Diffuse Multipath · Kulmer's Effective-Roughness Model

From a wall to the receiver

Top view · metres

SpecularDiffuseSelected patch q

Drag Tx, Rx, or the blue wall patch. Thin rays show representative in-plane paths; the power spectrum integrates the wall in 3D.

At the selected patch q

Scattering lobe

ER lobeLocal specular direction

The receiver samples one direction of this lobe. Radius uses a dB display; the S = 1 peak at each α_R is the reference, not an absolute power calibration.

Specular path delay τ_k60.13 ns

Local deviation ψ0 degrees

Power scaling from S alone0.09 × S = 1

The core scattering law · click a factor

dP_r =

For touch screens, drag the nodes or the blue patch. The roughness and attenuation sliders also work with the arrow keys.

The joint angle–delay spectrum

At Rx · elevation integrated

Every wall patch contributes power at its own arrival angle and delay. The orange cross locates the specular component; it is a reference, not part of this diffuse-power image.

Angular power spectrum

Integrate over delay

Delay power spectrum

Integrate over angles

Numerical RMS angular spread— degrees

Mean excess delay γ— ns

Mean angle − specular angle— degrees

Same power, new coordinates

f_ER(θ_DM,k) dA = p_DM,k(θ) dθ ; p_DM(θ) = ∑_k p_DM,k(θ)

The received peak need not point along the specular path. The local lobe is weighted by illumination, distance, and observed surface area. Swap Tx / Rx: the delay spectrum stays the same, but the angular spectrum generally changes.

One power pattern. Many random realizations.

1 realization

Sample-averaged |ν|²Expected bin power

Relative L1 power error: —Seed 21

Deterministic + stochastic

h(θ) = ∑_k [α_k δ(θ − θ_k) + ν_k(θ)]

α_k is a specular path's complex amplitude. It is not the roughness parameter α_R. θ contains directions at both nodes and propagation delay.

The diffuse field's moments

E[ν_k(θ)] = 0
E[ν_k(θ)ν_k*(θ′)] = p_DM,k(θ) δ(θ − θ′)

Different angle–delay coordinates are uncorrelated in this idealized model; so are different surfaces. Zero-mean field does not mean zero power.

Try 1 → 256 realizations. The fluctuating power approaches the same geometry-dependent average. This is a finite-bin complex-Gaussian illustration, not a measured waveform or the literal continuous white process. A Gaussian DM process is used in included paper [T2], §2.1; Chapter 3 itself specifies the moments.

Only under the thesis's far-field assumptions: an effectively infinite planar surface, with transceiver separation small compared with their distance to that surface. These are separate analytic curves, not a fit to the finite-wall lab.

Angular concentration

Azimuth offset from specular

p^ff_DM,k(φ^a) ∝ cos^2αR+3(φ^a − φ^a_k)

Eq. (3.17), on the illuminated angular support. Dashed: Gaussian comparison using the width reported in Eq. (3.24).

Delay tail

Nothing before τ_k

p^ff_DM,k(τ) ∝ τ_k^2αR+2τ^2αR+5 u(τ − τ_k)

Appendix A, Eq. (A.15). Solid: derived power-law tail. Dashed: exponential comparison, not the exact scattering law.

Reported Gaussian width, Eq. (3.24)— degrees

γ = τ_k / (2α_R + 3), Eq. (3.27)— ns

Tail exponent 2α_R + 513 power law

Less roughness → narrower angles and a shorter delay tail. Increasing α_R concentrates diffuse power near the specular direction. Increasing τ_k increases γ. Here γ is the exponential time constant (the thesis calls it a “decay rate,” but its units are ns).

Shape plots use peak normalization at each α_R and τ_k, followed by S² scaling. They do not compare absolute scattered power between different roughness values.

What you see	Dissertation source	Printed / PDF page
Deterministic plus stochastic spreading function	Chapter 3, Eq. (3.2)–(3.3)	18 / 34
Zero-mean, uncorrelated diffuse process	§3.2.2, Eq. (3.8a–c)	21 / 37
Effective-roughness scattering law and lobe	Eq. (3.9)–(3.11), Figs. 3.2–3.3	22–24 / 38–40
Angle–delay power and sums over surfaces	Eq. (3.12)–(3.16), Figs. 3.4–3.7	25–30 / 41–46
Far-field APS and Gaussian width	Eq. (3.17), (3.24)	28, 31 / 44, 47
Far-field delay spectrum and γ	Eq. (A.15), (3.27)	63, 32 / 79, 48
Reciprocity, offset angular peaks, limitations	Included paper [T1], §§2.1–2.4, 3.4	93–103 / 109–119
Gaussian diffuse-process illustration	Included paper [T2], §2.1	109 / 125
Position information in diffuse statistics left for future work	Chapter 6	57 / 73

One wall. Many paths.

From a wall to the receiver

At the selected patch q

The joint angle–delay spectrum

Angular power spectrum

Delay power spectrum

One power pattern. Many random realizations.

Angular concentration

Delay tail

Model, sources & boundaries

What is computed here

Normalization is deliberately explicit

Random-process illustration

Do not over-read the model

Included paper [T1]