, and asks what signal point processes result when paired powers are kept separate or recombined. It uses an idealized one-wall setting to isolate this geometry from programmable-surface effects.

I. INTRODUCTION

  • Keeping paths separate superposes their signal processes, whereas recombination adds each transmitter’s direct and reflected powers before recording one signal point.A threshold on inverse received power selects transmitter locations, allowing Campbell’s theorem to reduce mean measures to geometric area calculations.Source: §I · p. 1

A. Related work

  • The fixed-wall model complements earlier reflection models by giving every transmitter an exact, deterministic mirror image and retaining its paired direct and reflected powers.Unlike the related wave-level model of Keeler [1], it studies received powers from a random field of transmitters rather than coherent reflections from one transmitter between two walls.Source: §I · p. 1, A. Related work · p. 2

II. GEOMETRIC MODEL AND MEAN MEASURES

  • The model uses an infinite flat specular wall with constant power reflection coefficient κ and power-law path loss as a standard case.The reflected path is represented by the straight path from the virtual transmitter, whose image distance is never shorter than the direct distance; powers are combined noncoherently or after phase averaging, not as phase-locked fields.Source: §II · pp. 2–3, Figure 1 · p. 3

A. Random transmitter field

The transmitter field has constant density on the half-plane beside the wall, without an initial Poisson assumption. Reflection determines a coupled image field, and inverse received powers form the direct signal process whose mean measure is obtained from threshold-selected regions.

A. Random transmitter field

  • Constant density λ means each bounded region B has expected transmitter count λ|B|; the mean-measure results here do not require Poisson transmitters.The Poisson assumption is introduced later when distributional statements are needed.Source: A. Random transmitter field · p. 3
  • Each real transmitter determines one mirror image across the wall, so the image process has the same constant density but is strongly coupled to the real process.Independent positive fading marks are attached to transmitters, and the direct received power is the mark times the path-loss function.Source: A. Random transmitter field · p. 3

B. Direct signals

  • Mapping each direct received power to its inverse creates a positive-real-line signal process, where stronger signals lie closer to zero.Its threshold count has a mean measure derived by counting transmitters in the threshold-selected region; for power-law path loss, the relevant radius is determined by the fading mark and exponent.Source: B. Direct signals · p. 4

1) Transmitters in the half-plane:

For transmitters restricted to the half-plane, the direct-signal mean measure is obtained by averaging the area of the receiver-centered threshold region clipped by the wall. This result applies to constant-density transmitter processes without requiring Poisson locations.

1) Transmitters in the half-plane:

  • The mean measure is λ times the expected area of the receiver-centered disk clipped to the transmitter half-plane, evaluated at the fading-dependent threshold radius.Campbell’s theorem gives this expected-count result for constant-density processes with independent, identically distributed positive marks, without a Poisson assumption.Source: 1) Transmitters in the half-plane · p. 4
  • With constant fading, the general mean-measure expression simplifies; under power-law path loss, the inverse path-loss radius is the threshold-scaled fading mark raised to 1/β.The power-law mean measure therefore uses the clipped-area expression with that radius.Source: 1) Transmitters in the half-plane · pp. 4–5
  • The clipped disk’s area depends on the wall position: when the wall passes through the receiver the region is a half-disk, and moving the wall away increases the area until the full disk lies on the transmitter side.The complementary area on the other side of the wall is given by a closed-form expression.Source: 1) Transmitters in the half-plane · pp. 4–5

C. Reflected signals and virtual transmitters

Mirror reflection maps each real transmitter to a paired image process with the same density, while the fading marks on paired paths may be dependent. Campbell’s theorem gives the reflected signal process’s mean measure without requiring Poisson transmitters.

C. Reflected signals and virtual transmitters

  • Mirror reflection preserves area, so the image transmitters have constant density λ on the opposite half-plane, although their process is determined by the real transmitters.This mapping does not require a Poisson assumption.Source: C. Reflected signals and virtual… · p. 5
  • Fading-mark pairs are identically distributed across transmitters and independent of transmitter locations, but the two marks within a pair may be dependent.The reflection coefficient κ is kept separate from the reflected fading mark F′.Source: C. Reflected signals and virtual… · p. 5
  • The reflected inverse powers form a signal point process whose mean measure is obtained from the reflected threshold region and the transmitter density.For zero reflection coefficient, the reflected process is empty and its mean measure is zero.Source: C. Reflected signals and virtual… · p. 5

III. SEPARATE AND RECOMBINED SIGNALS

The section contrasts keeping direct and reflected paths as separate points with adding their received powers per transmitter before forming one point. These two rules produce distinct mean-measure geometries, while common fading also constrains which path can be strongest.

III. SEPARATE AND RECOMBINED SIGNALS

  • A reflected path that misses the threshold individually can still contribute when its paired recombined signal crosses the threshold.The two separate points may both lie beyond the threshold while the recombined inverse power falls within it.Source: §III · p. 5

A. Signals kept separate

  • Superposition retains both inverse-power points from each transmitter, and its mean measure is the sum of the direct and reflected mean measures even when the processes are dependent.For a lossless wall and equally distributed path marks, the occupied half-plane unfolds to the full-plane mean measure.Source: A. Signals kept separate · pp. 5–6
  • With common fading, a reflected path cannot be stronger than its paired direct path, so the largest individual path is direct, without requiring Poisson transmitters.The distribution of the smallest inverse received power is derived later under a Poisson assumption.Source: A. Signals kept separate · p. 6

B. Signals recombined

  • Recombination adds direct and reflected received powers per transmitter before recording one inverse-power point; it does not add the inverse powers.Campbell’s theorem gives the recombined mean measure without a Poisson assumption.Source: B. Signals recombined · p. 6
  • Under common fading, the recombined threshold region is Cassinian-type rather than a clipped disk, and its area is evaluated numerically.Its defining inequality involves the direct and image distances.Source: B. Signals recombined · p. 6

C. Bounds and asymptotics

The bounds use disk geometry to control the recombined mean measure, while the exact large-threshold ratio follows from a limiting half-disk geometry. Under common fading, the limit depends only on reflection coefficient and path-loss exponent.

C. Bounds and asymptotics

  • When F′≤F almost surely, the inverse recombined power lies between L/(1+κ) and L, giving simpler sandwich bounds for the mean measure.The general bound applies to arbitrary positive paired fading marks; the simpler one also holds under common fading.Source: C. Bounds and asymptotics · pp. 6–7
  • At large thresholds, the recombined-to-superposed mean-measure ratio converges to E[(F + κF′)^(2/β)] divided by E[F^(2/β)] + κ^(2/β)E[(F′)^(2/β)].The result assumes finite fractional moments of the positive paired marks; the marks within a pair need not be independent or identically distributed.Source: C. Bounds and asymptotics · p. 7
  • After rescaling, the recombined threshold region approaches a half-disk as wall distance tends to zero, and bounding it between clipped disks yields the asymptotic area.The bounds alone do not give the exact leading coefficient, even under common fading.Source: C. Bounds and asymptotics · p. 7, Figure 4 · p. 8
  • With common fading, the fractional fading moment cancels, leaving a limit determined only by κ and β; for κ = 1 and β = 3, it is 2^(-1/3) ≈ 0.794.Independent unit-mean exponential marks instead give 5/6 for β = 3 and κ = 1.Source: C. Bounds and asymptotics · p. 7
  • For unit fading, the ratio can exceed one at small thresholds although its common-fading limit is below one when β>2 and κ>0.With positive wall distance, a reflected path can miss the threshold individually while recombination enlarges the direct threshold region.Source: C. Bounds and asymptotics · p. 7

IV. POISSON TRANSMITTER FIELD

This section assumes a homogeneous Poisson transmitter field. Figure 6 plots the recombined-to-superposed mean-measure ratio against a rescaled threshold for two reflection coefficients.

IV. POISSON TRANSMITTER FIELD

  • The section begins after mean-measure, bound, and asymptotic results derived using constant density, stated mark assumptions, and Campbell’s theorem.Source: §IV · p. 8
  • Figure 6 plots the ratio of recombined to superposed mean measures against t/a^β for κ = 1 and κ = 0.5 at β = 3.The curves apply for every positive wall distance and cross one before tending to their limits.Source: Figure 6 · p. 8
  • The transmitter process is now assumed homogeneous Poisson with density λ, an assumption held throughout the section.Source: §IV · p. 8

A. Mapped processes and strongest received power

Under Poisson transmitters, direct, reflected, and recombined propagation processes are each Poisson, but their superposition is generally not. The strongest direct power follows from the direct process’s void probability, while common fading makes it strongest among separate paths.

A. Mapped processes and strongest received power

  • The direct, reflected, and recombined propagation processes are each Poisson under the stated Poisson and mark assumptions.For κ = 0, the reflected process is empty and the recombined process equals the direct process.Source: A. Mapped processes and strongest… · p. 9
  • The superposition of direct and reflected processes is generally not Poisson because each transmitter’s reflected image is spatially paired with it.When a = 0, κ = 1, and fading is common, points are repeated and superposed counts are even.Source: A. Mapped processes and strongest… · pp. 8–9
  • The smallest direct inverse power determines the strongest direct received power, whose distribution follows from the direct process’s void probability.Source: A. Mapped processes and strongest… · p. 9
  • With common fading, the strongest direct path is also the strongest path when powers are kept separate; without common fading, a reflected path may win.Source: A. Mapped processes and strongest… · p. 9

B. Paired signal counts

For Poisson transmitters, common fading partitions qualifying transmitters into independent one-path and two-path classes, yielding paired-count fluctuation results. For general paired marks, the observed superposed-to-recombined count ratio converges to the corresponding mean-measure ratio.

B. Paired signal counts

  • Figure 8 illustrates marking probabilities under unit, common, and independent fading; only independent fading permits reflected-only paths.Source: B. Paired signal counts · pp. 9–10, Figure 8 · p. 10
  • Under common fading, the independent one-path and two-path transmitter counts are Poisson, and their sum gives the direct count.A qualifying transmitter contributes either one direct path or both paths; reflected-only qualification cannot occur.Source: B. Paired signal counts · pp. 9–10, Figure 7 · p. 9
  • The reflected and superposed counts are formed from the one-path and two-path classes, whose count identities yield their covariance and variance.The zero-, one-, and two-path classes partition the marked Poisson transmitter process.Source: B. Paired signal counts · pp. 9–10
  • With common fading, reflected qualification is a subset of direct qualification, so the reflected mean gives the two-path count mean and the direct mean determines the one-path count mean by subtraction.Source: B. Paired signal counts · p. 10
  • Common fading adds excess variance 2ΛR(t) to the superposed count, placing its variance-to-mean ratio between one and two.The lower endpoint occurs at κ = 0; the upper endpoint occurs for a lossless wall through the receiver.Source: B. Paired signal counts · p. 10
  • For general positive paired marks, the superposed-count variance is Λ∪(t) + 2m2(t) and is at most 2Λ∪(t).Here m2(t) is the mean number of transmitters for which both paths qualify.Source: B. Paired signal counts · p. 10
  • As the threshold grows, each count divided by its mean converges in probability to one, enabling the observed count ratio to converge to its mean-measure ratio.Source: B. Paired signal counts · p. 10
  • As t tends to infinity, the observed ratio P converges to E[(F + κF′)^(2/β)] / (E[F^(2/β)] + κ^(2/β)E[(F′)^(2/β)]).Under common fading, the limit reduces to the universal constant in expression (30).Source: B. Paired signal counts · p. 10

C. The strongest individual path

The strongest individual path is the maximum of the paired direct and reflected powers, and the paper gives the probability that its winner is direct. At a wall through the receiver, the probability simplifies and the strongest power has an exact Fréchet distribution.

C. The strongest individual path

  • The strongest individual power is the maximum across direct and reflected paths, not the sum of transmitter contributions or the strongest recombined transmitter.Source: C. The strongest individual path · p. 10
  • The direct-winner probability is defined by retaining each transmitter’s stronger path, with ties assigned to the direct path.The total and direct-winner intensity measures are H(t) and HD(t), respectively.Source: C. The strongest individual path · p. 11
  • At a = 0, the stronger-path and direct-winner intensity measures reduce to expressions involving E[max(F, κF′)^(2/β)] and E[F^(2/β)1(F ≥ κF′)].The corresponding spatial regions are half-disks.Source: C. The strongest individual path · p. 11
  • The probability that the strongest individual path is direct has a closed form when the wall passes through the receiver.Source: C. The strongest individual path · p. 11
  • At a = 0, the strongest individual received power has an exact Fréchet distribution.The Poisson void probability supplies its exponential factor, not an assumption of exponential fading.Source: C. The strongest individual path · p. 11
  • The direct-winner probability is obtained from the direct-winner intensity and the Poisson void probability for the stronger-path process.Source: C. The strongest individual path · p. 11
  • For β = 3 and κ = 0.8, pD ≈ 0.579 at a = 0, versus pD ≈ 0.991 when λa² = 1 under the same independent marks.Under common fading, pD is one for these parameters.Source: C. The strongest individual path · p. 12

V. CONCLUSION

The paper characterizes signal processes created when a fixed reflecting wall pairs each transmitter with a mirror image. It derives large-threshold mean-measure behavior, Poisson count results, and strongest-path results, while treating the wall as a fixed specular reflector rather than a programmable surface.

V. CONCLUSION

  • The large-threshold ratio of recombined to superposed mean measures is independent of transmitter density and wall distance, but generally depends on joint fading; common fading leaves a constant set by reflection coefficient and path-loss exponent.The two processes respectively keep the paths separate or add their powers transmitter by transmitter.Source: §V · p. 12
  • For Poisson transmitters, the paper derives paired count fluctuations and convergence of the observed ratio; it also calculates the probability that the strongest individual path is direct.When the wall passes through the receiver, the direct-path probability has a closed form and the strongest received power has an exact Fréchet distribution.Source: §V · p. 12
  • The model is an infinite flat specular reflector with constant power reflection, not a programmable surface with phase control or beam steering.It isolates dependence created by mirror geometry and provides a reference case for reflected wireless signals.Source: §V · p. 12

APPENDIX PROOF OF PROPOSITION III.2

The proof rescales transmitter locations and wall distance so that the large-threshold limit becomes a zero-distance geometry. It bounds the qualifying region and extends the area limit to general fixed paired marks, yielding the asymptotic mean-measure ratio.

APPENDIX PROOF OF PROPOSITION III.2

  • Rescaling transmitter coordinates by t^(-1/β) makes the wall distance ε = at^(-1/β), which tends to zero for fixed wall distance as the threshold grows.The scaled direct and image distances are defined in the rescaled coordinates.Source: APPENDIX PROOF OF PROPOSITION III.2 · pp. 12–13
  • As the rescaled wall distance tends to zero, the direct and image distances coincide and the threshold region becomes the left half of a disk.Its radius in the unit-mark case is (1 + κ)^(1/β); zero rescaled distance is a limiting geometry, not necessarily a wall at physical distance zero.Source: APPENDIX PROOF OF PROPOSITION III.2 · p. 12
  • The proof bounds the positive-distance qualifying region between inner and outer sets, whose area bounds converge to the half-disk area as rescaled wall distance vanishes.The outer set is contained in a left half-disk together with a vertical strip of vanishing width.Source: APPENDIX PROOF OF PROPOSITION III.2 · pp. 12–13
  • For general fixed positive marks, the limiting qualifying region is a left half-disk of radius (f + κf′)^(1/β), and its area is integrable under the moment assumptions.Dominated convergence applied to the averaged half-disk areas proves the asymptotic formula.Source: APPENDIX PROOF OF PROPOSITION III.2 · p. 13
  • The direct and reflected mean measures have large-threshold coefficients πλE[F^(2/β)]/2 and πλκ^(2/β)E[(F′)^(2/β)]/2, respectively, whose sum yields the ratio without requiring independent or identically distributed paired marks.Source: APPENDIX PROOF OF PROPOSITION III.2 · p. 13
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