2. Echo Through an RIS: The Four-Hop Budget

Why Count Every Leg?

A communication receiver needs the outward BS–RIS–user path. A monostatic sensing receiver, located at the transmitter, also needs the target echo to return. Here we isolate BS–RIS–target–RIS–BS. Mixed direct/reflected paths require different budgets.

Let $P_t$ denote transmitted power in watts, $G_t$ and $G_r$ dimensionless BS transmit and receive gains, $\lambda$ wavelength, $A$ panel area, and $\eta$ power efficiency per reflection. Target radar cross section $\sigma$, in square metres, describes backscattered power in the observed direction. It is not simply the target’s physical area. Distances $d_1$ and $d_2$ are the two outward legs.

References for this section1

• A. Aubry, A. De Maio and M. Rosamilia, Reconfigurable Intelligent Surfaces for N-LOS Radar Surveillance, 2021. See the radar-equation derivation in Section IV.

Derive the Budget in Four Steps

For an ideal broadside aperture, the incident power density at the RIS is $S_1=P_tG_t/(4\pi d_1^2)$. Approximate the panel’s capture area by $A$ and its reradiation directivity by $4\pi A/\lambda^2$.

1. Illuminate the target: $S_T=S_1\eta A^2/(\lambda^2d_2^2)$.
2. Scatter back to the panel: $S_R=S_T\sigma/(4\pi d_2^2)$.
3. Redirect toward the BS: $S_B=S_R\eta A^2/(\lambda^2d_1^2)$.
4. Collect at the receiver: $P_r=S_BG_r\lambda^2/(4\pi)$.

Each density $S$ is in W/m². Multiplying these steps gives this teaching model:

$$P_r=\frac{P_tG_tG_r\sigma\eta^2A^4}{(4\pi)^3\lambda^2d_1^4d_2^4}.$$

This is an ideal aperture specialization, not a replacement for measured element patterns or the more detailed NLoS radar models in the references.

References for this section1

• A. Aubry, A. De Maio and M. Rosamilia, Reconfigurable Intelligent Surfaces for N-LOS Radar Surveillance, 2021. See the radar-equation derivation in Section IV.

A Budget You Can Calculate

The lab fixes $P_t=1$ W, $G_t=G_r=100$ (20 dBi each) and frequency at 3 GHz. At the default distances 50 m and 30 m, panel area 0.64 m², efficiency 0.7 and cross section 1 m², the echo is approximately −80.9 dBm.

For bandwidth $B=20$ MHz and receiver noise figure $F_{\rm dB}=6$ dB, the room-temperature noise approximation is:

$$N_{\rm dBm}=-174+10\log_{10}(B/1\,\mathrm{Hz})+F_{\rm dB}\approx-95.0\ \mathrm{dBm}.$$

SNR, the signal-to-noise ratio, is $P_{r,\rm dBm}-N_{\rm dBm}$, approximately 14.1 dB here. Detection also depends on clutter, integration, a detection threshold and acceptable false alarms.

Can a four-hop echo reach the receiver?

1. Formula and symbols

$$P_r=\frac{P_tG_tG_r\sigma\eta^2A^4}{(4\pi)^3\lambda^2d_1^4d_2^4}$$

• $P_t$: Transmitted power in watts
• $P_r$: Received echo power in watts
• $G_t$: Linear transmit gain at the BS
• $G_r$: Linear receive gain at the BS
• $\sigma$: target cross section
• $\eta$: aperture efficiency
• $A$: panel area
• $\lambda$: wavelength
• $d_{1}$: Distance between the BS and RIS
• $d_{2}$: Distance between the RIS and target

This is the stated broadside far-field aperture model.

BS–RIS distance

50 m

RIS–target distance

30 m

Panel area

0.64 m²

Power efficiency per reflection

70%

Target radar cross section

1 m²

2. Work with the current inputs

$$P_r=\frac{P_tG_tG_r\sigma\eta^2 A^4}{(4\pi)^3\lambda^2d_1^4d_2^4}$$

$$P_r=\frac{1\times100^2\times1.0\times0.70^2\times0.64^4}{(4\pi)^3(0.09993)^2\times50^4\times30^4}\ \mathrm{W}\approx8.194\times10^{-12}\ \mathrm{W}$$

$$P_{r,\mathrm{dBm}}=10\log_{10}(1000P_r/1\,\mathrm{W})=-80.9\ \mathrm{dBm}$$

$$N=-174+10\log_{10}(20\times10^6)+6=-95.0\ \mathrm{dBm}$$

3. Result

Echo power-80.9 dBm

Single-observation SNR14.1 dB

Far-field boundary25.6 m

References for this section1

• A. Aubry, A. De Maio and M. Rosamilia, Reconfigurable Intelligent Surfaces for N-LOS Radar Surveillance, 2021. See the radar-equation derivation in Section IV.

Distance, Area and Efficiency Are Different Levers

• Double either distance: echo falls by a factor of 16, approximately 12 dB.
• Double both distances: echo falls by 256, approximately 24 dB.
• Halve area at fixed wavelength: echo falls by 16 in this model.
• Halve efficiency: echo falls by four because the same loss occurs twice.

These ratios hold with other quantities fixed. At fixed physical area, adding more simulation elements does not create an unlimited physical aperture gain.

References for this section1

• A. Aubry, A. De Maio and M. Rosamilia, Reconfigurable Intelligent Surfaces for N-LOS Radar Surveillance, 2021. See the radar-equation derivation in Section IV.

Know Where the Formula Stops

For a square panel, conservatively use its diagonal $D=\sqrt{2A}$ as the maximum aperture dimension. A conventional far-field check is $d_1,d_2>2D^2/\lambda$. The lab keeps both distances beyond this boundary, increasing their minimum values when panel area grows. The model assumes calibrated coherent phases, broadside effective apertures and a reciprocal route. It omits clutter, blockage on the chosen legs, polarization mismatch and hardware saturation.

Near-field focusing requires spherical-wave modeling; changing frequency can also change antenna gain, efficiency and target cross section. Keep these dependencies explicit when comparing designs.

References for this section1

• A. Aubry, A. De Maio and M. Rosamilia, Reconfigurable Intelligent Surfaces for N-LOS Radar Surveillance, 2021. See the radar-equation derivation in Section IV.

Takeaway

Write the route before writing a gain claim. Next, split the available surface between data and sensing and see the price of sharing it.

References for this section1

• A. Aubry, A. De Maio and M. Rosamilia, Reconfigurable Intelligent Surfaces for N-LOS Radar Surveillance, 2021. See the radar-equation derivation in Section IV.