Exact Bistatic Detection With Dual-Use AF Relays in SAGINs

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Accepted for publication in IEEE Wireless Communications Letters, Copyright transform to IEEE

Exact Bistatic Detection With Dual-Use AF Relays in SAGINs Changhao He, Graduate Student Member, IEEE, and Xiaojuan Zhang, Senior Member, IEEE

arXiv:2610.38771v1 [cs.NI] 28 Oct 2026

Abstract We study bistatic UAV detection using dual-use amplify-and-forward (AF) relays in satellite–aerial–ground integrated networks. The target reflects the pilot, payload, and relay noise, allowing the exact finite-sample Neyman–Pearson detection probability to be derived in noncentral chi-square form under Gaussian payload signaling. The AF cascade introduces a variance signature even without payload transmission and leads to a rate-constrained pilot allocation, distinct relay-power limits, and phase-dependent variance changes under residual direct-path interference. For a decorrelated or suppressed direct path, we further derive a phase-only generalized likelihood ratio test with exact single-integral error probabilities at an assumed target magnitude; the correlated case retains a known-coefficient detection law. A 15-GHz low-Earth-orbit link budget and Monte Carlo simulations validate the analysis.

Index Terms Amplify-and-forward relay, bistatic sensing, integrated sensing and communications, intruder UAV detection, SAGIN.

I. I NTRODUCTION Low-altitude networks need to detect unauthorized unmanned aerial vehicles (UAVs), but deploying dedicated radar systems can be costly. Integrated sensing and communications (ISAC) offers an alternative by reusing communication infrastructure and waveforms [1], [2]. Recent studies have considered networked low-altitude ISAC [3], hybrid deterministic/Gaussian bistatic waveforms [4], [5], and amplification-assisted detection using an active intelligent reflecting surface (IRS) [6]. Amplify-andforward (AF) UAV relays can also support satellite–aerial–ground integrated networks (SAGINs) when direct satellite-to-ground links are blocked [7]. The closest model is [4], which derives an exact Neyman–Pearson (NP) detector for a direct deterministic/Gaussian waveform without AF gain normalization. Reference [5] considers a similar waveform but uses a generalized likelihood ratio test (GLRT) for unknown target gain with asymptotic analysis. Here, we retain exact finite-L performance while maximizing over the unknown echo phase at an assumed magnitude. Reference [6] considers target-reflected active-IRS amplifier noise but has neither an AF payload nor a served user. Our contribution is the AF-cascade specialization and its communication–sensing consequences, rather than a new Gaussian detector. Specifically, let Eµ denote the coherent pilot-echo energy over an L-symbol dwell, ∆ the target-induced per-symbol variance increment and β the relay gain. The AF mapping is Eµ ≜ L|a|2 β 2 |g|2 pd , ∆ ≜ |a|2 β 2 (|g|2 pg + σr2 ), Pr β2 ≜ 2 , (1) |g| Ps + σr2 where a is the target coefficient, g the satellite–relay channel, pd and pg the pilot and payload powers, Ps and Pr the satellite and relay transmit powers, and σr2 the relay-noise variance. Unlike direct transmission, the relay noise is also reflected by the target, giving ∆ > 0 even for pg = 0. Moreover, β couples Ps , Pr , the user signal-to-interference-plus-noise ratio and the sensing performance. This leads to a rate-constrained pilot split, distinct relay-power limits for matched-filter and full NP detection, and directpath interference whose correlation with the target echo can reverse the optimal decision tail. We also derive an exact phase-only GLRT at an assumed target magnitude. Fig. 1 shows the geometry, with illumination provided by a low-Earth-orbit (LEO) satellite through the AF relay; the analysis starts after range–Doppler cell formation, conditional on a coherent-cell target coefficient. C. He is with King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia (e-mail: changhao.he@kaust.edu.sa). X. J. Zhang is with the Institute of Advanced Intelligence and Computing, Agency for Science, Technology and Research (A*STAR), Singapore (e-mail: xiaojuanzhang@ieee.org). This work was supported by the Agency for Science, Technology and Research (A*STAR), Singapore under the Semiconductor RIE Flagship Programme (Integrated Circuit Design & Systems), grant No. H26-MSE1535.

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dual-use LEO satellite

xs , g

AF relay

illumination

h

target UAV echo, a

ρ̄b

served user

sensing receiver

Fig. 1. Dual-use AF-relay bistatic geometry; solid: communication, dashed: illumination/echo, dotted: residual relay–receiver direct path.

II. AF C ASCADE AND E XACT D ETECTOR We test target absence (H0 ) against target presence (H1 ). Over a coherent dwell of L symbols, the satellite transmits √ √ xs [ℓ] = pd c[ℓ] + pg s[ℓ] with pd + pg = Ps , where |c[ℓ]| = 1 is the known pilot and s[ℓ] ∼ CN (0, 1) is the Gaussian payload. Finite-alphabet signaling makes H1 a Gaussian mixture, so the derived law becomes approximate, although H0 and its false-alarm calibration remain exact under ideal DPI suppression. At the setting of Fig. 2(b), 106 -trial simulations show that quadrature phase-shift keying (QPSK) agrees with the Gaussian law within ±0.001; QPSK markers are also shown. All channels and a remain constant over the dwell, residual delay and Doppler errors are neglected, and payload and noise samples are independent across ℓ. The relay uses a real gain β ≥ 0 given by (1) and forwards xr [ℓ] = β(gxs [ℓ] + nr [ℓ]) with nr [ℓ] ∼ CN (0, σr2 ). The served user, with channel h and noise variance σu2 , treats the pilot as interference, giving |hβg|2 pg , (2) |hβg|2 pd + |hβ|2 σr2 + σu2 and Rc = τ log2 (1 + γc ) with resource fraction τ ∈ (0, 1]; the instantaneous (full-duplex) relaying in (3) gives τ = 1. After subtracting the known direct-path pilot, we first model the residual as decorrelated from the target-cell waveform. In Table I, the target lies 500 m off the relay–receiver baseline, giving an excess path of 236 m compared with the total-path resolution c/B = 2998 m (c is the speed of light), or 0.08 cell. The decorrelated residual is therefore an idealized cancellation model; Section III-A considers the correlated case. With δ0 = 0, δ1 = 1,
√ √ y[ℓ] = δi aβ g pd c[ℓ] + g pg s[ℓ] + nr [ℓ]
√ + ρ̄b β g pg se[ℓ] + n er [ℓ] + ne [ℓ], (3) γc =

under Hi , where ne ∼ CN (0, σe2 ) and the residual direct-path interference (DPI) coefficient ρ̄b ≜ ρb represents the relay–receiver channel b scaled by the suppression residual ρ. The tilded variables represent the residual waveform after range–Doppler projection or cancellation. They are independent of the target-cell waveform but have the same payload-plus-relay-noise variance. Thus, H0 : y[ℓ] ∼ CN (0, σ02 ) and H1 : y[ℓ] ∼ CN (µℓ , σ12 ), with v = |g|2 pg + σr2 , √ µℓ = aβg pd c[ℓ], σ02 = σe2 + |ρ̄b β|2 v, σ12 = σ02 + ∆,

∆ = |aβ|2 v > 0.

(4)

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Fading: The detection law is conditional on (a, g, ρ̄b ). If the relay tracks |g| over each dwell, as assumed in (1), feeder-link fading has little effect: Eµ → L|a|2 pd Pr /Ps and ∆ → |a|2 Pr pg /Ps for |g|2 Ps ≫ σr2 , satisfied by 56.45 dB in Table I. Varying |g|2 over ±20 dB changes PD by less than 9.4 × 10−5 . Target-path fluctuation is examined in Fig. 2(h). Operating regime: (21) gives a nominal |a| for an assumed radar cross section (RCS) but provides no phase information. The following analysis therefore serves as a conditional known-coefficient benchmark; Section V extends it to an unknown phase using a GLRT Pand examines magnitude uncertainty for both detectors. From (1), Eµ = ℓ |µℓ |2 = L|aβg|2 pd . Ignoring terms independent of the observations, the log-likelihood ratio is monotone in ∆ X 2 X T = 2 2 |y[ℓ]|2 + 2 ℜ{µ∗ℓ y[ℓ]}, (5) σ0 σ1 σ1 ℓ

ℓ

which combines received-signal energy and coherent pilot correlation. With w = ∆/(σ02 σ12 ) and κ = σ02 /∆, completing the square gives L X |y[ℓ] + κµℓ |2 . (6) T = wU − wκ2 Eµ , U= ℓ=1 −1 Define F ν,λ (x) = Pr{χ′2 ν (λ) > x}, with inverse F ν,λ in x.

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Theorem 1 (Exact finite-L probability): Under Hi , 2U ∼ χ′2 2L (λi ), σi2

λi =

2σi2 Eµ , i ∈ {0, 1}, ∆2

(7)

and for false-alarm probability PFA = α,  σ02 −1 (α) . (8) F σ12 2L,λ0 Proof: For zℓ = y[ℓ] + κµℓ , the means are E[zℓ | H0 ] = κµℓ and E[zℓ | H1 ] = (1 + κ)µℓ = (σ12 /∆)µℓ , with variances σ02 and σ12 , respectively. Since the shifted observations remain independent proper complex Gaussian, their squared norm follows a noncentral chi-square distribution with 2L degrees of freedom and the noncentralities in (7). Setting the threshold under H0 gives (8). □ A Gaussian approximation is obtained by taking U ≈ N (mi , vi ), where q0 = κ, q1 = 1 + κ, 

PD = F 2L,λ1

mi = Lσi2 + qi2 Eµ , vi = Lσi4 + 2σi2 qi2 Eµ , (9) √ G −1 giving PD = Q((m0 + v0 Q (α) − m1 )/ v1 ), with P Q(·) denoting the Gaussian tail function. A matched-filter (MF) benchmark and, for Fig. 2(c), the energy statistic Z = ℓ |y[ℓ]|2 have the exact detection probabilities ! p 2Eµ σ0 −1 MF PD = Q , (10) Q (α) − σ1 σ1   2 σ0 −1 E (α) , (11) F PD = F 2L,2Eµ /σ12 σ12 2L,0 which characterize the two components combined by the NP test. √

III. R ATE , P OWER , AND D IRECT-PATH C ONSEQUENCES A minimum rate Rmin requires γc ≥ γmin = 2Rmin /τ − 1. With A = |hβg|2 , B = |hβ|2 σr2 + σu2 and pg = Ps − pd , γc (pd ) = A(Ps − pd )/(Apd + B) decreases strictly with pilot power. Thus, the rate requirement is feasible if and only if APs γmin ≤ γc (0) = , (12) B and the maximum feasible pilot power is APs − γmin B . (13) pmax = d A(1 + γmin ) (2) treats the pilot as interference, providing a conservative bound. With perfect pilot cancellation, γc = A(Ps − pd )/B and pmax = Ps − γmin B/A; only the feasible pilot-power limit changes, not the allocation rule. The optimization holds Ps , Pr , g, d h, ρ̄b and η fixed, and hence also β. Proposition 1 (Kullback–Leibler-optimal pilot allocation): Assume (12) holds for a nondegenerate link with g ̸= 0, a ̸= 0, Pr > 0, σe2 > 0. Under the decorrelated residual-DPI model in (3), the Kullback–Leibler (KL) divergence D(P1 ∥P0 ) between max the distributions Pi of {y[ℓ]}L ℓ=1 under Hi increases strictly with pd for any residual DPI coefficient ρ̄b ∈ C. Hence, pd = pd is KL-optimal. Proof: Let G = |g|2 , K = |aβ|2 , C = |ρ̄b β|2 , N0 = σe2 , v̄ = GPs + σr2 and v = v̄ − Gpd ; then σ02 = N0 + Cv, σ12 = N0 + (C + K)v and Eµ /L + ∆ = K v̄. This gives D(P1 ∥P0 ) = L{K v̄/(N0 + Cv) − ln[1 + Kv/(N0 + Cv)]}, whose derivative is   LGK N02 + N0 C(v̄ + v) + C(C + K)v̄v ∂D = > 0. (14) ∂pd (N0 + Cv)2 [N0 + (C + K)v] The strict positivity proves the claim. □ Proposition 2 (Matched-filter asymptote): For g ̸= 0, a ̸= 0, pd > 0 and σe2 > 0, the coherent matched-filter deflection d2MF = 2Eµ /σ12 increases strictly with Pr and approaches lim d2MF =

Pr →∞

|a|2 2Lpd , 2 2 |ρ̄b | + |a| pg + σr2 /|g|2

(15)

which reduces to 2Lpd /(pg + σr2 /|g|2 ) under ideal DPI suppression, ρ̄b = 0. Proof: Substituting (1) and σ12 = σe2 + β 2 (|ρ̄b |2 + |a|2 )v shows that 2Eµ /σ12 increases strictly with β 2 , and hence with Pr . Dividing by β 2 and taking Pr → ∞ gives (15). □ Corollary 1 (Ideal-DPI limit of the full NP detector): Under ideal DPI suppression (ρ̄b = 0), with σe2 > 0 and |a|2 v > 0, the exact NP detector achieves PD → 1 as Pr → ∞. −1 Proof: From (8), σ02 = σe2 remains fixed while ∆, Eµ ∝ β 2 . Thus, σ02 /σ12 → 0 and λ0 = 2σ02 Eµ /∆2 → 0. Since F 2L,λ0 (α) remains finite, the threshold argument x → 0, giving PD ≥ F 2L,0 (x) → 1. □

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Proposition 3 (Residual-DPI ceiling): For fixed g, ρ̄b ̸= 0, pd , pg , σr2 , with a ̸= 0 and v > 0, the full NP detector approaches the following limit as Pr → ∞:   c0 −1 ∞ PD = F 2L,λ∞ F (α) , (16) ∞ 1 c1 2L,λ0 2 where c0 = |ρ̄b |2 v, c1 = (|ρ̄b |2 + |a|2 )v, e = L|a|2 |g|2 pd , c∆ = |a|2 v, and λ∞ i = 2ci e/c∆ . 2 2 2 Proof: Dividing (σ0 , σ1 , Eµ , ∆) by β in (8) and taking Pr → ∞ gives the stated limit. □ A. Partially Correlated Residual Direct Path Model (3) assumes decorrelated presidual DPI, whereas practical cancellation may retain part of the target-cell waveform. We model the residual as u e = ηu + 1 − η 2 u⊥ , where u contains the target-cell payload and relay noise, u⊥ is an independent copy (proper complex Gaussian, variance v, white over ℓ), and η ∈ [0, 1] is the post-processing correlation, not determined by the 0.08-cell offset. The null variance σ02 in (4) is unchanged, while
∆(η) = β 2 v |a|2 + 2η ℜ{aρ̄∗b } , (17) which reduces to the decorrelated model at η = 0 and the fully unresolved coherent model at η = 1, where u e = u. Proposition √ 1 does not follow by substitution: with ϕ = arg a − arg ρ̄b , correlation changes the variance increment to Kη = K + 2η cos ϕ KC but leaves Eµ ∝ |a|2 , invalidating Eµ /L + ∆ = K v̄. Using S = N0 + Cv and Sη = N0 + (C + Kη )v gives   ∂D CK v̄ − (Kη − K)N0 Kη N0 = LG + , (18) ∂pd S2 S Sη which reduces to (14) when Kη = K. To establish positivity, let K̃ = Kv,pC̃ = Cv, θ = v̄/v ≥ 1 (v ≥ σr2 > 0) and m = η cos ϕ. Multiplying (18) by S 2 Sη v/(LG) gives K̃J, where J = −4N0 C̃m2 + 2 K̃ C̃p (C̃θ − N0 )m + N02 + N0 C̃(1 + θ) + C̃ 2 θ + K̃ C̃θ is concave in m. Minimizing over m ∈ [−1, 1] at an endpoint and then over K̃ gives J ≥ N0 (θ − 1)(N0 /θ + C̃), positive for pd > 0 and nonnegative at pd = 0. Hence, D is strictly increasing on [0, pmax ], making pmax KL-optimal at fixed C for any d d correlation. This fixed-C model follows from the independence of ρ̄b and β from pd . In contrast, Figs. 2(f)–(g) fix the residual power Cv, hence σ02 , as pd varies; at the +5 dB level of Fig. 2(g), monotonicity holds only for −0.81 ≤ η cos ϕ ≤ 0.75. The completion in (6) remains valid for ∆(η) ̸= 0, but the relative phase may cause ∆(η) to cross zero, with a negative gap reversing ′2 the decision tail. For ∆(η) ̸= 0, define s = sgn ∆(η) and Gs,ν,λ (x) = Pr{χ′2 ν (λ) > x} for s = +1 and Pr{χν (λ) < x} for s = −1,   2 σ0 −1 2σ 2 Eµ (α) , λi = i 2 . (19) PD,η = Gs,2L,λ1 G s,2L,λ0 2 σ1 ∆(η) At ∆(η) = 0, the NP detector reduces to matched filtering. This occurs at η0 = −|a|2 /(2ℜ{aρ̄∗b }) = |a|/(2|ρ̄b || cos ϕ|) for cos ϕ < 0 andp η0 ≤ 1; |a|/(2|ρ̄b |) applies only at ϕ = π. Since the pilot mean remains, PD reaches the matched-filter value Q(Q−1 (α) − 2Eµ /σ02 ), not PFA : 0.0154, or 15PFA , in Fig. 2(g). Beyond η0 , detection recovers on the reversed tail. Thus, residual-DPI correlation determines both the variance-gap sign and the optimal decision tail. IV. L INK B UDGET AND R ESULTS We use the reproducible free-space link budget in Table I instead of normalized channel gains; the 600-km satellite–relay range is consistent with the LEO class in [8]. We choose σb = 0.035 m2 (−14.56 dBsm) as an illustrative bistatic-equivalent RCS: [9] reports −14.24 to −15.03 dBsm for three small drones at 15 GHz, based on monostatic measurements with strong aspect dependence. Fig. 2(h) therefore varies σb over ±10 dB around this value. For a one-way link of length d with antenna gains Gt , Gr and loss Lx , and for the bistatic echo, 2  λc |q(d)|2 = 10(Gt +Gr −Lx )/10 , (20) 4πd 10(Gr,t +Gs,r −Le )/10 λ2c σb |a|2 = , σn2 = kT0 B10F/10 . (21) (4π)3 d2rt d2ts Here λc is the wavelength; Gr,t and Gs,r are the relay-transmit and sensing-receive gains; Le is the echo loss; drt and dts are the two bistatic path lengths; k is Boltzmann’s constant; and F is the noise figure of the relay, user or sensing receiver for σr2 , σu2 or σe2 . Applying (20) to dsr , dru and drs gives |g|2 , |h|2 and |b|2 , respectively. This budget gives Ps |g|2 = −62.52 dBm at the relay, |a|2 Pr = −126.44 dBm at Pr = 40 W, and a sensing-noise power of −119.98 dBm. The uncancelled relay–sensor direct-path signal is 77.01 dB above the sensing-noise floor, already accounting for the 121.99 dB free-space loss over drs and the 15 and 20 dBi antenna gains. Thus, a residual-DPI/noise level of x dB in Fig. 2(f) requires 77.01 − x dB of additional isolation and cancellation, or 47–77 dB for x ∈ [0, 30] dB. The echo and direct path arrive 26.6◦ apart against a 3-dB beamwidth near 17◦ ; angular discrimination may help but is not credited. Analog suppression must precede digital cancellation to avoid receiver saturation, which is not modeled. The in-band full-duplex relay

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PD

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

Exact Gauss. MC

0.2 0.0

0

50

100

MF ceiling

NP, 12:8 NP, 2:18 MF, 2:18

0.8 0.6

150

Sensing length L

(e)

1.0

PD

(b)

Exact Gauss. MC, QPSK MC, Gauss.

0.2 0.0

0.8

10−4

10−3

10−2

10−1

(f)

L = 64 L = 128 MC

0.2

0.2 0

50

100

150

Relay power Pr (W)

200

0.0

0.4 0.2

1.0

−20

−10

0

10

20

Residual DPI/noise (dB)

30

0.50

0.75

Deterministic fraction pd /Ps

1.00

(g)

0.8 0.6

0.4

0.6

0.2 0.25

0.0

1.0

ϕ=0 ϕ = π/2 ϕ=π

2

0.4 0.2 0.50

0.75

DPI correlation η

1.00

4

6

Minimum rate Rmin (bit/s/Hz)

(h)

0.6

0.2 0.25

0

0.8

0.4

0.0 0.00

NP, 15 W NP, 40 W MF, 15 W

0.8

0.4

0.0 0.00

(d)

1.0

NP, genie GLRT Energy MC

False-alarm probability PFA

0.6

0.4

0.0

0.4

(c)

0.0 −25

Nonfluct. Rician, K=10 dB Rayleigh (Sw.-I) Log-normal, 5 dB

Table I

(a)

1.0

−20

−15

−10

Bistatic-equivalent RCS (dBsm)

−5

Fig. 2. Link-budget analysis and raw AF-cascade simulation; Monte Carlo (MC) markers carry 95% Wilson intervals and every threshold is analytic. (a) Finite-L validation at PFA = 0.02. (b) Low-PFA accuracy at L = 32, with QPSK markers. (c) Unknown echo phase: GLRT (23), genie bound (8) and energy detector. (d) Rate tradeoff at L = 64. (e) Relay power at L = 8 for pd :pg of 12:8 and 2:18 W; MF ceiling (15). (f) Decorrelated residual DPI. (g) Partially correlated residual DPI, (19), at +5 dB; at ϕ = π the gap crosses zero and PD floors at the matched-filter value. (h) Fluctuating target over a ±10 dB RCS sensitivity range, GLRT at pd /Ps = 0.5 matched to the mean |a|2 on the abscissa. Defaults: Pr = 40 W, PFA = 10−3 , L = 32, ideal DPI suppression, pd = pmax at Rmin = 2 bit/s/Hz. d TABLE I R EPRODUCIBLE 15-GH Z SAGIN S ETTING ; F IG . 2( E ) SWEEPS Pr TO 200 W. Parameter Carrier, bandwidth B, temperature T0 Satellite power; relay power Satellite-transmit / relay-receive gains Relay-transmit / user / sensing-receive gains dsr , dru , drt = dts , drs Target offset from baseline Relay / user / sensing noise figures Link losses: sr, ru, echo, direct Illustrative bistatic-equivalent UAV RCS σb

Value 15 GHz, 100 kHz, 290 K 20 W; 40 W (nominal) 38 / 30 dBi 15 / 3 / 20 dBi 600 km, 2, 1.118, 2 km 500 m (26.6◦ split) 5 / 5 / 4 dB 2 / 2 / 4 / 2 dB 0.035 m2 (−14.56 dBsm)

is assumed to suppress its own self-interference before (3). We assume low clutter: Fig. 2(h) examines target fluctuation, not ground multipath or urban clutter, which cannot generally be modeled by increasing σe2 . Monte Carlo markers use raw samples from (3), not the derived chi-square law: 4.5 × 104 trials for Fig. 2(a), 6 × 104 for (c) and (f), and 1.8 × 105 and 4 × 105 for Gaussian and QPSK markers in (b). Analytic thresholds are used throughout; the widest 95% Wilson interval is ±0.0046. Separate 106 -trial checks at the setting of Fig. 2(b) confirm detection and false-alarm performance, with PFA ∈ [0.97, 1.02] × 10−3 for Gaussian, QPSK, and 16-ary quadrature amplitude modulation (16-QAM). Figs. 2(a)–(b) confirm (8) within the 95% Wilson intervals; at Pr = 40 W, L = 32 and PFA = 10−4 , the Gaussian approximation overestimates PD by 0.056. Fig. 2(c) shows that the NP detector combines payload-echo energy and coherent pilot correlation. Fig. 2(d) illustrates (12) and the KL-optimal allocation (13)–(14): a higher minimum rate reduces PD , although the payload echo remains useful. Fig. 2(e) shows that the MF detector remains below its asymptotic limit over the simulated power range. At 200 W, its deflection is 0.90 and σ0 /σ1 = 0.70, compared with the limiting values of 1.78 and 0. Thus, MF (10) gives PD = 0.11, versus the limiting 0.91. Under ideal DPI suppression, the full NP detector has no such ceiling. The extended power sweep illustrates these trends rather than a practical 200 W relay design. Fig. 2(f) shows the effect of residual DPI: the leakage level at which PD falls to 0.2 increases from −1.6 dB at L = 64 to +3.3 dB at L = 128, indicating about 5 dB more leakage tolerance when the dwell length is doubled. A. Receiver Implementation −1

No online optimization is needed: for fixed (L, α, σ02 , σ12 , Eµ ), the threshold uα = 12 σ02 F 2L,λ0 (α) can be tabulated offline, while evaluating (5) requires one energy accumulation and one pilot correlation, PN giving O(L) complexity with constant memory. Two practical issues remain. First, σ02 must be estimated. With σ b02 = N −1 n=1 |yn |2 computed from N independent target-free reference samples, one per cell, ξ = σ b02 /σ02 satisfies 2N ξ ∼ χ22N . The estimate affects both the statistic and its threshold:

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TABLE II R EALIZED PFA W ITH N I . I . D . R EFERENCE S AMPLES (L = 32, λ0 = 126) N 16 32 64 128 512 αN /α, α = 10−3 22.7 7.5 3.2 1.9 1.2 αN /α, α = 10−4 102 21.7 6.0 2.6 1.3 cN , α = 10−4 1.53 1.18 1.07 1.03 1.01

b = P |y[ℓ] + ξκµℓ |2 , whose actual null law, normalized by the true σ 2 , is with ∆, Eµ fixed, the plug-in statistic is U 0 ℓ 2 ′2 b /σ 2 | ξ ∼ χ′2 (ξ 2 λ0 ). The receiver, however, calibrates under its assumed model, in which 2U b /b 2U σ ∼ χ (ξλ ). Thus, 0 0 0 2L 2L  −1 i h 2 (22) αN = Eξ F 2L,ξ λ0 ξ F 2L,ξλ0 (α) . The two effects partly offset, but false-alarm inflation remains. Table II shows αN /α = 102 at N = 16, α = 10−4 , falling −1 b02 F 2L,ξλ0 (α) to restore αN = α, at some cost in PD . below 3 only for N ≥ 128. The factor cN scales the plug-in threshold 12 σ This applies to the known-phase plug-in detector, not the GLRT in Section V. Second, scanning Nc approximately independent cells requires α = 1 − (1 − αscan )1/Nc ; correlated cells require an effective-cell correction. V. TARGET-C OEFFICIENT U NCERTAINTY A. Unknown Echo Phase With B = 100 kHz, the range resolution is c/2B = 1499 m, corresponding to 7.5 × 104 wavelengths at 15 GHz, so the range cell alone does not determine arg a. We model the echo phase as an unknown deterministic parameter without prior phase information and maximize the likelihood over it; phase estimates from previous observations could also be used. The magnitude is better constrained: (21) gives |a|2 for an assumed RCS, although the actual RCS may vary. Since ∆, w and κ depend only on |a|, maximizing the P log-likelihood ratio over the phaseP affects only the coherent term in (5),√replacing ℜ{·} by its modulus. T Defining W = L−1/2 ℓ c∗ [ℓ]y[ℓ], R = |W | and V = ℓ |y[ℓ]|2 − R2 , projection onto c/ L, with p c = [c[1], . . . , c[L]] , 2 makes V independent of W . Completing the square as in (6) then gives UG = (R + ω) + V , ω = κ Eµ . This derivation requires ∆, w, κ to depend only on |a|, which is not the case in (17) when η ρ̄b ̸= 0. In that case, ∆(η) depends on arg a, so maximizing over the phase also changes the variance and normalization. Thus, (19) remains a known-coefficient benchmark. Proposition 4 (Exact unknown-phase GLRT):pFor L ≥ 2, η ρ̄√ b = 0, unknown echo phase and assumed |a|, under Hi , R follows a Rice distribution with noncentrality δ E , scale σ / 2 and density fRi , while V ∼ (σi2 /2)χ22L−2 is independent i µ i √ 2 of R. Thus, for u > ω and r⋆ = u − ω,
! Z r⋆ 2 u − (r + ω)2 fRi (r) dr Pr{UG > u} = F 2L−2,0 σi2 0
2 2 + F 2,λW 2r /σ , ⋆ i i

λW i =

2δi2 Eµ , σi2

(23)

and Pr{UG > u} = 1 for u ≤ ω 2 . Setting the threshold using (23) under H0 to achieve α and evaluating the same expression under H1 gives the exact PD for a Gaussian payload, given |a| and the true σ02 . Both tail probabilities involve single integrals, so the finite-L exactness of (8) is retained, with (8) serving as the genie bound. The performance loss arises entirely from the coherent component and vanishes without a pilot: Fig. 2(c) shows that the two detectors coincide with the energy detector at pd = 0 but diverge as the pilot power increases. B. Plug-In Mismatch and Target Fluctuation b = ϱ2 ∆ > 0 and κ b while the actual target-present Consider a detector using b a = ϱejψ a, giving µ bℓ = ϱejψ µℓ , ∆ b = σ02 /∆, P 2 mis distribution remains CN (µℓ , σ1 ). Calibrating the false-alarm probability exactly for U = ℓ |y[ℓ] + κ bµ bℓ |2 gives  2  σ0 −1 mis PD = F 2L,ζ1 F (α) , (24) σ12 2L,ζ0 where ζ0 = 2b κ2 ϱ2 Eµ /σ02 and ζ1 = 2Eµ (1 + κ b2 ϱ2 + 2b κϱ cos ψ)/σ12 . Mismatch changes only the noncentrality parameters of the chi-square distributions, and ϱ = 1, ψ = 0 recovers (8). For the known-phase detector, Fig. 3 gives a matched detection probability of 0.3758: a ±6 dB magnitude error reduces it by at most 0.016, whereas a 60◦ phase error alone causes a reduction b reduces κ of 0.30. A magnitude overestimate can even improve detection at large ψ: increasing ∆ b and gives less weight to the mismatched coherent term. The GLRT is insensitive to phase error, and over the same ±6 dB magnitude range its PD varies by at most 0.010 around 0.230, as shown by the flat band in Fig. 3(b). These results support estimating the phase while using an assumed magnitude, rather than treating the magnitude as perfectly known. For target fluctuations with |a|2 fixed within each dwell, PD is averaged over |a|2 , while the false-alarm rate under H0 remains unchanged. Fig. 2(h) compares non-fluctuating, Rician (10-dB factor), Rayleigh (Swerling-I) [10], [11], and log-normal

7

PD

100

100

(a)

10−2

10−2

10−4

10−4 pd /Ps = 0.2 pd /Ps = 0.5 pd /Ps = 0.8

10−6 10−8

0

45

(b)

magnitude

−6 dB +0 dB +6 dB GLRT, ϱ ∈ [−6, 6] dB

10−6

90

135

Phase mismatch (degree)

180

10−8

0

45

90

135

Phase mismatch (degree)

180

Fig. 3. Exact plug-in mismatch of the known-phase detector, (24), at Pr = 40 W, L = 32, PFA = 10−3 , ideal DPI suppression. (a) Phase error; pilot-heavy operation is more phase-sensitive. (b) Joint error at pd /Ps = 0.5; dashed band: unknown-phase GLRT over the same ±6 dB magnitude error, flat in ψ.

models fitted to 15-GHz drone measurements [9]. All use the same linear mean |a|2 ; the log-normal standard deviation is 5 dB for 10 log10 |a|2 . Detector weights and thresholds are set at the nominal mean, not adjusted for each realization. VI. C ONCLUSION This letter investigated bistatic UAV detection in SAGINs, where an AF relay supports communication while also illuminating a target. We derived exact finite-sample detection probabilities by accounting for the target-reflected pilot, payload, and relay noise. The analysis showed that reflected relay noise provides useful variance information even without payload transmission, while relay gain normalization couples communication and sensing performance. We further characterized the effects of relay power and residual direct-path interference, and developed an exact phase-only GLRT for an assumed target magnitude. The results provide a basis for practical dual-use AF sensing in SAGINs, while joint magnitude–phase estimation and clutter remain open challenges. R EFERENCES [1] F. Liu, Y. Cui, C. Masouros, J. Xu, T. X. Han, Y. C. Eldar, and S. Buzzi, “Integrated sensing and communications: Toward dual-functional wireless networks for 6G and beyond,” IEEE J. Sel. Areas Commun., vol. 40, no. 6, pp. 1728–1767, Jun. 2022. [2] Y. Xiong, F. Liu, Y. Cui, W. Yuan, T. X. Han, and G. Caire, “On the fundamental tradeoff of integrated sensing and communications under Gaussian channels,” IEEE Trans. Inf. Theory, vol. 69, no. 9, pp. 5723–5751, Sep. 2023. [3] G. Cheng, X. Song, Z. Lyu, and J. Xu, “Networked ISAC for low-altitude economy: Coordinated transmit beamforming and UAV trajectory design,” IEEE Trans. Commun., vol. 73, no. 8, pp. 5832–5847, Aug. 2025. [4] X. Song, X. Yu, J. Xu, and D. W. K. Ng, “Integrated sensing and communications for low-altitude economy with deterministic sensing and Gaussian information signals,” arXiv:2604.19040, Apr. 2026. [5] L. Xie, F. Liu, S. Song, and S. Jin, “Bistatic target detection by exploiting both deterministic pilots and unknown random data payloads,” arXiv:2508.18728, Aug. 2025. [6] X. Song, X. Qin, X. Yu, J. Xu, and D. W. K. Ng, “Active-IRS-enabled target detection,” IEEE Wireless Commun. Lett., vol. 14, no. 9, pp. 2763–2767, Sep. 2025. [7] Bhola, Y.-J. Chen, A. Balakrishnan, S. De, and L.-C. Wang, “Adaptive multi-UAV relay deployment framework in satellite aerial ground integrated systems,” IEEE Trans. Veh. Technol., early access, 2026. [8] 3GPP, “NR; Satellite access node conformance testing,” 3GPP TS 38.181, V17.1.0, Release 17, Jul. 2023. [9] M. Ezuma, M. Funderburk, and I. Guvenc, “Compact-range RCS measurements and modeling of small drones at 15 GHz and 25 GHz,” in Proc. IEEE Radio Wireless Symp. (RWS), San Antonio, TX, USA, Jan. 2020, pp. 313–316. [10] P. Swerling, “Probability of detection for fluctuating targets,” IEEE Trans. Inf. Theory, vol. 6, no. 2, pp. 269–308, Apr. 1960. [11] X. Song, W. D. Blair, P. Willett, and S. Zhou, “Dominant-plus-Rayleigh models for RCS: Swerling III/IV versus Rician,” IEEE Trans. Aerosp. Electron. Syst., vol. 49, no. 3, pp. 2058–2064, Jul. 2013.

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