{"id":"e16b5a63-b541-4401-b6e0-73b9d1478584","arxiv_id":"1908.05308","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"This translated dissertation presents a maximum-likelihood method, a stochastic-approximation implementation, and a sequential test for resolving closely spaced targets with antenna arrays.","lead":"An English translation of Ulrich Nickel's 1982 PhD dissertation derives maximum-likelihood angular superresolution for closely spaced radar targets using antenna arrays. It develops a stochastic-approximation algorithm and a sequential multihypothesis test that estimates the number of targets while controlling overestimation errors.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The sequential multihypothesis test's claimed Type-1 error control rests on Γ̂Ab≈0 and on exactly known noise covariance; the paper's own Figs. 7-4/7-5 show the claim fails under 1-dB noise mismatch.","rationale":"The reader's weakest assumption identified the same load-bearing issue: the multihypothesis test threshold depends on known noise statistics and on the residual approximation Γ̂Ab≈0. This stress-test confirms and sharpens that concern by locating it in Eqs. (7.1-3)-(7.1-6) and connecting it to the paper's own simulations in Sec. 7.3 and Summary item 9. The concern does not reveal an internal inconsistency under the exact model; the derivations of the Q-function minimization, the uniqueness theorems, and the stochastic approximation are coherent, and the paper is transparent about the noise-model sensitivity. Therefore a rejection is not warranted. However, the central claim about approximately maintaining α should not be read as unconditional: it requires known noise first/second moments and sufficiently high SNR and iteration count. The paper already states much of this in Sec. 8, item 9, which is why the existing CONDITIONAL verdict is the right level. The proposed Monte Carlo test would quantify the degradation under the exact mismatch the paper flags as most critical, and would settle whether the condition is merely a caveat or a genuine boundary of the claim. Until that quantification is available, the verdict should remain conditional rather than accept.","tokens_in":55941,"tokens_out":12280,"duration_ms":130103,"concrete_test":"Run a Monte Carlo replication of the Fig. 7-1 scenario (ELAN 21L, two targets 0.55BW apart, Signal Model 4, true M=2, nominal α=5%, K=2, 17 iterations) with 10^5 trials. Compute the empirical Type-1 error P(φ_2=1) under three noise settings: (i) exact R=σ²I; (ii) R=σ²diag(1+ε_i), ε_i uniform ±25% (1 dB), with the same threshold η from (7.1-6); (iii) exact R but with Q̄ evaluated at the true directions instead of the estimated directions. If (i) matches 5% within Monte Carlo error, the asymptotic approximation is adequate under the ideal model; if (ii) exceeds 5% by more than the simulation error, the known-noise condition is load-bearing; the comparison (i) vs (iii) isolates the effect of the Γ̂Ab≈0 approximation. This directly tests the weakest link in the claimed α control.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing claim is that the sequential test in (3.3-1) determines M while approximately maintaining a prescribed α. The threshold η in (7.1-6) is derived from the approximation (7.1-3), Γ̂Ab_i≈0, so that KQ̄ is treated as σ² times a χ² variable with K(2N−2M) degrees of freedom, and from an exactly known σ². Both ingredients are approximate. The convergence theorem (5.1-6/5.1-8) guarantees Γ̂Ab_i→0 only asymptotically and only if the stochastic approximation reaches the attraction region; the paper verifies the attraction region by inspecting cuts of Q rather than by proof. In the operating regime actually used in Chapter 7 (17 iterations, SNR 3–7 dB, K=2–4), no bound on Γ̂Ab_i is given. The paper itself flags the consequence: Sec. 7.3 and Summary item 9 state that deviations from the noise model are the most critical, and Figs. 7-4/7-5 show that 1-dB receiver-noise fluctuations make the test miss level α and that an unmodeled jammer roughly halves the detection probability. The Type-1 guarantee is therefore conditional on noise moments that are not known in many radar applications and on a convergence that is only asymptotic. Without making these conditions explicit, the headline claim is stronger than what is established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript, an authorized translation of Ulrich Nickel's 1982 doctoral dissertation, addresses maximum-likelihood angular superresolution of closely spaced far-field point targets and the estimation of the number of targets present. It derives the criterion Q(ω)=z*Γz from the Gaussian likelihood, introduces strong and weak M-regularity conditions for identifiability and uniqueness, characterizes estimation accuracy through the curvature of Q and the Cramér-Rao bound, proposes a Robbins-Monro stochastic-approximation algorithm using sum and difference beams, and constructs a sequential multihypothesis test based on the averaged statistic Q̄ with a chi-square/normal threshold. Analytic detection probabilities for Swerling-II targets are derived via the distribution of averaged Hermitian forms, and extensive simulations examine robustness to coupling, extended targets, quantization, and noise mismatch.","tokens_in":56143,"tokens_out":8020,"duration_ms":74078,"significance":"If the results hold, the paper is a rigorous early treatment of ML superresolution and model-order selection, and several elements remain valuable: the Q-function formulation, the CRLB-based resolution curves, the stochastic-approximation implementation, and the closed-form detection probabilities of Theorem 7.2-2. The appendices contain complete proofs of the principal theorems and careful moment calculations, and the analytic detection formula is a parameter-free prediction in terms of the eigenvalues of A*ΓAB and σ², tested by simulation. The principal limitation, acknowledged in the text, is that the Type-1 error control of the sequential test is approximate and depends on exact knowledge of the noise covariance and on asymptotic convergence; the simulations show degradation under 1-dB noise mismatch and jamming. With these conditions made explicit, the paper is a useful archival contribution.","major_comments":[{"comment":"The claimed approximate Type-1 error control of the sequential test rests on the approximations Γ̂Ab_i≈0 (7.1-3) and exactly known σ²I noise, and the convergence supporting the first approximation is only asymptotic (Corollary 5.1-8). The operating regime of Chapter 7 (17 iterations, K=2–4 samples, SNR 3–7 dB) is finite, and no bound on ‖Γ̂Ab_i‖ is provided for that regime. The paper's own simulations show that 1-dB receiver-noise fluctuations make the test miss level α and that an unmodeled jammer roughly halves the detection probability (Figs. 7-4, 7-5; Summary item 9). The headline claim should be restated as conditional on exact noise moments and sufficiently converged estimates; as written it is stronger than what is proved.","section":"§7.1, Eqs. (7.1-3)–(7.1-6); §7.3, Figs. 7-4/7-5"},{"comment":"The Ljung theorem requires that all trajectories of ˙ω=−grad E{Q} starting in Ω2 remain in Ω1, and the text states that this cannot generally be verified because the location of the minimum in Ω1 is unknown. The assertion that 'all considered cuts' show M(ωg) to be part of the attraction region is an empirical statement, not a proof. Since convergence of the stochastic approximation is the estimation step on which the subsequent test relies, this missing verification is load-bearing; the paper should either prove the attraction-region condition for the arrays used or explicitly present the convergence result as conditional on the Ljung hypotheses, with the simulations as supporting evidence.","section":"§5.1, Eq. (5.1-7) and Corollary (5.1-8)"},{"comment":"The factorization of the total error probability assumes independence of the individual tests ϕi, which the text itself says requires new data for each test. The implementation in Chapter 7 does not state clearly whether the Q̄_M statistics for different M are computed from independent K-sample sets or from the same data; if they are computed from the same data, the product formula (3.3-3) does not apply and no alternative bound is given. Please clarify the data-reuse policy and, if data are reused, provide an analysis of the dependence or a conservative bound.","section":"§3.3, Eq. (3.3-3); §7.2 and §7.3"}],"minor_comments":[{"comment":"The definition of Γ omits the inverse on (A*R^{-1}A); compare with the correct form in Eq. (3.2-1). This should be corrected.","section":"Eq. (3.2-3)"},{"comment":"Several typographical and OCR artifacts remain in the translation, e.g., 'maping' in the symbol list, 'Kroneker' for 'Kronecker', and 'fiior' in §4.4; a thorough proofreading pass is needed.","section":"Symbol list and §4.4"},{"comment":"The figure captions in the provided text are minimal and do not always identify which of the four correction-vector variants, which SNR, and which target model are plotted; adding complete captions would improve reproducibility.","section":"§5.2, Figs. 5-6 to 5-9"},{"comment":"The normal approximation to the chi-square variable should state explicitly the range of N and K for which it is intended; the simulations use N=21 and K=2–4, and the text notes that the approximation is already 'sufficiently accurate' there, but no quantitative criterion is given.","section":"§7.1, Eq. (7.1-5)"},{"comment":"The text refers to '(7.2-5)' for the unequal-noise moments, but the displayed formulas for E_n{Q̄} and var_n{Q̄} are not numbered in the manuscript; the equations should be labeled.","section":"§7.3, paragraph before Eq. (7.2-5)"}],"recommendation":"major_revision","confidential_remarks":"This is a translation of a 1982 doctoral dissertation. The contribution is primarily archival and historical; the editorial decision should weigh whether the journal wishes to publish translations of dissertations and whether the lack of a modern literature review is acceptable. The citation list is contemporary to the original work, which is appropriate for a translation but should be flagged for readers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this arXiv posting is an English translation of Nickel's 1982 PhD dissertation, and it does not claim to be new research. The reader's verdict of CONDITIONAL is fair. If you read it as a 2019 preprint offering novel results, it would be a desk reject. If you read it as making a 1982 thesis accessible to a modern radar audience, it is a solid, useful piece of work.\n\nWhat is genuinely good: the core derivation is coherent and mostly complete. The Q-function reduction of ML direction estimation, the M-regularity conditions for uniqueness, and the stochastic approximation implementation are all laid out carefully. The paper also does something rare: it is explicit about its own limitations. Section 7.3 and Summary item 9 state plainly that deviations from the assumed noise model are the most critical weakness, and Figures 7-4/7-5 show the sequential test missing its level under 1-dB noise mismatch. That honesty deserves credit.\n\nSoft spots, in proportion. The Type-1 error control of the multihypothesis test rests on two approximations: the estimator is good enough that Gamma-hat A b is nearly zero, and the noise covariance is known exactly. The first is only asymptotic, and the attraction region of the stochastic approximation is verified by inspecting cuts of Q, not by proof. The second is acknowledged by the paper itself, but it is still load-bearing because radar noise moments are often not known to the required accuracy. The chi-squared-to-normal approximation also introduces a small but visible discrepancy, as the paper's own figures show. These are not fatal flaws in a 1982 dissertation; they are honest boundaries of what was established. But a modern reader should not take the headline claim of approximate Type-1 control as a blanket guarantee.\n\nThe citation pattern is fine: the dissertation cites the relevant 1970s literature, and the translation adds no new claims, so there is no self-citation inflation or missing modern context. The math is mostly checkable; the OCR rendering has some garbled equations, but the structure survives.\n\nWho is this for? Someone working on radar superresolution who wants the historical foundation of ML-based angular resolution and a clear derivation of the Q-function and stochastic approximation. It is not for someone looking for new results. I would not cite it in my own current work, but I might bring it to a reading group as a historical benchmark.\n\nRecommendation: send it to peer review as a historical/expository contribution, not as a novel method. A serious referee can verify the translation quality and check the original 1982 claims. The paper deserves referee time, with the clear expectation that the review focuses on historical accuracy and clarity, not novelty.","headline":"A clean translation of a 1982 ML superresolution dissertation that still reads well, but it is a historical document, not a new result, and its Type-1 error control is honestly conditional on noise-model knowledge.","tokens_in":56732,"tokens_out":893,"would_cite":false,"duration_ms":12979,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F03","62F05","62F12","62L20","94A12"],"pacs":["84.40.Xb","02.50.-r"],"model":"deepseek-v4-flash","headline":"Maximum-likelihood resolution of closely spaced radar targets reduces to minimizing one residual-energy function, and a sequential test on the averaged residual counts the targets while holding the false-alarm probability at a chosen level.","keywords":["angular superresolution","maximum likelihood estimation","direction-of-arrival estimation","antenna arrays","stochastic approximation","multihypothesis testing","target-number determination","Cramér–Rao bound"],"falsifier":"Simulate the sequential multihypothesis test with no targets present: with known $\\sigma^2 I$ noise, the empirical frequency of declaring one or more targets should sit at the chosen $\\alpha$ up to the $\\chi^2$-to-normal approximation the paper quotes for $N = 21$. The same simulation with the noise-model violations the paper tested, namely 1 dB per-channel power fluctuations or a sidelobe jammer of comparable power while the un-corrected threshold is used, should show the false-alarm rate and detection probability moving away from the predicted values, exactly the sensitivity documented in the paper's own figures.","tokens_in":55654,"feed_emoji":"📡","tokens_out":19902,"duration_ms":170518,"temperature":0.7,"pith_summary":"This dissertation shows that angular superresolution of closely spaced point targets with an antenna array is one well-posed optimization problem: the maximum-likelihood directions are the minimizers of $Q(\\omega) = z^*\\Gamma(\\omega)z$, the data energy left after projecting out the assumed target directions, and the curvature of $Q$ at the minimum equals the inverse of the direction block of the Fisher information matrix, tying estimation accuracy to the shape of this surface. Uniqueness of the solution is tied to an explicit algebraic condition on the array geometry, termed strong or weak $M$-regularity of the element positions, and a stochastic-approximation algorithm using only the current sum-and-difference beam outputs is shown to converge almost surely with covariance close to the Cramér–Rao bound. For the companion detection problem, deciding how many targets are present, the paper builds a sequential multihypothesis test on the averaged residual $\\bar Q$, whose $\\chi^2$ null distribution fixes the threshold so that the probability of overestimating the target count stays near a prescribed level $\\alpha$. If the claims hold, a radar can resolve targets well inside a beamwidth: two targets separated by half a beamwidth at roughly 17 dB SNR are resolved with direction errors near 0.04 beamwidths after 17 iterations, using ordinary sum-and-difference beams rather than supergain weighting. The same machinery transfers to other linear Gaussian problems, and the paper explicitly notes the spectral-line (time-sampled) case as an application.","feed_headline":"One projection resolves radar targets below a beamwidth","feed_subtitle":"Direction finding becomes one residual-energy minimization; a sequential test counts targets at a set error level.","key_machinery":"The load-bearing object is the projection residual $Q(\\omega) = z^*\\Gamma(\\omega)z$, where $\\Gamma = I - A(\\omega)(A(\\omega)^*A(\\omega))^{-1}A(\\omega)^*$ is the orthogonal projector onto the orthogonal complement of the signal subspace spanned by the $M$ steering vectors; $Q$ is literally the residual energy after the best fit of the assumed target directions is removed. Four structural facts about this object carry the argument. Minimizing $Q$ over the directions is equivalent to maximum-likelihood estimation, and the Hessian of $Q$ at the minimum is the inverse of the direction block of the Fisher information matrix (Theorem 4.3-2), connecting resolution accuracy to the local shape of the $Q$-surface. Uniqueness of the global minimum reduces to the purely algebraic strong or weak $M$-regularity conditions on the element positions, which tell a designer whether a given array can resolve $M$ targets at all. After the estimates are good, the normalized statistic $(2/\\sigma^2)K\\bar Q$ is $\\chi^2$-distributed with $K(2N-2M)$ degrees of freedom when exactly $M$ targets are present, so a chosen false-alarm probability $\\alpha$ becomes a closed-form threshold. Under the alternative, $K\\bar Q$ is an averaged positive-definite Hermitian form in complex Gaussian variables, and Theorem 7.2-2 gives its probability density in closed form, converting the threshold into detection probabilities.","core_discovery":"The paper's central claim is that resolving $M$ closely spaced point targets with an antenna array is a maximum-likelihood problem whose solution is the global minimum of a single scalar function: $Q(\\omega) = z^*\\Gamma(\\omega)z$ with $\\Gamma = I - A(A^*A)^{-1}A^*$, the squared residual of the data after its component in the span of the $M$ assumed steering vectors is removed. Minimizing $Q$ is exactly ML direction estimation, and Theorem 4.3-2 states that the second-derivative (curvature) matrix of $Q$ at the true minimum is the inverse of the direction block of the Fisher information matrix, so the accuracy of the estimate is governed by the curvature of this surface. Existence and uniqueness of the global minimum are guaranteed by the array geometry: a strongly $M$-regular layout, meaning every $2M$ steering vectors $a(\\omega_1), \\dots, a(\\omega_{2M})$ are linearly independent (requiring $N \\geq 2M$), suffices for a single spatial sample, while with several temporal samples and a regular covariance of the complex amplitudes, only weak $M$-regularity, meaning every $M+1$ steering vectors are independent (requiring $N \\geq M+1$), is needed. For detection, the paper proves that the averaged statistic $\\bar Q = \\frac{1}{K} \\sum_{k=1}^K \\|\\hat\\Gamma z_k\\|^2$ is the maximally invariant data reduction, that $(2/\\sigma^2)K\\bar Q$ is approximately $\\chi^2$-distributed with $K(2N-2M)$ degrees of freedom under the hypothesis of $M$ targets, and that a sequential multihypothesis test built on this statistic keeps the probability of overestimating the number of targets near a prescribed level $\\alpha$; detection probabilities are computed exactly for Swerling-II targets through the density of averaged positive-definite Hermitian forms (Theorem 7.2-2).","pith_inferences":["The threshold formula $\\eta = \\sigma^2(\\sqrt{(N-M)/K}\\, U_\\alpha + N-M)$ depends on no nuisance parameters besides the noise power $\\sigma^2$, so the same test could be applied to spectral-line counting in time-sampled data, a transfer the introduction mentions but the paper does not develop, by estimating $\\sigma^2$ from the residual at the largest fitted $M$.","The paper's sensitivity results suggest a concrete extension it does not test: whiten the data with an estimated noise covariance and then apply the unchanged $\\chi^2$ threshold; if the noise moments are recovered well, the nominal $\\alpha$ should be restored in the correlated-noise cases where the paper's own simulations show it fails.","The invariance of $Q$ under common element responses implies a design rule the paper stops short of stating: arrays built from matched subarrays with equal phase centers preserve the resolution test exactly, not merely approximately.","The summary's direction-frequency ambiguity relation is qualitative; applying the same $Q$-curvature analysis to the time-sampled spectral-line formulation would make the trade-off quantitative and check whether the Cramér–Rao equivalence of Theorem 4.3-2 carries over to frequency estimation."],"forward_implications":["Two targets separated by half a beamwidth can be resolved in about 17 iterations of the stochastic approximation, with direction standard deviations near 0.04 beamwidths, provided the total SNR is on the order of 16–17 dB (slightly higher for in-phase than for quadrature targets).","Temporal sampling can substitute for spatial sampling: with several pulses, an array need only be weakly $M$-regular (a 3-element linear array suffices for two targets), and pulse-to-pulse phase fluctuations wash out the in-phase and opposite-phase configurations that defeat single-pulse estimation.","The multihypothesis test holds the probability of overestimating the target count at the chosen level $\\alpha$ (about 5–10% is recommended), and this control is essentially independent of the amplitude-fluctuation model; the Swerling-II detection curves serve as pessimistic lower bounds for the other models.","The recursion is radar-friendly: each iteration consumes only the current sum-and-difference beam outputs, no data vector is stored, and for planar arrays the operation count favors the stochastic approximation over an averaged grid search.","Single-pulse superresolution is not practical for radar, since it demands SNR above roughly 25 dB; the intended mode of use is the multi-pulse sequential procedure."],"supporting_citations":[{"why":"First formulation of ML direction estimation as minimization of the Q function for a single sample; the paper's estimation half starts from here.","marker":"[11]"},{"why":"The most extensive earlier study of the Q function with the two extreme signal models; the paper corrects its high-SNR claim and adds a level-alpha sequential test.","marker":"[20]"},{"why":"Supplies the statistical foundations: asymptotic optimality of ML, likelihood-ratio asymptotics, sufficiency, and invariance results used for the test construction.","marker":"[4]"},{"why":"Ljung's convergence theorem for recursive stochastic algorithms, quoted as Theorem 5.1-6 to prove almost-sure convergence of the direction estimator.","marker":"[6]"},{"why":"Source of Fabian's asymptotic-normality theorem (5.1-9), used to show the stochastic approximation is nearly Cramér–Rao efficient.","marker":"[24]"},{"why":"Goodman's complex Gaussian distribution, used in Appendix A.5 to derive the characteristic function behind Theorem 7.2-2's density of averaged Hermitian forms.","marker":"[5]"},{"why":"Stuller's generalized-likelihood signal resolution, which supplies the decomposition of a multihypothesis test into pairwise 2-hypothesis tests used in Chapter 3.","marker":"[3]"},{"why":"Skolnik's radar handbook, the source of the Swerling-II (Rayleigh) target fluctuation model used for the detection-probability computations and curves.","marker":"[1]"}],"fun_headline_variants":["One residual minimum resolves sub-beamwidth targets","Residual-energy test counts and locates closely spaced targets","ML angular superresolution via a single scalar minimization","Curvature of residual surface sets sub-beamwidth accuracy","Sequential residual test pins target count at set error"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire error-control machinery presumes that the noise is Gaussian with known covariance (normalized to $\\sigma^2 I$ in the derivation, or with its first two moments known); the paper itself reports that deviations from the assumed noise model are the most critical influence on the test, and without accurate noise moments the claimed level $\\alpha$ is not attained.","fun_headline_variants_meta":{"raw":{"variants":["One residual minimum resolves sub-beamwidth targets","Residual-energy test counts and locates closely spaced targets","ML angular superresolution via a single scalar minimization","Curvature of residual surface sets sub-beamwidth accuracy","Sequential residual test pins target count at set error"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000343,"raw_usage":{"total_tokens":1910,"prompt_tokens":997,"completion_tokens":913,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":838}},"tokens_in":613,"tokens_out":913,"duration_ms":9911,"temperature":1.0,"reasoning_tokens":838,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:17:50.693880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the sequential multihypothesis test with no targets present: with known $\\sigma^2 I$ noise, the empirical frequency of declaring one or more targets should sit at the chosen $\\alpha$ up to the $\\chi^2$-to-normal approximation the paper quotes for $N = 21$. The same simulation with the noise-model violations the paper tested, namely 1 dB per-channel power fluctuations or a sidelobe jammer of comparable power while the un-corrected threshold is used, should show the false-alarm rate and detection probability moving away from the predicted values, exactly the sensitivity documented in the paper's own figures.","supporting_citations":[{"cited_title":"A decision theoretic approach to the angular resolution and parameter estimation of multiple targets,","cited_arxiv_id":null,"evidence_quote":"First formulation of ML direction estimation as minimization of the Q function for a single sample; the paper's estimation half starts from here."},{"cited_title":"Parameter estimation of multiple signals,","cited_arxiv_id":null,"evidence_quote":"The most extensive earlier study of the Q function with the two extreme signal models; the paper corrects its high-SNR claim and adds a level-alpha sequential test."},{"cited_title":"Witting and G","cited_arxiv_id":null,"evidence_quote":"Supplies the statistical foundations: asymptotic optimality of ML, likelihood-ratio asymptotics, sufficiency, and invariance results used for the test construction."},{"cited_title":"Analysis of recursive stochastic algorithms,","cited_arxiv_id":null,"evidence_quote":"Ljung's convergence theorem for recursive stochastic algorithms, quoted as Theorem 5.1-6 to prove almost-sure convergence of the direction estimator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of Fabian's asymptotic-normality theorem (5.1-9), used to show the stochastic approximation is nearly Cramér–Rao efficient."},{"cited_title":"Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction),","cited_arxiv_id":null,"evidence_quote":"Goodman's complex Gaussian distribution, used in Appendix A.5 to derive the characteristic function behind Theorem 7.2-2's density of averaged Hermitian forms."},{"cited_title":"Generalized likelihood signal resolution,","cited_arxiv_id":null,"evidence_quote":"Stuller's generalized-likelihood signal resolution, which supplies the decomposition of a multihypothesis test into pairwise 2-hypothesis tests used in Chapter 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Skolnik's radar handbook, the source of the Swerling-II (Rayleigh) target fluctuation model used for the detection-probability computations and curves."}],"review_version":1}