Joint Pilot Subcarrier Allocation Optimization

• The paper demonstrates that joint optimization of pilot subcarrier allocation reduces mutual coherence, enhancing compressed sensing-based channel estimation in MIMO-OFDM systems.
• It establishes that cyclic difference sets and greedy algorithms can effectively approximate the Welch bound, optimizing pilot placement under resource constraints.
• The approach employs analytical gradient descent and block-sparse penalties via Wirtinger calculus to reduce pilot overhead and improve normalized mean-square error performance.

Joint optimization of pilot subcarrier allocation refers to the integrated design and assignment of pilot subcarriers and, where relevant, pilot sequences in multi-antenna and multi-carrier systems, targeting enhanced channel estimation accuracy under resource constraints and practical deployment scenarios. The topic encompasses both the classical minimization of submatrix coherence for compressed sensing-based channel recovery and more recent block-sparse and sequence optimization for MIMO-OFDM architectures with channel sparsity. This entry focuses on principles, problem formulations, solution approaches, and practical impacts arising from deterministic and joint pilot allocation strategies.

1. Principles of Coherence Minimization in Sparse Channel Estimation

In sparse channel estimation for OFDM and MIMO-OFDM, recovery of the sparse channel impulse response $\mathbf{h}$ from noisy pilot measurements is modeled as

$$\widehat{\mathbf{H}}_p = \mathbf{F}_p \mathbf{h} + \mathbf{n}_p,$$

where $\mathbf{F}_p$ is a partial DFT matrix extracted by selecting rows indexed by the pilot subcarriers. The performance of greedy sparse recovery algorithms is governed by the coherence

$$\mu(\mathbf{F}_p) = \max_{l \neq k} \left| \frac{\langle \mathbf{f}_l, \mathbf{f}_k \rangle}{\|\mathbf{f}_l\|_2 \cdot \|\mathbf{f}_k\|_2} \right|,$$

where $\mathbf{f}_l$ and $\mathbf{f}_k$ are columns of $\mathbf{F}_p$. For unit-norm DFT columns, this simplifies to

$$\mu(\mathbf{F}_p) = \frac{1}{N_p} \max_{r \neq 0} \left| \sum_{i=1}^{N_p} e^{-j2\pi P_i r/N} \right|,$$

with ${P_1, ..., P_{N_p}}$ being the indices of pilot subcarriers. Minimizing $\mu(\mathbf{F}_p)$ is thus central to optimal pilot allocation, as lower coherence directly improves sparse recovery fidelity for algorithms like Orthogonal Matching Pursuit (OMP) and iterative thresholding procedures (IMAT) (Pakrooh et al., 2011).

2. Cyclic Difference Sets and Greedy Algorithms for Pilot Placement

The optimal deterministic pilot allocation is achieved by arranging pilot subcarrier indices so that their cyclic differences—modulo the FFT size $N$—are uniformly distributed. Formally, a cyclic difference set has the property

$$a_1 = a_2 = \cdots = a_{N-1} = \frac{N_p(N_p - 1)}{N-1},$$

where $a_d$ is the number of occurrences of difference $d$ among pilot indices. When such sets exist for the pair $(N, N_p)$, coherence attains the Welch lower bound and recovery is maximally robust.

For $(N, N_p)$ pairs lacking cyclic difference sets, a greedy search strategy is adopted. The procedure builds the pilot index set incrementally, at each step selecting the candidate that minimizes the variance of the difference histogram among chosen pilots, thereby closely approximating the uniform difference property.

Allocation Strategy	Applicability	Coherence Behavior
Cyclic difference	Certain $(N, N_p)$ pairs	Achieves Welch bound
Greedy method	All $(N, N_p)$ pairs	Approximates uniformity
Uniform allocation	Interpolation-based only	Suboptimal with sparsity

3. Joint Pilot Allocation and Non-Orthogonal Sequence Design in MIMO-OFDM

Channel sparsity in the delay and angle domains motivates compressed sensing approaches in MIMO-OFDM, where both pilot subcarrier allocation (discrete) and pilot sequences (continuous) affect the sensing matrix's coherence. The direct joint optimization leads to an intractable mixed-integer nonlinear program (MINLP). To circumvent this, a block sparse penalty is introduced over all pilot sequences, effectively driving unneeded pilot powers to zero and enabling the use of continuous optimization methods (Arai et al., 22 Sep 2025).

Let $X = [X_1, X_2, ..., X_K]$ denote pilot matrices for all subcarriers. The block sparsity penalty is defined as

$$g(X) = \left( \sum_{k=1}^K \|X_k\|_F^q \right)^{1/q},\quad 0 < q \leq 1,$$

coupled in the new objective: $$\min_X f^{\Psi}(X) + \lambda g(X),\quad \text{subject to } \|X\|_F^2 = P_t,$$ where $f^{\Psi}(X)$ is a generalized coherence metric over the sensing matrix columns. The design exploits the Kronecker product structure of the MIMO-OFDM dictionary to compute coherences efficiently, and gradient descent (via Adam) is performed based on closed-form Wirtinger derivatives.

Numerical results confirm that, with properly chosen sparsity parameters, the solution yields a block-sparse pilot matrix, reducing pilot overhead and improving the mutual coherence/fidelity of CS-based channel estimation.

4. Algorithmic Implementation: Gradient Descent and Coherence Metric Differentiation

The minimization of the coherence metric over pilot subcarrier/sequence space leverages analytic gradient expressions to enable efficient descent:

• The sensing matrix is rewritten as $$\Psi = \left(X_{\text{blk}}^T \otimes I_{N_r}\right) (B(\tilde{\tau}) \otimes A_t^*(\tilde{\phi}) \otimes A_r(\tilde{\theta}))$$.
• The coherence is decomposed into terms dependent solely on pilot design and on the angular dictionary.
• The gradient w.r.t. pilot sequences is computed analytically using Wirtinger calculus (see Eq. [grad_obj] in (Arai et al., 22 Sep 2025)), enabling unconstrained optimization in normalized form: $X = (\sqrt{P_t}/\|\bar{X}\|_F) \bar{X}$.

This approach ensures computational tractability compared to exhaustive discrete search, with candidate sets numbering $10^{11}$ for typical MIMO-OFDM parameters.

5. Impact on Channel Estimation Performance and Overhead

Simulation evidence demonstrates that joint optimization with block sparsity penalty:

• Produces pilot matrices that are sparse in frequency, reducing the number of active pilot subcarriers (e.g., $Q=9$ out of $K=64$).
• Achieves a lower worst-case mutual coherence in the partial sensing matrix compared to random or separately optimized designs.
• Improves normalized mean-square error (NMSE) for CS recovery algorithms such as OMP and GAMP-SBL over a wide SNR range.
• Outperforms baselines in both random pilot allocation and decoupled sequence/subcarrier design approaches.

These results substantiate the approach's practical utility for advanced MIMO-OFDM systems seeking to minimize pilot overhead while maintaining robust sparse channel recovery.

6. Limitations, Assumptions, and Applicability

The methodology assumes:

• The underlying channel exhibits strong delay-angle sparsity, a condition typical in mmWave and large-scale MIMO scenarios.
• The angular and delay dictionaries are pre-defined and may induce grid mismatch effects.
• Hyperparameters governing the coherence norm, block-sparse penalty, and penalty weights require empirical tuning, with stability and convergence subject to problem-specific properties.

Offline computational complexity is significantly reduced compared to brute-force search but remains non-negligible, making the technique suitable for pre-deployment optimization rather than real-time pilot adaptation.

7. Summary and Contextual Relevance

Joint optimization of pilot subcarrier allocation and pilot sequence design is crucial for maximizing CS-based channel estimation performance, particularly as subcarrier and antenna counts grow in modern wireless systems. Innovations such as cyclic difference sets, greedy uniformity-driven allocation, and continuous block-sparse penalty optimization collectively advance the state of the art in pilot design. They provide a foundation for minimizing pilot overhead and coherence for reliable sparse recovery, directly impacting spectral efficiency and accuracy in MIMO-OFDM deployments (Pakrooh et al., 2011, Arai et al., 22 Sep 2025).