IFFT-C2VNN: Low-Complexity ISAC Sensing

• The paper presents IFFT-C2VNN as a deep learning-based estimator that regresses AoA, AoD, and ToA using CSI-derived delay representations.
• It employs IFFT preprocessing to expose delay peaks and extract localized complex windows, significantly reducing computational complexity.
• The architecture integrates complex-valued convolutional layers with CReLU activation, ensuring robust performance over diverse SNR conditions.

IFFT-C2VNN is a learning-based estimator for high-resolution sensing parameter estimation in communication-centric bistatic integrated sensing and communication (ISAC). In the formulation reported in "High-Resolution Sensing in Communication-Centric ISAC: Deep Learning and Parametric Methods" (Naoumi et al., 2 Sep 2025), it operates on channel state information (CSI) estimated from ordinary OFDM reference symbols rather than on a dedicated radar waveform, applies an IFFT/IDFT across subcarriers to expose coarse delay structure, extracts a local complex-valued window around each delay-domain peak, and uses a complex-valued convolutional neural network (CVNN) to regress the parameters of each target. In its base form, the network estimates angle of arrival (AoA), angle of departure (AoD), and time of arrival (ToA); in an extension, it also estimates Doppler frequency (Naoumi et al., 2 Sep 2025).

1. Definition and problem setting

IFFT-C2VNN is designed for a bistatic communication-centric ISAC system in which a base station (BS) transmits downlink OFDM frames and a separate passive radar unit listens to the reflected signals. The system is communication-centric in the precise sense that sensing reuses CSI estimated from communication reference or pilot symbols and known transmitted OFDM symbols, rather than relying on a dedicated sensing waveform (Naoumi et al., 2 Sep 2025).

The sensing task is joint super-resolution estimation of scatterer parameters from CSI-derived observations. For each scatterer, the main formulation targets three continuous-valued quantities: AoA $\theta_m$, AoD $\phi_m$, and ToA $\tau_m$. The received signal model at subcarrier $n$ and OFDM symbol $k$ is

$$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$

and the passive radar first computes least-squares CSI estimates rather than feeding raw waveforms directly to the estimator (Naoumi et al., 2 Sep 2025).

The underlying challenge is the multidimensional nature of the inference problem. The paper frames joint parameter estimation through a maximum-likelihood objective and emphasizes that brute-force MLE is computationally prohibitive, with complexity scaling as

$$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$

This motivates an amortized estimator that learns the CSI-to-parameter mapping offline and performs low-complexity inference online (Naoumi et al., 2 Sep 2025).

2. Signal model and CSI-derived sensing input

The system model uses a BS with a uniform linear array of $N_{\mathrm t}$ antennas and a passive radar receiver with a uniform linear array of $N_{\mathrm r}$ antennas. The transmitted OFDM frame has $N_{\mathrm P}$ active subcarriers, $\phi_m$0 OFDM symbols, subcarrier spacing $\phi_m$1, useful symbol duration

$\phi_m$2

cyclic-prefix duration

$\phi_m$3

and total OFDM symbol duration

$\phi_m$4

Each scatterer has AoA, AoD, ToA, Doppler frequency, and complex gain, and under near-perfect synchronization the frequency-domain channel admits the factorization

$\phi_m$5

For low-mobility scenarios with negligible Doppler, this simplifies to

$\phi_m$6

The steering vectors are ULA responses,

$\phi_m$7

These expressions are central because AoA and AoD are encoded in antenna-domain phase progression, while delay is encoded across the subcarrier axis (Naoumi et al., 2 Sep 2025).

What IFFT-C2VNN actually consumes is a stacked CSI matrix obtained by LS estimation. For subcarrier $\phi_m$8,

$\phi_m$9

and the per-subcarrier CSI estimates are assembled as

$\tau_m$0

The paper identifies this stacked CSI matrix as the sensing input for both IFFT-C2VNN and the model-based comparator PARAMING (Naoumi et al., 2 Sep 2025).

3. IFFT preprocessing and windowed target representation

The defining front-end operation in IFFT-C2VNN is an IDFT/IFFT across the subcarrier dimension. The paper rewrites the CSI matrix as

$\tau_m$1

where

$\tau_m$2

and the delay dependence is concentrated in $\tau_m$3. Applying the IDFT yields

$\tau_m$4

Because the subcarrier-dependent factor is $\tau_m$5, the IFFT transforms frequency-domain delay phase into a delay-domain representation with peaks around the true delays (Naoumi et al., 2 Sep 2025).

For a single delay column, the transformed entry is

$\tau_m$6

The paper interprets this expression in two cases. If the delay is aligned to a discrete delay bin, energy concentrates at one row; if the delay is fractional, the energy spreads in a sinc-like way over nearby rows. This gives a coarse delay localization stage that reduces the burden on the neural estimator, because the network can focus on a small neighborhood around each target peak rather than processing the entire CSI tensor (Naoumi et al., 2 Sep 2025).

The transformed array has shape

$\tau_m$7

so the IFFT is applied only along the subcarrier axis, not along the antenna dimensions. Under non-fractional delays, a target ideally occupies a row indexed by

$\tau_m$8

For each target $\tau_m$9, IFFT-C2VNN extracts a local complex window

$n$0

centered at $n$1. The paper gives this window explicitly as the stack of rows from $n$2 through $n$3 of $n$4. Its meaning is operationally important: the first dimension is a local delay neighborhood, while the second dimension contains the stacked antenna-pair responses. The window therefore preserves coarse delay support, fractional-delay phase structure, and joint angular information across antenna pairs (Naoumi et al., 2 Sep 2025).

This IFFT-centered design is distinct from earlier IFFT-based methods in OFDM literature. In low-complexity selected mapping for PAPR reduction, cyclic shifts are applied inside a decimation-in-frequency IFFT to generate alternative OFDM candidates with lower computational cost (Kim et al., 2012). In massive-MIMO OFDM precoding, recursive convolutional precoding uses one IFFT per user rather than one IFFT per transmit antenna (Liu et al., 2016). Those works are not sensing networks, but they show a broader research pattern: the IFFT is treated as a structured computational object whose intermediate or transformed outputs can be exploited to reduce complexity [(Kim et al., 2012); (Liu et al., 2016)]. A plausible implication is that IFFT-C2VNN belongs to this wider class of methods that use transform-domain structure to shrink the effective inference search space.

4. Complex-valued neural architecture and training objective

IFFT-C2VNN processes each target window with a complex-valued CNN. The paper specifies a compact architecture: two complex-valued convolutional layers, each followed by $n$5, followed by two complex linear layers. Each convolutional layer has 10 complex filters. The text does not specify kernel sizes, strides, padding, pooling, normalization, dropout, exact hidden dimensions, or output activation, and those details therefore remain unspecified in the reported formulation (Naoumi et al., 2 Sep 2025).

For complex input

$n$6

and complex filter

$n$7

the complex convolution is implemented as

$n$8

For complex linear layers with weights and biases

$n$9

the output is

$k$0

The activation is the complex ReLU

$k$1

with real and imaginary parts rectified separately (Naoumi et al., 2 Sep 2025).

The network is a direct regressor. In the base formulation, it outputs

$k$2

for one target-centered window at a time. The training objective is mean-squared error over the target parameters: $k$3 The paper emphasizes that target-wise windowing sidesteps the output-order ambiguity that would arise in a set-valued joint detector-regressor, because each network invocation is associated with a specific IFFT peak window rather than an unordered scene tensor (Naoumi et al., 2 Sep 2025).

The training data are synthetic and generated from the system model. For each example, $k$4 targets are randomly positioned, CSI is synthesized from the bistatic OFDM channel model, and noise is added at varying SNR. The exact distributions of $k$5, $k$6, $k$7, and $k$8 are not explicitly specified in the supplied text. The optimization hyperparameters are given: Adam optimizer, learning rate $k$9, batch size $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$0, and $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$1 epochs. The paper compares SNR-specific training with wide-range training and reports that training over $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$2 to $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$3 yields better robustness and generalization than training separate models at fixed SNR (Naoumi et al., 2 Sep 2025).

5. Inference pipeline, Doppler extension, and computational profile

The online inference pipeline has four main steps. First, the passive radar estimates CSI from received and known transmitted OFDM symbols: $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$4 Second, it applies the IDFT to the transpose of the stacked CSI matrix: $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$5 Third, it identifies a dominant delay row for each target and extracts the local window $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$6. Fourth, it feeds each complex-valued window into the CVNN to estimate the target parameters (Naoumi et al., 2 Sep 2025).

The paper states that the Doppler extension stacks multiple sub-frame CSI estimates into a 3D input tensor and replaces 2D complex convolutions with 3D complex convolutions. In that extension, the output per target becomes

$$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$7

The paper does not provide exact tensor dimensions for this extension, but it explicitly states that $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$8 sub-frames are stacked and a temporal dimension is added to the delay and spatial dimensions (Naoumi et al., 2 Sep 2025).

IFFT-C2VNN is positioned as a low-complexity alternative to MLE and to grid-based parametric baselines because it avoids exhaustive AoA/AoD/ToA searches, repeated likelihood evaluations, and large steering-vector dictionaries. The paper provides a layerwise complexity model rather than an exact architecture-specific FLOP count. For convolutional layers $$\mathbf{y}_{n,k} = \mathbf{H}_{n,k}\mathbf{s}_{n,k}+\mathbf{w}_{n,k}, \qquad \mathbf{w}_{n,k}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}),$$9, forward-pass complexity is

$$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$0

where the factor $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$1 accounts for real and imaginary operations in complex convolutions. Training complexity is

$$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$2

This distinction is central to the method’s role: the expensive learning stage is offline, whereas online inference consists mainly of CSI estimation, an IFFT across subcarriers, local window extraction, and a small CVNN forward pass (Naoumi et al., 2 Sep 2025).

The reported system parameters in the experiments are $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$3, $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$4, carrier frequency $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$5, $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$6, $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$7 subcarriers, $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$8 OFDM symbols per sub-frame, $$\mathcal{O}\!\left( G_\tau^M G_\theta^M G_\phi^M G_\alpha^{2M} \cdot (N_{\mathrm r}N_{\mathrm t}M^2N_{\mathrm P}^4 + N_{\mathrm r}N_{\mathrm P}^3K_{\mathrm P}) \right).$$9 sub-frames for the Doppler extension, $N_{\mathrm t}$0, $N_{\mathrm t}$1, and temporal resolution

$N_{\mathrm t}$2

Monte Carlo testing spans SNR from $N_{\mathrm t}$3 to $N_{\mathrm t}$4 (Naoumi et al., 2 Sep 2025).

6. Empirical behavior, comparison to PARAMING, and limitations

The paper reports that IFFT-C2VNN is effective and robust across a range of SNRs, but its performance profile differs from that of PARAMING, the model-based alternative introduced in the same work. For ToA estimation, the reported metric is normalized RMSE relative to the temporal resolution $N_{\mathrm t}$5. Bartlett, MUSIC, and DML behave similarly in the single-target setup and saturate at high SNR because of fixed ToA grid resolution $N_{\mathrm t}$6. IFFT-C2VNN outperforms those grid-based baselines across low and high SNR, and the $N_{\mathrm t}$7 variant performs better than the broader-input variant because it focuses on peak regions and reduces noise. PARAMING is best at high SNR and is closest to the CRB, while IFFT-C2VNN is robust but not as asymptotically precise (Naoumi et al., 2 Sep 2025).

For AoA and AoD, the reported metric is MSE versus SNR. Bartlett, MUSIC, and DML again show nearly identical grid-limited behavior; RootMUSIC-2D is competitive for angles and enters its waterfall region earlier at lower SNR; PARAMING is closest to the CRB; and IFFT-C2VNN remains robust across SNR but stagnates at very high SNR. The same section reports that all methods, including both proposed methods, remain above the CRB, and that achieving normalized ToA RMSE $N_{\mathrm t}$8 requires roughly $N_{\mathrm t}$9 dB more SNR than the CRB (Naoumi et al., 2 Sep 2025).

For the Doppler extension, the high-SNR speed-estimation MAE is reported as about $N_{\mathrm r}$0 for IFFT-C2VNN and about $N_{\mathrm r}$1 for PARAMING. This indicates that the extension is functional, but that the model-based estimator remains more accurate in the high-SNR regime (Naoumi et al., 2 Sep 2025).

The computational comparison favors IFFT-C2VNN. The paper concludes that MLE requires roughly $N_{\mathrm r}$2 times more operations than IFFT-C2VNN and roughly $N_{\mathrm r}$3 times more operations than PARAMING, while PARAMING is generally more complex than IFFT-C2VNN and the gap widens as $N_{\mathrm r}$4 and $N_{\mathrm r}$5 increase. Average execution times over 300 Monte Carlo trials on an HPC cluster are reported as $N_{\mathrm r}$6 s for IFFT-C2VNN, $N_{\mathrm r}$7 s for PARAMING, $N_{\mathrm r}$8 s for Bartlett, $N_{\mathrm r}$9 s for MUSIC, $N_{\mathrm P}$0 s for RootMUSIC-2D, and $N_{\mathrm P}$1 s for DML. Within the reported comparison set, IFFT-C2VNN is therefore the fastest method (Naoumi et al., 2 Sep 2025).

Several assumptions delimit the method’s scope. The paper assumes near-perfect synchronization,

$N_{\mathrm P}$2

known ULA geometry, and implicitly calibrated arrays. It also assumes resolvable scatterers in the delay domain: each selected IFFT peak is associated with one scatterer. If multiple paths collapse into one peak, the paper states that a preliminary classification step would be needed to determine the number of paths within that peak, but no such method is specified. Performance depends on the match between synthetic training data and deployment conditions, and the paper does not demonstrate cross-configuration generalization across changes in $N_{\mathrm P}$3, $N_{\mathrm P}$4, $N_{\mathrm P}$5, array geometry, subcarrier spacing, or carrier frequency (Naoumi et al., 2 Sep 2025).

In the broader IFFT literature, this combination of transform-based localization and low-complexity downstream estimation has parallels, though not direct architectural identity. A 2025 RIS-assisted localization paper uses a deterministic 2D-IFFT in the RIS spatial domain to obtain coarse angular localization before refinement, but contains no complex-valued neural network and is therefore only partially relevant as an IFFT baseline rather than as a direct antecedent of IFFT-C2VNN (Xia et al., 5 Apr 2025). This suggests that IFFT-C2VNN’s distinctive contribution is not merely the use of an IFFT, but the specific pairing of delay-domain IFFT localization with complex-valued learned regression for bistatic CSI-based super-resolution sensing (Naoumi et al., 2 Sep 2025).