Radar Target Detection and Localization Aided by an Active Reconfigurable Intelligent Surface

In this letter, we consider a monostatic radar and study the joint detection and localization of a prospective target when the receiver is assisted by a reconfigurable intelligent surface (RIS). To mitigate the multiplicative path loss in the target-RIS-radar hops, we resort to an active RIS, which allows both redirecting and amplifying the incident signal. Upon choosing the array gain factor of the radar transmitter and the RIS to uniformly cover the inspected region, the joint target detection and localization is formulated as a composite hypothesis testing problem, which is solved via a generalized likelihood ratio test (GLRT). Numerical examples are provided to show the merits of the proposed architecture.

implementation constraints. These limitations can be overcome by resorting to active RISs, which inherit the same hardware structure and full-duplex operation of the passive counterpart, except that active (rather than passive) load impedances are employed to cover longer source-RIS and/or RIS-destination hops [6], [7], [8], [21]. Active RISs make use of reflection amplifiers, wherein the additional DC power provided by a biasing source is exploited to boost the RF impinging signal [22], [23], [24], [25]; remarkably, tunnel diode based reflection amplifiers only have a sub-mW DC power consumption. An active RIS is somehow similar a network-controlled repeater (NCR), as both devices rely upon the amplify-and-forward concept [26], [27]: however, their different architectures pose unique challenges. Indeed, the RIS employs a single panel with amplifying and reflecting elements, whereby both the impinging and reflected waves must be in the same half-space [6], [8]. On the other hand, the NCR is made of two back-to-back analog arrays with a signal amplifier in between, whereby limiting the loop-back interference in a full-duplex operation may heavily constrain the reciprocal orientation of the two panels [28], [29].
Leveraging the architecture proposed in [21], in this study we consider a monostatic radar that is equipped with a digital array and whose receiver is assisted by a widely-spaced active RIS. The radar illuminates a given area and, in the presence of a target, receives two independent echoes, namely, the direct and the RIS-assisted (indirect) echo: this system mimics a distributed multi-static radar; however, rather than having a second receiver with an additional RF processing chain, the RIS redirects the indirect echo towards the same receiver observing the direct echo. While [21] has only focused on target detection, our goal here is to configure the RIS and elaborate the received signal to jointly detect and localize a prospective target; in addition, we remove the assumption that the radar receiver has two separate antennas pointing towards the target and the RIS, respectively. Upon introducing a convenient signal model, we design the array gain factor of the radar transmitter and the RIS to uniformly cover the inspected region; then, the joint detection and localization problem is solved through a generalized likelihood ratio test (GLRT). Numerical examples are provided to assess the achievable performance and to elicit the impact of the RIS amplification gain and of the size of the inspected region.
The remainder of this work is organized as follows. Section II contains the system description. 1 Section III illustrates the proposed system design. Section IV contains the numerical analysis. Finally, concluding remarks are given in Section V. 1 Column vectors/matrices are denoted by lower/uppercase boldface letters. The symbols { · }, ( · ) * , ( · ) T , ( · ) H , , and ⊗ denotes real part, conjugate, transpose, conjugate-transpose, convolution, and Kronecker product, respectively. C is the set of complex numbers. I M is the M × M identity matrix. diag{x} is a diagonal matrix with the elements of x on the main diagonal.

II. SYSTEM DESCRIPTION
We consider a monostatic radar that employs a carrier wavelength λ, an average radiated power P r , a pulse repetition time T r , and a uniform linear array with N r elements spaced of δ r . For simplicity, we assume a two-dimensional geometry, as shown in Fig. 1. The radar transmitter emits the baseband signal √ P r T r φ r (t)b * r , where b r ∈ C N r is a unit-norm spatial beamformer and φ r (t) is a unit-energy pulse with support [0, Φ r ] and bandwidth W r . The radar receiver is assisted by an active RIS containing N s elements arranged into a uniform square array with inter-element spacing δ s along each axis. We denote by √ B s b * s ∈ C N s the RIS response, where b s 2 = N s and B s is the average element gain. The inspected region X ⊂ R 2 is in the radar and RIS far-field; also, radar and RIS are in each other's far-field and sufficiently spaced to provide independent looks at a prospective target in X [21].
In the following, H andH denote the null hypothesis (i.e., no target is present) and its alternative (i.e., one target is present at an unknown location x ∈ X ), respectively. Also, p r ∈ R 2 is the position of the radar phase center; d r (p) ≥ 0 and θ r (p) ∈ (−π/2, π/2) are the distance and angle of a point p ∈ R 2 with respect to the radar phase center, respectively; Θ r ⊂ (−π/2, π/2) is the angular sector containing the inspected region, as seen from the radar phase center; u r (θ) ∈ C N r and G r (θ) ≥ 0 are the array manifold vector and the element gain of the radar towards the angular direction θ ∈ (−π/2, π/2), respectively. For the RIS, are the array gain factors of the radar transmitter and the RIS, respectively, with θ ∈ (−π/2, π/2). We assume that the RIS does not obstruct the radar view, i.e., θ r (p s ) / ∈ Θ r , and that the RIS is not illuminated by the radar transmitter, i.e., Γ r (θ r (p s )) 0 (more on this in Section III). UnderH, the target scatters part of the incident signal back towards the radar receiver (direct echo) and part towards the RIS that in turn amplifies and redirects it towards the radar receiver (indirect echo); accordingly, the received signal is where h 1 (t; x) ∈ C N r and h 2 (t; x) ∈ C N r are the spacetime signatures of the direct and indirect echoes under H, respectively, α 1 (x) ∈ C and α 2 (x) ∈ C are the corresponding amplitudes, and n(t) ∈ C N r is the Gaussian noise. Assuming a narrowband signal [30], we have h 1 )/c, and c is the speed of light. Also, assuming a freespace propagation [31], we have α 1 (x) = γ 1 (x)α 1 , whereα 1 ∈ C is the unknown (monostatic) target response, while Similarly, α 2 (x) = γ 2 (x)α 2 , whereα 2 ∈ C is the unknown (bistatic) target response and Following [6], [7], [8], [21], the noise is the sum of two independent contributions, namely, n(t) = n r (t) + n s (t)u r (θ r (p s )).
Here, n r (t) is the noise of the radar receiver, and its entries are modeled as independent white Gaussian processes with power spectral density (PSD) σ 2 r,n . Instead, n s (t) is the noise coming from the RIS that is amplified together with the signal of interest and redirected towards the radar receiver; in particular, we have while the n-th entry ofñ s (t) ∈ C N s is the dynamic noise of the n-th RIS element, accounting for both the input and the internal noise, assumed independent of the reflected signal. The entries ofñ s (t) are modeled as independent white Gaussian processes with PSD σ 2 s,ñ , so that n s (t) is itself a white Gaussian process with PSD σ 2 s,n = g 2 s b s 2 σ 2 s,ñ . We assume that both σ 2 r,n and σ 2 s,n are known; for example, they can be estimated by resorting to secondary data.
Finally, the power consumed by the radar transmitter can be modeled as P tot r = ρ r + η −1 r,all P r , where ρ r is the static power [32] and η r,all is the overall efficiency of the RF amplifier [33,Eq. (10)]. Instead, the power consumed by the active RIS can be modeled as P tot s = ρ s + η −1 s,PAE P s,in (B s − 1) N s , where ρ s = ρ s,c + ρ s,dc is the static power per each element [6], [24], with ρ s,c being the switch and control circuit power and ρ s,dc the DC biasing power, η s,PAE is the power-added efficiency (PAE) of the amplifier [33, Eq. (12)], Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
is the input power (i.e., the sum of the dynamic noise power and incident power), and P s,in (B s − 1) is the difference between the output and input power; the term η −1 s,PAE P s,in (B s − 1) is often much smaller than ρ s,dc . Notice that P tot s = N s ρ s,c for a passive RIS and P tot s = 0 if no RIS is present. Hence, for the same power budget 2 P tot = P tot r + P tot s , the radar radiates more power when a passive RIS or no RIS is employed.

III. SYSTEM DESIGN
In this section, we first illustrate the design of the radar transmitter and RIS; then, we consider the radar receiver.
Given the prior uncertainty on the target location, we propose to choose b r and b s so that Γ r (θ) and Γ s (θ) approximate an ideal rectangular function that is non-zero only if θ ∈ Θ r and θ ∈ Θ s , respectively, so as to ensure a uniform converge of the inspected region. Many matching criteria are available in the literature: see [30] and references therein. For example, in Section IV we resort to a least-square solution with a Taylor tapering to reduce the level of the sidelobes.
The received signal y(t) is sent to a low-pass filter with impulse response ψ r (t) = √ Ψ r Rect(t/Ψ r ), where Rect(t) is a unit-amplitude rectangular pulse with support [0,1] and 1/Ψ r ≥ W r ; the filter output is then sampled at rate 1/Ψ r in the time interval [τ min , τ max + Φ r + Ψ r ], where τ min = min i∈{1,2}, x∈X τ i (x) and τ max = max i∈{1,2}, x∈X τ i (x). The samples taken at the time epochs {τ min + Ψ r } F r −1 =0 are organized into a vector y ∈ C M r , where F r = (τ max − τ min + Φ r + Ψ r )/Ψ r + 1 and M r = N r F r . Let φ r,i (x) ∈ C F r be the vector containing the samples of the filtered waveform φ r (t − τ i (x)) ψ r (t), for i = 1, 2; then, we have , and h 2 (x) = u r (θ r (p s )) ⊗ φ r,2 (x), while n contains the samples of the filtered noise. Under the considered assumptions, H(x) is full column-rank for any x ∈ X , and n is a circularlysymmetric complex Gaussian vector with covariance matrix C = (σ 2 r,n I N r + σ 2 s,n u r (θ r (p s ))u H r (θ r (p s ))) ⊗ I F r . We solve the binary test in (6) by resorting to a GLRT, wherein we treat the target location x and the corresponding signal amplitudes α 1 (x) and α 2 (x) underH as unknown deterministic parameters [34]. Given x and α(x), the log-likelihood ratio betweenH and H is (7) which is maximized when α(x) is chosen as Accordingly, the GLRT becomes 1 H(x)) −1/2 and η is a threshold set to have a specified probability of false alarm (PFA). Notice that Π H (x)y 2 is the energy of the whitened observation C −1/2 y falling in the two-dimensional column space of C −1/2 H(x). As customary, the maximization over X is undertaken by a grid search. IfH is declared, the argument of the maximum provides an estimate of the target location. The following remarks are now in order.
this occurs when the steering vectors towards the prospective target and the RIS are orthogonal or if the delay offset between the direct and indirect echoes exceeds the duration of φ r (t) ψ r (t), i.e., |d Remark 2: When no RIS is employed, the GLRT has still the form in (9) Since Π H (x)y 2 is now the energy of the whitened observation C −1/2 y falling in the one-dimensional subspace spanned by C −1/2 h 1 (x), for a fixed PFA, the GLRT needs here a lower threshold as compared to the proposed case.
Remark 3: The matrices {Π(x)} x∈X can be computed offline with a computational complexity that scales quadratically with the length M r of the observation vector and linearly with the number |X | of grid points. Instead, given the current measurement y, the in-line implementation of the GLRT in (9) is dominated by the multiplication of Π H (x) by y for each grid point, so that the computational complexity scales linearly with both M r and |X |. Interestingly, when no RIS is present, the above asymptotic costs remain the same.
Remark 4: When multiple targets are present in X , we may resort to standard approaches, such as for example subspacebased iterative procedures that extract one target at a time after eliminating the previously-detected signal components [35].

IV. PERFORMANCE ANALYSIS
We consider an X-band system with λ = 3 cm, G r (θ) = G s (θ) = 2 cos(θ)Rect(θ/π + 1/2), T r = 10 µs, φ r (t) = √ Φ r Rect(t/Φ r ), Φ r = Ψ r = 0.1 µs, W r = 10 MHz, N r = 16, N s = 3600, δ r = δ s = λ/2, and σ 2 r,n = σ 2 s,ñ = 1.6 · 10 −20 W/Hz. As to the power consumption, we assume ρ r = 30 W, η r,all = 0.9, and P r = 8 W for the radar transmitter, which gives a system efficiency P r /P tot r of about 20%, and ρ s,c = 0.1 mW, ρ s,dc = 0.32 mW, and η s,PAE = 0.8 for the active RIS. As to the target,α 1 andα 2 are generated as independent realizations of a circularly-symmetric Gaussian random variable with variance σ 2 t . With reference to Fig. 1, the radar is at the origin of the coordinate system and inspects the range interval [500 − Δ r /2, 500 + Δ r /2] m in the angular sector Θ r = [−5 • , 5 • ], while the RIS is at (300,300) m. Two values of Δ r are considered, namely, 15 and 150 m, with the former corresponding to the range resolution c/(2W r ). The top plot in Fig. 2 shows the array gain factor Γ r (θ) of the radar transmitter; here, the mainbeam covers the angular sector Θ r , and a null is placed at θ = θ r (p s ) = π/4. The middle and bottom plots in Fig. 2 show the array gain factor Γ s (θ) of the RIS for Δ r = 15, 150 m, respectively; notice here that a wider mainbeam is required to cover the region of interest when Δ r increases, at the price of a lower array gain.
We define the SNR of the direct echo as SNR 1 = N r γ 2 1 (x)σ 2 t /σ 2 r,n , wherex is a reference point with range 500 m and azimuth 0 • . Fig. 3 shows the detection probability (PD) and the root mean square error (RMSE) in the target localization of the proposed architecture versus SNR 1 when B s = 40 dB. The reported values are computed over 10 5 independent snapshots, while SNR 1 is varied by increasing the radar target crosssection σ 2 t . In each snapshot, the target is randomly dropped in X underH. The performance of the proposed architecture is contrasted with that of a radar helped by a passive RIS or operating alone, when keeping the same consumed power P tot ; in all cases, the detection threshold is chosen to have PFA = 10 −5 .
It is seen that helping the radar receiver with an active RIS can improve both the detection and localization performance of an existing radar. Instead, using a passive RIS may be even detrimental in the considered scenario. To explain these results, recall that an RIS-aided system radiates less power to maintain the same power consumption (see Section II), as compared to a radar operating alone; also, the corresponding GLRT presents a larger threshold to maintain the same PFA (see Remark 2); accordingly, processing the indirect echo becomes beneficial only if a signal amplification (in addition to the array gain) is introduced at the RIS to compensate for the multiplicative pathloss along the target-RIS-radar hops. To better illustrate this point, Table I reports the values of PD and RMSE for different amplification gains when SNR 1 = 15 dB, along with the SNR of the indirect echo, defined as SNR 2 = N r γ 2 2 (x)σ 2 t /(σ 2 r,n + σ 2 s,n ): it is seen that a help RIS is advantageous only if B s is sufficiently large to make SNR 2 comparable to or larger than SNR 1 . Finally, for the same amplification gain, both Fig. 3 and Table I indicate that the  active RIS becomes less rewarding when the inspected region X is enlarged: this is indeed due to the fact that the RIS array gain decreases, as previously shown in Fig. 2.

V. CONCLUSION
In this letter, we have used an active RIS to get a second observation of the region inspected by a monostatic radar equipped with a digital array. We have designed the array gain factor of the radar transmitter and the RIS to uniformly cover the inspected region and solved the target detection and localization problem via a generalized likelihood ratio test. The numerical analysis shows that, for the same power budget, the active RIS improves the radar performance if a sufficiently large amplification gain can be guaranteed. Future studies should account for the clutter presence and should investigate the radar and RIS scanning policy when multiple regions need to be inspected, each one possibly containing multiple targets.