Zak-OTFS for Identification of Linear Time-Varying Systems

TL;DR

This paper addresses identifying linear time-varying (LTV) radar channels by comparing traditional chirp waveforms with a Zak-OTFS waveform. It shows that Zak-OTFS yields a self-ambiguity function localized on a DD-domain lattice rather than a line, enabling higher-resolution localization of multiple targets under the crystallization condition where the delay spread is less than $\tau_p$ and the Doppler spread is less than $\nu_p$, with lower complexity $O(BT\log(BT))$ for cross-ambiguity computations. The authors derive the DD-domain formulation, analyze ambiguity functions, and demonstrate through simulations that Zak-OTFS outperforms chirps in range and velocity estimation, while avoiding ghost targets. The work provides a practical framework for high-resolution LTV system identification using lattice-based ambiguity in the delay-Doppler domain, with implications for radar waveform design and fast DD-domain processing.

Abstract

Linear time-varying (LTV) systems model radar scenes where each reflector/target applies a delay, Doppler shift and complex amplitude scaling to a transmitted waveform. The receiver processes the received signal using the transmitted signal as a reference. The self-ambiguity function of the transmitted signal captures the cross-correlation of delay and Doppler shifts of the transmitted waveform. It acts as a blur that limits resolution, at the receiver, of the delay and Doppler shifts of targets in close proximity. This paper considers resolution of multiple targets and compares performance of traditional chirp waveforms with the Zak-OTFS waveform. The self-ambiguity function of a chirp is a line in the delay-Doppler domain, whereas the self-ambiguity function of the Zak-OTFS waveform is a lattice. The advantage of lattices over lines is better localization, and we show lattices provide superior noise-free estimation of the range and velocity of multiple targets. When the delay spread of the radar scene is less than the delay period of the Zak-OTFS modulation, and the Doppler spread is less than the Doppler period, we describe how to localize targets by calculating cross-ambiguities in the delay-Doppler domain. We show that the signal processing complexity of our approach is superior to the traditional approach of computing cross-ambiguities in the continuous time / frequency domain.

Figures (9)

• Figure 1: Heat map illustrating the intensity of the cross-ambiguity function for two filtered chirp signals, an up-chirp with slope $2B/T = 4 \times 10^8$ Hz$^2$ and a down-chirp with slope $-2B/T$. The bandwidth is $B = 4$ MHz, the time duration $T=20$ ms, and the target location is $\tau_1 = 1.25 \mu s$ and $\nu_1 = -350$ Hz. The up-chirp is transmitted in the first $T/2$ seconds, the down-chirp is transmitted in the subsequent $T/2$ seconds, and the radar receiver computes the cross-ambiguity separately for the two intervals. The Doppler spread of each band is roughly $2/T = 100$ Hz, and the delay spread is roughly $1/B = 0.25 \mu s$. The intensity pattern is typical of Gaussian pulse shaping.
• Figure 2: Heat map illustrating the intensity of the cross-ambiguity function for two filtered chirp signals, an up-chirp with slope $2B/T = 4 \times 10^8$ and a down-chirp with slope $2B/T$. The bandwidth is $4$ MHz, the time duration $T=20$ ms, and there are four targets located at $(1 \mu s, -400 \hbox{\scriptsize{Hz}})$, $(3.125 \mu s, 175 \hbox{\scriptsize{Hz}})$, $(2.375 \mu s, -550 \hbox{\scriptsize{Hz}})$ and $(4.25 \mu s, -600 \hbox{\scriptsize{Hz}})$. The up-chirp is transmitted in the first $T/2$ seconds, the down-chirp is transmitted in the subsequent $T/2$ seconds, and the radar receiver computes the cross-ambiguity separately for the two intervals. There are four bands/lines with positive slope and four bands with negative slope. The intersection points are marked by circles, where blue circles indicate true targets and red circles indicate non-targets (ghosts). The intensity pattern is characteristic of Gaussian pulse shaping.
• Figure 3: Heat map illustrating the intensity of the self-ambiguity function $A_{p,p}(\tau, \nu)$ for the Zak-OTFS probe. The bandwidth is $B = 4$ MHz, the time duration $T = 20$ ms, the delay period $\tau_p = 100 \mu s$ and the Doppler period $\nu_p = 10$ KHz. The intensity pattern is characteristic of Gaussian pulse shaping. As an example, if the delay and Doppler shifts of the targets satisfy $\tau_{min} = 0$, $\tau_{max}=90 \mu s$, and $-\nu_{min} = \nu_{max} = 4$ KHz, then the crystallization condition is satisfied, and target locations can be estimated within the rectangle with the red border.
• Figure 4: Heat map illustrating the intensity of the cross-ambiguity function $A_{y, p}(\tau, \nu)$ for the Zak-OTFS probe. The bandwidth is $B = 4$ MHz, the time-duration $T = 20$ ms, the delay period $\tau_p = 100 \mu s$, and the Doppler period $\nu_p = 10$ KHz. There are three targets located at $(0.6 \mu s, -220)$ Hz, $(0.95 \mu s, -220)$ Hz and $(0.6 \mu s, -290)$ Hz. Noise-free radar processing. Three peaks corresponding to the three targets are separable/resolvable since any two targets are well separated along either delay or Doppler axis.
• Figure 5: Heat map illustrating the intensity of the cross-ambiguity function $A_{y, p}(\tau, \nu)$ for the Zak-OTFS probe. The bandwidth is $B = 4$ MHz, the time-duration $T = 20$ ms, the delay period $\tau_p = 100 \mu s$, and the Doppler period $\nu_p = 10$ KHz. There are four targets located at $(0.125 \mu s, 50)$ Hz, $(0.25 \mu s, 75)$ Hz, $(0.375 \mu s, -25)$ Hz and $(0.5 \mu s, -100)$ Hz. Noise-free radar processing. It is not possible to resolve the four peaks corresponding to the four targets since the targets are not well separated in both delay and Doppler.

Theorems & Definitions (2)

• Lemma 1
• Theorem 1