Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters

Arpan Das, Fathima Jesbin, Ananthanarayanan Chockalingam

TL;DR

This work addresses the complexity of evaluating Zak-OTFS in high-mobility channels by deriving discrete, closed-form end-to-end I/O expressions and noise covariances for delay-Doppler domain filtering. It covers sinc and Gaussian transmitter filters and three receiver configurations: identical, matched, and channel-matched filtering, providing exact or accurate closed-form results (with an approximate form for sinc identical filtering). The proposed expressions significantly reduce computation time for performance analysis and enable direct insight into how TX/RX filtering choices shape the effective channel taps $h_{\mathrm{eff}}[k,l]$ and noise statistics, as demonstrated by BER results on Veh-A channel models, where channel-matched filtering yields the best performance. The findings facilitate faster Zak-OTFS simulations and contribute practical tools for receiver design and evaluation, while outlining future work on additional filters and multiuser scenarios.

Abstract

The transceiver operations in the delay-Doppler (DD) domain in Zak-OTFS modulation, including DD domain filtering at the transmitter and receiver, involve twisted convolution operation. The twisted convolution operations give rise to multiple integrals in the end-to-end DD domain input-output (I/O) relation. The I/O relation plays a crucial role in performance evaluation and algorithm development for transceiver implementation. In this paper, we derive discrete DD domain closed-form expressions for the I/O relation and noise covariance in Zak-OTFS. We derive these expressions for sinc and Gaussian pulse shaping DD filters at the transmitter (Tx). On the receiver (Rx) side, three types of DD filters are considered, viz., $(i)$ Rx filter identical to Tx filter (referred to as `identical filtering'), $(ii)$ Rx filter matched to the Tx filter (referred to as `matched filtering'), and $(iii)$ Rx filter matched to both Tx filter and channel response (referred to as `channel matched filtering'). For all the above cases, except for the case of sinc identical filtering, we derive exact I/O relation and noise covariance expressions in closed-form. For the sinc identical filtering case, we derive approximate closed-form expressions which are shown to be accurate. Using the derived closed-form expressions, we evaluate the bit error performance of Zak-OTFS for different Tx/Rx filter configurations. Our results using Vehicular-A (Veh-A) channel model with fractional DDs show that, while matched filtering achieves slightly better or almost same performance as identical filtering, channel matched filtering achieves the best performance among the three.

Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters

TL;DR

This work addresses the complexity of evaluating Zak-OTFS in high-mobility channels by deriving discrete, closed-form end-to-end I/O expressions and noise covariances for delay-Doppler domain filtering. It covers sinc and Gaussian transmitter filters and three receiver configurations: identical, matched, and channel-matched filtering, providing exact or accurate closed-form results (with an approximate form for sinc identical filtering). The proposed expressions significantly reduce computation time for performance analysis and enable direct insight into how TX/RX filtering choices shape the effective channel taps and noise statistics, as demonstrated by BER results on Veh-A channel models, where channel-matched filtering yields the best performance. The findings facilitate faster Zak-OTFS simulations and contribute practical tools for receiver design and evaluation, while outlining future work on additional filters and multiuser scenarios.

Abstract

The transceiver operations in the delay-Doppler (DD) domain in Zak-OTFS modulation, including DD domain filtering at the transmitter and receiver, involve twisted convolution operation. The twisted convolution operations give rise to multiple integrals in the end-to-end DD domain input-output (I/O) relation. The I/O relation plays a crucial role in performance evaluation and algorithm development for transceiver implementation. In this paper, we derive discrete DD domain closed-form expressions for the I/O relation and noise covariance in Zak-OTFS. We derive these expressions for sinc and Gaussian pulse shaping DD filters at the transmitter (Tx). On the receiver (Rx) side, three types of DD filters are considered, viz., Rx filter identical to Tx filter (referred to as `identical filtering'), Rx filter matched to the Tx filter (referred to as `matched filtering'), and Rx filter matched to both Tx filter and channel response (referred to as `channel matched filtering'). For all the above cases, except for the case of sinc identical filtering, we derive exact I/O relation and noise covariance expressions in closed-form. For the sinc identical filtering case, we derive approximate closed-form expressions which are shown to be accurate. Using the derived closed-form expressions, we evaluate the bit error performance of Zak-OTFS for different Tx/Rx filter configurations. Our results using Vehicular-A (Veh-A) channel model with fractional DDs show that, while matched filtering achieves slightly better or almost same performance as identical filtering, channel matched filtering achieves the best performance among the three.

Paper Structure

This paper contains 25 sections, 3 theorems, 101 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Zak-OTFS System Model
  3. Identical Filtering
  4. Matched Filtering
  5. Channel Matched Filtering
  6. Closed-form Expressions for Identical Filtering
  7. $h_{\mathrm{eff}}[k,l]$ for sinc filter
  8. Noise covariance for sinc filter
  9. $h_{\mathrm{eff}}[k,l]$ for Gaussian filter
  10. Noise covariance for Gaussian filter
  11. Closed-form Expressions for Matched Filtering
  12. $h_{\mathrm{eff}}[k,l]$ for sinc filter
  13. Noise covariance for sinc filter
  14. Expressions for Gaussian filter
  15. Closed-form Expressions for Channel Matched Filtering
  16. ...and 10 more sections

Key Result

Theorem 1

For identical filtering with sinc filter, the effective channel $h_{\mathrm{eff}}[k,l]$ in approximate closed-from is given by where the function $P_{i,k}(f)$ is given by

Figures (9)

  • Figure 1: Transceiver signal processing in Zak-OTFS.
  • Figure 2: BER performance of Zak-OTFS for identical filtering with sinc filter using approximate closed-form and exact I/O relation expressions.
  • Figure 3: BER performance of Zak-OTFS for identical filtering, matched filtering, and channel matched filtering with sinc filter.
  • Figure 4: BER performance of Zak-OTFS for identical filtering, matched filtering, and channel matched filtering with Gaussian filter.
  • Figure 5: SNR performance of identical, matched, and channel matched filtering as a function of normalized delay ($\tau/\tau_\text{p}$) at $\nu=\nu_1$ for (a) sinc filter and (b) for Gaussian filter.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3