Short-Length Code Designs for Integrated Sensing and Communications: A Deep Learning Approach

I. INTRODUCTION

• ISAC waveform design must reconcile communication codebooks with sensing waveforms that favor predetermined signals with suitable correlation properties.Communication is evaluated by rate and block error rate, whereas sensing estimates channel parameters such as delay.Source: §I · p. 1
• Short codewords of 10–200 symbols make asymptotic coding analysis less applicable, while sensing constraints and noncoherent detection complicate code design.Pilot-based channel estimation can be inefficient because of its overhead in short transmissions.Source: §I · p. 2

1) Signal Design for Noncoherent Communications:

• Prior noncoherent communication designs span block- and fast-fading settings, including differential, orthogonal, and amplitude-based signaling approaches.Examples include pilot-free polar-coded systems and sparse signaling for severe channel uncertainty.Source: 1) Signal Design for Noncoherent… · p. 2

2) Signal Design for Sensing:

• Sensing waveform research includes constant-modulus, covariance-based, and ambiguity-function designs, often optimized for sidelobe criteria.The cited approaches use methods including nonconvex optimization, semidefinite quadratic programming, cyclic algorithms, and gradient descent.Source: 2) Signal Design for Sensing · p. 2

3) Communication Codeword Design based on Neural Encoders:

• Neural encoders have established an end-to-end approach for learning communication codewords across channel models, but this paper targets joint ISAC design.The paper frames short-length noncoherent ISAC code design as a distinct challenge not addressed by existing studies.Source: §I · pp. 1–2, 3) Communication Codeword Design based… · p. 2

B. Contributions

• The paper derives an optimal communication decoding principle and a modified Cramér–Rao bound, then uses their analysis to motivate codeword structures and trade-offs.The analysis indicates that both communication and sensing designs use uneven symbol powers; the sensing bound is invariant to a common phase rotation.Source: B. Contributions · p. 2
• The proposed method outperforms conventional coding and modulation schemes under short block lengths, with a tunable parameter balancing communication and sensing performance.The paper also reports that its results align with heuristic insights from the theoretical analysis.Source: B. Contributions · pp. 2–3
• The framework learns ISAC waveforms from derived performance metrics and introduces a tunable parameter to balance communication and sensing.Its reported results outperform conventional schemes under short block lengths.Source: B. Contributions · p. 2

C. Notations

• The notation defines matrix Hermitian and vector conjugation, identity and complex-number symbols, probability and expectation operators, and elementwise multiplication.It also specifies the gradient, Gaussian-distribution, and norm notation used in the analysis.Source: C. Notations · p. 3

II. SYSTEM MODEL

• The NN encoder maps K-bit messages to length-N power-normalized complex codewords, and communication and sensing receivers use separate channel outputs.The model focuses on delay estimation and leaves joint delay–Doppler estimation for future work.Source: §II · p. 3

A. Communication Model

• The communication channel is frequency-flat Rayleigh fading with diagonal, time-varying coefficients whose correlation is characterized by covariance Rc.Slow, fast, and partially correlated fading are represented as limiting or intermediate cases; fD and Ts are assumed known for design.Source: A. Communication Model · p. 3
• The time-varying channel correlation follows the Jake and Clark model, whose autocorrelation uses the order-zero modified Bessel function J0.The model relates Doppler frequency to receiver speed, light speed, and carrier frequency.Source: A. Communication Model · p. 3

B. Sensing Model

• The sensing receiver estimates P target delays from a single-carrier channel whose matrix has Toeplitz structure.The target count and delay statistics are assumed known during code design, and extending the design to unknown or time-varying P is left for future work.Source: B. Sensing Model · p. 4

III. THEORETICAL ANALYSIS

• The theoretical analysis derives the noncoherent communication ML receiver and the sensing MCRB to justify and guide the waveform-design metrics.These metrics are used in training the deep-learning-based waveform design.Source: §III · p. 4

A. Maximum Likelihood Detection for Noncoherent Communication

• Because channel coefficients can vary over time, symbol-wise noncoherent detectors do not directly apply; the paper derives a frame-wise optimal decoder.For equiprobable messages, the decoder selects the codeword with maximum likelihood, while BER and BCE are used for evaluation and training because likelihood over all hypotheses is not scalable.Source: A. Maximum Likelihood Detection for… · p. 4

B. Cram´er-Rao Lower Bounds Analysis for Sensing

• The MCRB simplifies the Fisher information matrix before inversion, trading tightness for analytical tractability.The passage says it is less tight than the Miller-Chang lower bound.Source: B. Cram´er-Rao Lower Bounds Analysis… · pp. 4–5
• Averaging over nuisance parameters can make the standard CRB’s Fisher information matrix non-diagonal, with an inverse that generally lacks a closed form.The delay vector is the parameter of interest; random fading and the random input are nuisance parameters.Source: B. Cram´er-Rao Lower Bounds Analysis… · p. 5
• Theorem 1 gives the MCRB expression for each target delay, and its off-diagonal Fisher-information terms are zero.This yields a diagonal Fisher information matrix for the estimation problem.Source: B. Cram´er-Rao Lower Bounds Analysis… · p. 5
• The resulting MCRB is a diagonal matrix that can be computed element-wise for the delay-estimation problem.The bound concerns the delay vector while treating fading and the random input as nuisance parameters.Source: B. Cram´er-Rao Lower Bounds Analysis… · p. 5

IV. DESIGN PRINCIPLES FOR CODE OPTIMIZATION

• The section frames these principles as heuristic insights from special examples because optimal short-blocklength ISAC code design remains open.The examples serve as a basis for the paper’s performance evaluation.Source: §IV · p. 5

herent Detection with Non- and Fully-Correlated Coefficients

• With uncorrelated fading, the likelihood depends only on symbol amplitudes, so equal-energy PSK symbols are indistinguishable under this rule.The passage therefore identifies PSK-type signals as poor performers in this case.Source: herent Detection with Non- and… · p. 5
• For independent fading, codewords with both large and small symbols resist erasure by deep fades, and distinct locations for strong components improve pairwise separation.When strong components overlap, their contributions can cancel and reduce distinguishability.Source: herent Detection with Non- and… · pp. 5–6
• For fully correlated fading and constant-power codewords, maximum-likelihood detection reduces to maximizing the squared received/codeword inner product.Good spherical codes should have low maximum squared correlation between codewords; Welch bound equality sets are one cited construction.Source: herent Detection with Non- and… · p. 6
• The two limiting cases yield different design rules, and the paper conjectures that intermediate-correlation codes should mix their characteristics.The paper describes this mixed design as difficult to derive analytically and turns to learning-based optimization.Source: herent Detection with Non- and… · p. 6

B. Implications of MCRB on Code Design for Sensing

• Under the power constraint, minimizing the MCRB reduces to maximizing x^H V x, whose solution is the principal eigenvector of V.Because identical row sums are generally not guaranteed, the resulting sequence generally allocates different energies across symbols.Source: B. Implications of MCRB on Code Design… · p. 6
• The sensing-optimal sequence is not unique under common phase rotation, and unknown target delay makes an analytic optimum difficult to obtain in advance.The paper says its neural network can find a good sensing waveform without knowing the delay.Source: B. Implications of MCRB on Code Design… · p. 6

C. Comparison of the Code Design for Communication and Sensing

• Both communication- and sensing-optimal codewords generally allocate power unevenly, but their optimal codeword relationships may not align.The comparison identifies constant-amplitude designs as suboptimal for both objectives.Source: C. Comparison of the Code Design for… · p. 6
• Sensing tolerates common phase-rotated versions of an optimum, whereas communication needs distinguishable codewords, suggesting a potential trade-off.The paper cautions that the trade-off is not guaranteed because some sensing-optimal sets may also perform well for communication.Source: C. Comparison of the Code Design for… · p. 6

V. DEEP LEARNING-BASED CODE DESIGN

• A tunable weight γ combines communication and sensing losses, with γ = 1 selecting communication loss and γ = 0 selecting sensing loss.Communication uses binary cross-entropy, while sensing uses the MCRB averaged over targets.Source: §V · pp. 6–7
• Training assumes known messages at the receiver and known target count and delays at the transmitter; inference uses channel output for communication decoding.The sensing procedure assumes the target count is estimated accurately before delay estimation, and training uses a fixed target count.Source: §V · p. 7

VI. VALIDATION USING A VERY SHORT BLOCK LENGTH

• The evaluation uses three autoencoder architectures and trains over uniformly sampled SNRs from 0 to 30 dB in an example with K = 6 and N = 9.The architectures are CNN, Turbo-CNN, and Turbo-RNN.Source: §V · p. 7, §VI · p. 7
• The validation studies communication-, sensing-, and joint-optimal designs in a representative very-short-blocklength setting.It compares communication encoders across fading conditions and evaluates sensing with MCRB and a complementary MSE-based assessment.Source: §VI · p. 7

A. Communication Optimal Design

• The communication AE is trained separately for each channel-correlation scenario because optimal code design depends on correlation.The paper uses a rate-2/3 convolutional code as a classical short-blocklength baseline and compares DBPSK, BPSK, and BASK modulation schemes.Source: A. Communication Optimal Design · p. 8
• The three AE architectures have similar BER and outperform conventional schemes combined with convolutional coding in most cases.In fully correlated fading, their BER nearly matches CC-DBPSK; in partially correlated and uncorrelated fading, they provide significant gains, while CNNs offer comparable performance to turbo AEs at lower complexity.Source: A. Communication Optimal Design · p. 8, Figure 2 · p. 8
• DBPSK degrades as channel correlation decreases because of error propagation, while BPSK and BASK are less sensitive to correlation.BASK performs relatively better in the uncorrelated high-SNR regime.Source: A. Communication Optimal Design · p. 9
• BER is lowest when training and testing Doppler match; mixed-condition training improves worst-case performance but remains suboptimal across tested Doppler values.The results indicate that optimal code design depends strongly on channel correlation.Source: A. Communication Optimal Design · p. 9

B. Sensing Optimal Design

• All AE designs have identical, lowest MCRBs, followed by ZC, D/BPSK, and 1H-OOK.The MCRB comparison uses P = 5 paths; Fig. 2 also reports MCRB visualizations.Source: Figure 2 · p. 8, B. Sensing Optimal Design · p. 9
• For a single target, the MCRB and CRLB overlap; non-outage MSE is similar for CNN, ZC, and BPSK, while 1H-OOK is higher.A gap remains between non-outage MSE and the bounds for all sequences, attributed to short blocklength and detector resolution limits.Source: Figure 2 · p. 8, B. Sensing Optimal Design · p. 9
• Sensing-optimized codewords have PSK-like structures with varying symbol amplitudes, allocating power unevenly to improve estimation rather than enforcing constant modulus.The designs are unconstrained; constant-modulus waveforms are often preferred for PAPR and power-amplifier efficiency, and the authors identify hardware-aware constraints as future work.Source: B. Sensing Optimal Design · p. 9

C. Joint Design

• At 0 dB SNR, a representative MCRB is reported because MCRB decreases linearly with SNR; sensing loss is additionally scaled by 1000 to compensate for its small scale.Source: C. Joint Design · pp. 9–10

VII. SIMULATION RESULTS FOR A MODERATELY SHORT BLOCK LENGTH

• At N=90, BER grows as γ decreases and the trade-off strengthens with less correlated fading; CNN MCRBs remain below ZC for γ<1.Source: Figure 4 · p. 10, §VII · p. 11
• For K=60, N=90, and fD=0.1, the Pareto curve decreases monotonically as γ varies, while DBPSK and ZC lie outside it.Source: Figure 5 · p. 10, §VII · p. 11

C. Joint Design

• Across all three fD cases, varying γ produces a clear BER–MCRB trade-off that becomes more pronounced under less correlated fading.Source: C. Joint Design · p. 10

VII. SIMULATION RESULTS FOR A MODERATELY SHORT BLOCK LENGTH

• For N=90, γ=0.5 achieves BER close to the optimal curve while maintaining a lower MCRB than the ZC sequence; the trade-off is milder than at N=9.Source: §VII · p. 11

VIII. CONCLUSION

• Learned waveforms outperform conventional designs and provide a controllable trade-off between BER and sensing accuracy, especially for short blocklength and partially or uncorrelated fading.Source: §VIII · p. 11

• The channel representation Hsx=X′Γ(τ)hs separates delay and fading, while each channel-matrix integral becomes a shifted Nyquist pulse.The pulse peaks when the sample-time difference matches a target delay.Source: A. Proof of Theorem 1 · p. 11
• For θ=(τ, hs, x), the received signal is complex Gaussian, and differentiating its mean with respect to a delay leaves only that path's contribution.Source: A. Proof of Theorem 1 · pp. 11–12
• Taking the expectation over fading cancels the FIM's off-diagonal entries, making the matrix diagonal and simplifying its inversion.Source: A. Proof of Theorem 1 · p. 12
• The MCRB is obtained by inverting the FIM; with a diagonal FIM, each bound uses the reciprocal of its corresponding expected diagonal entry.Source: A. Proof of Theorem 1 · p. 12

• The conditional covariance is decomposed as a diagonal matrix plus a rank-one outer-product term, with ρc distinguishing uncorrelated and fully correlated cases.Source: B. Likelihood of Correlated Fading… · p. 12
• The diagonal component A has entries N0,c+(1−ρc)|xi|^2 and is used to evaluate the covariance determinant and likelihood exponent.Source: B. Likelihood of Correlated Fading… · p. 12
• The determinant is evaluated using the matrix determinant lemma, while the inverse needed for the exponent uses the Sherman–Morrison formula.Source: B. Likelihood of Correlated Fading… · p. 12
• These closed forms yield an explicit likelihood for correlated fading, including intermediate correlation values ρc∈(0,1).Source: B. Likelihood of Correlated Fading… · p. 12