Pith. sign in

REVIEW 3 major objections 5 minor 30 references

Angular Resolution of Closely-Spaced Targets with Antenna Arrays

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict A clean translation of a 1982 ML superresolution dissertation that still reads well, but it is a historical document, not a new result, and its Type-1 error control is honestly conditional on noise-model knowledge. read the letter →

arxiv 1908.05308 v1 pith:YEOL7LCF submitted 2019-08-14 eess.SP

Ulrich Nickel , David F Crouse This is my paper · ORCID
classification eess.SP MSC 62F0362F0562F1262L2094A12 PACS 84.40.Xb02.50.-r
keywords angularsuperresolutionmaximumlikelihoodestimationdirection-of-arrivalantennaarraysstochasticapproximationmultihypothesistestingtarget-numberdeterminationCramér–Raobound
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§7.1, Eqs. (7.1-3)–(7.1-6); §7.3, Figs. 7-4/7-5] 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.
  2. [§5.1, Eq. (5.1-7) and Corollary (5.1-8)] 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.
  3. [§3.3, Eq. (3.3-3); §7.2 and §7.3] 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.
minor comments (5)
  1. [Eq. (3.2-3)] 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.
  2. [Symbol list and §4.4] 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.
  3. [§5.2, Figs. 5-6 to 5-9] 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.
  4. [§7.1, Eq. (7.1-5)] 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.
  5. [§7.3, paragraph before Eq. (7.2-5)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained: Q-minimization and detection thresholds follow from the stated Gaussian model with no fitted parameter renamed as a prediction.

full rationale

No significant circularity is present. The central estimation result, minimization of Q(ω)=z*[I−A(A*A)^−1A*]z, is obtained by algebraically eliminating the complex amplitudes b from the Gaussian likelihood (Secs. 3.2 and 4.1), so it is a derived consequence of the model rather than an input. The uniqueness theorems (4.2-2 and 5.1-2) are proved directly from the stated strong/weak M-regularity assumptions using linear independence of the array manifold, with proofs included in the text and appendices. The accuracy claims are benchmarked against the Cramér-Rao lower bound, which is an external information-theoretic standard, not a fitted quantity. The multihypothesis test threshold η in (7.1-4)/(7.1-6) is derived from an explicitly stated approximation, Γ̂Ab≈0, which makes KQ̄ approximately chi-squared or normal under the assumed known noise covariance; the paper then validates this threshold by simulation. The paper itself flags in Sec. 7.3 and Summary item 9 that deviations from the noise model are the most critical limitation and shows in Figs. 7-4/7-5 that the claimed level α fails under 1-dB noise mismatch. This is an honest conditional claim with exposed assumptions, not circular reasoning: the guarantee is stated for the model in which it is derived, and its failure outside that model confirms it is a substantive prediction. The heuristic constants μ and η are tuned by simulation (Secs. 5.1, 5.2), but they affect convergence speed and correction-vector damping rather than the identifiability, CRLB, or Type-1 error claims, and they are not presented as derived predictions. External mathematical results cited, such as Ljung's stochastic-approximation theorem, Fabian's asymptotic-normality theorem, and Goodman's distribution theory, are standard independent results, not self-citations carrying the argument. No step in the paper reduces, by construction or by self-citation, to its own output.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation is a parameterized statistical model with no fitted constants in the Q-function minimization itself; the two heuristic algorithm parameters mu and eta are tuned by simulation but do not enter the theoretical identifiability or CRLB results. The main burden is the model assumption of known Gaussian noise and the regularity conditions on the array.

free parameters (2)
  • Stochastic-approximation step-size coefficient mu = mu = delta/(ae) * BW/2 with delta=0.9 for linear arrays, delta=1.8 for planar arrays
    Eq. (5.1-10): the step size is estimated from the first three measurements and the constants are chosen through simulation, not derived.
  • Correction-vector saturation bound eta = eta = 0.8 * ae
    Section 5.2, Version 3: the bound is a heuristic damping value chosen from simulations; the paper says it proved good through simulation.
assumptions (6)
  • domain assumption Far-field point-target plane-wave signal model z = Ab + n with additive Gaussian noise, where targets are infinitely distant point scatterers (Eq. 2.1-1, 2.1-2).
    The entire derivation uses this parametric model; the paper states point targets are assumed for radar applications.
  • domain assumption The receiver noise is independent, zero-mean complex Gaussian with known covariance, normalized to a multiple of the identity for the main derivation (Section 2.2, and normalization sigma^2=1).
    Identifiability and the test threshold both require the noise distribution and variance; Chapter 8 item 9 says noise-model deviations are the most critical.
  • domain assumption Array layouts satisfy strong M-regularity (4.2-1) or weak M-regularity (5.1-1) in the search region.
    Uniqueness of the Q-function minimum and consistency of ML estimation are proven only under these regular-layout assumptions.
  • standard math Ljung's theorem on recursive stochastic algorithms (Theorem 5.1-6) applies; the paper checks conditions including Lipschitz continuity and the existence of fourth moments.
    The convergence of the stochastic approximation is imported from Ljung (1977) and used as the convergence proof engine.
  • domain assumption After a good estimate, the residual projection satisfies Gamma Ab approximately 0, making Q_bar approximately chi-squared distributed with 2(N-M)K degrees of freedom (Eq. 7.1-3).
    This approximation sets the detection threshold and is described as asymptotically valid; the paper's simulations show small deviations from the predicted detection curve.
  • domain assumption Ergodic averaging: time averages of zz* and bb* converge to expected values, so K-sample ML minimization is replaced by expected Q (Eq. 3.2-4).
    Used to justify the stochastic approximation and the averaged grid search.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Angular Resolution of Closely-Spaced Targets with Antenna Arrays." pith.science (2026). https://pith.science/paper/YEOL7LCF

@misc{pith2026190805308,
  author       = {Pith},
  title        = {Pith review of: Angular Resolution of Closely-Spaced Targets with Antenna Arrays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YEOL7LCF}},
  note         = {Machine review of arXiv:1908.05308}
}
read the original abstract

This is an English translation of Ulrich Nickel's PhD dissertation with the original title "Winkelaufl\"osung eng benachbarter Ziele mit Gruppenantennen." It describes maximum-likelihood angular superresolution of closely-spaced targets. It also discusses estimating the number of targets present.

Figures

Figures reproduced from arXiv: 1908.05308 by the authors.

Figure 2-1
Figure 2-1. Layout of the receiver antenna arrays. The diameter of the antennas is 8 times the wavelength. [PITH_FULL_IMAGE:figures/full_fig_p007_2-1.png] view at source ↗
Figure 2-2
Figure 2-2. Layout of the receiver antenna array ELAN 192. ELAN 192 consists of [PITH_FULL_IMAGE:figures/full_fig_p007_2-2.png] view at source ↗
Figure 4-1
Figure 4-1. Q-surface for two targets using ELAN 7 L. Define γ :=    ω ∈ V M [PITH_FULL_IMAGE:figures/full_fig_p018_4-1.png] view at source ↗
Figures from the paper (42 more)
Figure 4-2
Figure 4-2. Figure 4-2: Q-surface for two targets using ELAN 6. displayed. These are shown in Figs. 4-2, 4-3, and 4-4 for ELAN 6. Because one is primarily interested in the minimum of Q, the Q function has been flipped and is shown from below so that the minimum appears as the maximum. At t…
Figure 4-3
Figure 4-3. Figure 4-3: Q-surface for two targets using ELAN 6. This means that because a ∗ 1a1 = a ∗ 2a2 = N and |a ∗ 1a2| 2 = N2 |f(ω2 − ω1)| 2 , if f(ω) is the directional response of the element arrangement, then |f(ω2 − ω1)| 2 6= 1, which means that the antenna array cannot have a seco…
Figure 4-4
Figure 4-4. Figure 4-4: Q-surface for two targets using ELAN 6. b1 = 1. Then a(ω1) = Xr i=1 a(ωi)bi − X M i=2 a(ωi)ci for an appropriate ci ∈ C. Consequently, a(˜ω1), . . . , a(˜ωr), a(ω1), . . . , a(ωr) are linearly independent, which is a contradiction! – For (ii): One can say that Q = sΓ…
Figure 4-5
Figure 4-5. Figure 4-5 [PITH_FULL_IMAGE:figures/full_fig_p025_4-5.png]
Figure 4-6
Figure 4-6. Figure 4-6 [PITH_FULL_IMAGE:figures/full_fig_p025_4-6.png]
Figure 4-7
Figure 4-7. Figure 4-7: amplitudes rotate the principal curvature directions in a direction that is more parallel to the coordinate system. In order to find an ordering of array elements that is good for target resolution, one can maximize the mean curvature independently of the random phas…
Figure 4-8
Figure 4-8. Figure 4-8: where Lemma 1 of Appendix 1 was used. Using the rule that ∂|R(ϑ)| ∂ϑi = |R|trace  R−1 ∂R ∂ϑi  [PITH_FULL_IMAGE:figures/full_fig_p027_4-8.png]
Figure 4-9
Figure 4-9. Figure 4-9: The curves hardly change if instead of ELAN 11 L, linear arrays having a large number of elements with λ/2 spacing are used and the total SNR is held constant (the mean SNR on the individual elements decreases by 10 log N). For the curves for ELAN 11 L, one obtains a…
Figure 4-10
Figure 4-10. Figure 4-10: The assessment of the quality of an estimation method as in Figs. 4-10 and 4-11 is somewhat problematic. The dispersion of the minimum ωˆ is in this instance on the order of the entire search area. By quantizing such a strongly clipped random variable, the usual def…
Figure 4-11
Figure 4-11. Figure 4-11: be unrealistic due to the high-SNR demands. 5 STOCHASTIC APPROXIMATION Summary of Chapter 5 The question of whether a computationally feasible direction-estimation procedure exists is the decisive question for the application of a superresolution procedure. In this …
Figure 4-12
Figure 4-12. Figure 4-12: Thereafter, the procedure is tested through simulation on various deviations from the signal and noise models. It turns out to be relatively insensitive to model mismatches. The SNR of a single pulse is generally not sufficient to resolve multiple targets, as in Cha…
Figure 5-1
Figure 5-1. Figure 5-1: Generalized difference beams for two targets. [PITH_FULL_IMAGE:figures/full_fig_p032_5-1.png]
Figure 5-2
Figure 5-2. Figure 5-2: This limit value exists according to (3.2-4) and 1 n Xn k=1 grad Q(ω, zk) → grad E{Q(ω)} (n → ∞) Thus, A2 holds. 4) The Lipschitz-constant always has the form Kkzk 4 . Thus, for A3, the existence of the fourth moment of z is required. This is a limitation on the all…
Figure 5-3
Figure 5-3. Figure 5-3: Asymptotic dispersion of ML and the stochastic approximation [PITH_FULL_IMAGE:figures/full_fig_p038_5-3.png]
Figure 5-4
Figure 5-4. Figure 5-4: with asterisks is the average dispersion according to the CRLB, that is  1 λ1 + 1 λ2  /2 and curves 1, 2, 3, 4, 5 are the average dispersion of the stochastic approximation (M1 + M2)/2 over (u1 − u2)/BW for µ = 0.6¯µ, 0.8¯µ, µ, ¯ 1.2¯µ, 1.4¯µ (µ¯ = 1 λmin ). One re…
Figure 5-5
Figure 5-5. Figure 5-5: the same configuration for Signal Model 2 with σ(∆ϕ) = 2π/8, and Signal Models 3 and 4. The plots show that it is good to demand phase fluctuations. That is, a wider bandwidth or a longer sampling period, so that the worst-case phase relation ∆ϕ = 180◦ is averaged ou…
Figure 5-6
Figure 5-6. Figure 5-6: As in 1, 2, it is questionable whether E{G} exists. For a constant input, the differential equation (5.1-5) does not have a global solution, because the right side of the differential equation is not continuous. For an arbitrary starting point, a local solution that …
Figure 5-7
Figure 5-7. Figure 5-7: estimator bias with a particular number of iterations exists (Array and target configurations are the same as in Figs. 5-6 and 5-7). The behavior given a poor initial estimate when using Versions 3 and 1 with a poor initial estimate and bounding (η = 0.8æ) and also o…
Figure 5-10
Figure 5-10. Figure 5-10: There, the estimate after 17 iterations (residual noise) is shown as a function of the initial direction estimate u0 = (ug − 0.9BW, ug + 0.9BW). Given target positions u ex = (−0.45BW, 0.45BW), both targets are also in the search region with ug = −0.45BW. The curves…
Figure 5-8
Figure 5-8. Figure 5-8: 5.3 Application to Planar Antenna Arrays Though in the mathematical formulation, no difference needs to be made between linear and planar antenna arrays, significant differences become evident through simulation. 5.3.1 Limiting the Search Area: The necessary restrict…
Figure 5-9
Figure 5-9. Figure 5-9: the strong M-regularity of the ordering is poorly fulfilled. For example if ∆ϕ = 0◦ , the order ω1 = (−u, u, 0, 0) can be barely distinguished from the ordering ω1 = (0, 0, −v, v). That means that kgrad |E{Q}k (ω2) is small for kgrad E{Q}k(ω1) = 0. This is shown in …
Figure 5-10
Figure 5-10. Figure 5-10: In any case, a limitation of the correction vector should be applied. Whether Version 1, 2, or 4 should be used depends on the desired properties of the estimate. These three procedures provide an unbiased estimate after a small number of iterations. An example of t…
Figure 5-11
Figure 5-11. Figure 5-11: Convergence of the stochastic approximation with two targets. [PITH_FULL_IMAGE:figures/full_fig_p048_5-11.png]
Figure 5-12
Figure 5-12. Figure 5-12: Convergence of the stochastic approximation with two targets. [PITH_FULL_IMAGE:figures/full_fig_p049_5-12.png]
Figure 5-13
Figure 5-13. Figure 5-13: 5.4.3 Correlated Noise (External Interference) and Amplifier Fluctuations: If the noise vector n has the properties E{nn∗ } = W 6= I but E{n} = 0, E{n ∗ s} = 0, then one gets a displacement of the expected value of the estimate so that E{Q} = traceΓABA∗ + traceΓW an…
Figure 5-14
Figure 5-14. Figure 5-14: fluctuations of 1 dB are assumed and σ 2 i ∼ R σ 2 (1 − 1/4), σ2 (1 + 1/4) is chosen. The convergence in the u direction for the same target configuration and antenna array as in [PITH_FULL_IMAGE:figures/full_fig_p051_5-14.png]
Figure 5-15
Figure 5-15. Figure 5-15: Convergence of the stochastic approximation with three targets. [PITH_FULL_IMAGE:figures/full_fig_p052_5-15.png]
Figure 5-16
Figure 5-16. Figure 5-16: ωˆ k is quite large. Moreover, one can hope that with phase fluctuations, the worst case scenario (for example, ∆ϕ = 0◦ , 180◦ when M = 2) averages out. The same situations as in [PITH_FULL_IMAGE:figures/full_fig_p053_5-16.png]
Figure 5-17
Figure 5-17. Figure 5-17: stochastic approximation averaged grid search 2i(M2 + M) linear array +i real roots m [PITH_FULL_IMAGE:figures/full_fig_p054_5-17.png]
Figure 5-18
Figure 5-18. Figure 5-18: For a linear antenna with, for example two largets and five beams, the average grid search with 60m operations is the same complexity as the stochastic approximation with i12 operations for m = 3, i = 15 and m = 4, i = 20, and so on. By this number of iterations, on…
Figure 5-19
Figure 5-19. Figure 5-19: The averaged grid search will only be of interest in special cases. The computational complexity with linear arrays is with the same higher accuracy as that of the stochastic approximation. With planar arrays, the computational complexity is significantly higher. 7 …
Figure 6-1
Figure 6-1. Figure 6-1: calculated without considerable computational effort. This test value measures the residual power after the subtraction of the estimated signal. In order to have a multihypothesis test like that described in Chapter 3, the test must hold to an approximate probability…
Figure 7-1
Figure 7-1. Figure 7-1: Detection characteristics for two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p065_7-1.png]
Figure 7-2
Figure 7-2. Figure 7-2: Detection characteristics for two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p065_7-2.png]
Figure 7-3
Figure 7-3. Figure 7-3: Detection characteristics for two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p066_7-3.png]
Figure 7-4
Figure 7-4. Figure 7-4: Detection characteristics for two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p066_7-4.png]
Figure 7-5
Figure 7-5. Figure 7-5: Detection characteristics for two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p067_7-5.png]
Figure 7-6
Figure 7-6. Figure 7-6: Detection characteristics for two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p067_7-6.png]
Figure 7-7
Figure 7-7. Figure 7-7: The resolution of two targets using the ELAN 21L array with [PITH_FULL_IMAGE:figures/full_fig_p068_7-7.png]
Figure 7-8
Figure 7-8. Figure 7-8: The resolution of two targets with du = 0.5BW. parameters. For ELAN 21 L, [PITH_FULL_IMAGE:figures/full_fig_p068_7-8.png]
Figure 7-9
Figure 7-9. Figure 7-9: Detection characteristics for two targets using the ELAN 25 array with [PITH_FULL_IMAGE:figures/full_fig_p069_7-9.png]
Figure 7-10
Figure 7-10. Figure 7-10: The resolution of two targets with du = 0.5BW. for ELAN 25 and 29. For ELAN 39, with the highest element density, the estimation is better, but the loss due to stronger coupling is larger at about 1 dB. In all, the theoretical detection probabilities provide the abi…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    M. I. Skolnik, Radar Handbook. McGraw Hill, 1970

    work page 1970

  2. [2]

    Über eine Klassifizierungsmöglichkeit von Radarzielen mittels kohärent gemessener Echosignale,

    K. von Schlachta, “Über eine Klassifizierungsmöglichkeit von Radarzielen mittels kohärent gemessener Echosignale,” Ph.D. dissertation, TU Berlin, 1977

    work page 1977

  3. [3]

    Generalized likelihood signal resolution,

    J. A. Stuller, “Generalized likelihood signal resolution,” IEEE Trans. IT , vol. 21, no. 3, May 1975

    work page 1975

  4. [4]

    Witting and G

    H. Witting and G. Nölle, Angewandte mathematische Statistik . Stuttgart: Teubner, 1970

    work page 1970

  5. [5]

    Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction),

    N. R. Goodman, “Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction),” Ann. of Math. Stat. , vol. 34, 1963

    work page 1963

  6. [6]

    Analysis of recursive stochastic algorithms,

    L. Ljung, “Analysis of recursive stochastic algorithms,” IEEE Transactions on Automatic Control , vol. 22, no. 4, Aug. 1977

    work page 1977

  7. [7]

    Data adaptive spectral analysis,

    R. T. Lacoss, “Data adaptive spectral analysis,” Geophysics, vol. 36, pp. 661–675, Aug. 1971

    work page 1971

  8. [8]

    Maximum entropy spectral analysis and autoregressive decomposition,

    T. J. Ulrych and T. N. Bishop, “Maximum entropy spectral analysis and autoregressive decomposition,” Rev. Geophysics and Space Phys. , vol. 13, pp. 183–200, Feb. 1975

    work page 1975

Show all 30 references
  1. [9]

    Optimum space-time signal processing and parameter estimation,

    G. O. Young, “Optimum space-time signal processing and parameter estimation,” IEEE Transactions on Aerospace and Electronic Systems , vol. 4, no. 3, May 1968

    1968

  2. [10]

    Applications of space-time decision and estimation theory to antenna processing system-design,

    G. O. Young and J. E. Howard, “Applications of space-time decision and estimation theory to antenna processing system-design,” Proceedings of the IEEE, vol. 38, no. 5, May 1970

    1970

  3. [11]

    A decision theoretic approach to the angular resolution and parameter estimation of multiple targets,

    A. A. Ksienki and R. B. McGhee, “A decision theoretic approach to the angular resolution and parameter estimation of multiple targets,”IEEE Transactions on Aerospace and Electronic Systems , vol. 4, no. 3, May 1968

    1968

  4. [12]

    Entdeckung und Parameterschätzung bei Zielen mit geringem Winkelabstand,

    W. D. Wirth, “Entdeckung und Parameterschätzung bei Zielen mit geringem Winkelabstand,” FFM, Tech. Rep. 192, Oct. 1972

    1972

  5. [13]

    Low-angle radar tracking in the presence of multipath,

    W. D. White, “Low-angle radar tracking in the presence of multipath,” IEEE Transactions on Aerospace and Electronic Systems , vol. 10, no. 6, Nov. 1974

    1974

  6. [14]

    Angular tracking of two closely spaced radar targets,

    G. E. Pollon and G. W. Lank, “Angular tracking of two closely spaced radar targets,” IEEE Transactions on Aerospace and Electronic Systems , vol. 4, no. 4, Jul. 1968

    1968

  7. [15]

    Array aperture sampling technique for multipath compensation,

    F. G. Willwerth and I. Kupiec, “Array aperture sampling technique for multipath compensation,” Microwave Journal, Jun. 1976

    1976

  8. [16]

    Direkt-lösende Algorithmen für die Interferenzanalyse Teil I und II,

    K. Bauer, “Direkt-lösende Algorithmen für die Interferenzanalyse Teil I und II,” Frequenz, vol. 30, no. 4 and 5, 1976

    1976

  9. [17]

    Multiple tone parameter estimation from discrete-time observations,

    D. C. Rife and R. R. Boorstyn, “Multiple tone parameter estimation from discrete-time observations,” Bell Syst. Techn. J. , vol. 55, no. 9, Nov. 1976

    1976

  10. [18]

    Signal processing techniques for resolving individual pulses in a multipath signal,

    J. E. Ehrenberg, T. E. Ewart, and R. D. Morris, “Signal processing techniques for resolving individual pulses in a multipath signal,” J. Acoust. Soc. Am. , vol. 63, no. 6, Jun. 1978

    1978

  11. [19]

    Maximum likelihood estimation of the number, directions, and strengths of point radio sources from variable baseline interferometer data,

    I. N. El Behery and R. H. McPhie, “Maximum likelihood estimation of the number, directions, and strengths of point radio sources from variable baseline interferometer data,” IEEE Trans. AP , vol. 26, no. 2, Mar. 1978

    1978

  12. [20]

    Parameter estimation of multiple signals,

    R. A. Birgenheier, “Parameter estimation of multiple signals,” Ph.D. dissertation, University of California, Los Angeles, 1972

    1972

  13. [21]

    T. W. Anderson, An introduction to Multivariate Statistical Analysis . J. Wiley, 1958

    1958

  14. [22]

    Klingenberg, Eine Vorlesung über Differentialgeometrie

    W. Klingenberg, Eine Vorlesung über Differentialgeometrie . Heidelberger Taschenbücher, Springer, 1973

    1973

  15. [23]

    Riemannsche Geometrie im Großen,

    D. Gromoll, W. Klingenberg, and W. Meyer, “Riemannsche Geometrie im Großen,” Lecture Notes in mathematics , vol. 55, 1975

    1975

  16. [24]

    M. T. Wasan, Stochastic Approximation. Cambridge University Press, 1969

    1969

  17. [25]

    Multiple Gaussian targets: the track-on-jam problem,

    I. Kanter, “Multiple Gaussian targets: the track-on-jam problem,” IEEE Transactions on Aerospace and Electronic Systems , vol. 13, no. 6, Nov. 1977

    1977

  18. [26]

    The ratio of functions of random variables,

    ——, “The ratio of functions of random variables,” IEEE Transactions on Aerospace and Electronic Systems , vol. 13, no. 6, Nov. 1977

    1977

  19. [27]

    Robust estimation via stochastic approximation,

    R. D. Martin and C. J. Masreliez, “Robust estimation via stochastic approximation,” IEEE Transactions on Information Theory , vol. 21, no. 3, May 1975

    1975

  20. [28]

    Ein rechnergesteuertes Nahfeldvermessungsverfahren für elektronisch steuerbare Gruppenantennen,

    G. Hüschelrath, “Ein rechnergesteuertes Nahfeldvermessungsverfahren für elektronisch steuerbare Gruppenantennen,” Ph.D. dissertation, TH Aachen, Feb. 1978

    1978

  21. [29]

    Sum of exponential random variables,

    J. C. Demaret and A. Garcet, “Sum of exponential random variables,” AEÜ, vol. 31, no. 11, 1977

    1977

  22. [30]

    Stoer, Einführung in die Numerische Mathematik I

    J. Stoer, Einführung in die Numerische Mathematik I . Springer, 1973

    1973

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.