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Continuous-Phase Frequency-Hopping MIMO Signaling and Processing for ISAC

Yi Wang, Xianxiang Yu, Jing Yang, Tao Fan, Linlong Wu, Xiangrong Wang, Guolong Cui

arXiv:2610.22956v1eess.SP

TL;DR

FH-MIMO-ISAC faces challenges in phase-continuous transmission, demodulation without accurate channel-phase information, and coherent processing of multi-frequency echoes. The paper proposes a CP-FH waveform with matched communication and radar processing; its analysis describes how the approach addresses demodulation and radar-processing issues, including grating-lobe detections and Doppler mismatch.

  • Problem

    FH-MIMO designs can require accurate channel-phase information, suffer phase discontinuities and spectral leakage, and conventional pulse-Doppler processing can produce periodic range grating lobes that impair detection.

    Source: §I · p. 2

  • Method

    The paper jointly embeds information in frequency combinations and CPM symbols, using projection-energy demodulation for communications and structure-matched coherent processing for radar echoes.

    Source: §I · p. 2

  • Results

    The proposed radar processor reduces Doppler-mismatch loss and extra detections from periodic range grating lobes while preserving frequency-aperture range and full-aperture angular resolution under the stated conditions.

    Source: §I · p. 2, §VI · p. 15

  • Takeaways & Limitations

    The analysis characterizes target-discrimination limits under periodic ambiguity and shows that output noise power is equal under the stated branch-noise conditions.

    Source: §I · p. 2

Abstract

from arXiv · show

Frequency-hopping MIMO integrated sensing and communications (FH-MIMO-ISAC) supports agile multidimensional signaling but faces challenges in phase-continuous transmission, demodulation without channel-phase information, and coherent processing of multi-frequency echoes. We propose a continuous-phase frequency-hopping (CP-FH) MIMO ISAC waveform with matched communication and radar processing. Information is conveyed jointly by frequency combinations and continuous-phase-modulation symbols. CPM phase accumulation and frequency-hopping linear-frequency-modulation phase compensation maintain phase continuity and reduce spectral leakage. We derive exact inter-element orthogonality conditions, residual cross-correlation bounds, and a bound on averaged transmit-beampattern ripple. For communication, projection-energy demodulation requires no channel-phase information and approximates least-squares detection under weak inter-branch correlation. It separates frequency-combination search from CPM-symbol decisions, reducing complexity while retaining direct joint-search decisions. For radar, structure-matched coherent processing combines space-time-frequency matched filtering, velocity-consistent Doppler focusing, and frequency-aperture focusing to form a three-dimensional coarse-range--velocity--fine-range response. It reduces Doppler-mismatch loss and extra detections from periodic range grating lobes in conventional pulse-Doppler processing. We further derive the joint range response, output signal and noise powers, and receive beampattern. The analysis characterizes target-discrimination limits under periodic ambiguity and shows that the proposed processor preserves frequency-aperture range resolution and full-aperture angular resolution, with equal output noise power under the stated conditions.

I. Introduction

FH-MIMO-ISAC faces challenges from phase discontinuities, channel-phase-dependent detection, and periodic range grating lobes; the paper proposes a continuous-phase waveform with matched communication and radar processing. Its construction embeds information in frequency offsets and CPM symbols, while the proposed processors address demodulation and coherent multi-frequency echo processing.

I. Introduction

  • Phase discontinuities in subpulse-wise frequency and phase switching can aggravate sidelobe modulation and spectral leakage, while conventional pulse-Doppler processing may yield range grating lobes and unreliable detection.The paper identifies the need for phase-continuous transmission, demodulation without accurate channel-phase information, and processing that uses the carrier-dependent phase structure across frequency branches.Source: §I · p. 2
  • The proposed waveform jointly conveys information through frequency combinations and CPM symbols, while PED and SMC provide matched communication and radar processing.PED approximates least-squares detection under weak inter-branch correlation; SMC forms a three-dimensional coarse-range–velocity–fine-range response and reduces extra detections from periodic range grating lobes.Source: §I · p. 2

A. Signal Model

  • The signal model assumes CPI-wide phase coherence and assigns each antenna a distinct frequency offset within each subpulse.The carrier-offset pool has K candidates and K ≥ N.Source: A. Signal Model · pp. 2–3

1) Element-wise CP-FH waveform:

  • CPM phase accumulation and recursively chosen FH–LFM phase compensation preserve phase continuity across subpulse boundaries.Frequency offsets and CPM symbols carry information through frequency hopping and CPM phase modulation, respectively.Source: 1) Element-wise CP-FH waveform · p. 3

2) Directional Transmit Signal:

B. Waveform Properties

  • The waveform-properties analysis connects inter-element residual correlation to orthogonality conditions and the averaged transmit beampattern.The normalized inter-element correlation is defined per subpulse, with subpulse energy given by the unit-modulus waveform property.Source: B. Waveform Properties · p. 3

1) Subpulse-Level Orthogonality Conditions:

The section establishes conditions for exact or bounded inter-element orthogonality and connects residual correlations to averaged transmit-beampattern ripple. It also describes joint frequency–CPM information embedding and a projection-energy demodulator that separates frequency selection from symbol decisions.

1) Subpulse-Level Orthogonality Conditions:

  • The inter-element correlation is governed by the combined frequency-spacing and CPM-symbol-difference term; sufficient spacing bounds the residual correlation.Increasing ΔfΔt, or choosing the modulation parameters to avoid cancellation of frequency spacing, reduces residual correlation.Source: 1) Subpulse-Level Orthogonality… · pp. 3–4

2) Averaged Transmit-Beampattern Behavior:

  • Exact orthogonality makes the transmit beampattern averaged over V subpulses isotropic; increasing ΔfΔt or K suppresses average residual correlation and ripple.The general ripple bound retains a term for cross-subpulse residual coupling.Source: 2) Averaged Transmit-Beampattern… · p. 4

A. Communication Information Embedding

  • Information is embedded jointly through ordered frequency-combination codewords and CPM-symbol codewords, with each subpulse conveying NF + NΦ bits.The frequency and CPM-symbol codebooks have one-to-one bit-labeling maps; the common antenna permutation preserves their pairing and is not an information-bearing index.Source: A. Communication Information Embedding · p. 4

B. Projection-Energy Demodulation

  • Projection-energy demodulation eliminates unknown branch coefficients and does not require accurate branch-phase prior information.When inter-branch correlations are small relative to template energy, the Gram matrix is approximated as diagonal, allowing frequency-combination selection to separate from branchwise symbol decisions with the same optimal metric value.Source: B. Projection-Energy Demodulation · p. 5
  • The separated search has per-subpulse complexity O(KML + |Γ|N), versus O(KML + |Γ|M^N N) for direct enumeration under the diagonal metric.The separated procedure reuses the maximizing symbol indices when recovering the selected frequency combination.Source: B. Projection-Energy Demodulation · p. 5

A. Receive Signal Model

The receive model describes point-target echoes across pulses and frequency branches. The proposed SMC processor applies joint matched filtering, velocity-consistent Doppler focusing, and frequency-aperture focusing, then repeats the procedure over an angle grid.

A. Receive Signal Model

  • The receive model represents each point target by azimuth, CPI-start range, radial velocity, and complex scattering coefficient, with positive velocity directed toward the array.Its instantaneous two-way delay is specified for each pulse relative to the CPI-start delay.Source: A. Receive Signal Model · p. 5

B. Proposed Structure-Matched Coherent Processor

  • SMC combines space–time–frequency joint matched filtering, velocity-consistent Doppler focusing, and frequency-aperture focusing for each azimuth hypothesis.Angular search repeats this procedure over the angle grid.Source: B. Proposed Structure-Matched Coherent… · p. 5

1) Space–Time–Frequency Joint Matched Filtering:

  • Under Proposition 1’s conditions, distinct active frequency branches have normalized correlation bounded by ε0 at the matched delay under the fixed-delay model.For the typical configuration P = 64, V = 30, N = 4, and K = 8, the probability of complete frequency coverage is approximately 0.9999995232 per PRI and numerically indistinguishable from one over the CPI.Source: 1) Space–Time–Frequency Joint Matched… · p. 6

2) Velocity-Consistent Doppler Focusing:

The processor aligns Doppler focusing on a common physical-velocity grid, then focuses across frequency branches to form a coarse-range–velocity–fine-range response. Its range analysis characterizes periodic ambiguity and how branch weighting shapes fine-range resolution.

2) Velocity-Consistent Doppler Focusing:

  • Velocity-consistent Doppler focusing integrates slow time on a common physical-velocity grid while retaining branch-dependent phase for subsequent frequency-aperture focusing.Different carrier frequencies map the same physical velocity to different Doppler frequencies across branches.Source: 2) Velocity-Consistent Doppler Focusing · pp. 6–7

3) Frequency-Aperture Focusing:

  • Frequency-aperture focusing compensates branch-dependent phase and produces a three-dimensional response cube over coarse range, velocity, and fine range.For equally spaced carriers, the operation is DFT-like; nonuniform carriers use a nonuniform Fourier sum.Source: 3) Frequency-Aperture Focusing · p. 7

1) Joint Coarse–Fine Range Discrimination:

  • A larger weighted frequency variance makes the fine-range response decrease faster near its peak; small weight perturbations can therefore narrow or widen the 3-dB mainlobe.A decrease in response at the reference response’s descending half-power point corresponds to a narrower mainlobe.Source: 1) Joint Coarse–Fine Range… · p. 8
  • Periodic fine-range replicas remain after coarse-range weighting, which can attenuate them but does not generally eliminate them.The analysis defines a range-ambiguity region using half the central peak power.Source: 1) Joint Coarse–Fine Range… · p. 8
  • The number of ambiguity intervals within the coarse-range mainlobe is 2W + 1, with W set by the fine-range period and the coarse-envelope half-power width.Outer intervals are narrower than the central interval when 1/2 < Ac(wTf) < 1.Source: 1) Joint Coarse–Fine Range… · p. 8

2) Output-SNR Comparison With PD Processing:

The analysis compares SMC with conventional pulse-Doppler processing in signal and noise behavior, and characterizes the receiver’s angular selectivity. Waveform validation also examines inter-channel correlation and averaged transmit-beampattern ripple.

2) Output-SNR Comparison With PD Processing:

  • With independent pulse noise, orthogonal actual branch templates, rectangular slow-time weights, and unit-modulus compensation, SMC and PD output noise powers are equal even with nonuniform or absent branch energies.The signal-side difference arises because single-reference PD leaves frequency-dependent Doppler mismatch, while SMC compensates branch-dependent target phase.Source: 2) Output-SNR Comparison With PD… · pp. 8–9

3) Effective Receive Beampattern:

  • SMC processing restores the full-aperture coherent spatial factor at the receiver, so average transmit-beampattern behavior alone does not determine final angular selectivity.The result assumes a colocated array and matched beamformer.Source: 3) Effective Receive Beampattern · p. 9

1) Correlation of Frequency-Indexed Waveform Components:

  • Full-pulse inter-channel correlation remains below the subpulse-level bound and prescribed tolerance, supporting approximate inter-channel separability over the full pulse.The numerical subpulse correlation agrees with the derived dependence on effective normalized spacing.Source: 1) Correlation of Frequency-Indexed… · p. 9

2) Transmit-Beampattern Characteristics:

  • A larger frequency pool reduces transmit-beampattern deviation, whereas a larger array increases residual ripple through additional off-diagonal antenna-pair terms.Increasing ΔfΔt also reduces the Monte Carlo deviation estimate.Source: 2) Transmit-Beampattern Characteristics · pp. 9–10

3) Spectral Leakage:

The section evaluates CP-FH spectral leakage, communication performance, and single-target radar response. Its reported results cover phase-continuity variants, PED performance as frequency spacing changes, and SMC coarse- and fine-range focusing.

3) Spectral Leakage:

  • CP-FH has the lowest mean spectral leakage in both tested configurations: 1.49% for (h̄ = 0.5, M = 4) and 2.02% for (h̄ = 0.4, M = 2).It combines CPM phase accumulation with FH–LFM phase compensation to preserve continuity at subpulse boundaries.Source: 3) Spectral Leakage · p. 10

B. Communication Performance

  • CP-FH carries 10 bits per subpulse and has BER of the same order as rate-matched POCS at 0 dB; its 1.94% angular-averaged leakage is below FH-PSK, FHSP, HIM, and POCS.Unlike POCS, CP-FH enables noncoherent reception without accurate channel-phase prior information.Source: B. Communication Performance · pp. 11–12
  • At normalized spacings ΔfΔt = 4, 3, and 2, PED and LS BERs fall as SNR increases; at 1.5, PED’s loss grows markedly, with BER 2.10 × 10^-3 at 20 dB versus no observed LS errors.At ΔfΔt = 1.5, maximum branch-correlation magnitude and Gram-matrix condition number are 0.234 and 2.29, respectively.Source: B. Communication Performance · p. 11

1) Single-Target SMC-Domain Response:

  • SMC preserves PD’s coarse range–velocity localization while removing its observed range grating lobes; measured and theoretical fine-range resolutions agree within 0.66%.The measured 3-dB fine-range mainlobe width is 4.1805 m.Source: 1) Single-Target SMC-Domain Response · p. 12

2) Two-Target Responses and Discrimination:

The evaluation shows how SMC handles periodic range ambiguity, coherent-integration mismatch, detection, and angular focusing. Across the tested scenarios, it suppresses grating-lobe-induced extra detections while retaining the array’s full-aperture angular response.

2) Two-Target Responses and Discrimination:

  • At 15 m separation, SMC resolves two targets despite overlapping coarse-range responses; at one fine-range period it cannot, but at six periods separated coarse-range responses distinguish them.PD resolves the targets in these scenarios but exhibits periodic range grating lobes; the paper identifies unresolved coarse-range and coincident fine-range coordinates as a discrimination limit for SMC.Source: 2) Two-Target Responses and… · pp. 12–13

3) Range-Domain Performance Analysis:

  • Cases A and B have one and five 3-dB unresolved intervals, respectively, despite comparable local widths; these target-referenced profiles do not establish global cycle-identification performance.Random branch-energy allocations generally narrow the mainlobe and increase PSLR as normalized weighted frequency variance rises; alternating omissions shorten the fine-range period to 7.5 m.Source: 3) Range-Domain Performance Analysis · p. 13

4) Sensitivity of Coherent Integration:

  • At Q/P = 4, nearest-bin velocity compensation has mean and sample-maximum power losses of approximately 0.074 dB and 0.223 dB, respectively.The maximum normalized coherent-sum error under half-bin mismatch decreases with Q and remains below the stated bound.Source: 4) Sensitivity of Coherent Integration · p. 14
  • For stationary targets with aligned coarse delay, at half-bin fine-delay offset, mean power loss is approximately 0.221 dB for L/K = 4 and 0.055 dB for L/K = 8.Source: 4) Sensitivity of Coherent Integration · p. 14

6) Detection Performance:

  • Single-target detection probabilities nearly coincide for SMC and PD at each tested false-alarm probability, while SMC produces substantially fewer extra detections than PD.Approximately 97.4% of PD’s extra detections lie within matching tolerances of nonzero periodic replicas; two-target detection probabilities are similar at small separations.Source: 6) Detection Performance · p. 15

7) Receive Beampattern:

  • For array sizes N = 2 to 6, simulated and theoretical receive beampatterns agree, and the mainlobe narrows as N increases.Source: 7) Receive Beampattern · p. 15

VI. Conclusion

  • SMC combines frequency-aperture fine-range focusing with coarse-range constraints on periodic ambiguity, and simulations show fewer extra target reports from periodic range responses.The conclusion states that the processor needs no target-range or velocity priors and recovers the full-aperture coherent spatial response under the analyzed colocated-array and matched-beamforming conditions.Source: §VI · p. 15

Appendix A Proofs for CP-FH Waveform Properties

The appendix proves waveform orthogonality and beampattern-ripple results, then considers residual-correlation moments under random frequency–phase assignment. Its moment analysis assumes frequency spacing dominates the bounded CPM perturbation.

A. Proof of Proposition 1

  • When the CPM-symbol difference satisfies the stated nonzero-integer condition, the sinc term vanishes and the inter-element correlation is zero.The proof cancels the common LFM quadratic term in the inter-element product, leaving a frequency-like term from the current CPM-symbol difference.Source: A. Proof of Proposition 1 · p. 17

B. Proof of Proposition 2

  • When the exact-orthogonality condition holds for all n ≠ m and all v, E = 0 and ∆P(θ) = 0 for all θ, establishing the isotropic-beampattern result.Source: B. Proof of Proposition 2 · p. 17

C. Scaling of the Residual-Correlation Moments

  • The residual-correlation moment analysis uses random frequency–phase assignments and assumes FH spacing dominates the bounded CPM perturbation.The supplied derivation relates the moments to a normalized summation involving the harmonic number H_(K−1).Source: C. Scaling of the Residual-Correlation… · pp. 17–18

Appendix B Auxiliary Derivations for the SMC Processor

The appendix bounds full-pulse inter-channel correlation under a fixed-delay model and derives a matched-filter peak model for the SMC processor. The peak model supports branchwise Doppler compensation while retaining phase needed for frequency focusing.

A. Full-Pulse Inter-Channel Correlation Bound

  • The full-pulse inter-channel correlation bound applies to nonzero-energy branches after downconversion at the matched delay under the fixed-delay model, not to arbitrary delay and Doppler offsets.If either branch has zero energy, the unnormalized cross-correlation is zero and the normalized coefficient is undefined.Source: A. Full-Pulse Inter-Channel… · p. 18

B. Pulse-Compression Peak Model

  • When intrapulse Doppler is small, the peak model replaces the branch-dependent phase with an approximately branch-independent phase.This approximation follows when 2π|f_d,k|T is much less than 1 and the branch-dependent perturbation is small.Source: B. Pulse-Compression Peak Model · pp. 18–19

C. Coherent Loss Due to Residual Phase Differences

The appendix relates coherent-integration loss to residual phase differences and bounds the effect of velocity-grid mismatch. These results specify conditions for approximating coherent sums without exact phase alignment.

C. Coherent Loss Due to Residual Phase Differences

  • For small residual phase differences, coherent loss is related to the amplitude-weighted phase variance across pulses and frequency branches.The phase expression connects that variance to normalized velocity and carrier-weighted normalized energy centroids.Source: C. Coherent Loss Due to Residual Phase… · p. 19

1) Velocity-Grid Mismatch:

  • Nearest-grid velocity mismatch has a controlled absolute error in the slow-time coherent sum under the peak model.The phase bound depends on the velocity-grid spacing and the maximum slow-time index; the proof gives a sufficient grid choice for a prescribed phase tolerance.Source: 1) Velocity-Grid Mismatch · p. 19

2) Fine-Delay-Grid Mismatch:

The analysis bounds fine-delay grid loss and characterizes the joint fine-range response, including its periodicity and the one-branch limit.

2) Fine-Delay-Grid Mismatch:

  • The grid-peak power is no lower than the power at the nearest grid point, with a bound for offsets across the peak neighborhood.For a uniform grid with spacing Δτf = 1/(LΔfeff), the nearest-point offset is at most Δτf/2.Source: 2) Fine-Delay-Grid Mismatch · p. 20

1) Joint Range Response:

  • The joint fine-range response combines the coarse-range envelope with the fine-range response to characterize its 3-dB unresolved region.Source: 1) Joint Range Response · p. 20

2) Periodicity of the Fine-Range Response:

3) Branch-Energy Distribution and Mainlobe Width:

The analysis relates branch-energy distribution to fine-range mainlobe width and ambiguity, and compares structure-matched compensation with conventional pulse-Doppler processing.

3) Branch-Energy Distribution and Mainlobe Width:

4) Range Discrimination With Uniform Branch Weights:

  • With identical accumulated CPI energy across branches, the weights are uniform and the fine-range response has peaks at integer multiples of Tf.The uniform-weight analysis neglects residual intrapulse Doppler effects.Source: 4) Range Discrimination With Uniform… · pp. 20–21

2. The fine-range 3-dB full width is therefore

  • When the coarse-range envelope varies slowly across the central fine-range mainlobe, its 3-dB full width is approximately ΔRfine,3dB.Source: §2 · p. 21
  • The analysis bounds retained ambiguity branches by the coarse-range half-width, with Nbranch = 2W + 1 when |w|Tf < δc.Source: §2 · p. 21
  • Under the local approximation, outer retained ambiguity branches have narrower unresolved widths than the central branch.Source: §2 · p. 21

1) Signal-Side Output Behavior:

  • SMC compensates branch-dependent Doppler phases, whereas single-reference PD retains mismatch proportional to frequency offset, target velocity, and slow time.Source: 1) Signal-Side Output Behavior · p. 21
  • The branch-energy analysis allows unequal and zero branch energies rather than requiring complete frequency coverage or equal branch amplitudes.Source: 1) Signal-Side Output Behavior · p. 21

2) Noise-Side Output Behavior:

The analysis specifies when matched-filter noise outputs are correlated or equal in power and derives the SMC receive beampattern from full-aperture spatial factors.

2) Noise-Side Output Behavior:

  • White receiver noise alone does not make matched-filter branch outputs uncorrelated; their covariance depends on template inner products.Unequal branch energies yield unequal matched-filter noise variances, and absent branches contribute zero covariance rows and columns.Source: 2) Noise-Side Output Behavior · p. 22
  • Equal output noise powers do not require equal branch energies, but depend on the stated covariance and combining conditions.For approximately orthogonal templates, the powers are approximately equal; in the general correlated case, fine-delay-averaged SMC noise cannot stand in for a particular target cell.Source: 2) Noise-Side Output Behavior · p. 22

C. Proof of Proposition 4: SMC-Domain Receive Beampattern

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