1. Introduction
Motivated by challenges encountered in wireless communication over time-variant channels, such as Doppler dispersion or equalization, a new modulation technique termed orthogonal time frequency space (OTFS) was introduced in [
1]. The driving idea behind OTFS is to utilize the delay-Doppler (DD) domain to represent information-carrying symbols. The interaction of the corresponding OTFS waveform with a time-frequency (TF) dispersive channel results in a two-dimensional convolution of the symbols in the DD domain ([
2], [Section III-A]). OTFS utilizes the time-invariant channel interaction in the DD domain and outperforms orthogonal frequency division multiplexing (OFDM) in high-mobility scenarios, as shown in [
1,
2,
3,
4,
5,
6], making it an ideal waveform candidate for 6G.
Most of the literature on OTFS considers OTFS as a pre- and postprocessing technique for OFDM systems, as described in [
3,
5,
7]. However, the
continuous Zak transform provides a more fundamental relationship between the DD and time domain, as pointed out in [
2] and studied in [
8]. In principle, OTFS describes a time domain signal by its DD representations in a similar way to OFDM, which defines a signal in the TF domain. The difference between the DD and TF domains is that the TF domain allows a continuous-time signal to be described by a discrete number of coefficients in the TF domain [
9]. On the other hand, the
continuous Zak transform maps a continuous-time signal to continuous values in the Zak domain. In [
8], a discretization of the Zak representation was achieved using time and bandwidth limitations on the signal, represented by a point in the DD domain. However, depending on the domain of the signal under study, different variants of the Zak transform exists. The discrete-time version is referred to as the discrete-time Zak transform (DTZT) and the discrete (and finite) version is the discrete Zak transform (DZT) [
10]. The DTZT is discrete in the delay and continuous in the frequency domain, while the DZT is discrete in both the delay and Doppler domains. Thus, an alternative description of OTFS can be provided by the DZT, as we show in this work.
Another motivation for using the DZT can be found by considering OFDM. The fundamental concept of OFDM, that is, mapping symbols onto a set of orthogonal signals in the frequency domain, dates back to 1966 [
11]. The success of OFDM is based on its efficient
digital implementation to compute the discrete Fourier transform (DFT) [
12]. Equivalently, OTFS can be efficiently implemented using the
discrete Zak transform (DZT). The DZT itself is based on the DFT, which allows for efficient implementation as well. Implementations of OTFS which resemble the DZT have been studied previously, in [
13], for example. However, the proposed systems is based on OFDM that adds a cyclic prefix (CP) to every OFDM symbol. The CP adds additional signaling overhead and results in a different channel interaction in the DD domain.
DZT-based OTFS is closely related to radar processing in a pulse Doppler radar. A pulse radar transmits a pulse train with uniformly spaced and identical pulses. Target motion introduces a phase shift for each pulse, which is utilized at the receiver to extract the velocity information of a radar target. To this end, the sampled signal is arranged in a two-dimensional grid, and a DFT is applied along the so-called slow time to extract the velocity information of a target; see ([
14], [Chapter 17]) or ([
15], [Chapter 3]) for details. This variant of Doppler processing is equivalent to the DZT. Similarly, the radar transmitter of such a pulse Doppler radar can be described by the inverse DZT, as demonstrated in [
16]. The close connection to radar makes OTFS an ideal waveform for joint communication and sensing, which has been explored by [
6], among others.
A fundamental treatment of OTFS based on the DZT is currently absent from the literature. The aim of this work is to close this gap in the literature by providing a complete treatment of OTFS based solely on the DZT. Therefore, we discuss the DZT and its properties, then we derive the input–output relationship for TF dispersive channels in the DD using the DZT and its properties. Our DZT-based approach provides an intuitive understanding of OTFS and drastically simplifies its analysis. Based on our analysis, we further show that the capacity in the DD domain is equivalent to the capacity of the time-variant channel in the time domain (Parts of this work were presented at the 2022 IEEE International Conference on Communications Workshops (ICC Workshops) [
17]).
The remainder of the paper is organized as follows. In
Section 2, we provide an introduction to the DZT covering all properties needed for OTFS. The signal model based on the
is described in
Section 3. Based on the presented signal model, we further establish the input–output relationship of OTFS based on the DZT in
Section 4. In
Section 5, we establish the connection between the DD and the TF domain, which allows the implementation of OTFS by an OFDM system. In
Section 6, we demonstrate that operating in the DD incurs no loss in capacity. Finally, our conclusions are presented in
Section 7.
4. Delay Doppler Input–Output Relationship
To express the input–output relationship in the DD domain for the system presented in
Figure 5, we first note that the DZT is a linear transform; as such, we can write the DZT of (
44) as
where is the DZT of sequence
described in (
45) and
is the DZT of the noise. The elements of
are i.i.d. zero-mean Gaussian random variables with variance
. This follows from the fact that the DZT is a unitary transform ([
10], Section VI).
For the signal model of a single reflector in (
45), we provide the following result for the input–output relationship in the DD domain for the OTFS system described in
Section 3.
Theorem 1.
Considering the fundamental rectangle of complex symbols in the DD domain, the input–output relation for OTFS transmission over a time-frequency selective channel for a single reflector iswhere and are the delay and Doppler spreading functions, respectively. The delay spreading function is the DZT of the shifted and sampled impulse in (43), and the Doppler spreading functions is provided as follows: Proof. To illustrate the spreading of a single symbol in the DD domain, we consider the following example. Let
and
The fundamental rectangle with the only nonzero element is presented in
Figure 6a. Furthermore, assume that
and
. Note that this example causes the maximum spread of a single symbol in the DD domain. We can visualize the spreading of the symbol defined in (
49) in two steps. Therefore, we define
as the DZT resulting from the inner convolution in (
47), presented in
Figure 6b, with respect to the Doppler index
k. The resulting spread of the nonzero symbol is visualized in
Figure 6c. Finally, the symbol that has been spread in the Doppler domain is spread in the delay domain by the delay spreading function
, which is illustrated in
Figure 6d. Note that due to the limited support of
(see (
43)), the magnitude of
is independent of the index
k. The resulting spread of the nonzero symbol in the DD domain is shown in
Figure 6e.
For the particular case of
with
and
with
,
simplifies to
i.e., the received symbols
are in the DD domain displaced symbols
.
Theorem 1 shows that the channel interaction with the symbols in the DD domain is time-invariant, neglecting the additional phase terms due to the quasi-periodicity and modulation. The invariance is helpful in the detection of the symbols. Consider a TDL-C channel with a delay spread of 300 ns, a carrier frequency of 4 GHz, and a maximum velocity of 120 kmph. Furthermore, assume an OTFS system with
and
and
MHz. The channel response
in the DD domain is illustrated in
Figure 7a. The magnitude of this channel stays approximately constant throughout the entire transmission of an OTFS frame.
Figure 7b illustrates the equivalent OFDM channel. The variation of the channel along the subcarrier index k as well along the time index n can be seen. To keep track of the channel, additional pilots need to be used, and these cannot be used for communication.
In addition to constant channel interaction, OTFS offers the advantage of a concise and sparse channel description compared to OFDM. In an OFDM system, the channel coefficient for each subcarrier must be estimated for subsequent symbol detection. In contrast, for symbol detection in an OTFS system, knowledge of the interference introduced by each reflector is sufficient. The sparsity can be seen in
Figure 7; the support of
is limited to a small area, while the channel transfer function changes with each subcarrier and time index, that is,
l and
m, respectively.
Remark 4.
The discrete two-dimensional convolution in (46) can be equivalently expressed in the formwhere , , and are the vectorized DTZs , , and , respectively. The vectors are all of length . The matrix describes the intersymbol interference in the DD domain. Because and have small support in the DD domain, the corresponding matrix is sparse. The matrix-vector formulation of the input–output relationship is the basis for many works on OTFS; for example, see [5,6]. 5. OTFS Overlay for OFDM
Currently, orthogonal frequency division multiplexing (OFDM) is the dominant modulation scheme in wireless communication. For example, it is used in 5G and in several 802.11 standards. This section shows that DFT-based ODFM can be used for OTFS modulation and demodulation. In this context, OTFS is considered a pre- and postprocessing step for the OFDM system.
To derive the pre- and postprocessing step, we first derive an alternative way to compute the DZT. For this purpose, we consider (
27). If we choose the sequence
y such that its DZT
, then we can obtain the DZT
through the right-hand side of (
27). The
N periodic sequence
y with DZT
is
With this particular choice of
y, we recognize the inner product on the right-hand side of (
27) as
which is the scaled
L-point DFT of the samples
for
. If we define
for
and
, then the DZT of
x is obtained through
i.e., by the SFFT of the coefficients
. Note that the set
represents the Gabor expansion coefficients for the choice of a rectangular analysis window (see [
25], Section 4), and thus a mixed TF representation of the sequence
x.
The coefficients
, on the other hand, are obtained from
using (
25):
The samples of the sequence
x for
are obtained as follows:
which is the
L-point IDFT of the coefficients
am,l for a fixed
m. Thus, the DZT (IDZT) can be implemented by consecutive execution of the DFT (IDFT) and the SFFT (ISFFT).
The above-described two-step approach for the calculation of the DZT and IDZT can be used to implement OTFS using OFDM hardware, which is typically based on the IDFT/DFT (see ([
26], Section 19.3), ([
23], Section 6.4.2), ([
27] Section 12.4.3), or ([
28], Section 4.6)) by extending the transmitter and receiver by the ISFFT and SFFT, respectively. The coefficients
am,l then represent the coefficient in the TF domain. The index
m refers to the
mth OFDM symbol in the time domain, and
l is the corresponding subcarrier index. Note that for the DZT, the parameter
L the grid size in the delay domain. For DFT-SFFT implementation, on the other hand,
L defines DFT size, which defines the number of points in the frequency domain. Thus, an
grid in the DD domain translates to a
grid in the TF domain.
Remark 5.
In CP-OFDM, a CP is added for each OFDM symbol by copying the last O samples of an OFDM symbol and inserting them in front of the corresponding OFDM symbol with length L. This symbol-wise CP is not required in the OFDM implementation of OTFS. Instead, a single CP is added by copying the last O samples of the entire sequence and inserting them in front of the sequence.
6. DD Channel Capacity
The input–output relationship in (
41) is equivalently expressed as
where
is the time-variant multi-tap channel response at time instance
n and
is the support of
in
m. This channel response is deterministic and periodic (considering
) with some finite period
M, i.e.,
for any
and
. Upon using the channel
N times, the input output relationship can be written in the following vector form:
where
is the input block,
is the corresponding output block,
is the block of noise samples (all column vectors), and
is the channel (convolution) matrix constructed from the time-varying channel response
.
The above channel can be shown to be
information-stable (see Section 3.9 in [
29]); hence, its capacity is provided by the following multi-letter limiting expression [
30]:
where
is the multi-letter input distribution for block length
N. For each block length
N, the corresponding mutual information term in (
60) is maximized by a Gaussian input [
31]; hence, the capacity is provided by
Let
be the SVD of
. Then, the optimal input covariance matrix is provided by
, where
is a diagonal matrix obtained using water-filling [
31]. The capacity-achieving strategy is characterized by a sequence
.
In case we do not wish to use the channel response matrix in the construction of input sequences, we may add the restriction that the multi-letter input distribution must be isotropic. In this case, we simply have
, and the capacity is provided by
It is evident that
is achieved by any input of the form
, where
is a set of orthonormal basis (i.e.,
) and
is a vector of zero-mean i.i.d. Gaussian symbols with covariance
. As shown in
Section 2, the set of sequence
forms an orthonormal basis. Thus, the capacity of the DD channel is provided by (
62).
7. Conclusions
In this work, we have presented an OTFS based on the discrete Zak transform. The discrete Zak transform-based description allows for an efficient digital implementation of OTFS. Furthermore, we derived the input–output relation for the symbols in the delay-Doppler domain solely based on discrete Zak transform properties, which provides a concise description of OTFS compared to the pre- and postprocessing approaches for OFDM.
Our presented discrete Zak transform approach can be used to study and evaluate OTFS from a different perspectives, potentially leading to OTFS performance improvements. For example, considering Nyquist pulses with larger roll-off factors allows the interference in the delay domain to be controlled. Additionally, applying windows to the subsampled sequences of the DZT reduces the interference in the Doppler domain.