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Placing the Beamspace Window on a Hopping Carrier: Bin Drift, Exact Capture and the Cost of Rounding
Moris S. Mukwayi, Michael A. van Wyk, Andre M. McDonald
TL;DR
Carrier hopping moves a radar look direction across spatial-FFT bins, so a beamspace processor must choose where to place its window at each hop. The paper compares window-placement rules and finds that exact placement guarantees full look capture, while jammer loss depends on interference geometry and can favor another placement.
Problem
Carrier hopping moves a radar look direction across spatial-FFT bins, leaving the placement of a fixed-size beamspace window as an open question.
Source: §I · p. 1Method
The paper analyzes beamspace window placement using its span invariance, derives rectangular-array capture loss, and evaluates placement rules against jammers.
Source: §I · p. 1, B. An orthonormal window depends only… · p. 2Results
At us = 0.55, the frozen window loses 39 to 60 dB at the worst carrier; the exact window beats nearest-bin placement by 0.31 to 0.65 dB, but a nine-beam window near broadside is 0.61 dB worse.
Source: B. Results · p. 3Takeaways & Limitations
Exact placement guarantees full look-vector capture, but the loss-minimising centre depends on jammer geometry and need not be the look-direction centre.
Source: §V · p. 4, §VI · p. 5
Abstract
from arXiv · showhide
A frequency-agile radar with a large digital array moves the spatial frequency of its look direction across the beams of a spatial FFT every time it changes carrier, so a beamspace adaptive processor must decide where to put its window at each hop. Because the beamspace transform is orthonormal, the output of the processor depends only on the span of the window, and a hop changes nothing but where the window sits. Centring the window exactly on the fractional bin of the look direction keeps the look vector fully inside it at every carrier, for the price of one phase ramp before the FFT. On a 32 by 32 array over 4 to 8 GHz the look bin drifts by 1.6 to 6.9 bins for look directions from u=0.2 to 60 degrees. Rounding to the nearest bin costs up to 1.34, 0.74 and 0.50 dB of look-direction capture for 3 by 3, 5 by 5 and 7 by 7 windows. In a three-jammer scenario with 41 carriers, a window frozen at the lowest carrier loses 39 to 60 dB at the worst carrier for a look direction of u=0.55, the exact window beats nearest-bin placement by 0.31 to 0.65 dB there, and the advantage reverses for a nine-beam window near broadside, where nearest-bin was 0.61 dB better.
I. INTRODUCTION
Frequency hopping moves a target across spatial-FFT bins, making window placement a central question for beamspace adaptive processing. Prior work studies beamspace windows and related wideband effects, but the paper addresses where to place a fixed-size window when radar hops carriers.
I. INTRODUCTION
- Prior beamspace work centers windows on target FFT bins and studies window size and zero padding, but not placement when a radar carrier hops.The paper states that, to the authors’ knowledge, prior work does not answer this hopping-radar placement question.Source: §I · p. 1
- A hopping carrier moves the look direction across several FFT bins, making the window’s placement the central question.The spatial-wideband effect also moves interference nulls, which matters for interference suppression after a hop.Source: §I · p. 1
- The paper studies beamspace MVDR window placement under hopping, exact rounded-window capture loss on rectangular arrays, and performance against jammers.It also uses the processor’s dependence on the window span and measures placement rules against random jammer laydowns.Source: §I · p. 1
A. Array and bin drift
On the 32 × 32 array, carrier changes shift the spatial-FFT location of a look direction. The shift grows linearly with carrier and can exceed the half-width of a practical small window.
A. Array and bin drift
- The modeled array is a 32 × 32 uniform rectangular array with 1,024 elements spaced 18.75 mm apart.The element spacing is half a wavelength at 8 GHz.Source: A. Array and bin drift · p. 2
- A spatial FFT places the arriving plane wave at a fractional bin, whose location grows linearly with carrier.The input is a plane wave arriving from direction cosines (u, v) at carrier f.Source: A. Array and bin drift · p. 2
- The look-bin drift is ∆p = usNxd(fmax − fmin)/c, or 8.0 us bins over 4–8 GHz for this array.At the edge of the ±60° sector, the look bin moves almost seven bins—more than the half-width of any window small enough to be worthwhile.Source: A. Array and bin drift · p. 2
B. An orthonormal window depends only on where it sits
A beamspace window’s columns are orthonormal even when its centre is fractional. Under unitary changes of basis, processor SINR is invariant, so a carrier hop matters through the window’s position.
B. An orthonormal window depends only on where it sits
- The window columns are orthonormal for every real centre because any two columns differ by integer bin offsets.The window has W × W columns, and beamspace MVDR adapts within that window.Source: B. An orthonormal window depends only… · p. 2
- For a unitary basis change, output SINR is unchanged for both true-covariance processing and sample matrix inversion using the same training and diagonal loading.This invariance motivates treating the hop as a window-placement problem rather than a change in processor output caused by the basis itself.Source: B. An orthonormal window depends only… · p. 2
- A hop changes the processor only through the window span and centre, so the placement rules vary where the window sits relative to the carrier-dependent look bin.The compared rules include fractional-bin, nearest-bin, frozen-at-lowest-carrier, and zero-padded-grid placement.Source: B. An orthonormal window depends only… · p. 2
- The section evaluates the look-direction capture loss from rounding to the FFT grid and from a zero-padded grid for the same window size.The supplied table caption establishes the comparison’s scope but does not provide cell values.Source: Table II · p. 2
- Fractional placement captures the look vector completely at every carrier, using a phase ramp before the unchanged FFT.The ramp adds 1,024 complex multiplications per snapshot to an FFT requiring about 5,120.Source: B. An orthonormal window depends only… · p. 2
III. THE COST OF ROUNDING
Rounding the look bin to the FFT grid loses some look-vector energy, while centring the window on the fractional bin captures it exactly. Zero padding reduces rounding loss but changes the processor and increases computational cost.
III. THE COST OF ROUNDING
- 1.34 dB is the worst capture loss for a 3×3 window; the corresponding 7×7-window loss is 0.50 dB.The loss is for the look vector alone, before interference.Source: §III · p. 2
- At 5.91 GHz, the look direction leaves the frozen 5×5 window, while the fractional window has no look-capture loss.The figure shows the nearest-bin window frozen at 4 GHz.Source: Figure 1 · p. 3
- P ≥ 2 reduces Table II capture loss to 0.014 dB or less, but P = 2 costs about 4.8 times the base FFT, versus 20% for the ramp.Keeping the same angular extent requires P^2B beams, and adaptive-stage cost approaches a P^6 factor for large B.Source: §III · pp. 2–3
A. Setup
The evaluation uses three 30 dB noise jammers and 41 carrier values from 4 to 8 GHz. It reports worst-carrier loaded-SMI loss and validates the testbench against a separate implementation.
A. Setup
- The scenario uses three 30 dB-per-element noise jammers, 0 dB signal-to-noise ratio per element, and no clutter.The look directions are u_s = 0.20, 0.55, and 0.85; 41 carriers are evenly spaced from 4 to 8 GHz, with 500 dwells each.Source: A. Setup · p. 3
- Table III reports worst-carrier loaded-SMI loss in dB for K = 2B, with gain defined as fractional minus nearest-bin loss.Its final column is the share of carriers where frozen-window clairvoyant subspace loss exceeds 3 dB.Source: Table III · p. 3
- The testbench was built in Simulink from library blocks, and every environment model was validated against a separate implementation.The stored mean intervals have half-widths at most 0.04 dB for fractional and nearest-bin rules and 0.05 dB for frozen placement.Source: A. Setup · p. 3
B. Results
A frozen window can incur very large worst-carrier loss away from broadside, while fractional placement beats nearest-bin placement in most tested cases. The advantage is not universal: nearest-bin can perform better with a small window near broadside.
B. Results
- 39 to 60 dB is the frozen-window worst-carrier loss at u_s = 0.55; near broadside, frozen 5×5 and 7×7 windows stay within 0.15 dB of the other rules.At u_s = 0.55, frozen-window clairvoyant subspace loss exceeds 3 dB on 95%, 66%, and 41% of carriers for 9, 25, and 49 beams, respectively.Source: B. Results · p. 3
- 0.65, 0.36, and 0.31 dB are the fractional-window gains over nearest-bin at u_s = 0.55 for 9, 25, and 49 beams.Fractional placement beats nearest-bin in seven of the nine cases, but at u_s = 0.20 the nine-beam fractional window is 0.61 dB worse.Source: B. Results · p. 3
- For 25 beams at u_s = 0.55, exact-window loss peaks at 0.36 dB versus 0.72 dB for nearest-bin; for nine beams at u_s = 0.20, exact loss is larger at 38 of 41 carriers.The frozen 25-beam window exceeds 3 dB from 5.4 GHz and reaches 49 dB at 8 GHz.Source: Figure 2 · p. 4
C. Random laydowns and jammer-aware placement
Across random three-jammer laydowns, exact placement usually has lower loss than nearest-bin placement, while jammer-aware oracle placement offers its largest gains for nine beams. The results use a defined simulation scope, and the oracle benchmark is not an implementable rule.
C. Random laydowns and jammer-aware placement
- The random-laydown evaluation scores 200 three-jammer laydowns across 41 carriers per look direction, yielding 8,200 cases per row with the true covariance.Jammers are drawn within a 0.5-radius disc around the look direction, at least 0.06 away; Table IV reports median loss, win/loss shares, and oracle gain.Source: C. Random laydowns and jammer-aware… · p. 4, Table IV · p. 4
- The exact window has lower loss than nearest-bin placement in 70 to 95% of random-laydown cases, with median loss of 0.15 dB or less for 25- and 49-beam windows.The lead is smallest for nine beams, at 70 to 75%.Source: C. Random laydowns and jammer-aware… · p. 4
- Oracle placement improves median loss over the exact window by 0.45 to 0.65 dB for nine beams, 0.06 dB for 25 beams, and 0.01 dB for 49 beams.For 25 beams or more, look-direction centring is within 0.44 dB of the best centre in nine cases out of ten; the largest gain is 4.2 dB for nine beams.Source: C. Random laydowns and jammer-aware… · p. 4
V. DISCUSSION
The discussion recommends tracking the look bin across hops rather than freezing the window, while noting that exact placement removes rounding loss at a phase-ramp cost. Its simulation evidence is limited to specified idealized settings and excludes several operational effects.
V. DISCUSSION
- A frozen window is unsuitable when the look direction is far from broadside unless the window is large relative to the bin drift.Nearest-bin placement incurs 0.1 to 0.65 dB loss in the discussed cases and up to 1.34 dB of look-capture loss; exact placement removes that penalty at the cost of a phase ramp.Source: §V · p. 4
- The best subspace-loss window centre depends on jammer geometry, but look-centred exact placement is within 0.44 dB of the best centre in nine cases out of ten for 25 beams or more.The oracle gain reaches 4.2 dB in the worst of 660 cases for the nine-beam window; choosing from training data is not covered.Source: §V · p. 4
- The simulations use one array size, one loading level, an ideal array, square windows, and three look directions with vs = 0.Sampled-covariance results use one jamming laydown; random-laydown results use the true covariance, three equal-power jammers within 0.5 of the look direction, and one drawing scheme.Source: §V · p. 4
- Land clutter, detection, array errors, multipath, and simulated comparison with a zero-padded grid are outside the study’s scope.The reported smallest differences, 0.07 and 0.10 dB, exceed stored interval widths but are described as too small to matter in practice.Source: §V · p. 4
VI. CONCLUSION
Because the beamspace transform is orthonormal, hopping changes the window’s position rather than the processor’s relevant span. Exact fractional-bin placement captures the look vector across carriers, whereas frozen and rounded placement have distinct costs and trade-offs.
VI. CONCLUSION
- A hop affects the windowed beamspace processor only through the window’s position, because the beamspace transform is orthonormal.The relevant processor change is the span of the window.Source: §VI · p. 5
- Exact fractional-bin placement fully captures the look vector at every carrier for the cost of one phase ramp.Rounding to the nearest bin costs up to 1.34 dB of look capture and 0.1 to 0.65 dB of loss against three jammers, except that small windows near broadside did slightly better.Source: §VI · p. 5
- A window frozen at one carrier loses 33 to 60 dB at the worst carrier for look directions us = 0.55 and 0.85.Across random jammer laydowns, exact placement has lower loss in 70 to 95% of cases; oracle movement gains a median of at most 0.65 dB for nine beams and 0.06 dB for 25 beams or more.Source: §VI · p. 5