Orthogonal Time Frequency Space (OTFS) Modulation for Wireless Communications
School of Electrical Engineering and Telecommunications Shuangyang Li Doctor
of Philosophy August 2022 Jinhong Yuan

## Abbreviations

AMP | Approximate message passing
---|---
AoA | Angle-of-arrival
AoD | Angle-of-departure
AWGN | Additive white Gaussian noise
BCH | Bose-Chaudhuri-Hocquenghem
BER | Bit error rate
BPSK | Binary phase shift keying
BS | Base station
B5G | Beyond the fifth-generation
CDMA | Code-division multiple access
CP | Cyclic prefix
CSI | Channel state information
CSIT | Channel state information at transmitter
DD | Delay-Doppler
DDA | Delay-Doppler-angular
DFT | Discrete Fourier transform
DoF | Degrees of freedom
DTFT | Discrete-time Fourier transform
DZT | Discrete Zak transform
E-UTRA | Evolved Universal Terrestrial Radio Access
EXIT | Extrinsic information transfer
FER | Frame error rate
FFT | Fast Fourier transform
GFDM | Generalized frequency-division multiplexing
IDZT | Inverse Discrete Zak transform
IFFT | Inverse fast Fourier transform
ICI | Intercarrier interference
ISAC | Integrated sensing and communications
ISFFT | Inverse symplectic finite Fourier transform
ISI | Intersymbol interference
IZT | Inverse Zak transform
KL | Kullback-Leibler
LoS | Line-of-sight
LTV | Linear time-varying
MAP | Maximum _a posteriori_
MCA | Mobile communications on board aircraft
MIMO | Multiple-input multiple-output
MISO | Multiple-input single-output
ML | Maximum-likelihood
MLSE | Maximum-likelihood sequence estimation
mmWave | Millimeter wave
MRC | Maximum ratio combining
MSE | Mean square error
LEO | Low-earth-orbit
LDPC | Low-density parity-check
L-MMSE | Linear minimum mean square error
LTE | Long term evolution
OAMP | Orthogonal approximate message passing
ODDM | Orthogonal delay-doppler division multiplexing
OFDM | Orthogonal frequency-division multiplexing
OSTF | Orthogonal short time Fourier
PAPR | Peak-to-average power ratio
PEP | Pairwise-error probability
PIC | Parallel interference cancellation
PSD | Power spectral density
QPSK | quadrature phase shift keying
RCS | Radar cross section
RMS | Root-mean-square
SFFT | Symplectic finite Fourier transform
SIC | Successive interference cancellation
SINR | Signal-to-interference-plus-noise ratio
SNR | Signal-to-noise ratio
SOMP | Structured orthogonal matching pursuit
SPA | Sum-product algorithm
TD | Time-delay
TDMA | Time division multiple access
TDS | Time-delay-spatial
TF | Time-frequency
UAMP | AMP with a unitary transformation
UAV | Unmanned aerial vehicle
UE | User equipment
US | Uncorrelated scattering
VOFDM | Vector OFDM
WSS | Wide-sense stationary
WSSUS | Wide-sense stationary uncorrelated scattering
ZF | Zero forcing
ZT | Zak transform
2D | Two-dimensional
3D | Three-dimensional
3GPP | 3rd Generation Partnership Project
5G | 5-th generation
16-QAM | 16-quadrature amplitude modulation

## List of Notations

Scalars, vectors, and matrices are written in italic, boldface lower-case and
upper-case letters, respectively, e.g., $x$, $\mathbf{x}$, and $\mathbf{X}$.

$x^{\mathrm{*}}$ | Conjugate of $x$
---|---
$\mathbf{X}^{\mathrm{T}}$ | Transpose of $\mathbf{X}$
$\mathbf{X}^{\mathrm{H}}$ | Hermitian transpose of $\mathbf{X}$
$\left\lceil\cdot\right\rceil$ | Round up operation
$\textrm{vec}\left({\bf{X}}\right)$ | Vectorization of $\mathbf{X}$
${\left[{\cdot}\right]_{x}}$ | Modulo operation with $x$
$\delta\left(x\right)$ and $\delta\left[x\right]$ | Dirichlet function with a continuous/discerte variable $x$
$\otimes$ | Kronecker product operator
$\Pr\\{\cdot\\}$ | Probability of an event
${\cal X}\\{\cdot\\}$ | Returns the cardinality of a set
$\propto$ | Both sides of the equation are multiplicatively connected to a constant
$\mathbf{I}_{N}$ | $N$ dimension identity matrix
${\mathbb{E}}\left\\{\cdot\right\\}$ | Statistical expectation
${\mathbb{Z}}$ | Set of all integers
$\mathcal{CN}(\mu,\sigma^{2})$ | Circularly symmetric complex Gaussian random variable:
| the real and imaginary parts are i.i.d. $\mathcal{N}(\mu/2,\sigma^{2}/2)$
$\ln(\cdot)$ | Natural logarithm
$\max\left\\{\cdot\right\\}$ | Return the maximum element of its input
$\min\left\\{\cdot\right\\}$ | Return the minimum element of its input
${\cal O}\left(\cdot\right)$ | Asymptotical Description for the order of computational complexity

## Abstract

The orthogonal time frequency space (OTFS) modulation is a recently proposed
multi-carrier transmission scheme, which innovatively multiplexes the
information symbols in the delay-Doppler (DD) domain instead of the
conventional time-frequency (TF) domain. The DD domain symbol multiplexing
gives rise to a direct interaction between the DD domain information symbols
and DD domain channel responses, which are usually quasi-static, compact,
separable, and potentially sparse. Therefore, OTFS modulation enjoys appealing
advantages over the conventional orthogonal frequency-division multiplexing
(OFDM) modulation for wireless communications.

In this thesis, we investigate the related subjects of OTFS modulation for
wireless communications, specifically focusing on its signal detection,
performance analysis, and applications. In specific, we first offer a
literature review on the OTFS modulation in Chapter 1. Furthermore, a summary
of wireless channels is given in Chapter 2. In particular, we discuss the
characteristics of wireless channels in different domains and compare their
properties. In Chapter 3, we present a detailed derivation of the OTFS concept
based on the theory of Zak transform (ZT) and discrete Zak transform (DZT). We
unveil the connections between OTFS modulation and DZT, where the DD domain
interpretations of key components for modulation, such as pulse shaping, and
matched-filtering, are highlighted.

The main research contributions of this thesis appear in Chapter 4 to Chapter
7. In Chapter 4, we introduce the hybrid maximum a posteriori (MAP) and
parallel interference cancellation (PIC) detection. This detection approach
exploits the power discrepancy among different resolvable paths and can obtain
near-optimal error performance with a reduced complexity.

In Chapter 5, we propose the cross domain iterative detection for OTFS
modulation by leveraging the unitary transformations among different domains.
After presenting the key concepts of the cross domain iterative detection, we
study its performance via state evolution. We show that the cross domain
iterative detection can approach the optimal error performance theoretically.
Our numerical results agree with our theoretical analysis and demonstrate a
significant performance improvement compared to conventional OTFS detection
methods.

In Chapter 6, we investigate the error performance for coded OTFS systems
based on the pairwise-error probability (PEP) analysis. We show that there
exists a fundamental trade-off between the coding gain and the diversity gain
for coded OTFS systems. According to this trade-off, we further provide some
rule-of-thumb guidelines for code design in OTFS systems.

In Chapter 7, we study the potential of OTFS modulation in integrated sensing
and communication (ISAC) transmissions. We propose the concept of spatial-
spreading to facilitate the ISAC design, which is able to discretize the
angular domain, resulting in simple and insightful input-output relationships
for both radar sensing and communication. Based on spatial-spreading, we
verify the effectiveness of OTFS modulation in ISAC transmissions and
demonstrate the performance improvements in comparison to the OFDM
counterpart.

A summary of this thesis is presented in Chapter 8, where we also discuss some
potential research directions on OTFS modulation. The concept of OTFS
modulation and the elegant theory of DD domain communication may have opened a
new gate for the development of wireless communications, which is worthy to be
further explored.

## Publications and Presentations

### List of Publications

#### Journal Papers:

  1. J-1.

S. Li, W. Yuan, Z. Wei, and J. Yuan, “Cross Domain Iterative Detection for
Orthogonal Time Frequency Space Modulation,” to appear in _IEEE Transactions
on Wireless Communications ,_ vol. 21, no. 4, pp. 2227-2242, Apr. 2022.

  2. J-2.

S. Li, W. Yuan, Z. Wei, J. Yuan, B. Bai, D. W. K. Ng, and Y. Xie, “Hybrid MAP
and PIC detection for OTFS modulation,” _IEEE Transactions on Vehicular
Technology_ , vol. 70, no. 7, pp. 7193-7198, Jul. 2021.

  3. J-3.

S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, “Performance
Analysis of Coded OTFS Systems over High-Mobility Channels,” _IEEE
Transactions on Wireless Communications_ , vol. 20, no. 9, pp. 6033-6048, Apr.
2021.

  4. J-4.

S. Li, W. Yuan, C. Liu, Z. Wei, J. Yuan, B. Bai, and D. W. K. Ng, “A Novel
ISAC Transmission Framework based on Spatially-Spread Orthogonal Time
Frequency Space Modulation,” to appear in _IEEE Journal on Selected Areas in
Communications ,_ 2022.

#### Conference Papers:

  1. C-1.

S. Li, W. Yuan, Z. Wei, R. He, B. Ai, B. Bai, and J. Yuan, “A Tutorial to
Orthogonal Time Frequency Space Modulation for Future Wireless
Communications,” in _IEEE International Communication Conference in China
(ICCC) Workshop ,_ Xiamen, China, 2021. pp. 439-443.

  2. C-2.

S. Li, W. Yuan, J. Yuan, and Giuseppe Caire, “On the Potential of Spatially-
Spread Orthogonal Time Frequency Space Modulation for ISAC Transmissions,” to
appear in _IEEE International Conference on Acoustics, Speech and Signal
Processing (ICASSP) ,_ Singapore, 2022.

  3. C-3.

S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, “On the Performance
of Coded OTFS Modulation over High-Mobility Channels," in _IEEE International
Communication Conference (ICC) Workshop ,_ Montreal, Canada, 2021. pp. 1-6.

The following papers, which are not included in the thesis, have also been
published by the author.

#### Journal Papers:

  1. J-1.

S. Li, Z. Wei, W. Yuan, J. Yuan, B. Bai, D. W. K. Ng, and L. Hanzo, “Faster-
than-Nyquist Asynchronous NOMA Outperforms Synchronous NOMA,” to appear in
_IEEE Journal on Selected Areas in Communications ,_ 2022.

  2. J-2.

S. Li, W. Yuan, J. Yuan, B. Bai, D. W. K. Ng, and L. Hanzo, “Time-Domain vs.
Frequency-Domain Equalization for FTN Signaling,” _IEEE Transactions on
Vehicular Technology_ , vol. 69, no. 8, pp. 9174-9179, Aug. 2020.

  3. J-3.

S. Li, J. Yuan, B. Bai, and N. Benvenuto, “Code based Channel Shortening for
Faster-than-Nyquist Signaling: Reduced-Complexity Detection and Code Design,”
_IEEE Transactions on Communications_ , vol. 68, no. 7, pp. 3996-4011, Jul.
2020.

  4. J-4.

S. Li, B. Bai, J. Zhou, P. Chen, and Z. Yu, “Reduced-Complexity Equalization
for Faster-than-Nyquist Signaling: New methods based on Ungerboeck Observation
Model,” _IEEE Transactions on Communications ,_ vol. 66, no. 3, pp. 1190-1204,
Mar. 2018.

  5. J-5.

S. Li, B. Bai, J. Zhou, Q. He, and Q. Li, “Superposition Coded Modulation
based Faster-than-Nyquist Signaling,” _Wireless Communications and Mobile
Computing ,_ vol. 2018, Article ID 4181626, 10 pages, 2018.

  6. J-6.

Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “Off-grid Channel Estimation
with Sparse Bayesian Learning for OTFS Systems,” to appear in _IEEE
Transactions on Wireless Communications._ 2022.

  7. J-7.

W. Yuan, S. Li, Z. Wei, J. Yuan, and D. W. K. Ng, “Data-Aided Channel
Estimation for OTFS Systems with A Superimposed Pilot and Data Transmission
Scheme,” _IEEE Wireless Communications Letters ,_ vol. 10, no. 9, pp.
1954-1958, Sept. 2021.

  8. J-8.

B. Liu, S. Li, Y. Xie, and J. Yuan, “A Novel Sum-Product Detection Algorithm
for Faster-than-Nyquist Signaling: A Deep Learning Approach,” _IEEE
Transactions on Communications ,_ vol. 69, no. 9, pp. 5975-5987, Sept. 2021.

  9. J-9.

R. Chong, S. Li, J. Yuan, and D. W. K. Ng, “Achievable Rate Upper-Bounds of
Uplink Multiuser OTFS Transmissions,” to appear in _IEEE Wireless
Communications Letters_ , vol. 11, no. 4, pp. 791-795, Apr. 2022.

  10. J-10.

W. Yuan, S. Li, L. Xiang, and D. W. K. Ng, “Distributed Estimation Framework
for Beyond 5G Intelligent Vehicular Networks,” _IEEE Open Journal of Vehicular
Technology._ vol. 1, pp. 190-214, Nov. 2020.

  11. J-11.

P. Kang, K. Cai, X. He, S. Li, and J. Yuan, “Generalized Mutual Information-
Maximizing Quantized Decoding of LDPC Codes with Layered Scheduling,” to
appear in _IEEE Transactions on Vehicular Technology ,_ 2022.

  12. J-12.

M. Liu, S. Li, C. Zhang, B. Wang, B. Bai, “Coded Orthogonal Time Frequency
Space Modulation,” _ZTE Communications_ , vol. 19, no. 4, pp. 54-62, Dec.
2021.

  13. J-13.

Y. K. Enku, B. Bai, F. Wan, C. U. Guyo, I. N. Tiba, C. Zhang, and S. Li, “Two-
dimensional Convolutional Neural Network based Signal Detection for OTFS
Systems,” _IEEE Wireless Communications Letters_ , vol. 10, no. 11, pp.
2514-2518, Nov. 2021.

  14. J-14.

W. Yuan, Z. Wei, S. Li, J. Yuan, and D. W. K. Ng, “Integrated Radar Sensing
and Communication-assisted Orthogonal Time Frequency Space Transmission for
Vehicular Networks,” _IEEE Journal of Selected Topics in Signal Processing ,_
vol. 15, no. 6, pp. 1515-1528, Nov. 2021.

  15. J-15.

Z. Wei, W. Yuan, S. Li, J. Yuan, B. Ganesh, R. Hadani, and L. Hanzo,
“Orthogonal Time-Frequency Space Modulation: A Promising Next Generation
Waveform,” _IEEE Wireless Communications ,_ vol. 28, no. 4, pp. 136-144,
August 2021.

  16. J-16.

Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “Transmitter and Receiver
Window Designs for Orthogonal Time Frequency Space Modulation,” _IEEE
Transactions on Communications_ , vol. 69, no. 4, pp. 2207-2223, Apr. 2021.

  17. J-17.

D. Shi, W. Yuan, S. Li, N. Wu, and D. W. K. Ng, “Cycle-Slip Detection and
Correction for Carrier Phase Synchronization in Coded Systems,” _IEEE
Communications Letters._ vol. 25, no. 1, pp. 113-116, Jan. 2021.

  18. J-18.

W. Yuan, C. Liu, F. Liu, S. Li, and D. W. K. Ng, “Learning-based Predictive
Beamforming for UAV Communications with Jittering,” _IEEE Wireless
Communications Letters._ vol. 9, no. 11, pp. 1970-1974, Nov. 2020.

  19. J-19.

J. Zhang, B. Bai, S. Li, M. Zhu, and H. Li, “Tail-Biting Globally-Coupled LDPC
Codes,” _IEEE Transactions on Communications ,_ vol. 67, no. 12, pp.
8206-8219, Dec. 2019.

  20. J-20.

J. Zhang, B. Bai, M. Zhu, S. Li, and H. Li, “Protograph-based Globally-Coupled
LDPC Codes over the Gaussian Channel with Burst Erasures,” _IEEE Access ,_
vol. 7, pp. 153853-153868, 2019.

#### Conference Papers:

  1. C-1.

S. Li, Z. Wei, W. Yuan, J. Yuan, B. Bai, and D. W. K. Ng, “On the Achievable
Rates of Uplink NOMA with Asynchronized Transmission,” in _IEEE Wireless
Communications and Networking Conference (WCNC) ,_ Nanjing, China, 2021, pp.
1-7.

  2. C-2.

S. Li, J. Yuan, and B. Bai, “Code based Channel Shortening for Faster-than-
Nyquist Signaling,” in _IEEE International Communication Conference (ICC).
_Dublin, Ireland, 2020, pp. 1-6.

  3. C-3.

S. Li, J. Zhang, B. Bai, X. Ma, and J. Yuan, “Self-Superposition Transmission:
A Novel Method for Enhancing Performance of Convolutional Codes,” in _IEEE
International Symposium on Turbo Codes _&_ Iterative Information Processing
(ISTC)_, Hong Kong, 2018, pp. 1-5.

  4. C-4.

H. Wen, W. Yuan, and S. Li, “Downlink OTFS Non-Orthogonal Multiple Access
Receiver Design based on Cross-Domain Detection" to appear in _IEEE
International Conference on Communications (ICC) Workshop ,_ Seoul, 2022.

  5. C-5.

R. Chong, S. Li, W. Yuan, and J. Yuan, “Outage Analysis for OTFS-based Single
User and Multi-User Transmissions" to appear in _IEEE International Conference
on Communications (ICC) Workshop ,_ Seoul, 2022.

  6. C-6.

Y. K. Enku, B. Bai, S. Li, M. Liu, and I. N. Tiba, “Deep-Learning based Signal
Detection for MIMO-OTFS Systems" to appear in _IEEE International Conference
on Communications (ICC) Workshop ,_ Seoul, 2022.

  7. C-7.

Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “Performance Analysis and
Window Design for Channel Estimation of OTFS Modulation,” in _IEEE
International Conference on Communications (ICC) ,_ Montreal, Canada, 2021,
pp. 1-5.

  8. C-8.

Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “A New Off-grid Channel
Estimation Method with Sparse Bayesian Learning for OTFS Systems,” in _IEEE
Global Communication Conference (GlobeCom)_ , Madrid, Spain, 2021, pp. 1-7.

  9. C-9.

W. Yuan, S. Li, Z. Wei, J. Yuan, and D. W. K. Ng, “Bypassing Channel
Estimation for OTFS Transmission: An Integrated Sensing and Communication
Solution,” in _IEEE Wireless Communications and Networking Conference (WCNC)
Workshop ,_ Nanjing, China, 2021, pp. 1-6.

  10. C-10.

B. Liu, S. Li, Y. Xie, and J. Yuan, “Deep Learning Assisted Sum-Product
Detection Algorithm for Faster-than-Nyquist Signaling,” in _IEEE Information
Theory Workshop (ITW)_ , Visby, Sweden, 2019, pp. 1-5.

  11. C-11.

M. Liu, S. Li, Q. Li, and B. Bai, “Faster-than-Nyquist Signaling based
Adaptive Modulation and Coding,” in _IEEE International Conference on Wireless
Communications and Signal Processing (WCSP)_ , Hangzhou, 2018, pp. 1-5.

### List of Presentations

#### Poster presentations:

  * •

“Self-superposition transmission: A novel method for enhancing performance of
convolutional codes," in _Australia Information Theory School (AusITS)_ ,
Sydney, 2019.

###### Contents

  1. Abbreviations
  2. List of Notations
  3. Abstract
  4. Publications and Presentations
  5. 1 Introduction
    1. 1.1 Main Contributions and Organization of the Thesis
    2. 1.2 Literature Review
      1. 1.2.1 OTFS Concept and Implementation
      2. 1.2.2 Channel Estimation
      3. 1.2.3 Signal Detection
      4. 1.2.4 Performance Analysis
      5. 1.2.5 Applications
    3. 1.3 Organizations of the Thesis
  6. 2 Wireless Channel Revisit
    1. 2.1 Wireless Channel Backgrounds
      1. 2.1.1 Path Loss and Fading
      2. 2.1.2 Flat Fading Channels
      3. 2.1.3 Time Dispersive Channels
      4. 2.1.4 Frequency Dispersive Channels
      5. 2.1.5 Doubly Selective Channels
    2. 2.2 Mathematical Representations of Wireless Channels: Deterministic Description
      1. 2.2.1 Time-Delay Representation of LTV Channels
      2. 2.2.2 Time-Frequency Representation of LTV Channels
      3. 2.2.3 Delay-Doppler Representation of LTV Channels
    3. 2.3 Mathematical Representations of Wireless Channels: Stochastic Description
      1. 2.3.1 WSSUS Channels: Descriptions and Properties
      2. 2.3.2 WSSUS Channels: Delay and Doppler Profiles and Important Parameters
      3. 2.3.3 Underspread WSSUS Channels
      4. 2.3.4 Extensions to Non-WSSUS Channels
    4. 2.4 Sampling and Reconstruction
      1. 2.4.1 Sampling in the TF Domain
      2. 2.4.2 Sampling in the DD Domain
    5. 2.5 Summary of the Chapter
  7. 3 OTFS Concepts and Principles
    1. 3.1 A Summary on the Zak Transform and Discrete Zak Transform
      1. 3.1.1 Gabor Expansion and Gabor Coefficients
      2. 3.1.2 Definitions of Zak Transform and Discrete Zak Transform
      3. 3.1.3 Properties and Relations of the Zak Transform
      4. 3.1.4 Properties and Relations of the Discrete Zak Transform
    2. 3.2 Concepts of OTFS Modulation: DZT-based OTFS
      1. 3.2.1 Reduced-CP Frame Structure
      2. 3.2.2 Descriptions of DZT-based OTFS: Transmitter Side
      3. 3.2.3 Descriptions of DZT-based OTFS: Receiver Side
      4. 3.2.4 Descriptions of DZT-based OTFS: Equivalent Model, Pulse Shaping, and DD Domain Channel Sampling
    3. 3.3 Matrix Form Representation of the DZT-based OTFS
    4. 3.4 Parameter Selection
    5. 3.5 Summary of the Chapter
  8. 4 Hybrid MAP and PIC Detection for OTFS Modulation
    1. 4.1 Near-Optimal Symbol-Wise MAP Detection
    2. 4.2 Hybrid MAP and PIC Detection
      1. 4.2.1 Error Performance Analysis
      2. 4.2.2 Complexity Analysis and Parameter Selection
    3. 4.3 Numerical Results
    4. 4.4 Summary of the Chapter
  9. 5 Cross Domain Iterative Detection for OTFS Modulation
    1. 5.1 Cross Domain Iterative Detection for OTFS Modulation
      1. 5.1.1 Module A: L-MMSE Estimator for TD Domain OTFS Signal
      2. 5.1.2 Cross Domain Message Passing: from TD Domain to DD Domain
      3. 5.1.3 Module B: Symbol-by-symbol Detection for DD Domain Symbols
      4. 5.1.4 Cross Domain Message Passing: from DD Domain to TD Domain
    2. 5.2 Performance Analysis
      1. 5.2.1 MSE Performance Analysis via State Evolution
      2. 5.2.2 Analysis of Effective DD Domain SNR
      3. 5.2.3 Analysis of Detection Complexity
    3. 5.3 Numerical Results
    4. 5.4 Summary of the Chapter
    5. 5.5 Appendices of the Chapter
      1. 5.5.1 Justification of Problem Formulation (5.8)
      2. 5.5.2 Proof of Proposition 5-1
      3. 5.5.3 Proof of Lemma 5-2
      4. 5.5.4 Proof of Theorem 5-2
  10. 6 Performance Analysis of Coded OTFS Systems over High-Mobility Channels
    1. 6.1 Error Performance Analysis
      1. 6.1.1 Error Performance Analysis for Coded OTFS systems
      2. 6.1.2 Error Performance Analysis for Large Values of $P$
      3. 6.1.3 Code Design Issues
    2. 6.2 Numerical Results
    3. 6.3 Summary of the Chapter
    4. 6.4 Appendices of the Chapter
      1. 6.4.1 Proof of Lemma 6-3
      2. 6.4.2 Proof of Theorem 6-1
      3. 6.4.3 Proof of Theorem 6-2
      4. 6.4.4 Proof of Theorem 6-3
  11. 7 OTFS for Integrated Sensing and Communications
    1. 7.1 System Model
      1. 7.1.1 Transmitter Structure
      2. 7.1.2 Communication Model
      3. 7.1.3 Radar Model
      4. 7.1.4 Model Simplifications with Spatial Spreading and De-spreading
    2. 7.2 Radar Sensing Designs based on SS-OTFS Modulation
      1. 7.2.1 Beam Tracking
      2. 7.2.2 AoA Estimation
      3. 7.2.3 Power Allocation for Radar Sensing
    3. 7.3 Sensing-Assisted Communication Design
      1. 7.3.1 Pair-wise Error Probability Analysis
      2. 7.3.2 Precoding Design
      3. 7.3.3 Power Allocation for Communication
    4. 7.4 Numerical Results
      1. 7.4.1 Beam Tracking Performance
      2. 7.4.2 Precoding Performance
      3. 7.4.3 Radar Sensing and Communication Performances with Power Allocations
    5. 7.5 Summary of the Chapter
    6. 7.6 Appendices of the Chapter
      1. 7.6.1 Proof of Lemma 7-1
      2. 7.6.2 Proof of Corollary 7-1
      3. 7.6.3 Proof of Theorem 7-1
      4. 7.6.4 Proof of Lemma 7-3
      5. 7.6.5 Proof of Proposition 7-1
  12. 8 Conclusions and Future Work
    1. 8.1 Future Work
      1. 8.1.1 Exploiting DD Domain Channel Characteristics
      2. 8.1.2 Fundamental Limits for DD Domain Communications
      3. 8.1.3 DD Domain Communication Designs

###### List of Figures

  1. 2.1 TD domain representation of the LTV channel, where the channel parameters are given in Table 2.2. For a better clarification, only the real part of the complex channel response is shown.
  2. 2.2 TD domain representation of the LTV channel, where the channel parameters are given in Table 2.2. For a better clarification, only the imaginary part of the complex channel response is shown.
  3. 2.3 TF domain representation of the LTV channel, where the channel parameters are given in Table 2.2.
  4. 2.4 DD domain representation of the LTV channel, where the channel parameters are given in Table 2.2.
  5. 2.5 Channel scattering function with exponential power-delay profile and Jakes power-Doppler profile, where the channel parameters are given in Example 2-4.
  6. 3.1 An illustration for Gabor expansions with critical sampling, oversampling, and undersampling.
  7. 3.2 A graphical interpretation of the ZT.
  8. 3.3 A diagram of the CP insertion for DZT-based OTFS transmissions.
  9. 3.4 A transmitter diagram of the DZT-based OTFS modulation according to [6, 25].
  10. 3.5 The diagram of the transformation between the DD domain and the TF domain.
  11. 3.6 An illustration of DD domain symbols and their corresponding TF domain OTFS symbols.
  12. 3.7 An illustration of DD domain symbols and their corresponding TD domain OTFS signals.
  13. 3.8 A receiver diagram of the DZT-based OTFS modulation according to [6, 25].
  14. 3.9 A diagram of the equivalent model for DZT-based OTFS transmission.
  15. 3.10 TD domain effective channel matrix for OTFS transmissions with integer delay and Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  16. 3.11 TF domain effective channel matrix for OTFS transmissions with integer delay and Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  17. 3.12 DD domain effective channel matrix for OTFS transmissions with integer delay and Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  18. 3.13 TD domain effective channel matrix for OTFS transmissions with fractional delay and integer Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  19. 3.14 TF domain effective channel matrix for OTFS transmissions with fractional delay and integer Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  20. 3.15 DD domain effective channel matrix for OTFS transmissions with fractional delay and integer Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  21. 3.16 TD domain effective channel matrix for OTFS transmissions with integer delay and fractional Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  22. 3.17 TF domain effective channel matrix for OTFS transmissions with integer delay and fractional Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  23. 3.18 DD domain effective channel matrix for OTFS transmissions with integer delay and fractional Doppler indices, where the corresponding channel parameters are given in Table 3.1.
  24. 4.1 The probabilistic graphical model of the near-optimal _symbol-wise_ MAP algorithm.
  25. 4.2 The probabilistic graphical model of the Hybrid-MAP-PIC algorithm.
  26. 4.3 BER performance of the proposed Hybrid-MAP-PIC algorithm for uncoded OTFS systems with $P=4$, compared with the message passing algorithm in [25], the variational Bayes algorithm in [41], and the near-optimal _symbol-wise_ MAP algorithm.
  27. 4.4 FER performance of the proposed Hybrid-MAP-PIC algorithm for convolutionally coded OTFS systems with $P=4$, comparing with the message passing algorithm in [25] and the near-optimal _symbol-wise_ MAP algorithm.
  28. 5.1 The diagram of the proposed OTFS detector.
  29. 5.2 TD domain MSE performance for OTFS modulation with $P=4$, where the frame contains $2048$ DD domain QPSK or 16-QAM symbols. In specific, the SNR for the QPSK case is ${E_{s}}/{N_{0}}=12$ dB, while that for the 16-QAM case is ${E_{s}}/{N_{0}}=17$ dB.
  30. 5.3 Effective DD domain SNRs for OTFS modulation with different number of iterations, where the frame contains $2048$ DD domain QPSK or 16-QAM symbols. In specific, the SNR for the QPSK case is ${E_{s}}/{N_{0}}=14$ dB, while that for the 16-QAM case is ${E_{s}}/{N_{0}}=17$ dB. The considered wireless channel contains $P=4$ paths with different delay indices for each path.
  31. 5.4 Effective DD domain SNRs for OTFS modulation with different number of iterations, where the frame contains $2048$ DD domain QPSK or 16-QAM symbols. In specific, the SNR for the QPSK case is ${E_{s}}/{N_{0}}=14$ dB, while that for the 16-QAM case is ${E_{s}}/{N_{0}}=17$ dB. The considered wireless channel contains $P=4$ paths and the first two paths share the same delay index.
  32. 5.5 BER performance for OTFS modulation with $P=4$, where the error performance of the proposed algorithm is compared with that of DD domain MRC and MMSE detections, as well as the SPA detection with integer Doppler shifts.
  33. 5.6 BER performance for OTFS modulation with fractional Doppler shifts, where $P=10$.
  34. 5.7 SNR gains vs. number of iterations, where both QPSK and 16-QAM signals are considered with $P=4$ and $P=6$.
  35. 6.1 Average coding gain for error sequences with $P=2$ and ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ in terms of various delay and Doppler indices, comparing with the coding gain bound.
  36. 6.2 Average coding gain for error sequences with ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ in terms of different numbers of independent resolvable paths, comparing with the coding gain bounds (in dashed lines), where the maximum delay and Doppler indices are set to be $l_{\rm{max}}=2$ and $k_{\rm{max}}=4$, respectively.
  37. 6.3 Average coding gain for error sequences with ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ in terms of different numbers of independent resolvable paths, comparing with the coding gain bounds (in dashed lines), where the maximum delay and Doppler indices are set to be $l_{\rm{max}}=2$ and $k_{\rm{max}}=2$, respectively.
  38. 6.4 FER performance for OTFS modulation with different codes for $P=2$, where the relative user equipment (UE) speed is $250$ km/h.
  39. 6.5 FER performance for OTFS modulation with different codes for $P=8$, where the relative UE speed is $250$ km/h.
  40. 6.6 FER performance of code D for OTFS modulation with $P=3$ and $P=8$, comparing with that of uncoded OTFS systems, where the relative UE speed is $250$ km/h.
  41. 6.7 FER performance of code A for QPSK/16-QAM mapped OTFS systems with different number of paths and the relative UE speed of $250$ km/h.
  42. 6.8 FER performance of code A for OTFS modulation with $P=3$ and $P=8$, compared to an OFDM system, where the relative UE speed is $250$ km/h.
  43. 6.9 FER performance of code A for OTFS modulation and OFDM modulation with $P=4$, where the values of the relative UE speed are $150$ km/h, $250$ km/h, and $300$ km/h, respectively.
  44. 6.10 FER performance of 5G LDPC code and 3GPP Turbo code for OTFS modulation with $P=4$ and a relative UE speed of $250$ km/h, where the codeword length is $512$ bits.
  45. 7.1 A brief diagram of the considered ISAC transmission.
  46. 7.2 The block diagram of the SS-OTFS-enabled ISAC transmitter, where the two arrows represent different beam directions.
  47. 7.3 The domain transformations the SS-OTFS enabled ISAC transmitter.
  48. 7.4 Absolute values of ${\bf{\tilde{H}}}_{i,p}^{\rm{A}}$ without spreading and de-spreading.
  49. 7.5 Absolute values of ${\bf{\tilde{H}}}_{i,p}^{\rm{A}}$ with spreading and de-spreading.
  50. 7.6 Average determinant values of ${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$ with (in dash-dotted lines) and without precoding (in solid lines for random delay indices and in dotted lines for different delay indices), comparing with the determinant bound in (7.54) (in dashed lines), where $M=8,N=8,l_{\rm max}=2$, and $k_{\rm max}=2$, respectively.
  51. 7.7 AoA estimation performance with various beam widths for $K=4$ UEs, where each UE has $P=2$ paths.
  52. 7.8 FER comparisons between the proposed ISAC transmissions with and without precoding, where (7, 5) convolutionally coded BPSK signals are considered with $P=8$.
  53. 7.9 FER performance comparisons between SS-OTFS, BF-OTFS, and BF-OFDM for the first UE with $P=4$.
  54. 7.10 Miss-detection probability for radar sensing with radar/communication power allocation for different number of UEs and paths.
  55. 7.11 Comparisons between the FER performances for a specific UE with communication power allocation and with radar power allocation.

###### List of Tables

  1. Abbreviations
  2. List of Notations
  3. 2.1 A Summary of Possible Conditions for Different Types of Wireless Channels
  4. 2.2 Channel Parameters for Example 2-1, 2-2, and 2-3
  5. 2.3 WSSUS Channel Characteristics in Different Domains
  6. 3.1 Channel Parameters for Effective Channels
  7. 3.2 Parameter Selection for OTFS Modulation
  8. 4.1 Detection Complexity Analysis
  9. 5.1 Notations of Parameters for Cross Domain Iterative Detection
  10. 6.1 Code Parameters
  11. 6.2 Code Parameters for Fig. 6.10
  12. 7.1 List of Notations for ISAC Transmitter
  13. 7.2 List of Notations for ISAC Channels
  14. 7.3 List of Notations for ISAC Receivers

##  Chapter 1 Introduction

Beyond the fifth-generation (B5G) wireless communication systems are required
to accommodate various emerging applications in high-mobility environments,
such as mobile communications on board aircraft (MCA), low-earth-orbit (LEO)
satellites, high-speed trains, and unmanned aerial vehicles (UAVs) [1, 2, 3,
4]. On top of that, ultra-higher data rate requirements push mobile providers
to utilize higher frequency bands, such as millimeter wave (mmWave) bands,
where a huge chunk of the spectrum is available. However, wireless
communications in high-mobility scenarios at high carrier frequencies suffer
from severe Doppler spread effect, caused by the relative motion between
transceivers. Consequently, the currently deployed orthogonal frequency
division multiplexing (OFDM) modulation may not be able to support efficient
and reliable communications in such scenarios [5]. Therefore, as a potential
solution to supporting heterogeneous requirements of B5G wireless systems, the
recently proposed orthogonal time frequency space (OTFS) modulation has
attracted substantial attention [6].

In high-mobility scenarios, wireless channels are usually doubly-dispersive in
the time-frequency (TF) domain [7, 8]. In specific, the time dispersion is
caused by the effect of multi-path, while the frequency dispersion is caused
by the Doppler shifts. Conventionally, OFDM modulation can efficiently
mitigate the intersymbol interference (ISI) induced by the time dispersion by
introducing a cyclic prefix (CP). However, the success of OFDM modulation
relies deeply on maintaining the orthogonality among all the sub-carriers.
Note that perfect orthogonality is highly impractical at the receiver,
especially in high-mobility environments, due to the exceedingly large
frequency dispersion, and consequently, the performance of conventional OFDM
systems is unsatisfactory in such scenarios [5]. On the other hand, by
invoking the two-dimensional (2D) symplectic finite Fourier transform (SFFT),
OTFS modulates the information symbols in the delay-Doppler (DD) domain, where
the channel parameters are relatively stable compared to those in the TF
domain [7]. More importantly, it can be shown that by modulating the
information symbols in the DD domain rather than the TF domain, each symbol
principally experiences the whole fluctuations of the TF channel over an OTFS
frame, and thus OTFS modulation offers the potential to exploiting the full
channel diversity, achieving a better error performance compared to that of
the conventional OFDM modulation in a high-mobility environment [6].

Although OTFS modulation has shown great potentials for future wireless
communications, there are still many theoretical and practical issues of OTFS
modulation that need to be solved. From a theoretical point of view, the
intrinsic connections between OTFS modulation and DD domain signal processing
have not been fully understood. OTFS modulation is built upon the elegant
mathematical theory of Zak transform (ZT) [6]. But the interpretation of OTFS
modulation and demodulation in the DD domain by using ZT has long been missing
in the literature. Furthermore, the theoretical error performance advantages
of OTFS systems over the conventional OFDM systems have not been thoroughly
studied yet, especially for coded cases. We note that there are several
previous works [9, 10, 11] on the error performance analysis of OTFS systems.
However, these works mainly considered the uncoded case and their analysis may
not be directly extended to the coded cases. Consequently, the error
performance of coded OTFS systems has not been fully studied. As commonly
recognized, channel coding is an efficient tool to combat fading and channel
impairments and thus is a key enabler for reliable communications between
users with high-mobility [12]. For OTFS modulation, the transformation from
the TF domain to the DD domain provides the potential of exploiting the full
TF diversity. In this case, a good channel code needs to couple the coded
symbols to the 2D OTFS modulation, in order to exploit the full diversity and
in the meantime maximize the coding gain. However, it is still unknown what is
the key coding parameter determining the coding gain for OTFS modulation.

From a practical point of view, OTFS transmission usually requires advanced
and complex detection methods. The rationale behind this observation is that
the DD domain channel usually has few non-zero responses as the result of the
separability of DD domain channels. Consequently, a sequence-wise equalization
is needed for OTFS to achieve a good error performance, whereas the OFDM
counterpart only requires a symbol-wise detection (single-tap equalization) as
the channel response in the TF domain can be characterized by a point-wise
multiplication. Furthermore, the combination between OTFS modulation and
emerging wireless technologies is also of great practical importance. For
example, radar sensing is expected to be an important service for future
wireless networks. Therefore, a joint design for integrated sensing and
communication (ISAC) transmission using OTFS waveforms is worth to be
investigated.

In this thesis, we will study the related aspects of OTFS modulation. It is
worthwhile to point out that OTFS modulation is a relatively new concept and
its development is still in infancy. There are some concepts and
interpretations of OTFS modulation that are still unclear in the literature.
However, the authors would like to explain the OTFS modulation in a systematic
way to the best of their capability.

###  1.1 Main Contributions and Organization of the Thesis

Throughout this thesis, we will discuss several related subjects of OTFS
modulation, and the main contributions of this thesis can be summarized as
follows.

  * •

We provide a literature review of the recent advances of OTFS modulation. The
review includes the related works in OTFS concepts, channel estimation, signal
detection, performance analysis, and applications.

  * •

We present the derivation on OTFS modulation by using the concepts of ZT and
discrete Zak transform (DZT). We show the intrinsic connections between DD
domain signal processing and conventional modulation components, such as pulse
shaping, and derive the corresponding input-output relationships for OTFS
modulation. Several examples are given to show the properties of the OTFS
waveform and a summary of the OTFS system model in the vector form is
presented with various channel conditions.

  * •

We study the hybrid maximum _a posteriori_ (MAP) and parallel interference
cancellation (PIC) (Hybrid-MAP-PIC algorithm) detection for OTFS modulation.
The Hybrid-MAP-PIC algorithm is motivated by the fact that different paths may
have highly diverse channel gains, such that it is possible to select the
paths with large gains for MAP detection, while performing PIC detection for
the remaining paths, without scarifying too much on the error performance. We
give detailed descriptions of the Hybrid-MAP-PIC algorithm and also provide
the error performance analysis and parameter selection. Our numerical results
agree with our analysis and show a marginal performance loss (less than 1 dB)
to the optimal detection. The related contents of the hybrid MAP and PIC
detection have been presented in

    * –

S. Li, W. Yuan, Z. Wei, J. Yuan, B. Bai, D. W. K. Ng, and Y. Xie, “Hybrid MAP
and PIC detection for OTFS modulation,” _IEEE Trans. Veh. Technol._ , vol. 70,
no. 7, pp. 7193-7198, Jul. 2021.

  * •

The cross domain iterative detection for OTFS modulation is investigated. The
core idea of the cross domain iterative detection is to perform the message
passing detection in different domains and pass the extrinsic information
iteratively via the unitary transformation. In particular, we apply two basic
estimation/detection algorithms in different domains and show that the cross
domain iterative detection enjoys promising error performance even in very
severe and complex fractional Doppler cases. State evolution is derived to
characterize the convergence performance of the algorithm. Our numerical
results agree with our theoretical analysis and demonstrate a significant
performance improvement compared to conventional OTFS detection methods. The
related contents of the cross domain iterative detection have been presented
in

    * –

S. Li, W. Yuan, Z. Wei, and J. Yuan, “Cross domain iterative detection for
orthogonal time frequency space modulation,” _IEEE Trans. Wireless Commun._ ,
vol. 21, no. 4, pp. 2227-2242, Apr. 2022.

  * •

We provide the analysis of the error performance of coded OTFS systems based
on the study of pairwise error probability (PEP). By leveraging advanced
bounding techniques, we show that the OTFS transmission has a fundamental
trade-off between the diversity gain and coding gain. In particular, we find
that the diversity gain of OTFS systems improves with the number of resolvable
paths, while the coding gain declines. Rule-of-thumb guidelines for code
design in OTFS systems are also given. Those analytical results are explicitly
verified by our numerical results. The related contents of the performance
analysis for coded OTFS systems have been presented in

    * –

S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, “Performance
analysis of coded OTFS systems over high-mobility channels," _IEEE Trans.
Wireless Commun._ , vol. 20, no. 9, pp. 6033-6048, Apr. 2021.

    * –

S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, “On the performance
of coded OTFS modulation over high-mobility channels," _IEEE Int. Commun.
Conf. Workshop_ , pp. 1-6, 2021.

  * •

We consider the application of OTFS modulation in ISAC transmissions. To
facilitate the system design, we extend the OTFS idea to the spatial domain by
introducing the concept of spatial-spreading. The key novelty of this concept
is the discretization of the angular domain, which results in simple and
insightful input-output relation for both radar sensing and communication
functionalities. We derive the system model based on the spatially-spread OTFS
(SS-OTFS) modulation for both radar sensing and communication and propose
simple and direct estimation, power allocation, and precoding schemes. Our
numerical results verify the effectiveness of the considered SS-OTFS framework
and demonstrate the performance improvements in comparison to the OFDM
counterpart. The related contents of the ISAC designs using OTFS waveforms
have been presented in

    * –

S. Li, W. Yuan, C. Liu, Z. Wei, J. Yuan, B. Bai, and D. W. K. Ng, “A novel
ISAC transmission framework based on spatially-spread orthogonal time
frequency space modulation,” to appear in _IEEE J. Sel. Areas Commun._ , 2022.

    * –

S. Li, W. Yuan, J. Yuan, and G. Caire, “On the potential of spatially-spread
orthogonal time frequency space modulation for ISAC transmissions,” to appear
in _Proc. - ICASSP IEEE Int. Conf. Acoust. Speech Signal Process._ , pp. 1-5,
2022.

###  1.2 Literature Review

The technology of OTFS modulation has been developed rapidly ever since the
pioneering paper of Hadani, _et. al._ [6], and several summaries about the
developments of OTFS modulation have also appeared in the literature [13, 14,
15]. In this section, we aim to provide a literature review of the recent
advances of OTFS modulation. In particular, we will summarize the related
works of OTFS modulation based on different subjects of the topic, including
OTFS concept and implementation, channel estimation, signal detection,
performance analysis, and applications.

####  1.2.1 OTFS Concept and Implementation

The concept of OTFS modulation was built upon the elegant mathematical theory
of ZT [16, 17], but the intrinsic connections between OTFS modulation and ZT
were not discussed in detail in the first paper of OTFS [6]. In [18], the
author provided an interesting interpretation of OTFS from the ZT point of
view, where a rigorous derivation of OTFS modulation from the first principles
is given. In particular, some important and interesting properties of OTFS
modulation have been discussed and explained, such as the twisted convolution,
and DD domain localization. A recent work [19] also gives a direct
implementation of OTFS modulation based on the DZT [20, 21], where the authors
have shown that the OTFS modulation can be simply implemented by a DZT
transformation. More importantly, this paper also considered the pulse shaping
issue of OTFS modulation from the DD domain interpretations, where insightful
conclusions are also obtained. A recent book chapter also explains the OTFS
modulation using ZT [22], where the authors have shown that bandlimiting
filters and finite-time windows can be used to formulate OTFS modulation with
DD domain Nyquist pulse shaping.

Although the OTFS modulation can be directly implemented in the DD domain via
ZT/DZT, the implementation of OTFS modulation via the TF domain has also been
widely considered in the literature [6, 23]. The idea of the TF domain
implementation is first to transform the DD domain information symbols into
the TF domain by using the inverse symplectic finite Fourier transform (ISFFT)
and then modulates the TF domain symbols based on conventional TF domain
multi-carrier modulators, such as OFDM modulator [13]. Furthermore, depending
on how to insert the cyclic prefix (CP), there are two types OTFS that
appeared in the literature, namely ZT/DZT-based OTFS [18, 19] and CP/OFDM-
based OTFS [24]. In specific, the ZT/DZT-based OTFS only requires one CP in
the time-delay (TD) domain and consequently, there will be interference in
both the TF domain and DD domain at the receiver side. Some derivations on the
ZT/DZT-based OTFS, including the input-output relationship among different
domains, could be found in [25] and [26]111Strictly speaking, the work of [25]
considered the zero-padding technique instead of inserting CP. However, it has
been shown that the resultant input-output relationship is similar to the
ZT/DZT-based OTFS [27]. . On the other hand, the CP/OFDM-based OTFS appends
the CP similar to the conventional CP insertion for OFDM modulation, e.g.,
each OFDM symbol contains one CP. Therefore, after the CP removal, the
resultant TF domain symbols will have no intersymbol interference (ISI) [24].
A concise system model for CP/OFDM-based OTFS has been demonstrated in [24].
It can be shown that these two implementations have distinct properties, and
we will provide more detailed discussions on DZT-based OTFS in Chapter 3.

The comparisons and connections between OTFS modulation and conventional
multi-carrier modulation schemes have also been discussed in the literature.
For example, the connection between OTFS modulation and the generalized
frequency-division multiplexing (GFDM) modulation was discussed in [28], where
the authors have shown that OTFS modulation could be implemented based on the
GFDM framework with a simple permutation. In particular, the authors have
shown that the permutation is the key ingredient for achieving the promising
performance over practical wireless channels. A recent DD domain communication
scheme called orthogonal delay-Doppler division multiplexing (ODDM) has been
studied [29, 30]. The most interesting feature is that the information symbols
can be modulated with respect to practical Nyquist pulses with full separation
without violating the uncertainty principle [29]. Furthermore, it has been
shown that ODDM can enjoy a better out-of-band emission performance than the
conventional OTFS modulation. The connections and comparisons between OTFS
modulation and other schemes, such as vector OFDM (VOFDM) modulation, and
orthogonal short time Fourier (OSTF) modulation, have also been discussed
before, and we refer the interested readers to [31] and [32] for more details.

####  1.2.2 Channel Estimation

The channel estimation is an important issue for OTFS systems. In comparison
to the conventional channel estimation in the TF domain, OTFS modulation
enables DD domain channel estimation approaches that could capture the DD
domain channel response with a reduced signaling overhead. Perhaps, the most
famous channel estimation scheme is the one proposed in [33]. In [33], an
embedded pilot scheme for OTFS channel estimation has been proposed, where a
sufficiently large guard interval is applied around the pilot to improve the
acquisition of delay and Doppler responses. As the DD domain interaction
between the transmitted signal and the channel response can be characterized
by the twisted convolution, such a scheme can result in direct channel
estimation by simply checking the received signal’s value around the DD grid
of the embedded pilot. This approach has soon been extended to the multiple-
input multiple-output (MIMO) case in [34], where several pilot symbols are
inserted for different transmit antennas with sufficient guard spacing in the
DD domain. Although [33] has offered a very simple and direct channel
estimation framework, it may require a large guard space to perform well,
especially when the DD domain channel sparsity does not hold, e.g., fractional
Doppler case. To overcome this issue, a superimposed pilot scheme has been
considered in [35], where there is no guard interval around the pilot, such
that the signaling overhead is minimized. Consequently, an iterative receiver
is adopted to recover the superposition between pilot and information symbols,
and the simulation results showed a similar error performance to that of [33]
at an expense of higher computation complexity. Meanwhile. several compressed
sensing-based approaches have been studied in the literature. For instance,
the authors in [36] proposed a three-dimensional (3D) structured orthogonal
matching pursuit (SOMP) algorithm to estimate the delay-Doppler-angle domain
channel via exploiting the 3D structured sparsity. A 3D Newtonized orthogonal
matching pursuit (NOMP) algorithm was proposed in [37], which exploits the
fractional component in the Doppler and angle domains via Newton’s method.
Furthermore, some recent works on channel estimation based on sparse Bayesian
learning approaches have also appeared in [38, 39, 40], which have
demonstrated better error performances compared to the orthogonal matching
pursuit-based approaches.

####  1.2.3 Signal Detection

The signal equalization for OTFS has also been a widely studied subject. Many
existing studies focused on the low-complexity detection for OTFS modulation.
In [25], the authors developed an iterative receiver based on the classic
message passing algorithm (MPA), where the interference from other information
symbols is treated as a Gaussian variable to reduce the detection complexity.
However, due to the short cycles on the probabilistic graphical model, MPA may
fail to converge and results in performance degradation. To solve this issue,
the authors of [41] proposed a convergence guaranteed receiver based on the
variational Bayes framework. The basic idea of this detector is to approximate
the corresponding _a posteriori_ distribution of the optimal detection by
exploiting the Kullback-Leibler (KL) divergence such that the message passing
algorithm can be implemented based on a simpler graphical model. A hybrid
detection scheme is proposed in [42], where both _maximum a posteriori_ (MAP)
detection and PIC detection are considered according to the channel
coefficients. Simulation results show that this hybrid detection can approach
the error performance of near-optimal symbol-wise MAP detection, especially
for coded OTFS systems, but only requires a reduced complexity. An OTFS
detection approach based on approximate message passing (AMP) with a unitary
transformation (UAMP) developed in [43], which not only enjoys a strong
resilience to the error propagation due to the short cycles on the
probabilistic graphical model but also enables an efficient implementation.
Two DD domain low-complexity linear equalizers for OTFS modulations were
studied in [44], where the authors proposed simplified implementation of
linear minimum mean squared error (L-MMSE) and zero-forcing (ZF) detections
with only a logarithmic complexity by exploiting the DD domain channel
properties. A Rake-receiver-based OTFS detection was proposed in [45], where
the zero-padding technique was also adopted to shape the corresponding channel
matrix to assist the application of DD domain maximum ratio combining. An
iterative MMSE detection zero-padded OTFS modulation was proposed in [46],
where the MMSE detection was applied in a window-by-window manner with the
help of successive interference cancellation (SIC).

It should be noticed that most of the existing works on the OTFS detection
take advantage of the DD domain channel sparsity to reduce the detection
complexity. However, when an OTFS frame duration is not sufficiently long, the
resultant DD domain effective channel can be dense due to the insufficient
resolution of the Doppler frequency, i.e., fractional Doppler [25]. In such a
case, conventional detection methods may experience a very high detection
complexity since the channel sparsity no longer holds. In light of this, a
cross domain iterative detection was proposed in [47], where the extrinsic
information is passed between the TD domain and DD domain via the
corresponding unitary transformations in order to improve the error
performance. This method has shown a robust error performance in the case of
fractional Doppler without introducing additional complexity.

####  1.2.4 Performance Analysis

The unique properties of OTFS modulation has given rise to various advantages
for the practical system design. For example, it has been reported that OTFS
modulation enjoys the potential of achieving full channel diversity [6, 23].
In specific, the diversity performance of uncoded OTFS signals has been
studied in [11], where the authors have shown that the diversity gain of
uncoded OTFS systems can be one but the full diversity can be obtained by
suitable precoding schemes. The _effective_ diversity was discussed in [9],
where the authors have shown that the full diversity can be achieved almost
surely for the case of two resolvable paths when the frame size is
sufficiently large, even for uncoded OTFS modulation systems.

The error performance of coded OTFS systems has been investigated in [48, 49],
where the authors have shown that there is a trade-off between the diversity
gain and the coding gain of OTFS systems. In particular, it has been shown in
[48, 49] that the diversity gain of OTFS systems improves with the number of
resolvable paths, while the coding gain declines, which provided some
insightful code design guidelines. Furthermore, the correctness of this
conclusion has been verified in [50, 51], where extensive simulations with
state-of-art channel coding schemes are demonstrated. The error performance of
OTFS systems with carefully designed TF domain windows has been studied in
[52, 53], where the authors have shown that the TF domain windowing could
effectively mitigate the Doppler effect from the channel and improve the
channel estimation performance. The achievable rate for CP/OFDM-based OTFS
systems was studied in [24]. In particular, it is shown that both CP/OFDM-
based OTFS and conventional OFDM systems share the same achievable rate, due
to the unitary transformation between TF and DD domains. The pragmatic
capacities of both OTFS and OFDM were compared in [54], where the achievable
rates of both OTFS and OFDM are computed under practical channel estimation
and signal detection schemes. In particular, it is shown in [54] that the
pragmatic capacity of OTFS outperforms the OFDM counterpart, especially in the
presence of severe Doppler effects. In [55], the peak-to-average power ratio
(PAPR) of OTFS modulation was investigated, which has shown that OTFS
modulation has a better PAPR performance than that of OFDM modulation and GFDM
modulation. In particular, the authors have shown that the PAPR of OTFS
waveform is proportionate to the number of Doppler bins/time slots instead of
the number of delay bins/sub-carriers. Therefore, it generally enjoys a
relatively low PAPR in practical systems.

####  1.2.5 Applications

The success of OTFS has attracted many applications. In this section, we will
mainly review some recent works on those applications, such as multiple access
transmissions and ISAC.

#####  1.2.5.1 Multiple Access Transmissions

In light of the advantages of OTFS modulation over the conventional OFDM
modulation [14], it is natural to investigate the multiple access schemes
based on OTFS modulation to realize robust multi-user transmissions in high-
mobility environments. For instance, a path-division multiple access scheme
was proposed in [37], where the inter-user interference for downlink
transmission is significantly suppressed by a low complexity beamforming
design. Also, a study on DD domain multiple access schemes was reported in
[56], where the authors have shown the bit error rate (BER) advantages of the
DD domain multiple access schemes over the conventional OFDM counterpart via
simulations. The achievable rate upper-bounds for DD domain multiple access
schemes have been studied in [57], where the authors derived upper-bounds for
both delay division multiple access scheme and Doppler division multiple
access scheme. Furthermore, it is shown in [57, 58] that the OTFS modulation
has an achievable rate upper-bound that is independent from the delay and
Doppler distributions, thanks to the “channel hardening” effect. However, the
achievable rate upper-bound for the OFDM counterpart depends highly on the
delay and Doppler distributions, due to the inevitable TF domain superposition
among resolvable paths. Consequently, it is shown that DD domain multiple
access schemes enjoy an improved achievable rate performance compared to the
OFDM counterpart222It should be noted that this conclusion does not contradict
to the ones from [24], because the unitary transformation between different
domains may not hold in the multiple access transmission, because each user
only occupies limited resources [57]..

#####  1.2.5.2 Integrated Sensing and Communications

ISAC transmission has been widely recognized as an efficient approach to deal
with the foreseeable coexistence between communication and radar [59, 60]. A
key motivation for ISAC transmission designs is that both radar sensing and
communication naturally have similar channel characteristics which can be
exploited. For example, let us consider a common downlink scenario in a mobile
network, where the antennas for radar sensing and communication are co-located
at the BS. It is not hard to notice that the physical channel between the BS
and user equipments (UEs) is the same for both radar sensing and
communication, despite the fact that radar sensing is operated based on the
received echoes at the BS after the _round-trip_ signal propagation, while
signal detection for communication is based on the _one-way_ transmission from
the BS to UEs. It is worthwhile to notice that radar sensing carries out
parameter estimations based on the delay, Doppler, and angular features
associated to resolvable paths, whose core idea aligns perfectly with the OTFS
modulation for communication purposes. The DD domain symbol multiplexing
enables the direct interactions between the information symbols and the DD
domain channel, whose channel response can be potentially inferred from the
radar estimates in practice [61]. The synergistic ecosystem established by the
needs for communication and radar sensing has motivated us to consider the
ISAC transmission design based on OTFS modulation. Various lines of research
works have been conducted in the literature. For example, the effectiveness of
OTFS modulation for ISAC transmission has been evaluated in [62], where the
authors have shown that the estimation error lower bounds for radar sensing
can be achieved by using OTFS signals while maintaining a satisfactory
communication performance. This work has then been extended to the case of
MIMO transmissions [63], where a hybrid digital-analog beamformer is devised
for both radar sensing and communication. In addition, the authors in [63]
have also developed an efficient maximum-likelihood (ML) algorithm to
facilitate target detection and parameter estimation. In addition, a novel
OTFS-based matched-filter algorithm for target range and velocity estimation
for radar sensing has been proposed in [64]. Specifically, the proposed
matched-filter algorithm takes advantage of the structures of the DD domain
effective channel matrix and has shown better estimation performance compared
to the OFDM counterpart. Furthermore, an ISAC-assisted OTFS system has been
proposed in [61, 65], where both uplink and downlink communications are
considered. In particular, the authors proposed a novel DD domain channel
estimation algorithm and introduced a message-passing-based detection
algorithm for uplink transmission. On the other hand, the downlink
communication transmission is designed based on the channel state information
(CSI) obtained from radar sensing, such that it can bypass the need of channel
estimation and equalization at the receiver side.

The application of OTFS modulation in ISAC transmissions will be discussed in
detail in Chapter 7 of this thesis.

###  1.3 Organizations of the Thesis

This thesis is organized as follows. A review of wireless channels is provided
in Chapter 2. Furthermore, the detailed derivations for OTFS modulation appear
in Chapter 3. Two OTFS signal detection approaches are introduced in Chapter 4
and Chapter 5, namely, the hybrid MAP and PIC detection and the cross domain
iterative detection. In Chapter 6, we discuss the error performance of coded
OTFS systems. The application of OTFS modulation in ISAC transmissions is
studied in Chapter 7. Finally, the conclusion of this thesis is presented in
Chapter 8, and some future research directions are also highlighted.

##  Chapter 2 Wireless Channel Revisit

Wireless channels play a fundamental role in communications over a wireless
scenario. Typical wireless communication systems may involve one or several
transmitter(s) and receiver(s), which are equipped with at least one antenna
capable of radiating or/and receiving electromagnetic waves. Information is
modulated onto the electromagnetic wave with a specific carrier frequency that
is transmitted over the wireless channel between the transmitter(s) and
receiver(s), thereby enabling communication between transmitter(s) and
receiver(s). From a physical point of view, the propagation of electromagnetic
wave is determined by Maxwell’s equations according to the underlying wireless
environment. Unfortunately, solving Maxwell’s equations is usually infeasible
due to the interactions between the electromagnetic wave and the scatterers
(such as reflection, transmission, scattering, and diffraction [7]), except
for the ideal free-space propagation. However, from a communication point of
view, it is usually sufficient to describe the wireless channel in a
stochastic manner with some reasonable simplifications.

In this chapter, we will introduce some fundamentals of wireless channels from
the communication point of view. The ideas and discussions of this chapter
follow the classic textbook for wireless channels, e.g., Ref. [7, 8, 66]. In
particular, we are mainly interested in the radio channels, despite of the
fact that some of the related discussions are also relevant to acoustic
channels.

###  2.1 Wireless Channel Backgrounds

Signals transmitted over wireless channels are generally affected by power
attenuation, distortions, and the additive white Gaussian noise (AWGN). In
particular, the signal distortions may come from various channel impairments
[7, 8], such as the multi-path effect and the Doppler effect, which are
introduced as follows.

####  2.1.1 Path Loss and Fading

The received signal power is an important parameter determining the overall
performance of wireless communications. In particular, the channel impairments
may cause severe fluctuations of the receiver power, which can be
characterized by three phenomena, namely, the pass loss, the large-scale
fading, and the small-scale fading, respectively [7, 66].

The path loss is caused by the power decay of the transmission, which is
usually distance-dependent. In particular, the path loss is usually modeled by
an exponential distribution. Let $\alpha$ be the path loss exponent111The
typical value of $\alpha$ is between $2$ to $4$. For example, the 3rd
Generation Partnership Project (3GPP) suggests $\alpha=3.76$ for evolved
universal terrestrial radio access (E-UTRA) [67].. Then, the pass loss in
decibels is given by
${\rm{PL}}\buildrel\Delta\over{=}{\rm{10}}\alpha{\rm{lo}}{{\rm{g}}_{10}}\left(d\right)$,
where $d$ denotes the distance the electromagnetic wave has propagated. The
path loss can be mitigated by transmit power control that is relevant to the
link budget. The transmit power control is usually enabled by a feedback loop
from the receiver to the transmitter.

Large-scale fading usually comes from the block or attenuation of the
propagation paths. In specific, these two channel impairments are usually
referred to as the _shadowing_ and _absorption loss_ , respectively [7]. Many
experimental tests have shown that the large-scale fading can be accurately
modeled by a random variable with the log-normal distribution that is related
to the geographic characteristics [66]. Similar to the path loss, the large-
scale fading can be mitigated by the transmit power control, as the geographic
characteristics are relatively constant.

Small-scale fading is the consequence of the constructive and destructive
interference of multi-path transmission with respect to the corresponding
delay and Doppler shifts. Unlike path loss and large-scale fading, small-scale
fading varies fast and it is therefore usually modeled stochastically by
Gaussian distributed channel coefficients. Depending on whether there is an
LoS path, the magnitude of the channel coefficient (channel gain) can be
modeled by either Rayleigh distribution or Rice distribution. Transmit power
control cannot combat the small-scale fading as the received power fluctuates
rapidly. The most effective method to mitigate the small-scale fading is the
application of diversity techniques [8]. In the following subsections, we will
present common classifications for different types of small-scale fading.

####  2.1.2 Flat Fading Channels

We refer to a wireless channel as a flat fading channel, if the channel gives
a (roughly) constant response for all time and frequency components during the
transmission. A flat fading channel is neither time dispersive nor frequency
dispersive, and the received signal is merely the multiplication of the
transmitted signal with a specific fading coefficient in the noiseless regime.
A flat fading channel usually appears in indoor environments, where there are
strong line-of-sight (LOS) components and few weak multi-path components. For
flat fading channels, the excess delays associated to different paths are
usually small and the transmitter and receiver are relatively static or moving
slowly.

####  2.1.3 Time Dispersive Channels

We refer to a channel as a time dispersive channel if it only has the multi-
path effect. The multi-path effect arises when the transmitted signal is
picked up by the receive antenna after propagating through several different
paths, which usually appears in the presence of multiple scatterers with
relatively strong reflectivities [7, 8]. Consequently, the received signal is
a superposition of several delayed, power attenuated replicas of the original
transmitted signal. In other words, the original transmitted signal is
smeared-out (dispersive) in the time domain after being transmitted over the
wireless channels with multiple resolvable paths. Time dispersive channels are
frequency selective, which indicates that the strengths of channel responses
are different among different frequency components. The frequency domain
selectivity can be understood from the Fourier analysis, where the time domain
delays of different paths introduce different phase rotations to the frequency
domain responses associated to different paths and therefore, frequency
selectivity appears as the consequence of the non-coherent superposition of
those responses. A time dispersive channel could appear in large indoor halls
or outdoor environments, where there are strong multi-path components. For
time dispersive channels, the excess delay associated to different paths is
usually large and the transmitter and receiver are relatively static or moving
slowly.

####  2.1.4 Frequency Dispersive Channels

We refer to a channel as a frequency dispersive channel if it only has the
Doppler effect. The Doppler effects appear if there is a relative movement
between the transmitter and receiver, in which case the emitted
electromagnetic wave will experience frequency shifts. The Doppler effect may
cause different signal distortions subjected to the signal bandwidth and
relative velocity. In particular, in the case where the relative velocity $v$
is much lower than the speed of light $c_{0}=3*10^{8}$ and the signal
bandwidth is much smaller than the carrier frequency $f_{\rm c}$, the Doppler
effect can be approximated by a frequency shift $f_{\rm d}$, i.e., [7, 8]

$\displaystyle{f_{\rm{d}}}\buildrel\Delta\over{=}\frac{{v\cos\left(\theta\right)}}{{{c_{0}}}}{f_{\rm{c}}},$
(2.1)

where $\theta$ is angle-of-arrival (AoA) of the electromagnetic wave relative
to the direction of motion of the receiver. Due to the Doppler effect, the
received signal is a superposition of several frequency-shifted, power
attenuated replicas of the original transmitted signal, resulting in frequency
dispersion. Frequency dispersive channels are time selective, which again can
be understood from the Fourier analysis. A frequency dispersive channel could
appear in outdoor environments, where there are strong LOS components and few
weak multi-path components. For frequency dispersive channels, the excess
delay associated to different paths is usually small and the transmitter and
receiver are relatively moving at a high speed.

####  2.1.5 Doubly Selective Channels

We refer to a channel as a frequency dispersive channel if it has both the
multi-path effect and the Doppler effect. Doubly selective channels are also
called doubly dispersive channels or linear time-varying (LTV) channels, over
which the received signal suffers from both time and frequency dispersions and
selectivities relative to the transmitted signal. The doubly selectivity is a
direct consequence of signal transmission in the presence of the multi-path
effect with non-negligible delay and Doppler shifts. A doubly dispersive
channel could appear in outdoor environments, where there are strong multi-
path components. For doubly dispersive channels, the excess delay associated
to different paths is usually large and the transmitter and receiver are
relatively moving at a high speed.

Although doubly selective channels may impose great challenges for channel
estimation and equalization from a communication perspective, recent advances
in wireless communication have shown that both the time and frequency
selectivities can be exploited to improve the communication performance [7,
8]. In particular, doubly selectivity provides new degrees-of-freedom (DoFs),
which are commonly known as the delay/frequency diversity and Doppler/time
diversity. However, to fully exploit the diversity gain, the transmitted
signal needs to be carefully designed and OTFS waveforms are a type of signals
that can almost surely exploit the full time-frequency (TF) diversity.

We provide a table summarizing the aforementioned types of wireless channels.
More details of doubly selective channels will be discussed in the following
subsections.

Table 2.1: A Summary of Possible Conditions for Different Types of Wireless Channels Channel Type |  Scenario |  Multi-path |  LOS |  Delay |  Doppler
---|---|---|---|---|---
Flat fading |  Indoor |  Weak |  Yes |  Small |  Small
Time dispersive |  Indoor/Outdoor |  Strong |  Yes |  Large |  Small
Frequency dispersive |  Outdoor |  Weak |  Yes |  Small |  Large
Doubly dispersive |  Outdoor |  Strong |  No |  Large |  Large

###  2.2 Mathematical Representations of Wireless Channels: Deterministic
Description

In this section, we are interested in a deterministic scenario, where the
physical attributes of the channel remain roughly unchanged during the signal
transmission, including the number of scatterers, the relative velocity of the
scatterers, the distances between the transmitter and the receiver relative to
each scatterer, and the reflectivities associated to each scatterer,
respectively. In particular, the deterministic scenario can be viewed as a
“snapshot" of the real channel, and how long can this snapshot accurately
represent the channel genuinely depends on the real scenario [7, 66]. Some
relevant discussions on this point will be given in the section. Without loss
of generality, we are interested in the mathematical representations of
wireless channels in the equivalent complex baseband domain. In what follows,
we focus on general LTV channels in a deterministic scenario with $P$
resolvable paths, where $h_{p}$, $\tau_{p}$, $\nu_{p}$ are used to describe
the complex fading coefficient, the time delay, and the Doppler frequency
associated to the $p$-th path with $1\leq p\leq P$, respectively.

####  2.2.1 Time-Delay Representation of LTV Channels

Assume that a signal $s(t)$ is transmitted over an LTV channel. Then, with the
noiseless assumption, the received signal $r(t)$ can be modelled by222Here, we
assume that the channel takes the Doppler shift first and the delay shift
second. It should be noted that an equivalent interpretation could be derived
by taking the delay shift first and the Doppler shift second. Some further
discussions on this ordering issue could be found in [6] and in Section 2.2.3
and Section 3.1.3. [7]

$\displaystyle
r\left(t\right)=\sum\limits_{p=1}^{P}{{h_{p}}}s\left({t-{\tau_{p}}}\right){e^{j2\pi{\nu_{p}}t}}.$
(2.2)

Particularly, (2.2) can be interpreted as the signal transmission over a
wireless channel with $P$ distinct scatterers with respect to their own
physical attributes, including the reflectivity (fading coefficient), the time
delay, and the Doppler frequency. According to (2.2), it can be shown that the
TD domain channel response of an LTV channel can be represented by [7]

$\displaystyle{h_{{\rm{TD}}}}\left({t,\tau}\right)=\sum\limits_{p=1}^{P}{{h_{p}}}{e^{j2\pi{\nu_{p}}t}}\delta\left({\tau-{\tau_{p}}}\right),$
(2.3)

and the input-output relationship between $s(t)$ and $r(t)$ is given by

$\displaystyle
r\left(t\right)=\int_{-\infty}^{\infty}{s\left({t-\tau}\right){h_{{\rm{TD}}}}\left({t,\tau}\right)}{\rm{d}}\tau.$
(2.4)

Based on (2.3), we notice that the TD domain channel only has at most $P$
separable responses along the delay domain, while it has responses for all the
time domain components with respect to the Doppler frequency. For further
clarification, let us consider the following example.

Example 2-1 (_Time-Delay Representation of LTV Channels_): Let us consider the
following example of the LTV channel, where the related parameters are given
in Table 2.2. The corresponding TD representation of the channel is given in
Fig. 2.1 and Fig. 2.2.

Table 2.2: Channel Parameters for Example 2-1, 2-2, and 2-3 Number of paths $P$ |  $5$
---|---
Time delays (ms) |  $[0,0.1,0.3,0.4,0.5]$
Doppler frequency (kHz) |  $[-4,-2,0,2,4]$
Fading coefficients |  Rayleigh fading
Figure 2.1: TD domain representation of the LTV channel, where the channel
parameters are given in Table 2.2. For a better clarification, only the real
part of the complex channel response is shown. Figure 2.2: TD domain
representation of the LTV channel, where the channel parameters are given in
Table 2.2. For a better clarification, only the imaginary part of the complex
channel response is shown.

As indicated by both Fig. 2.1 and Fig. 2.2, it can be shown that the TD
representation of the channel is sparse in the delay domain, but dense in the
time domain. In particular, the channel only has responses at the given time
delays, while the channel response changes periodically with respect to the
corresponding Doppler frequency along the time domain.

####  2.2.2 Time-Frequency Representation of LTV Channels

The TF domain channel can be obtained by performing the Fourier transform
along the delay domain to (2.3), i.e.,

${h_{{\rm{TF}}}}\left({t,f}\right)=\int_{-\infty}^{\infty}{{h_{{\rm{TD}}}}\left({t,\tau}\right){e^{-j2\pi
f\tau}}}{\rm{d}}\tau=\sum\limits_{p=1}^{P}{{h_{p}}{e^{j2\pi{\nu_{p}}t}}}{e^{-j2\pi{\tau_{p}}f}}.$
(2.5)

The TF domain channel representation is the most known channel description in
the field of wireless communications thanks to the success of orthogonal
frequency-division multiplexing (OFDM). Some fundamentals of OFDM and its
connections with OTFS will be discussed in the next chapter. Similar to the TD
domain channel representation, we use the following example to further clarify
(2.5).

Example 2-2 (_Time-Frequency Representation of LTV Channels_): Let us consider
the following example of the LTV channel, where the related parameters are
given in Table 2.2. The corresponding TF domain representation of the channel
is given in Fig. 2.3.

Figure 2.3: TF domain representation of the LTV channel, where the channel
parameters are given in Table 2.2.

As can be observed from Fig. 2.3, the TF domain representation is dense over
the whole TF plane. This observation aligns with (2.5), where we can notice
that both delay and Doppler change the phases of the channel responses.
Although the TF domain channel response seems to be complex, there generally
exist regions within the TF plane, where the complex channel response remains
roughly constant [7, 66]. Such regions are usually referred to as the
_coherence region_ [8]. Some more discussions on the coherence region will
appear in Section 2.3.3.

####  2.2.3 Delay-Doppler Representation of LTV Channels

The delay-Doppler (DD) domain channel response can be obtained by performing
the Fourier transform along the time domain to (2.3), i.e.,

${h_{{\rm{DD}}}}\left({\tau,\nu}\right)=\int_{-\infty}^{\infty}{{h_{{\rm{TD}}}}\left({t,\tau}\right){e^{-j2\pi
t\nu}}}{\rm{d}}t=\sum\limits_{p=1}^{P}{{h_{p}}\delta\left({\tau-{\tau_{p}}}\right)}\delta\left({\nu-{\nu_{p}}}\right).$
(2.6)

Equivalently, the DD domain channel response can also be obtained via the so-
called _symplectic finite Fourier transform_ (SFFT) to the TF domain channel
(2.5) [6], i.e.,

${h_{{\rm{DD}}}}\left({\tau,\nu}\right)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{{h_{{\rm{TF}}}}\left({t,f}\right){e^{-j2\pi
t\nu}}}{e^{j2\pi
f\tau}}{\rm{d}}t{\rm{d}}f=\sum\limits_{p=1}^{P}{{h_{p}}\delta\left({\tau-{\tau_{p}}}\right)}\delta\left({\nu-{\nu_{p}}}\right).$
(2.7)

In particular, the SFFT can be viewed as the combination of a Fourier
transform to the time domain and an inverse Fourier transform to the frequency
domain [6]. As indicated by (2.7), the dense TF domain channel responses
become _sparse_ channel responses in the DD domain. Furthermore, (2.7) implies
that the responses associated to different paths are _separable_ , i.e., there
is no interference between different paths’ responses333It should be noted
that this only asymptotically holds given sufficiently large number of time
and frequency resources [7].. On top of that, the DD domain channel response
is _compact_ given the maximum delay and Doppler, which indicates that it only
has responses within the region constrained by the maximum time delay and
Doppler frequency. Similar to the previous subsections, let us consider the
following example.

Example 2-3 (_Delay-Doppler Representation of LTV Channels_): Let us consider
the following example of the LTV channel, where the related parameters are
given in Table 2.2. The corresponding DD domain representation of the channel
is given in Fig. 2.4.

Figure 2.4: DD domain representation of the LTV channel, where the channel
parameters are given in Table 2.2.

As can be observed from Fig. 2.4, the DD domain channel response only has five
peaks over the whole DD plane, whose coordinations align with the parameters
in Table 2.2. Furthermore, the channel responses from different paths are
sufficiently separated as shown in Fig. 2.4. Those observations are consistent
with our discussions above.

It is also of practical importance to discuss the DD domain input-output
relationships for the purposes of communications. In specific, given the input
signal $s\left({t}\right)$, the corresponding output signal
$r\left({t}\right)$ after transmitting over the DD domain channel
${h_{{\rm{DD}}}}\left({\tau,\nu}\right)$ can be written by [6, 25]

$r\left(t\right)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{h_{{\rm{DD}}}}\left({\tau,\nu}\right)}}{e^{j2\pi\nu\left({t-\tau}\right)}}s\left({t-\tau}\right){\rm{d}}\tau{\rm{d}}\nu,$
(2.8)

or

$r\left(t\right)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{h_{{\rm{DD}}}}\left({\tau,\nu}\right)}}{e^{j2\pi\nu{t}}}s\left({t-\tau}\right){\rm{d}}\tau{\rm{d}}\nu.$
(2.9)

The above two input-output relationships are the consequences of the ordering
issue that is either we taking the delay shift first and Doppler shift second,
or _vice versa_. Consequently, different orderings result in a phase
difference ${e^{j2\pi\nu\tau}}$ between (2.8) and (2.9). Despite this phase
difference, these two interpretations will lead to equivalent results [23], as
long as the notations are consistent.

###  2.3 Mathematical Representations of Wireless Channels: Stochastic
Description

While the deterministic description of the wireless channel focuses on the
“snapshot" of the real channel, the stochastic description focuses on the
statistical characteristics of the wireless channel, e.g., the second-order
statistics. For any given linear time-varying channel, its second-order
statistics can in general be represented by four different variables that are
time, frequency, delay, and Doppler, respectively [7, 68]. In 1963, Bello has
introduced the wide-sense stationary uncorrelated scattering (WSSUS)
assumption, which greatly simplifies the above complex dependence from four
variables to only two variables [68]. In what follows, we will introduce the
concept of WSSUS channels first and then briefly extend our discussions to
non-WSSUS channels.

####  2.3.1 WSSUS Channels: Descriptions and Properties

Without loss of generality, let us start our discussion from the TD domain
channel in (2.3). According to [7, 68], a _wide-sense stationary_ (WSS)
channel should satisfy

$\displaystyle{\mathbb{E}}\left\\{{{h_{{\rm{TD}}}}\left({t,\tau}\right)h_{{\rm{TD}}}^{*}\left({t^{\prime},\tau^{\prime}}\right)}\right\\}={C_{{\rm{TD}}}}\left({t-t^{\prime};\tau,\tau^{\prime}}\right),$
(2.10)

i.e., the channel taps are jointly WSS with respect to the time variable,
where ${C_{{\rm{TD}}}}$ denotes the correlation function of the TD domain
channel responses. On the other hand, if the LTV channel is said to feature
_uncorrelated scattering_ (US), the following equation must hold [7, 68]

$\displaystyle{\mathbb{E}}\left\\{{{h_{{\rm{TD}}}}\left({t,\tau}\right)h_{{\rm{TD}}}^{*}\left({t^{\prime},\tau^{\prime}}\right)}\right\\}={C_{{\rm{TD}}}}\left({t,t^{\prime};\tau}\right)\delta\left({\tau-\tau^{\prime}}\right).$
(2.11)

By combining both (2.10) and (2.11), we have

$\displaystyle{\mathbb{E}}\left\\{{{h_{{\rm{TD}}}}\left({t,\tau}\right)h_{{\rm{TD}}}^{*}\left({t^{\prime},\tau^{\prime}}\right)}\right\\}={C_{{\rm{TD}}}}\left({t-t^{\prime};\tau}\right)\delta\left({\tau-\tau^{\prime}}\right),$
(2.12)

which holds for general WSSUS channels. The rationale of the WSSUS assumption
is that in practical wireless channels, the channel impairments are introduced
by physical scatterers with different reflectivities and any two distinct
scatterers generally have uncorrelated reflectivities. In specific, the US
assumption implies that the different delay shifts associated with different
resolvable paths are uncorrelated. On the other hand, the WSS assumption
indicates that the different Doppler shifts associated with different
resolvable paths are uncorrelated such that the time domain channel response
is jointly WSS with respect to the time variable.

Now let us turn our attention from the TD domain to the DD and TF domains.
Based on the above discussions, let us define the
${C_{{\rm{DD}}}}\left({\tau;\nu}\right)$ as the _channel scattering function_
[7, 68, 66], which closely relates to the DD domain channel (2.6) and (2.7) by

$\displaystyle{\mathbb{E}}\left\\{{{h_{{\rm{DD}}}}\left({\tau,\nu}\right)h_{{\rm{DD}}}^{*}\left({\tau^{\prime},\nu^{\prime}}\right)}\right\\}={C_{{\rm{DD}}}}\left({\tau;\nu}\right)\delta\left({\tau-\tau^{\prime}}\right)\delta\left({\nu-\nu^{\prime}}\right).$
(2.13)

The physical meaning of ${C_{{\rm{DD}}}}\left({\tau;\nu}\right)$ can be
interpreted as the average strength of scatterers with time delay $\tau$ and
Doppler frequency $\nu$ for the given LTV channel. As discussed in [7], the
channel scattering function is a white process but it is non-stationary.

The _TF correlation function_ is defined by

$\displaystyle{\mathbb{E}}\left\\{{{h_{{\rm{TF}}}}\left({t,f}\right)h_{{\rm{TF}}}^{*}\left({t^{\prime},f^{\prime}}\right)}\right\\}={C_{{\rm{TF}}}}\left({t-t^{\prime};f-f^{\prime}}\right)={C_{{\rm{TF}}}}\left({\Delta
t;\Delta f}\right),$ (2.14)

which is a 2D stationary process with respect to the time variable and
frequency variable. This interpretation of this 2D stationary process is
straightforward. In specific, the WSS assumption implies that the channel is
stationary in time by its definition, while the US assumption suggests that
the channel is stationary in frequency as the Dopplers are uncorrelated [7].
By observing the relationship between the TF and DD domain channel responses
in (2.7), it can be shown that the TF correlation function and the channel
scattering function can be calculated from each other via the SFFT and ISFFT
[7], e.g.,

$\displaystyle{C_{{\rm{DD}}}}\left({\tau;\nu}\right)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{C_{{\rm{TF}}}}\left({\Delta
t;\Delta f}\right){e^{-j2\pi\left({\Delta t\nu-\Delta
f\tau}\right)}}}}{\rm{d}}\Delta t{\rm{d}}\Delta f.$ (2.15)

What’s more interesting is that (2.14) and (2.15) together imply that the
channel scattering function ${C_{{\rm{DD}}}}\left({\tau;\nu}\right)$ is
essentially the 2D power spectral density (PSD) of the 2D stationary process
${h_{{\rm{TF}}}}$ [7]. This relationship actually suggests that the channel
estimation of either TF or DD domain can be carried out in its dual domain if
the dual domain has more appealing properties, e.g., block-fading or a limited
number of delay and Doppler responses.

Based on the discussions in this subsection, let us summarize the WSSUS
channel characteristics in different domains in Table 2.3.

Table 2.3: WSSUS Channel Characteristics in Different Domains Domains |  Statistics |  Time |  Frequency |  Delay |  Doppler
---|---|---|---|---|---
TD |  ${C_{{\rm{TD}}}}\left({t-t^{\prime};\tau}\right)$ |  Stationary |  N/A |  Uncorrelated |  N/A
TF |  ${C_{{\rm{TF}}}}\left({\Delta t;\Delta f}\right)$ |  Stationary |  Stationary |  N/A |  N/A
DD |  ${C_{{\rm{DD}}}}\left({\tau;\nu}\right)$ |  N/A |  N/A |  Uncorrelated |  Uncorrelated

####  2.3.2 WSSUS Channels: Delay and Doppler Profiles and Important
Parameters

As both delay and Doppler are two attributes of the scatterer, it is sometimes
useful to focus on the “marginal function" instead of the second order
statistics. For example, the power-delay profile is defined by

$\displaystyle{C_{{\rm{Delay}}}}\left(\tau\right)\buildrel\Delta\over{=}\int_{-\infty}^{\infty}{{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\nu}.$
(2.16)

Meanwhile, the power-Doppler profile is defined by

$\displaystyle{C_{{\rm{Doppler}}}}\left(\nu\right)\buildrel\Delta\over{=}\int_{-\infty}^{\infty}{{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\tau}.$
(2.17)

Given the WSSUS assumption, ${C_{{\rm{Delay}}}}\left(\tau\right)$ can be
viewed as the mean power of the channel tap with delay $\tau$. Similarly,
${C_{{\rm{Doppler}}}}\left(\tau\right)$ can be viewed as the mean power of the
channel tap with Doppler $\nu$. Exponential power-delay profile [66] and the
Jakes power-delay profile [69] are two well-known models for
${C_{{\rm{Delay}}}}\left(\tau\right)$ and
${C_{{\rm{Doppler}}}}\left(\nu\right)$. In particular, the exponential power-
delay profile is developed based on the distance-dependent decay of the signal
power in the free space transmission, while the Jakes power-delay profile is
derived based on the assumption of uniform distributed AoAs [69].

Based on the definition of power-delay profile and power-Doppler profile, some
important parameters are ready to be discussed. For example, the pass loss is
an important parameter implies the average power attenuation of the underlying
channel, which directly links to the link budget. The pass loss
${\rho_{{\rm{PL}}}^{2}}$ can be derived by [7]

$\displaystyle{\rho_{{\rm{PL}}}^{2}}$
$\displaystyle=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\tau{\rm{d}}\nu}}=\int_{-\infty}^{\infty}{{C_{{\rm{Delay}}}}\left(\tau\right){\rm{d}}\tau}=\int_{-\infty}^{\infty}{{C_{{\rm{Doppler}}}}\left(\nu\right){\rm{d}}\nu}$
$\displaystyle={\mathbb{E}}\left\\{{{{\left|{{h_{{\rm{TF}}}}\left({t,f}\right)}\right|}^{2}}}\right\\}.$
(2.18)

Other than the pass loss, it is also useful for the practical system to have
some knowledge of the channel attenuation with respect to the delay and
Doppler. According to [7], the mean delay and mean Doppler shift can be
defined by

$\displaystyle\bar{\tau}\buildrel\Delta\over{=}\frac{1}{{{\rho_{{\rm{PL}}}^{2}}}}\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\tau{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\tau{\rm{d}}\nu}}=\frac{1}{{{\rho_{{\rm{PL}}}^{2}}}}\int_{-\infty}^{\infty}{\tau{C_{{\rm{Delay}}}}\left(\tau\right){\rm{d}}\tau},$
(2.19)

and

$\displaystyle\bar{\nu}\buildrel\Delta\over{=}\frac{1}{{{\rho_{{\rm{PL}}}^{2}}}}\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\nu{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\tau{\rm{d}}\nu}}=\frac{1}{{{\rho_{{\rm{PL}}}^{2}}}}\int_{-\infty}^{\infty}{\nu{C_{{\rm{Doppler}}}}\left(\nu\right){\rm{d}}\nu},$
(2.20)

respectively. The mean delay and mean Doppler shifts indicate the mean
absolute vales of the delay variable and Doppler variable that have a non-zero
channel response444Due to the causality of the LTV channel response,
${C_{{\rm{Delay}}}}\left(\tau\right)=0$ for $\tau<0$.. Furthermore, the _delay
spread_ and _Doppler spread_ are also two terminologies widely used in the
system design, which are typically defined by the root-mean-square (RMS)
widths of power-delay profile and power-Doppler profile [7], such as

$\displaystyle{\sigma_{\tau}}\buildrel\Delta\over{=}\frac{1}{{\sqrt{\rho_{{\rm{PL}}}^{2}}}}\sqrt{\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{\left({\tau-\bar{\tau}}\right)}^{2}}{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\tau{\rm{d}}\nu}}}=\frac{1}{{\sqrt{\rho_{{\rm{PL}}}^{2}}}}\sqrt{\int_{-\infty}^{\infty}{{{\left({\tau-\bar{\tau}}\right)}^{2}}{C_{{\rm{Delay}}}}\left(\tau\right){\rm{d}}\tau}},$
(2.21)

and

$\displaystyle{\sigma_{\nu}}\buildrel\Delta\over{=}\frac{1}{{\sqrt{\rho_{{\rm{PL}}}^{2}}}}\sqrt{\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{\left({\nu-\bar{\nu}}\right)}^{2}}{C_{{\rm{DD}}}}\left({\tau;\nu}\right){\rm{d}}\tau{\rm{d}}\nu}}}=\frac{1}{{\sqrt{\rho_{{\rm{PL}}}^{2}}}}\sqrt{\int_{-\infty}^{\infty}{{{\left({\nu-\bar{\nu}}\right)}^{2}}{C_{{\rm{Doppler}}}}\left(\nu\right){\rm{d}}\nu}},$
(2.22)

respectively. The delay and Doppler spread can be viewed as the “effective"
range of the delay and Doppler shifts that contains the most of the energy of
the channel response. For a better illustration of the discussions in this
subsection, let us consider the following example.

Example 2-4 (_Exponential Power-Delay Profile and Jakes Power-Doppler
Profile[7, 66]_): Let us discuss the WSSUS channel model considered in [7,
66], where the channel scattering function can be separated by

$\displaystyle{C_{{\rm{DD}}}}\left({\tau;\nu}\right)=\frac{1}{{\rho_{{\rm{PL}}}^{2}}}{C_{{\rm{Delay}}}}\left(\tau\right){C_{{\rm{Doppler}}}}\left(\nu\right).$
(2.23)

In particular, we consider the exponential power-delay profile and Jakes
power-Doppler profile, given by

${C_{{\rm{Delay}}}}\left(\tau\right)=\left\\{\begin{array}[]{l}\frac{{\rho_{\rm
PL}^{2}}}{{{\tau_{0}}}}{e^{-\frac{\tau}{{{\tau_{0}}}}}},\tau\geq 0\\\
0,\quad\quad\quad\tau<0\end{array}\right.$ (2.24)

and

${C_{{\rm{Doppler}}}}\left(\nu\right)=\left\\{\begin{array}[]{l}\frac{{\rho_{\rm
PL}^{2}}}{{\pi\sqrt{\nu_{\max}^{2}-{\nu^{2}}}}},\left|\nu\right|<{\nu_{\max}}\\\
0,\quad\quad\quad\quad\left|\nu\right|>{\nu_{\max}}\end{array}\right.$ (2.25)

respectively. In particular, the term $\tau_{0}$ is a delay parameter
characterizing the decay exponent, while the term ${\nu_{\max}}$ is the
maximum Doppler value. For example, if we set ${\tau_{0}}=5\;\mu s$,
${\nu_{\max}}=50\;{\rm Hz}$, and ${\rho_{\rm PL}^{2}}=1$, the corresponding
channel scattering function can be shown in Fig. 2.5.

Figure 2.5: Channel scattering function with exponential power-delay profile
and Jakes power-Doppler profile, where the channel parameters are given in
Example 2-4.

As can be observed in Fig. 2.5, the channel scattering function behaves like a
“bowel-shaped" spectrum in the Doppler domain, while having an exponential
decay in the delay domain. In particular, it can be observed that the
amplitudes of the channel scattering function tend to have large values for
small delays and large Doppler shifts, while tend to have small values for
large delays and small Doppler shifts. Furthermore, it can be shown that
$\bar{\tau}={\tau_{0}}=20\;\mu s$, ${\bar{\mu}=\nu_{\max}}=50\;{\rm Hz}$,
${\sigma_{\tau}}={\tau_{0}}=20\;\mu s$, and
$\bar{\nu}=\frac{{{\nu_{\max}}}}{{\sqrt{2}}}=35.36\;{\rm{Hz}}$, respectively.

####  2.3.3 Underspread WSSUS Channels

We focus on underspread WSSUS channels in the following. The underspread
property is of particular interest for most of the wireless system designs,
including both OFDM and OTFS systems [7]. For an underspread channel, there
exists a TF region that is no smaller than one, where the channel response is
roughly constant. Specifically, the corresponding time domain and frequency
domain intervals are commonly referred to as the _coherence time_ $T_{\rm c}$
and _coherence bandwidth_ $F_{\rm c}$, respectively. In particular, the value
of $T_{\rm c}$ is inversely proportional to the Doppler spread, while the
value of $F_{\rm c}$ is inversely proportional to the delay spread [7].
Typically, we have

$\displaystyle{T_{\rm{c}}}\buildrel\Delta\over{=}\frac{1}{{{\sigma_{\nu}}}},$
(2.26)

and

$\displaystyle{F_{\rm{c}}}\buildrel\Delta\over{=}\frac{1}{{{\sigma_{\tau}}}},$
(2.27)

respectively. The approximation accuracy corresponding to the coherence time
and bandwidth can be examined via the calculation of the mean squared
difference between ${h_{{\rm{TF}}}}\left({t,f}\right)$ and
${h_{{\rm{TF}}}}\left({t+\Delta t,f+\Delta f}\right)$, for $\left|{\Delta
t}\right|\leq T_{\rm c}$ and $\left|{\Delta f}\right|\leq F_{\rm c}$. In
specific, it can be shown that [7].

$\displaystyle\frac{1}{{\rho_{{\rm{PL}}}^{2}}}{\mathbb{E}}\left\\{{{{\left|{{h_{{\rm{TF}}}}\left({t+\Delta
t,f+\Delta
f}\right)-{h_{{\rm{TF}}}}\left({t,f}\right)}\right|}^{2}}}\right\\}\leq
2\pi\left[{{{\left({\frac{{\Delta
t}}{{{T_{\rm{c}}}}}}\right)}^{2}}+{{\left({\frac{{\Delta
f}}{{{F_{\rm{c}}}}}}\right)}^{2}}}\right].$ (2.28)

Meanwhile, the Taylor expansion of ${C_{{\rm{TF}}}}\left({\Delta t;\Delta
f}\right)$ and can also be used to show the approximation accuracy and some
relevant details can be found in [7].

According to the definitions of coherence time and coherence bandwidth, a
channel is said to be underspread, if ${T_{\rm{c}}}{F_{\rm{c}}}\geq 1$,
${T_{\rm{c}}}\geq{\sigma_{\tau}}$, and ${F_{\rm{c}}}\geq{\sigma_{\nu}}$ [7].
Alternative descriptions can also be given based on the delay and Doppler
spreads. For example, for general underspread channel,
${\sigma_{\tau}}{\sigma_{\nu}}\leq 1$ holds. A more relaxed but insightful
description of the underspread channel relying on the maximum delay $\tau_{\rm
max}$ and Doppler $\nu_{\rm max}$ values. In specific, the maximum Doppler
value is given by ${\nu_{\max}}=\frac{v}{c_{0}}{f_{c}}$, where $v$ is the
relative speed, $c_{0}$ is the speed of light, and $f_{c}$ is the carrier
frequency, respectively. The maximum delay shift is given by
${\tau_{\max}}=\frac{d_{\rm{max}}}{c_{0}}$, where $d_{\rm{max}}$ is the
maximum distance difference among the $P$ channel paths. Then, general
underspread channels always have
${\rm{4}}{\tau_{{\rm{max}}}}{\nu_{{\rm{max}}}}\leq{\rm{1}}$ [7].

The underspread condition implies that the channel cannot be both strong time
dispersion and strong frequency dispersion [7]. Equivalently, this indicates
that the channel cannot be both strongly time-selective and strongly
frequency-selective [7]. It should be noted that most of the real-world
wireless (radio) channels are virtually always underspread. However, these
underspread channels may not hold for some specific communication scenarios,
such as the underwater acoustic channels [7]. For a better understanding of
the underspread channel, let us consider the following example.

Example 2-5 (_Underspread Channel[7]_): Following the channel parameters and
profiles given in Example 2-4, it can be shown from (2.24) and (2.25) that
${T_{\rm{c}}}=\frac{{\sqrt{2}}}{{{\nu_{\max}}}}=28.28\;{\rm ms}$ and
${F_{\rm{c}}}=\frac{{1}}{{{\tau_{0}}}}=50.00\;{\rm kHz}$. Let us further
consider a signal of duration $1\;{\rm ms}$ and bandwidth $5\;{\rm kHz}$.
Then, based on (2.28), we have

$\displaystyle\frac{1}{{\rho_{{\rm{PL}}}^{2}}}{\mathbb{E}}\left\\{{{{\left|{{h_{{\rm{TF}}}}\left({t+\Delta
t,f+\Delta
f}\right)-{h_{{\rm{TF}}}}\left({t,f}\right)}\right|}^{2}}}\right\\}$
$\displaystyle\leq$ $\displaystyle 2\pi\left[{{{\left({\frac{{\Delta
t}}{{{T_{\rm{c}}}}}}\right)}^{2}}+{{\left({\frac{{\Delta
f}}{{{F_{\rm{c}}}}}}\right)}^{2}}}\right]$ $\displaystyle\leq$ $\displaystyle
2\pi\left[{{{\left({\frac{{1\times{{10}^{-3}}}}{{28.28\times{{10}^{-3}}}}}\right)}^{2}}+{{\left({\frac{{5\times{{10}^{3}}}}{{50\times{{10}^{3}}}}}\right)}^{2}}}\right]$
$\displaystyle\approx$ $\displaystyle 7.07\%,$ (2.29)

which implies that the mean squared difference of the TF domain channel
responses for any two TF grids is at most $7.07\%$. In fact, the underlying
can be viewed as a flat fading channel and it is strongly underspared, whose
TF domain channel response is roughly constant for the transmission of the
whole block, i.e., it is a block-fading channel [7].

####  2.3.4 Extensions to Non-WSSUS Channels

We have discussed the important characteristics of WSSUS channels in the
previous subsections. Now let us turn our attention to non-WSSUS channels
[70]. In contrast to the WSSUS channels, non-WSSUS channels have not been
widely studied in the literature, despite its practical interests. In
specific, the non-WSSUS channels require the time, frequency, delay, and
Doppler together to describe the channel characteristics. According to [7],
the _channel correlation function_ can be used to describe the correlations
between the DD domain parameters and between the TF domain parameters, which
is defined by

$\displaystyle{\cal R}\left({\Delta t,\Delta
f,\Delta\tau,\Delta\nu}\right)\buildrel\Delta\over{=}$
$\displaystyle\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{E\left\\{{{h_{{\rm{TF}}}}\left({t,f+\Delta
f}\right)h_{{\rm{TF}}}^{*}\left({t-\Delta
t,f}\right)}\right\\}}}{e^{-j2\pi\left({t\Delta\nu-f\Delta\tau}\right)}}{\rm{d}}t{\rm{d}}f$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{E\left\\{{{h_{{\rm{DD}}}}\left({\tau,\nu+\Delta\nu}\right)h_{{\rm{DD}}}^{*}\left({\tau-\Delta\tau,\nu}\right)}\right\\}}}{e^{j2\pi\left({\nu\Delta
t-\tau\Delta f}\right)}}{\rm{d}}\tau{\rm{d}}\nu.$ (2.30)

In particular, the channel correlation function can be viewed as a description
for the correlations of the multi-path components separated by $\Delta t$ in
time, $\Delta f$ in frequency, $\Delta\tau$ in delay, and $\Delta\mu$ in
Doppler. Further discussions regarding the channel correlation function can be
found in [7, 70].

In general WSSUS channels, the stationarity does not hold for the whole TF
plane. However, it can be shown that there exist regions in the TF plane,
described by the time and frequency intervals, where the stationarity of the
channel response roughly holds. Following the descriptions in [7], the
_stationarity time_ and _stationarity bandwidth_ are defined by

$\displaystyle{T_{\rm{s}}}\buildrel\Delta\over{=}\frac{1}{{\overline{\Delta\nu}}},$
(2.31)

and

$\displaystyle{F_{\rm{s}}}\buildrel\Delta\over{=}\frac{1}{{\overline{\Delta\tau}}},$
(2.32)

respectively, where ${\overline{\Delta\nu}}$ and ${\overline{\Delta\tau}}$ are
the delay and Doppler lags within which there are significant correlations
[7]. We are not going to introduce the details of stationarity time and
stationarity bandwidth here. But it can be shown that for general LTV
channels, the _stationarity region_ defined by the both the _stationarity
time_ and _stationarity bandwidth_ are much larger than the _coherence region_
[70]. The intuition of this fact is not hard to understand. The stationarity
region quantifies the TF region where the channel can be roughly approximated
by a WSSUS channel, while the coherence region describes the TF region where
the channel response is roughly constant for a WSSUS channel. Therefore, the
coherence region must be no larger than the region where the channel is
roughly WSSUS by definition and consequently, there could be several coherence
regions with potentially different channel responses within one stationarity
region.

Remarks 2-1: OTFS modulation essentially takes advantage of the deterministic
DD domain channel response given in (2.6), where it is generally assumed that
the channel response holds for at least one block transmission [6]. However,
the channel response may change after several blocks’ transmission in
practical scenarios. But how often will the DD domain channel change is still
not fully understood in the literature. An intuitive understanding of this
question may be developed based on the stationarity region of the non-WSSUS
channels. The multi-path transmission comes from the signal reflections of
certain scatterers. In practical systems, the reflectivity for each scatter
could hold for several blocks of transmissions due to the fact that the change
of physical geometry of the channel is so slow compared to the speed of light.
Therefore, the channel responses for adjacent blocks may be quite similar
because they are very likely to be generated from the same scatterers. This
observation can be viewed as a sufficient condition for having the
stationarity region in the non-WSSUS channel. If the physical attributes of
the scatterers remain unchanged, the delay and Doppler responses associated to
different paths are roughly the same and therefore independent, then the
channel can be approximated as a WSSUS channel. The duration of this WSSUS
approximation may be dependent on the stationarity region, which is much
larger than the coherence region.

###  2.4 Sampling and Reconstruction

In practical systems, sampling is of great importance, because the baseband
equivalent system model essentially looks at the discrete channel response
instead of the continuous one. Therefore, it is necessary to discuss how to
sample the channel response such that the actual response can be reconstructed
with acceptable errors.

####  2.4.1 Sampling in the TF Domain

Conventional sampling usually takes place in the TF domain. An example could
be the well-known OFDM modulation. Without loss of generality, let us consider
the sampling using the _sinc_ function. Thus, the corresponding interpolation
can be described by [7]

$\displaystyle{\hat{h}_{{\rm{TF}}}}\left({t,f}\right)=\sum\limits_{k=-\infty}^{\infty}{\sum\limits_{l=-\infty}^{\infty}{{h_{{\rm{TF}}}}\left({kT,lF}\right)}}{\rm{sinc}}\left({\frac{\pi}{T}\left({t-kT}\right)}\right){\rm{sinc}}\left({\frac{\pi}{F}\left({f-lF}\right)}\right),$
(2.33)

where $T$ and $F$ are the sampling time and frequency, respectively. In order
to achieve the perfect reconstruction in the mean squared sense, the channel
scattering function ${C_{{\rm{DD}}}}\left({\tau;\nu}\right)$ must have a
compact support within the range of
$\left[{-{\tau_{\max}},{\tau_{\max}}}\right]\times\left[{-{\nu_{\max}},{\nu_{\max}}}\right]$
and the sampling time and frequency need to satisfy
$F\leq\frac{1}{{2{\tau_{\max}}}}$ and $T\leq\frac{1}{{2{\nu_{\max}}}}$ [7]. On
the other hand, if the above condition is not met,
${\hat{h}_{{\rm{TF}}}}\left({t,f}\right)$ and
${h_{{\rm{TF}}}}\left({t,f}\right)$ will inevitably be different due to the
aliasing. However, it is be shown that the normalized mean squared
reconstruction error can be upper-bounded by [7]

$\displaystyle\frac{1}{{\rho_{{\rm{PL}}}^{2}}}{\mathbb{E}}\left\\{{{{\left|{{{\hat{h}}_{{\rm{TF}}}}\left({t,f}\right)-{h_{{\rm{TF}}}}\left({t,f}\right)}\right|}^{2}}}\right\\}\leq
2\left({\sigma_{\tau}^{2}{F^{2}}+\sigma_{\nu}^{2}{T^{2}}}\right).$ (2.34)

Some interesting insights can be obtained based on this upper-bound. Firstly,
we notice that the sampling error relates to both the RMSs of the power-delay
and power-Doppler profiles. This observation suggests that the sampling error
is generally small for underspread channels, where the RMSs are small.
Secondly, we also notice that smaller sampling time and frequency will also be
helpful for an accurate channel reconstruction. Furthermore, it can be shown
from (2.34) that the upper-bound is minimized by choosing

$\displaystyle\frac{T}{F}=\frac{{{\sigma_{\tau}}}}{{{\sigma_{\nu}}}},$ (2.35)

and we have

$\displaystyle\frac{1}{{\rho_{{\rm{PL}}}^{2}}}{\mathbb{E}}\left\\{{{{\left|{{{\hat{h}}_{{\rm{TF}}}}\left({t,f}\right)-{h_{{\rm{TF}}}}\left({t,f}\right)}\right|}^{2}}}\right\\}\leq
2TF{\sigma_{\tau}}{\sigma_{\nu}}.$ (2.36)

Based on the above discussions, we summarize that the TF sampling could be
more accurate if the sampling time and frequency are sufficiently small and
the channel is more underspread.

####  2.4.2 Sampling in the DD Domain

Similar to the sampling in the TF domain, the channel response can be sampled
in the DD domain by certain pulses. However, direct sampling in the DD domain
has not been widely discussed in the literature. To the best knowledge of the
author, the DD domain sampling has only received much attention recently since
the invention of OTFS modulation. Conventionally, the information is carried
in the TF domain for most of the communication schemes. In this case, the
transmitted signal directly faces the TF domain channel, where the equivalent
discrete system model relies on the TF domain sampling of the channel
response. On the other hand, for OTFS systems, the information is carried in
the DD domain instead of the TF domain. Thus, the OTFS signal faces the DD
domain channel and the corresponding equivalent discrete system model needs to
be developed based on the DD domain sampling. As the DD domain sampling has a
close relationship to the pulse shaping for OTFS transmissions, we leave the
related discussions in the following chapter.

###  2.5 Summary of the Chapter

In this chapter, we have briefly summarized the characteristics and properties
of general wireless channels. Furthermore, we have discussed the mathematical
descriptions of wireless channels including both the deterministic channel
description and the stochastic description, where the difference and
connections between these two types of descriptions are discussed. We have
also introduced some key properties of the wireless channels, including the
WSSUS channels, non-WSSUS channels, and underspread channels. The discussions
for the related channels are also provided with the help of some examples.
Finally, we have briefly discussed the sampling of the channels in different
domains.

##  Chapter 3 OTFS Concepts and Principles

In this chapter, we present the concepts and principles of OTFS modulation [6,
23, 14]. Although OTFS modulation was only formally introduced in 2017, it has
stimulated a wide range of research from different viewpoints. In specific,
this chapter will discuss some of the key components of OTFS modulation in a
systematic way. It should be noted that OTFS modulation inherits the merits of
ZT/DZT by multiplexing the information symbols in the DD domain (Zak domain)
[6, 23, 14]. Therefore, before formally introducing the definition of OTFS
modulation, we will first review the definitions and properties of ZT and DZT
in the coming section. After discussing the related contexts of ZT and DZT, we
will introduce the concepts of OTFS modulation. It should be noted that there
are several implementations of OTFS modulation in the literature [6, 18, 26,
25, 19, 24]. Without loss of generality, we will only focus on one of the
popular implementations, namely DZT-based OTFS [25], and discuss the
corresponding system models111There are also other popular choices of OTFS
implementation, such as the OFDM-based OTFS [24]. However, the details of
those implementations are out of the scope of this thesis. .

###  3.1 A Summary on the Zak Transform and Discrete Zak Transform

The ZT [16] (also known as Weil-Brezin mapping and Gelfand mapping) was
originally introduced in the field of solid state physics and then studied by
Janssen from a signal processing point of view [17]. Later, some advanced
results of ZT and its corresponding discrete form were developed by Bölcskei
_et. al._ in [20, 21]. The ZT has a close relationship to the related concepts
of Gabor expansion, which is introduced in the coming subsection.

####  3.1.1 Gabor Expansion and Gabor Coefficients

In the pioneering paper of Gabor in 1946 [71], he suggested expanding signals
into TF shifted versions of an elementary Gaussian function (pulse). By doing
so, the TF plane is partitioned into rectangles of equal sizes, which is then
referred to as the _Gabor expansion_. The local content on each TF grid point
is called the _Gabor coefficient_ or the _expansion coefficient_ , which
relates to the way of sampling for the underlying signal. Let $T$ and $F$ be
the time-shift and frequency-shift parameters. With respect to the choices of
$T$ and $F$, there are generally three types of sampling are of interest,
namely, the critical sampling for $TF=1$, oversampling for $TF<1$, and
undersampling for $TF>1$, as indicated in Fig. 3.1 [20]. Critical sampling is
the most commonly used sampling technique. However, it may have poor numerical
stability [20], while the oversampling could have a better numerical stability
at the cost of redundant and non-unique Gabor coefficients222An example of
oversampling signals could be the type of faster-than-Nyquist signals [72, 73,
74, 75, 76, 77]. However, it should be noted that the DoF improvement of
faster-than-Nyquist signaling comes from the excess bandwidth of the signaling
pulse and a higher transmit symbol rate, instead of the oversampling itself at
the receiver side [72, 78, 79].. On the other hand, the Gabor expansion
generally does not exist for arbitrary signals in the case of undersampling
[20].

Figure 3.1: An illustration for Gabor expansions with critical sampling,
oversampling, and undersampling.

A frame of Gabor type is commonly referred to as the _Weyl-Heisenberg_ frame
and the corresponding Gabor expansion is a common tool for filterbank analysis
[20]. For example, the impulse responses of individual filters in filterbank
analysis are simply the modulated versions of the same signaling pulse.
Consequently, the resultant spectral decomposition is a “constant-bandwidth”
decomposition [20]. Some more details regarding these could be found in [20,
21]. In the context of Gabor expansion, the ZT is a commonly used signal
transform to obtain the Gabor expansion, which will be introduced in the
coming subsection.

####  3.1.2 Definitions of Zak Transform and Discrete Zak Transform

The ZT is a certain operation that takes a function of one variable as input
and produces a function of two variables as output. Several definitions of ZT
and DZT have appeared in the literature. In this thesis, we focus on the ones
in [17, 18, 20, 21, 80] as stated below.

Definition 3-1 (_Zak Transform[20, 17, 18, 21]_): Let $x\left(t\right)$ be a
complex time-continuous function and $T$ be a positive constant. Then, the ZT
is defined by

$\displaystyle{{\cal
Z}_{x}}\left({\tau,\nu}\right)\buildrel\Delta\over{=}\sqrt{T}\sum\limits_{k=-\infty}^{\infty}{x\left({\tau+kT}\right){e^{-j2\pi
k\nu T}}},$ (3.1)

for $-\infty<\tau<\infty$ and $-\infty<\nu<\infty$.

Definition 3-2 (_Discrete Zak Transform[21, 20, 17, 18]_): Let
$x\left[n\right]$ be a function of integer variable $n$ and $M$ and $N$ be two
positive constants. Then, the DZT is defined by

$\displaystyle{\cal D}{{\cal
Z}_{x}}\left[{l,k}\right]\buildrel\Delta\over{=}\frac{1}{{\sqrt{N}}}\sum\limits_{n=0}^{N-1}{x\left[{l+nM}\right]{e^{-j2\pi\frac{n}{N}k}}},$
(3.2)

for $0\leq l\leq M-1$ and $0\leq k\leq N-1$.

Figure 3.2: A graphical interpretation of the ZT.

As implied by Fig. 3.2, the ZT may be interpreted as the discrete-time Fourier
transform (DTFT) of a periodically sampled signal with sample period $T$ and
time shift $\tau$. In the context of ZT, the so-called “ _fundamental
rectangle_ " is a range of size $\tau\in\left[{0,T}\right)$ and
$\nu\in\left[{0,\frac{1}{T}}\right)$, which is used to sufficiently describe
the characteristics of ZT of the signal $x\left(t\right)$ [20]. The rationale
of this comes from the quasi-periodicity property of the ZT, which will be
explained in the coming subsection.

On the other hand, the DZT is the corresponding discrete form (discrete in
both delay and Doppler) of ZT with $T=1$, which is defined in the fundamental
rectangle. For completeness, we may assume that $x\left[n\right]$ is a
periodic sequence with a period of $MN$ and thus the definition of DZT is also
applicable outside the fundamental rectangle333It should be noted that such a
periodic assumption is usually enabled by adding CP in communication systems.
Some of the related discussions can be found in Section 3.2.1. [20]. In
specific, in some references, the DZT with an $MN$-periodic sequence
$x\left[n\right]$ is called “ _extended DZT_ ” [21]. Other than the ZT and
DZT, the discrete-time ZT (discrete in delay and continues in Doppler) is also
of importance for some applications [21]. However, the details of discrete-
time ZT are out of the scope of this thesis and we refer the interested
readers to [21] for more details.

####  3.1.3 Properties and Relations of the Zak Transform

In this subsection, we will summarize some useful properties of ZT based on
the results from [20, 6].

Proposition 3-1 (_Quasi-Periodicity[20]_): The ZT is quasi-periodic along the
delay axis with period $T$ and periodic along the Doppler axis with period
${\frac{1}{T}}$, i.e.,

$\displaystyle{{\cal Z}_{x}}\left({\tau+T,\nu}\right)={e^{j2\pi T\nu}}{{\cal
Z}_{x}}\left({\tau,\nu}\right),$ (3.3)

and

$\displaystyle{{\cal Z}_{x}}\left({\tau,\nu+\frac{1}{T}}\right)={{\cal
Z}_{x}}\left({\tau,\nu}\right).$ (3.4)

Proof: The related proofs can be directly obtained by based on the definition
of ZT in Definition 3-1. $\blacksquare$

As implied by Proposition 3-1, the fundamental rectangle of size
$\tau\in\left[{0,T}\right)$ and $\nu\in\left[{0,\frac{1}{T}}\right)$ is
sufficient to describe the characteristics of signal $x\left(t\right)$ [20],
because the DD representations of the signal outside of the fundamental
rectangle are determined by periodically extending the representations within
the fundamental rectangle.

Proposition 3-2 (_Shift Properties[20]_): Given the original signal
$x\left(t\right)$, and its time-delayed version
${x_{1}}\left(t\right)=x\left({t-{\tau_{0}}}\right)$ and frequency-shifted
version ${x_{2}}\left(t\right)=x\left(t\right){e^{j2\pi{\nu_{0}}t}}$, the
following holds

$\displaystyle{{\cal Z}_{{x_{1}}}}\left({\tau,\nu}\right)={{\cal
Z}_{x}}\left({\tau-\tau_{0},\nu}\right),$ (3.5)

and

$\displaystyle{{\cal
Z}_{{x_{2}}}}\left({\tau,\nu}\right)={e^{j2\pi{\nu_{0}}\tau}}{{\cal
Z}_{x}}\left({\tau,\nu-{\nu_{0}}}\right).$ (3.6)

Proof: The related proofs can be directly obtained by based on the definition
of ZT in Definition 3-1. $\blacksquare$

Proposition 3-3 (_Inverse Zak Transform[20]_): Given a signal
$x\left(t\right)$, whose ZT is given by ${{\cal
Z}_{x}}\left({\tau,\nu}\right)$, the inverse ZT (IZT) is represented by

$\displaystyle x\left(t\right)={\cal IZT}\left({{{\cal
Z}_{x}}\left({t,\nu}\right)}\right)\buildrel\Delta\over{=}{\sqrt{T}}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({t,\nu}\right)}{\rm{d}}\nu.$ (3.7)

Proof:

$\displaystyle{\sqrt{T}}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({t,\nu}\right)}{\rm{d}}\nu=$ $\displaystyle
T\int_{0}^{\frac{1}{T}}{\sum\limits_{k=-\infty}^{\infty}{x\left({t+kT}\right){e^{-j2\pi
k\nu T}}}}{\rm{d}}\nu$ $\displaystyle=$ $\displaystyle
T\sum\limits_{k=-\infty}^{\infty}{x\left({t+kT}\right)\int_{0}^{\frac{1}{T}}{{e^{-j2\pi
k\nu T}}}}{\rm{d}}\nu.$ (3.8)

By noticing $\int_{0}^{\frac{1}{T}}{{e^{-j2\pi k\nu
T}}}{\rm{d}}\nu=\frac{1}{T}\delta\left[k\right]$, (3.7) holds. $\blacksquare$

Proposition 3-4 (_Connections to Fourier transform[20]_): Given a signal
$x\left(t\right)$, whose ZT and Fourier transform are given by ${{\cal
Z}_{x}}\left({\tau,\nu}\right)$ and $X\left(f\right)$, respectively, the
following holds

$\displaystyle X\left(f\right)=\frac{1}{{\sqrt{T}}}\int_{0}^{T}{{{\cal
Z}_{x}}\left({\tau,f}\right){e^{-j2\pi f\tau}}}{\rm{d}}\tau.$ (3.9)

Proof:

$\displaystyle\frac{1}{{\sqrt{T}}}\int_{0}^{T}{{{\cal
Z}_{x}}\left({\tau,f}\right){e^{-j2\pi f\tau}}}{\rm{d}}\tau$ $\displaystyle=$
$\displaystyle\int_{0}^{T}{\sum\limits_{k=-\infty}^{\infty}{x\left({\tau+kT}\right){e^{-j2\pi\left({\tau+kT}\right)f}}}}{\rm{d}}\tau=\int_{-\infty}^{\infty}{x\left(t\right){e^{-j2\pi
ft}}}{\rm{d}}t=X\left(f\right).$ (3.10)

$\blacksquare$

Proposition 3-5 (_Multiplication Property[19]_): Let
$z\left(t\right)=x\left(t\right)y\left(t\right)$ be the modulated signal of
the signal $x\left(t\right)$ with respect to $y\left(t\right)$. Then, the ZTs
of $z\left(t\right)$, $x\left(t\right)$, and $y\left(t\right)$ satisfy

$\displaystyle{{\cal
Z}_{z}}\left({\tau,\nu}\right)={\sqrt{T}}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({\tau,\nu^{\prime}}\right){{\cal
Z}_{y}}\left({\tau,\nu-\nu^{\prime}}\right)}d\nu^{\prime}.$ (3.11)

Proof:

$\displaystyle{{\cal Z}_{z}}\left({\tau,\nu}\right)$
$\displaystyle=\sqrt{T}\sum\limits_{n=-\infty}^{\infty}{x\left({\tau+nT}\right)y\left({\tau+nT}\right)}{e^{-j2\pi
n\nu T}}$
$\displaystyle=\sqrt{T}\sum\limits_{n=-\infty}^{\infty}{\sqrt{T}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({\tau+nT,\nu^{\prime}}\right)}d\nu^{\prime}y\left({\tau+nT}\right)}{e^{-j2\pi
n\nu T}}$
$\displaystyle=\sqrt{T}\sum\limits_{n=-\infty}^{\infty}{\sqrt{T}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({\tau,\nu^{\prime}}\right)}d\nu^{\prime}y\left({\tau+nT}\right)}{e^{-j2\pi
n\left({\nu-\nu^{\prime}}\right)T}}$
$\displaystyle=\sqrt{T}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({\tau,\nu^{\prime}}\right)}d\nu^{\prime}\sqrt{T}\sum\limits_{n=-\infty}^{\infty}{y\left({\tau+nT}\right)}{e^{-j2\pi
n\left({\nu-\nu^{\prime}}\right)T}}$
$\displaystyle=\sqrt{T}\int_{0}^{\frac{1}{T}}{{{\cal
Z}_{x}}\left({\tau,\nu^{\prime}}\right){{\cal
Z}_{y}}\left({\tau,\nu-\nu^{\prime}}\right)}d\nu^{\prime}$ (3.12)

$\blacksquare$

Proposition 3-6 (_DD Domain Twisted Convolution[6]_): Let ${{\cal
Z}_{r}}\left({\tau,\nu}\right)$ be the ZT of the received signal
$r\left(t\right)$ and ${{\cal Z}_{s}}\left({\tau,\nu}\right)$ be the ZT of the
transmitted signal $s\left(t\right)$. Then, with the DD domain channel
response ${h_{{\rm{DD}}}}\left({\tau,\nu}\right)$, the DD domain twisted
convolution can be derived according to (2.8) and (2.9) by,

$\displaystyle{{\cal
Z}_{r}}\left({\tau,\nu}\right)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{\cal
Z}_{s}}\left({\tau-\tau^{\prime},\nu-\nu^{\prime}}\right)}}{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}\left(\tau-\tau^{\prime}\right)}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}.$
(3.13)

or

$\displaystyle{{\cal
Z}_{r}}\left({\tau,\nu}\right)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{\cal
Z}_{s}}\left({\tau-\tau^{\prime},\nu-\nu^{\prime}}\right)}}{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}\tau}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}.$
(3.14)

Proof: According to (3.1) and (2.8), we have

$\displaystyle{{\cal Z}_{r}}\left({\tau,\nu}\right)$
$\displaystyle=\sqrt{T}\sum\limits_{k=-\infty}^{\infty}{r\left({\tau+kT}\right){e^{-j2\pi
k\nu T}}}$
$\displaystyle=\sqrt{T}\sum\limits_{k=-\infty}^{\infty}{\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}\left({\tau+kT-\tau^{\prime}}\right)}}s\left({\tau+kT-\tau^{\prime}}\right){e^{-j2\pi
k\nu T}}}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}}$
$\displaystyle=\sqrt{T}\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\sum\limits_{k=-\infty}^{\infty}{s\left({\tau-\tau^{\prime}+kT}\right)}}}{e^{-j2\pi
k\left({\nu-\nu^{\prime}}\right)T}}{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}\left({\tau-\tau^{\prime}}\right)}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}$
$\displaystyle=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{\cal
Z}_{s}}\left({\tau-\tau^{\prime},\nu-\nu^{\prime}}\right)}}{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}\left({\tau-\tau^{\prime}}\right)}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}.$
(3.15)

On the other hand, based on (3.1) and (2.8), we have

$\displaystyle{{\cal Z}_{r}}\left({\tau,\nu}\right)$
$\displaystyle=\sqrt{T}\sum\limits_{k=-\infty}^{\infty}{r\left({\tau+kT}\right){e^{-j2\pi
k\nu T}}}$
$\displaystyle=\sqrt{T}\sum\limits_{k=-\infty}^{\infty}{\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}\left({\tau+kT}\right)}}s\left({\tau+kT-\tau^{\prime}}\right){e^{-j2\pi
k\nu T}}}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}}$
$\displaystyle=\sqrt{T}\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\sum\limits_{k=-\infty}^{\infty}{s\left({\tau-\tau^{\prime}+kT}\right)}}}{e^{-j2\pi
k\left({\nu-\nu^{\prime}}\right)T}}{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}{\tau}}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}$
$\displaystyle=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{{{\cal
Z}_{s}}\left({\tau-\tau^{\prime},\nu-\nu^{\prime}}\right)}}{h_{{\rm{DD}}}}\left({\tau^{\prime},\nu^{\prime}}\right){e^{j2\pi\nu^{\prime}{\tau}}}{\rm{d}}\tau^{\prime}{\rm{d}}\nu^{\prime}.$
(3.16)

$\blacksquare$

Note that the two interpretations of the twisted convolution properties of the
ZT align with the two interpretations of the DD domain channel input-output
relationships in (2.8) and (2.9), where the channel performs the delay
operation and the Doppler operation in different orders.

####  3.1.4 Properties and Relations of the Discrete Zak Transform

In this subsection, we will summarize some useful properties of DZT based on
the results from [21, 20], where we assume that $x\left[n\right]$ is an
$MN$-periodic sequence for the completeness of definition. As the following
propositions are the discrete representations of the corresponding
propositions of ZT, we omit some of the related proofs here and refer the
interested readers to [21, 20] for more details.

Proposition 3-7 (_Quasi-Periodicity[21]_): The DZT is quasi-periodic along the
delay axis with period $M$ and periodic along the Doppler axis with period
$N$, i.e.,

$\displaystyle{\cal D}{\cal
Z}_{x}\left[{l+M,k}\right]={e^{j2\pi\frac{k}{N}}}{\cal D}{\cal
Z}_{x}\left[{l,k}\right],$ (3.17)

and

$\displaystyle{\cal D}{\cal Z}_{x}\left[{l,k+N}\right]={\cal D}{\cal
Z}_{x}\left[{l,k}\right].$ (3.18)

Proposition 3-8 (_Shift Properties_): Given the $MN$-periodic
$x\left[n\right]$, and its time-delayed version
${x_{1}}\left[n\right]=x\left[{n-{l_{0}}}\right]$ and frequency-shifted
version
${x_{2}}\left[n\right]=x\left[n\right]{e^{j2\pi\frac{{{k_{0}}}}{{MN}}n}}$, the
following holds

$\displaystyle{{\cal DZ}_{{x_{1}}}}\left[{l,k}\right]={{\cal
DZ}_{x}}\left[{l-l_{0},k}\right],$ (3.19)

and

$\displaystyle{{\cal
DZ}_{{x_{2}}}}\left[{l,k}\right]={e^{j2\pi\frac{{{k_{0}}l}}{{MN}}}}{{\cal
DZ}_{x}}\left[{l,k-{k_{0}}}\right].$ (3.20)

Proposition 3-9 (_Inverse Discrete Zak Transform[21]_): Given the
$MN$-periodic sequence $x\left[n\right]$, whose DZT is given by ${{\cal
DZ}_{x}}\left[{l,k}\right]$, the inverse DZT (IDZT) is represented by

$\displaystyle x\left[n\right]={\cal IDZT}\left({{{\cal
DZ}_{x}}\left[{n,k}\right]}\right)\buildrel\Delta\over{=}\frac{1}{{\sqrt{N}}}\sum\limits_{k=0}^{N-1}{{{\cal
DZ}_{x}}\left[{n,k}\right]}.$ (3.21)

Proof:

$\displaystyle\frac{1}{{\sqrt{N}}}\sum\limits_{k=0}^{N-1}{{{\cal
DZ}_{x}}\left[{n,k}\right]}$
$\displaystyle=\frac{1}{N}\sum\limits_{k=0}^{N-1}{\sum\limits_{l=0}^{N-1}{x\left[{n+lM}\right]{e^{-j2\pi\frac{l}{N}k}}}}$
$\displaystyle=\frac{1}{N}\sum\limits_{l=0}^{N-1}{x\left[{n+lM}\right]\sum\limits_{k=0}^{N-1}{{e^{-j2\pi\frac{l}{N}k}}}}$
$\displaystyle=\frac{1}{N}\sum\limits_{l=0}^{N-1}{x\left[{n+lM}\right]N\sum\limits_{k=-\infty}^{\infty}{\delta\left[{l-kN}\right]}}$
$\displaystyle=\sum\limits_{l=0}^{N-1}{\sum\limits_{k=-\infty}^{\infty}{x\left[{n+kMN}\right]}}=x\left[n\right].$
(3.22)

$\blacksquare$

Proposition 3-10 (_Connections to Discrete Fourier transform[21]_): Given a
$MN$-periodic $x\left[n\right]$, whose DZT and discrete Fourier transform
(DFT) are given by ${{\cal DZ}_{x}}\left[{l,k}\right]$ and $X\left[k\right]$,
respectively, the following holds

$\displaystyle X\left[k\right]=\sqrt{N}\sum\limits_{l=0}^{M-1}{{{\cal
DZ}_{x}}\left[{l,k}\right]}{e^{-j2\pi\frac{{kl}}{N}}},$ (3.23)

for $0\leq k\leq K-1$.

Proof: The proof relates to the decimation-in-time fast Fourier transform
(FFT) algorithm and the Cooley-Tukey algorithm [21]. Some details could be
found in [21] and [20]. $\blacksquare$

Proposition 3-11 (_Multiplication Property[19]_): Let
$z\left[n\right]=x\left[n\right]y\left[n\right]$ be the modulated signal of
the $MN$-periodic sequence $x\left[n\right]$ with respect to
$y\left[n\right]$. Then, the DZTs of $z\left[n\right]$, $x\left[n\right]$, and
$y\left[n\right]$ satisfy

$\displaystyle{\cal D}{{\cal
Z}_{z}}\left[{l,k}\right]=\frac{1}{{\sqrt{N}}}\sum\limits_{k^{\prime}=0}^{N-1}{{\cal
D}{{\cal Z}_{x}}\left[{l,k^{\prime}}\right]{\cal D}{{\cal
Z}_{y}}\left[{l,k-k^{\prime}}\right]}.$ (3.24)

Proof:

$\displaystyle{\cal D}{{\cal Z}_{z}}\left[{l,k}\right]=$
$\displaystyle\frac{1}{{\sqrt{N}}}\sum\limits_{n=0}^{N-1}{x\left[{l+nM}\right]}y\left[{l+nM}\right]{e^{-j2\pi\frac{n}{N}k}}$
$\displaystyle=$