Superimposed Sparse Pilot Structure

• Superimposed sparse pilot structure is a technique that overlays pilot symbols onto data carriers using sparse allocation in delay, Doppler, or angle domains to preserve spectral efficiency.
• It enables advanced channel estimation by leveraging compressed sensing and iterative interference cancellation, which is critical in high-dimensional MIMO, OTFS, and AFDM systems.
• The design optimizes pilot placement and power allocation to balance channel estimation accuracy, bit error rate, and spectral efficiency across diverse communication scenarios.

A superimposed sparse pilot structure refers to a class of pilot embedding techniques in which pilot symbols are added directly onto (superimposed with) data symbols on the same time-frequency-spatial resources, exploiting sparsity in one or more domains (e.g., delay, frequency, Doppler, angle, or code space). This design preserves spectral efficiency by minimizing the pilot overhead, enables compressed sensing-based channel estimation, and often supports high-dimensional MIMO, doubly-selective, or joint communication-sensing applications. The "superimposed" aspect enables pilot and data to coexist on the same channel uses, while "sparse" refers to the deliberate non-uniform allocation or structured code-domain separation of pilot energy, facilitating efficient and robust channel estimation with low pilot density.

1. Signal Model and Mathematical Formulation

In superimposed sparse pilot schemes, the transmitted signal at each user or antenna is formed by allocating pilot and data energy components within the same vectorized resource block. Abstracted to a single-user, single-stream scenario, the transmit signal can typically be represented as

$$\mathbf{x} = \sqrt{\rho_p} \mathbf{p} + \sqrt{\rho_d} \mathbf{d}$$

where $\mathbf{p}$ is the (possibly sparse) pilot sequence, $\mathbf{d}$ is the data sequence, and $\rho_p$, $\rho_d$ denote the power devoted to pilots and data respectively, with $\rho_p + \rho_d = 1$. In the multi-user and/or MIMO case, $\mathbf{x}$ may be an $N\times T$ matrix with independently superimposed pilots per antenna or resource.

In domains such as OTFS or AFDM, both pilot and data vectors are mapped into the delay-Doppler or DAFT domain; in the traditional MIMO-OFDM setting, pilots—potentially modulated by pseudo-random sequences to guarantee mutual incoherence—are overlaid on the same subcarriers across antennas (Gao et al., 2015, Gong et al., 2020). In all cases, sparsity is exploited: only a small subset of symbols or subcarriers carry nonzero pilot energy, mitigating the overhead increase that would otherwise scale with the number of transmit antennas or system dimension.

The received signal at the base station or receiver is written

$\mathbf{Y} = \mathbf{H}\mathbf{S} + \mathbf{N}$

where $\mathbf{S}$ is the superimposed pilot+data transmit matrix, $\mathbf{H}$ is the channel, and $\mathbf{N}$ is additive noise. This model naturally supports compressed-sensing-based or Bayesian channel estimation approaches.

2. Pilot Placement, Sparsity, and Design Criteria

A core aspect of the superimposed sparse pilot structure is the deliberate placement of pilots to maximize separability and minimize mutual interference. In AFDM, optimal placement is shown to be equally spaced with spacing at least $Q+1$, where $Q$ is the number of channel paths, to ensure orthogonality between pilot signatures after all Doppler and delay shifts (Zheng et al., 16 Apr 2024). In OTFS or OFDM-based MIMO, pilot subcarriers are shared across antennas with distinct pseudo-random or CAZAC sequences to enable code-domain discrimination while reducing overhead (Gao et al., 2015, Bergamasco et al., 19 Mar 2025).

For block-sparse or structured sparsity cases (e.g., large-scale or doubly-selective MIMO-OFDM), guard tones or blocks (zeroed data symbols near pilots) are inserted to ensure the measurement matrix is interference-free or nearly orthogonal, which is critical for compressive sensing-based channel recovery (Gong et al., 2020). Pilot positioning may be further optimized using discrete stochastic algorithms that minimize block-coherence of the measurement dictionary, improving support recoverability (Gong et al., 2020).

In Table 1, placement strategies and the domains in which sparsity is imposed are contrasted:

Domain	Placement Criterion	Sparsity Target
OTFS/DD	Separation $\geq l_{max}+1, 2k_{max}+1$	Delay-Doppler grid
AFDM/DAFT	Spacing $\geq Q+1$	DAFT bins
OFDM-MIMO	Shared subcarriers	Antenna/code domain
OFDM-BEM	Guard tones around pilot	Delay-tap blocks

Orthogonality or near-orthogonality among pilot signatures on the shifted/affected coordinates is a recurring requirement for minimax MSE design.

3. Channel Estimation and Sparse Recovery

Channel estimation under superimposed sparse pilot structures is fundamentally an interference-aware, often sparsity-exploiting, estimation problem. The standard approach is to use a linear MMSE or LS estimator, treating data/pilot interference (arising from data superposition) as colored Gaussian noise (Mishra et al., 2020, Zheng et al., 16 Apr 2024). Explicitly, in the compressed sensing setting, one solves

$$\min_{\mathbf{h}} \|\mathbf{y} - \mathbf{\Phi} \mathbf{h}\|_2^2 + \lambda \|\mathbf{h}\|_{2,1}$$

where $\mathbf{\Phi}$ is the pilot-coded measurement matrix and $\mathbf{h}$ is the sparse vector/matrix of channel parameters. In settings with common or structured sparsity (e.g., spatial-temporal joint sparsity in MIMO), mixed-norm minimization or greedy support-pursuit (SSP, BSOMP) algorithms are employed to reconstruct the entire multi-user/multi-antenna channel matrix with minimal pilot overhead (Gao et al., 2015, Gong et al., 2020).

Iterative refinement is prevalent in the presence of data interference. Here, after an initial channel estimate (possibly treating data as noise), the pilot-induced channel contribution is subtracted, data symbols are re-estimated (using e.g., message passing), and the process is repeated with updated channel and data estimates to progressively suppress interference and improve estimation NMSE and subsequent BER (Kanazawa et al., 27 Jan 2025, Mishra et al., 2020, Gan et al., 16 Feb 2024).

Bayesian and approximate message passing strategies have also been integrated, particularly for high-dimensional or joint communication-sensing ISAC scenarios where angle-delay sparsity is leveraged and hyperparameters are adaptively estimated (Gan et al., 16 Feb 2024).

4. Message Passing and Data Detection

Given the interference structure induced by superimposed pilots, detection of data symbols after (imperfect) channel estimation is often nontrivial. Message passing algorithms (MP), specifically structured for the sparsity and connectivity of OTFS or other time-frequency-modulated channels, enable efficient low-complexity symbol detection even when pilot/data interference remains (Mishra et al., 2020, Kanazawa et al., 27 Jan 2025).

Detection proceeds on a factor graph where variable nodes represent data symbols, factor nodes represent observations, and edges correspond to nonzero (often sparse) entries in the effective channel matrix. Gaussian approximation or belief propagation rules iteratively refine posterior symbol beliefs under the current channel estimate, with outputs feeding back into subsequent channel estimation rounds for interference cancellation (Kanazawa et al., 27 Jan 2025, Mishra et al., 2020).

Graph-based turbo equalization also appears in OTFS systems, where iterative message exchange exploits pilot design and channel sparsity for improved data/BER performance under superimposed pilot structures (Bergamasco et al., 19 Mar 2025).

5. Performance Metrics and Trade-offs

Superimposed sparse pilot structures primarily target three metrics: spectral efficiency (SE), channel estimation quality (NMSE), and bit error rate (BER).

• Spectral Efficiency: Because pilot and data symbols are non-orthogonally superposed (no time/frequency slots are solely dedicated to pilots), the pilot overhead is minimized and SE is significantly improved compared to embedded- or frame-based orthogonal pilot schemes. For instance, in OTFS and AFDM, SE gains of 20–57% over embedded-pilot designs are reported, with loss limited to the small pilot density $\rho = P/N$ (Zheng et al., 16 Apr 2024, Bergamasco et al., 19 Mar 2025).
• Channel Estimation NMSE: Increases with reduced pilot power or density. Orthogonal pilot placement achieves the minimum trace of the error covariance; iterative or Bayesian refinement mitigates residual interference.
• BER/Detection: A key trade-off exists between pilot power/density and detection performance. Excessively low pilot energy degrades channel estimation; excessively high pilot energy reduces data SINR. Iterative detection and cancellation improves BER, but single-shot non-iterative methods plateau at high SNR due to residual interference (Mishra et al., 2020, Kanazawa et al., 27 Jan 2025).
• PAPR: In superimposed/CAZAC-based designs (e.g., OTFS with superimposed Chu sequence pilots), PAPR can be reduced by distributing pilot energy, further suppressing temporal peaks (Bergamasco et al., 19 Mar 2025).

Practical guidance suggests optimal pilot/data splits $\alpha\in[0.3,0.6]$, orthogonal/sparse pilot positions, and 4–6 outer interference-cancellation iterations for near-optimal performance (Kanazawa et al., 27 Jan 2025, Bergamasco et al., 19 Mar 2025).

6. Applications in Massive MIMO, OTFS/AFDM, and ISAC

Superimposed sparse pilot structures have demonstrated benefit across several advanced wireless scenarios:

• Massive MIMO: In both uplink and downlink, superimposed pilots facilitate channel estimation in regimes where the number of antennas renders orthogonal pilots impractical. Exploiting spatial-temporal common sparsity and code-domain orthogonality, pilot overhead per antenna can be reduced to the order of $O(2K)$ (with $K$ the support size) rather than $O(M)$ (with $M$ antennas) (Verenzuela et al., 2017, Gao et al., 2015).
• Doubly-Selective Channels (AFDM, OTFS): Superimposed sparse pilots with optimal delay-Doppler placement or DAFT-bin allocation enable robust acquisition of sparse multipath channels, yielding accurate channel estimates and high SE without guard symbols or large pilot regions (Zheng et al., 16 Apr 2024, Kanazawa et al., 27 Jan 2025, Mishra et al., 2020).
• Joint Communication and Sensing (ISAC): By superimposing sparse pilots onto data signals, efficient channel estimation and high-precision localization (centimeter-level) are achieved using Bayesian learning or compressed sensing, attaining spectral efficiency >96% of pure communication designs and more than 133% throughput improvement over disjoint pilot-based ISAC schemes (Gan et al., 16 Feb 2024).

7. Limitations and Comparative Evaluation

Despite the benefits, several limitations and trade-offs characterize superimposed sparse pilot systems:

• Interference: Superposition introduces coherent and non-coherent data-pilot interference. In massive MIMO, while pilot contamination is mitigated, new interference terms may offset the gain when compared to length-optimized regular pilots (Verenzuela et al., 2017).
• Optimization: The pilot/data power split and pilot position/density must be numerically optimized for best SINR or throughput. No analytic universally-optimal split exists in general; numerical line search or quadratic solutions are used (Verenzuela et al., 2017, Bergamasco et al., 19 Mar 2025).
• Benefit Margins: In massive MIMO, when regular pilots are permitted longer sequences, the SE and EE advantage of superimposed sparse pilots becomes marginal (Verenzuela et al., 2017).
• Complexity: Iterative (Bayesian, message-passing) receivers are needed to fully exploit interference cancellation potentials. For non-iterative schemes, SE may be high but NMSE/BER can plateau, particularly at high SNR.

Comparatively, superimposed sparse pilot structures are most beneficial where pilot overhead is otherwise prohibitive (massive MIMO, large delay-Doppler grids, ISAC) or in regimes where spectral efficiency gain and modest localization/estimation accuracy are prioritized over absolute minimum NMSE.

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Key references for further paper:

• "Spectral and Energy Efficiency of Superimposed Pilots in Uplink Massive MIMO" (Verenzuela et al., 2017)
• "Structured Compressive Sensing Based Superimposed Pilot Design in Downlink Large-Scale MIMO Systems" (Gao et al., 2015)
• "A low-PAPR Pilot Design and Optimization for OTFS Modulation" (Bergamasco et al., 19 Mar 2025)
• "Channel Estimation for AFDM With Superimposed Pilots" (Zheng et al., 16 Apr 2024)
• "Superimposed Pilot-Based OTFS -- Will it Work?" (Kanazawa et al., 27 Jan 2025)
• "Block Distributed Compressive Sensing Based Doubly Selective Channel Estimation and Pilot Design for Large-Scale MIMO Systems" (Gong et al., 2020)
• "Bayesian Learning for Double-RIS Aided ISAC Systems with Superimposed Pilots and Data" (Gan et al., 16 Feb 2024)
• "OTFS Channel Estimation And Data Detection Designs With Superimposed Pilots" (Mishra et al., 2020)