---
title: 'FAOR: Fast OTFS Radar Algorithm'
url: https://www.emergentmind.com/topics/fast-algorithm-otfs-radar-faor
type: topic
---

# FAOR: Fast OTFS Radar Algorithm

Fast Algorithm OTFS Radar (FAOR) designates OTFS-based radar sensing procedures that estimate target range and velocity from structured peaks in the delay-Doppler (DD) domain. In the 2019 OTFS radar formulation, FAOR is an efficient OTFS-based matched filter algorithm for target range and velocity estimation [1901.09300]. In later OTFS-ISAC work, the same acronym denotes a low-complexity method that computes a two-dimensional cyclic correlation between a known transmitted DD-domain grid and a reordered received DD-domain grid, then forms a range-Doppler map by inverse FFT [2509.25846]. Both usages rely on the OTFS premise that moving reflectors are naturally represented as sparse DD shifts of a localized probing waveform, so sensing can be cast as delay-Doppler support recovery rather than dense time-frequency processing [2302.08696].

## 1. Delay-Doppler basis of FAOR

OTFS processes signals in the DD domain, where a doubly spread channel is modeled by a sparse spreading function
$$
h(\tau,\nu)=\sum_{i=1}^4 h_i\,\delta(\tau-\tau_i)\delta(\nu-\nu_i),
$$
with complex gains $h_i$, delays $\tau_i$, and Doppler shifts $\nu_i$. The corresponding noiseless received signal is
$$
y(t)=\int\!\!\int h(\tau,\nu)\,x(t-\tau)e^{j2\pi \nu(t-\tau)}\,d\tau\,d\nu
=\sum_{i=1}^4 h_i\,x(t-\tau_i)e^{j2\pi \nu_i(t-\tau_i)}.
$$
For radar, this is the classical point-target model: the scene is a sparse set of reflectors, and the sensing objective is to estimate $\{(\tau_i,\nu_i),h_i\}$ directly in the DD plane [2302.08696].

The same foundation describes the OTFS waveform as a quasi-periodic pulse in the DD domain, defined through the Zak transform. Its salient sensing property is localization over a small rectangle of size roughly $1/B$ along delay and $1/T$ along Doppler. When such a DD pulse interacts with a reflector at $(\tau_i,\nu_i)$, the response is shifted to a distinct DD location. In the illustrative derivation, transmitting at $(\tau_a,\nu_a)$ yields received replicas centered at
$$
(\tau_a+\tau_i,\nu_a+\nu_i),\qquad i=1,2,3,4.
$$
This is the core OTFS radar intuition: delay and Doppler separation occur simultaneously, so a sparse scene forms a sparse DD image rather than a rapidly time-varying time-frequency response [2302.08696].

A central regularity condition is the crystallization condition,
$$
\tau_p>(\max_i\tau_i-\min_i\tau_i),\qquad \nu_p>(\max_i\nu_i-\min_i\nu_i),
$$
with reciprocal periods $\nu_p=1/\tau_p$, equivalently
$$
(\max_i\tau_i-\min_i\tau_i)(\max_i\nu_i-\min_i\nu_i)<1.
$$
When this condition holds, DD replicas do not overlap, quasi-periodic aliasing is avoided, the channel response to one DD pulse predicts the response to another, and received power becomes invariant to transmit location in DD. FAOR exploits precisely this sparse, predictable, non-aliasing DD structure [2302.08696].

## 2. Discrete OTFS radar model and physical parameter mapping

In the discrete OTFS radar model, resources are sampled on a time-frequency grid
$$
\Lambda=\{(nT,m\Delta f),\; n=0,\ldots,N-1,\; m=0,\ldots,M-1\},\qquad \Delta f=\frac{1}{T},
$$
and on a DD grid
$$
\Gamma=\left\{\left(\frac{k}{NT},\frac{l}{M\Delta f}\right),\; k=0,\ldots,N-1,\; l=0,\ldots,M-1\right\}.
$$
An OTFS frame contains $NM$ symbols placed on the DD grid, mapped to the time-frequency domain by the ISFFT, and transmitted through pulse shaping and a Heisenberg transform. The continuous-time signal is described as
$$
s(t)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1} X[n,m]\, g_{\rm tx}(t-nT)\, e^{j2\pi m\Delta f (t-nT)}.
$$
The radar scene is modeled as $P$ targets with range $R_i$, velocity $V_i$, delay $\tau_i$, and Doppler frequency $\nu_i$, related by
$$
\frac{\tau_i}{2}=\frac{R_i}{c},\qquad \frac{\nu_i}{2}=f_c\frac{V_i}{c},
$$
or equivalently
$$
R_i=\frac{c\tau_i}{2},\qquad V_i=\frac{c\nu_i}{2f_c}.
$$
The received waveform is
$$
r(t)=\int\!\!\int h(\tau,\nu)\, s(t-\tau)\, e^{j2\pi \nu (t-\tau)}\, d\tau\, d\nu,
$$
and, after receive processing and SFFT, yields DD-domain samples used for sensing [1901.09300].

A later FAOR formulation writes the approximate received DD-domain signal as
$$
Y(m,n)\approx\sum_{p=1}^{P}h_p e^{j2\pi\left(\frac{m-l_p}{M}\right)\frac{k_p}{N}a_p(m,n)}
X(\langle m-l_p\rangle_M,\langle n-k_p\rangle_N)+\eta(m,n),
$$
where $l_p=\tau_p/T_s$, $k_p=Nf_p/\Delta f$, and $T_s=1/(M\Delta f)$. This notation makes the sensing structure explicit: each target induces a shifted DD-domain replica indexed by normalized delay and Doppler [2509.25846].

Once a DD peak is located, the discrete indices map directly to physical parameters. In the 2019 OTFS radar formulation,
$$
\tau_l=\frac{l}{M\Delta f},\qquad \nu_k=\frac{(k)_N}{NT},
$$
hence
$$
R=\frac{c}{2}\cdot \frac{l}{M\Delta f},\qquad
V=\frac{c}{2f_c}\cdot \frac{(k)_N}{NT}.
$$
FAOR therefore produces a two-dimensional DD image whose dominant peaks are interpreted as range-velocity estimates rather than as intermediate communication-domain quantities [1901.09300].

## 3. Core FAOR processing chain

The earliest FAOR formulation is an OTFS-domain matched-filter radar. It assumes the transmitted OTFS frame is known, observes the received DD symbols, and estimates the target-related DD channel coefficients. In vectorized form,
$$
\mathbf{y}=\widetilde{\mathbf{X}}\mathbf{h}+\mathbf{w},
$$
with $\mathbf{y}\in\mathbb{C}^{MN\times 1}$, $\mathbf{h}\in\mathbb{C}^{MN\times 1}$, and a structured matrix $\widetilde{\mathbf{X}}\in\mathbb{C}^{MN\times MN}$ formed from shifted versions of the transmit symbols. The matched filter computes
$$
\hat{\mathbf{h}}=\widetilde{\mathbf{X}}^{\rm H}\mathbf{y},
$$
and the gain matrix
$$
\mathbf{G}=\widetilde{\mathbf{X}}^{\rm H}\widetilde{\mathbf{X}}.
$$
For i.i.d. QPSK symbols with power $P_s$, the diagonal entries satisfy $G[i,i]=MNP_s$, while off-diagonal entries have mean zero and variance $\mathrm{var}[G[i,j]]=MNP_s^2$. After normalization, $(MNP_s)^{-1}\mathbf{G}$ becomes increasingly diagonal-dominant as $MN$ grows, so the matched filter concentrates target energy near the true DD cells. The stated matched-filter detection complexity is
$$
\mathcal{O}((MN)^2).
$$
In this form, FAOR is a structured correlation over all shifted versions of the known OTFS frame [1901.09300].

A later FAOR realization replaces explicit large-matrix correlation by FFT-based two-dimensional cyclic correlation. For monostatic radar, the correlation is
$$
Z(k,l)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}Y(m,n)X^*(\langle m-k\rangle_M,\langle n-l\rangle_N).
$$
Rather than evaluating this directly, the algorithm computes the 2D FFTs of $X(m,n)$ and $Y(m,n)$, forms
$$
\hat{Z}(k,l)=\hat{Y}(k,l)\hat{X}^*(k,l),
$$
and then applies a 2D IFFT to obtain the range-Doppler map. The stated complexity is
$$
O(MN{\rm log}_2(MN)),
$$
which is lower than the $\mathcal{O}((MN)^2)$ complexity reported for the earlier matched-filter form [2509.25846].

A practical prototype-oriented formulation expresses the same idea as an FFT-domain correlation-like product
$$
D(a,b)=Y(a,b)X^*(a,b),
$$
followed by zero-delay self-interference cancellation,
$$
\tilde{D}(a,b)=D(a,b)-\frac{1}{|S_{data}|}\sum_{a\in S_{data}}D(a,b),
$$
and a 2D IFFT
$$
R(m,n)=\frac{1}{\sqrt{MN}}\sum_{a=0}^{M-1}\sum_{b=0}^{N-1}\tilde{D}(a,b)
\exp\!\left(j\frac{2\pi ma}{M}\right)\exp\!\left(j\frac{2\pi bn}{N}\right).
$$
Target indices are then obtained by peak picking,
$$
(l_1,k_1)=\arg\max_{m,n}|R(m,n)|.
$$
This preserves the essential FAOR logic—correlate in the DD-compatible domain, form a range-Doppler map, and read off peaks—while adapting it to leakage-dominated hardware operation [2510.00668].

| Formulation | Core operation | Stated complexity or deployment |
|---|---|---|
| Matched-filter FAOR | $\hat{\mathbf h}=\widetilde{\mathbf X}^{\rm H}\mathbf y$ | $\mathcal{O}((MN)^2)$ |
| FFT-correlation FAOR | 2D cyclic correlation, then 2D IFFT | $O(MN{\rm log}_2(MN))$ |
| Prototype FAOR chain | $D(a,b)$, self-interference cancellation, then 2D IFFT | JRC hardware implementation |

## 4. Radar capability, processing gain, and comparison with OFDM

The OTFS radar argument against OFDM rests on cyclic-prefix overhead and Doppler robustness. OTFS uses one cyclic prefix for the whole OTFS frame, whereas OFDM radar requires one cyclic prefix per OFDM symbol, that is, $N$ cyclic prefixes. The stated transmission-time saving is $(N-1)L$ symbols, where $L$ is the cyclic-prefix length. The practical benefit reported is shorter total transmission duration, which means longer unambiguous sensing range for a given frame structure and/or faster update rate for target tracking [1901.09300].

The Doppler comparison is sharper. The 2019 paper states that OFDM can exactly detect Doppler only up to about 10% of the subcarrier spacing $\Delta f$, while OTFS can detect Doppler frequencies up to $\Delta f$ without interference. OFDM is degraded by inter-carrier interference because motion breaks subcarrier orthogonality; OTFS is described as inter-carrier interference-free in the DD representation and more suited for high-speed trains, vehicles, drones, flying cars, and supersonic platforms [1901.09300].

The numerical study in that paper uses $f_c=24$ GHz, $N_c=256$, $B=10$ MHz, $\Delta f=39.063$ kHz, $N_s=64$, range resolution $\Delta R=15$ m, velocity resolution $\Delta V=3.8125$ m/s, unambiguous range $R_{\max}=3840$ m, unambiguous velocity $V_{\max}=\pm 122$ m/s, and SNR $10$ dB. For a single target at $R=975$ m and $V=80$ m/s, the reported OFDM result detects range correctly but incurs a velocity error of about $19$ m/s, with lower PSLR and degradation from high inter-carrier interference because the Doppler is about 65% of $\Delta f$. The OTFS radar result detects both range and velocity correctly, with higher PSLR and more stable estimates [1901.09300].

The same study reports that, as target speed increases, OFDM velocity RMSE grows significantly, while OTFS RMSE stays essentially zero across the tested velocities. The OFDM error exceeds 25% at around $\pm 90$ m/s, whereas OTFS remains accurate. For OTFS radar, PSLR and image SNR are reported to be largely independent of target velocity; at low SNR, image SNR behaves close to $\sigma^2/(MNP_s)$, and at high SNR saturation occurs around $1/(MN)$ because of residual off-diagonal terms in the gain matrix. Within the scope of that experiment, FAOR is therefore presented not merely as a low-complexity implementation, but as a processing chain whose DD-domain correlation peaks remain reliable under larger Doppler shifts than OFDM radar [1901.09300].

## 5. FAOR in ISAC, bistatic sensing, and hardware prototypes

In OTFS-based ISAC, FAOR is extended from monostatic sensing with a fully known frame to scenarios in which only pilot symbols are known at the receiver. For monostatic radar, the whole transmitted DD-domain frame can be used in the cyclic correlation. For bistatic radar, the receiver uses only pilot symbols by defining
$$
X_1(m,n)=s_p(m,n)
$$
on the pilot grid and $X_1(m,n)=0$ elsewhere, then applying the same FFT-correlation-IFFT chain to obtain a bistatic range-Doppler map. The paper states that FAOR can be applied not only to monostatic radar but also to bistatic radar with a much lower computational complexity compared to state-of-the-art radar sensing technology [2509.25846].

The 2025 ISAC paper couples FAOR with pilot-aided channel estimation. After sensing detects the delays and Dopplers, the DD-domain observation model is rewritten as
$$
\boldsymbol{y}=\boldsymbol{X}_{c}\boldsymbol{h},
$$
with least-squares estimate
$$
\boldsymbol{h}=(\boldsymbol{X}_{c}^{\rm H}\boldsymbol{X}_{c})^{-1}\boldsymbol{X}_{c}^{\rm H}\boldsymbol{y}.
$$
The same paper proposes a multifunction pilot design that serves both bistatic radar sensing and channel estimation without any guard symbol added. In the reported comparison, a $55\times55$ pilot grid with $3025$ pilot symbols randomly assigned binary values $+1$ or $-1$ is contrasted with a conventional single-pilot design using one pilot symbol of value $55$ at the center of the $55\times55$ region. Both use the same total pilot power, but the reported PAPR is $44$ dB for the single-pilot design and $13$ dB for the proposed design, with pilot overhead
$$
\frac{55\times55}{4096\times100}=0.74\%<1\%.
$$
The reported sensing simulations use $f_c=60$ GHz, $M=4096$, $N=100$, subcarrier spacing $30$ kHz, bandwidth $122.88$ MHz, and 4-QAM. In monostatic sensing, target 1’s range and speed errors are reported to become minimized and stable from about $-36$ dB SNR, and target 2 from about $-26$ dB SNR; in bistatic sensing, target 2 remains stable only from around $-20$ dB because less information is available [2509.25846].

A further development is an OTFS-based joint radar and communication prototype in which FAOR is embedded in an SDR/mmWave platform. The OTFS waveform is generated in MATLAB, transmitted and received by an NI USRP2944, and upconverted and downconverted to mmWave with EVAL-ADMV1013 and EVAL-ADMV1014 boards. Reported parameters are 100 MHz bandwidth, 29 GHz carrier frequency, 30 kHz subcarrier spacing, 14 symbols per slot, 1 slot per subframe, 20 slots per frame, and 1 transmit and 1 receive antenna. Because the transmit and radar receive antennas are close together, self-interference can obscure the reflected signals; the paper explicitly states that the fast algorithm alone is not enough in the real prototype, and that the cancellation step is what enables multi-target separation and practical radar detection. In the reported static-human experiment at about 1.2 m, the target is not visible in the range-Doppler map without cancellation and is recovered with the proposed cancellation. The same platform also reports breathing and heartbeat extraction from repeated radar snapshots, walking-human speed detection with resolution up to 0.1 m/s, and human/non-human discrimination using both signal processing and a 1D CNN [2510.00668].

## 6. Related methods, limits, and nomenclature

FAOR should not be conflated with every low-complexity OTFS sensing method. A 2023 OTFS sensing paper explicitly states that it does not use the term FAOR, although its method resembles FAOR-like ideas because it operates in the DD domain, uses a correlation-based matched-filter style computation rather than a full exhaustive maximum-likelihood search, and employs closed-form fractional refinement rather than iterative optimization [2306.11223].

That paper also highlights a substantive limitation of integer-bin DD peak detection: real targets are generally off-grid. Its method first computes a 2D DD-domain correlation
$$
V[k,l]=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}Y_{\mathrm{DD}}^{*}[n,m]\,
X_{\mathrm{DD}}([n-k]_N,[m-l]_M),
$$
then applies a generalized likelihood ratio test with a CA-CFAR threshold to detect targets when the number of targets is unknown, and finally estimates fractional Doppler and delay from leakage ratios of the two strongest neighboring bins. The reported degradation scenarios are also informative for FAOR-style sensing more generally: targets with fractional parts near $0.5$ can have weaker main-lobe magnitudes under rectangular windows, and if two targets share the same delay or Doppler, the assumptions behind the fractional-estimation proof are violated [2306.11223].

A different branch of fast OTFS radar research appears in mono-static MIMO-OTFS joint communications and radar, where complexity is reduced by decoupling angle-of-arrival estimation from DD estimation. One 2024 method estimates AoA first by Root-MUSIC and then estimates range and velocity by a reduced 2D delay-Doppler grid maximum-likelihood search with an LMMSE front-end. Its reported complexity,
$$
\mathcal{O}\big(N_a^3 + (NM)^3 + (N_aNM)^2\big),
$$
is lower than the cited state-of-the-art joint approach with complexity
$$
\mathcal{O}\big((N_aNM)^3\big).
$$
This is related to FAOR in the broad sense of fast OTFS radar parameter estimation, but it is methodologically distinct: the complexity reduction comes from spatial-DD decoupling rather than from the cyclic-correlation range-Doppler-map construction characteristic of FAOR [2402.14612].

A plausible implication is that FAOR is best understood as a family of OTFS radar algorithms centered on DD-domain correlation, peak extraction, and structured complexity reduction, rather than as a single immutable procedure. Across its matched-filter, FFT-correlation, ISAC, bistatic, and prototype realizations, its defining feature remains the same: OTFS converts target echoes into sparse, resolvable DD-domain peaks from which delay and Doppler—and therefore range and speed—are read directly.

Source: https://www.emergentmind.com/topics/fast-algorithm-otfs-radar-faor