Dual-Band OFDM Delay Estimation

• Dual-band OFDM delay estimation is a method that combines two non-contiguous subbands to enhance delay resolution in integrated localization and communication systems.
• It employs a PSF-centric framework and an adapted RELAX algorithm to achieve robust multi-target delay estimation while managing elevated sidelobes and ambiguity.
• Simulation results demonstrate that this approach reduces RMSE by 30–50% over conventional methods and offers improved sidelobe suppression under practical noise conditions.

Dual-band OFDM delay estimation is an approach for resolving multipath signal delays using non-contiguous, fragmented spectrum allocations within integrated localization and communication (ILC) systems. Conventional delay estimation is constrained by available contiguous bandwidth, which is often infeasible due to spectrum fragmentation and hardware limitations. Aggregating multiple narrow bands increases the frequency span, offering the potential for enhanced delay resolution but introducing fundamental challenges including elevated sidelobes and ambiguity. The dual-band OFDM methodology utilizes a point-spread-function (PSF)-centric framework, analytically links band configuration to achievable resolution and ambiguity, and employs an adapted RELAX iterative algorithm for robust multi-target delay estimation under practical signal and noise conditions (Kou et al., 26 Jan 2026).

1. Signal Model and Point-Spread Function Framework

Dual-band OFDM delay estimation relies on subcarrier selection within a larger OFDM grid. Let $K$ denote the total number of subcarriers with spacing $\Delta f$ and overall span $B = K \Delta f$. A subset of $M$ active subcarriers indexed by $\mathcal{S}$ forms the binary subcarrier selection function (SCF): $$W[k] = \begin{cases} 1, & k \in \mathcal{S} \ 0, & \text{otherwise} \end{cases},\qquad k=0,\dots,K-1 \qquad \text{(1)}$$ For an $L$-path channel model with gains $\{\alpha_\ell\}$ and delays $\{\tau_\ell\}$, the observed channel frequency response (CFR) for subcarrier $k$ is

$$H[k] = W[k] \sum_{\ell=0}^{L-1} \alpha_\ell\, \exp(-j2\pi k\Delta f \tau_\ell) \qquad \text{(2)}$$

Applying the $K$-point IDFT gives the delay-domain profile,

$$h(\tau) = \frac{1}{K} \sum_{k=0}^{K-1} H[k]\, \exp(j2\pi k\Delta f \tau) \qquad \text{(3)}$$

Substituting (2) into (3) yields a convolutional observation: $$h(\tau) = \sum_{\ell=0}^{L-1} \alpha_\ell\, p(\tau-\tau_\ell) \qquad \text{(4)}$$ where the point-spread function is

$$p(\tau) = \frac{1}{K} \sum_{k=0}^{K-1} W[k]\, \exp(j2\pi k\Delta f \tau) \qquad \text{(5)}$$

This formalism models the observed profile $y(\tau)$ as a convolution of target impulses and the PSF: $$y(\tau) = (x \star p)(\tau) = \int x(u)\, p(\tau-u)\, du \qquad \text{(6)}$$

In the dual-band setting, two non-contiguous subbands of bandwidth $B_\mathrm{sub} = N \Delta f$ separated by a gap of $g$ subcarriers ($f_\mathrm{gap} = g\Delta f$) are selected: $$\mathcal{S} = \{0,1,\dots,N-1\} \cup \{g, g+1,\dots,g+N-1\} \qquad \text{(7)}$$ The closed-form dual-band PSF becomes \begin{align} p(\tau) &= (1+e^{j2\pi f_\mathrm{gap}\tau})\, D_N(\tau) \ D_N(\tau) &= \frac{e^{j\pi(N-1)\Delta f \tau}}{K}\frac{\sin(\pi N\Delta f\tau)}{\sin(\pi \Delta f \tau)} \qquad \text{(8,9)} \end{align} where $D_N(\tau)$ is the Dirichlet kernel envelope.

2. Resolution, Sidelobe Structure, and Ambiguity

The dual-band PSF structure controls estimation performance through its envelope and modulating cosine component. The main-lobe width for the contiguous-band envelope is governed by

$$\Delta\tau_{\rm env} \approx \frac{2}{N\Delta f} = \frac{2}{B_{\rm sub}}$$

The cosine modulation $\cos(\pi f_{\rm gap} \tau)$ induces oscillations of period

$T_{\rm osc} = \frac{1}{f_{\rm gap}}$

resulting in sharply defined "super-resolution" peaks with width approximately $1/f_\mathrm{gap}$.

The first Dirichlet sidelobe lies approximately $-13$ dB below the main lobe. Increasing $N$ (number of subcarriers per band) suppresses envelope sidelobes. Cosine modulation introduces numerous local maxima spaced by $T_\mathrm{osc}$, increasing ambiguity as $f_\mathrm{gap}$ grows.

In summary, delay resolution improves with larger $f_\mathrm{gap}$ due to narrower peaks, but ambiguity increases due to more pronounced sidelobes. The envelope width—proportional to $1/B_\mathrm{sub}$—determines the maximum resolvable span, while the tradeoff between sidelobe contrast and ambiguity is set by $N, \Delta f$, and $f_\mathrm{gap}$.

3. RELAX Algorithm for Multi-Target Dual-Band Delay Estimation

The dual-band RELAX algorithm provides robust multi-target delay estimation under the non-contiguous band scenario. Noisy CFR samples from active subcarriers are compiled into $\mathbf{y} \in \mathbb{C}^M$: $$\mathbf{y} = \sum_{\ell=1}^L \alpha_\ell\, \mathbf{a}_\mathcal{S}(\tau_\ell) + \mathbf{n},\qquad \mathbf{n}\sim\mathcal{CN}(0,\sigma^2\mathbf{I}) \qquad \text{(10)}$$ where the dual-band steering vector is

$$\mathbf{a}_\mathcal{S}(\tau) = [e^{-j2\pi k\Delta f \tau}]_{k\in\mathcal{S}}, \quad \mathbf{a}_\mathcal{S}(\tau) \in \mathbb{C}^M \qquad \text{(11)}$$

Maximum likelihood estimation of $\{(\alpha_\ell, \tau_\ell)\}$ seeks to minimize

$$\min_{\{\alpha,\tau\}} \|\mathbf{y} - \sum_\ell \alpha_\ell\, \mathbf{a}_\mathcal{S}(\tau_\ell)\|_2^2 \qquad \text{(12)}$$

This nonconvex $2L$-dimensional problem is tractably solved by the residual-based RELAX algorithm with cyclic single-target updates. The approach iteratively acquires new targets, refines each by maximizing steering correlation on a dense grid of candidate delays $\mathcal{T}$, and refreshes the residual. Termination occurs either when the global residual norm falls below a tolerance $\varepsilon$ or the number of estimated targets reaches $L_\mathrm{max}$.

Algorithmic Complexity

Each delay search across $\mathcal{T}$ incurs $O(M|\mathcal{T}|)$ cost; one RELAX cycle with $J$ refinements totals $O(JM|\mathcal{T}|)$; overall worst-case scaling is $O(L_\mathrm{max}^2 M|\mathcal{T}|)$.

4. Simulation Regime and Estimation Performance

Simulations are conducted at a carrier frequency of 5.2 GHz with $K=1024$, $\Delta f=312.5$ kHz, resulting in a 320 MHz grid. Two subbands, each $B_\mathrm{sub}=40$ MHz ($N=128$) and variable gap $f_\mathrm{gap}\in\{40,80,120,280\}$ MHz, form the dual-band allocation ($M=256$ active subcarriers). Three static targets ($L=3$) with delays $\tau\in\{66,100,133\}$ ns and Rayleigh gains are recovered over SNRs from $-10$ dB to $+35$ dB using 1000 Monte Carlo trials.

Delay estimation accuracy is quantified by root-mean-square error: $$\text{RMSE}_\tau = \sqrt{\mathbb{E}\left[\frac{1}{L}\sum_{\ell=1}^L (\hat\tau_\ell-\tau_\ell)^2\right]} \qquad \text{(13)}$$ Key findings include:

• Increasing $f_\mathrm{gap}$ reduces the high-SNR RMSE floor (approaching the CRB) but raises the threshold SNR at which strong sidelobes induce estimation errors.
• RELAX confers 30–50% RMSE reduction over OMP in multi-target scenarios for SNR = 0–20 dB.
• RELAX reconstructions exhibit >20 dB peak-to-sidelobe ratio compared to ~8 dB using raw IDFT.
• Estimation bias is sub-nanosecond at moderate to high SNR.

5. Practical Implementation Considerations, Assumptions, and Extensions

Achieving robust dual-band delay estimation in practice requires precise time-frequency synchronization across bands, including CFO and phase-noise correction. Band-dependent amplitude and frequency response mismatches necessitate equalization, and filter-bank imperfections must be accounted for. The grid resolution $|\mathcal{T}|$ directly trades off with search complexity; off-grid refinement, such as Newton interpolation, may alleviate bias. Real-time constraints and computational resources can further bound target number and RELAX iterations.

Assumptions underlying methodology include AWGN noise, perfect knowledge of $\Delta f$, and static channels with no Doppler or subband CFO. Under low SNR and with pronounced $f_\mathrm{gap}$, ambiguity from elevated PSF sidelobes dominates, suggesting that moderate gaps (100–150 MHz) provide optimal tradeoff between resolution and robustness. Extensions to joint delay–Doppler estimation (e.g., "2D RELAX," Editor's term) and application of robust subcarrier windowing for further sidelobe suppression are identified. Over-the-air validation considering hardware and multipath clustering remains an open direction (Kou et al., 26 Jan 2026).