Orthogonal Delay-Doppler Division Multiplexing (ODDM)

In digital communications, the role of a modulation scheme is to map digital information onto an analog waveform that matches the characteristics of the physical channel [1].

    Let $f_c$ and $\theta_c$ be the carrier frequency and phase, respectively. Typically, a real-valued passband digital modulation waveform with symbol index $i\in \mathbb Z$ is represented as: \begin{equation} x_{pb}(t)=\sum_i \sqrt{2} A_i \cos(2\pi (f_c+f_i(t))t+\theta_c+\theta_i)p_i(t), \end{equation} where the amplitude $A_i$, the phase $\theta_i$, the frequency (possibly time-varying) $f_i(t)$ and the finite duration pulse $p_i(t)$ can be used for modulation. The corresponding complex-valued baseband waveform (assuming ideal carrier synchronization) is given by $x(t)=\sum_i x_i(t)$, where the $i$th symbol \begin{equation}\label{x_it} x_i(t)=A_i e^{j\theta_i} e^{j2\pi f_i(t) t}p_i(t)\triangleq X[i]g_i(t) \end{equation} consists of two components: an information-bearing digital symbol (complex number) $X[i]=A_i e^{j\theta_i}$, drawn from a signaling alphabet, and a finite-energy, continuous-time function $g_i(t)=e^{j2\pi f_i(t) t}p_i(t)$, referred to as transmit (modulating) pulse. As $x_i(t)$ can be generated by feeding $X[i]\delta(t)$ into a filter with impulse response $g_i(t)$, $g_i(t)$ is also called a transmit filter. In the receiver, a correlator performing cross-correlation with a receive pulse $\gamma_i(t)$ is usually used to extract the $i$th digital symbol as $Y[i]=\int_{-\infty}^{+\infty} y(t)\gamma_i^*(t)dt$, where $y(t)$ is the received waveform. Similarly, because the correlator can be implemented by passing $y(t)$ through a matched filter $\gamma_i^\ast(-t)$ and then sampling the filter output at an appropriate time [1] [2], $\gamma_i^*(-t)$ is called a receive filter.

    Since $x(t)$ is composed of digitally modulated pulses, transmit pulses $\{g_i(t), i\in \mathbb Z\}$ play a fundamental role in waveform design and are essential to defining the waveform. To avoid inter-symbol interference (ISI), the pulses are required to be mutually orthogonal. Hence, designing a digital modulation waveform is essentially equivalent to determining a set of orthogonal pulses, also referred to as basis functions.

Naturally, the first step is to determine the channel characteristics to which the orthogonal pulses are intended to match. For a linear time-invariant (LTI) channel, the primary characteristic is the available bandwidth $\mathbb B$. In single-carrier (SC) modulation with $f_i(t)= 0$, each transmit pulse occupies the entire bandwidth and is multiplexed in time according to a symbol interval not less than $\frac{1}{\mathbb B}$ or equivalently a symbol rate not greater than $\mathbb B$ to achieve the orthogonality. Examples, including Nyquist and root Nyquist pulses whose Nyquist interval is the symbol interval, are essentially band-limited impulses. The channel’s time dispersion generally disrupts the orthogonality among these pulses and introduce ISI into the correlator’s output $Y[i]$. Consequently, a digital channel equalizer is required to recover $X[i]$.

Considering their own orthogonality and simple scaled channel output, a better choice of pulses may be the eigenfunctions of LTI channels, namely complex sinusoids $e^{j2\pi f t}, \forall f, -\infty < t < +\infty$, which obviously must be truncated for practical use. Given a truncation window function or equivalently a prototype pulse $g(t)$, to retain orthogonality among truncated complex sinusoids, we can set $f$ to be integer multiples of a fundamental frequency $\Delta F$, and obtain multiple pulses multiplexed in frequency according to $\Delta F$. Further considering a time interval $\Delta T$ for time-multiplexing, we can replace the symbol index $i$ with a time index $m$ and a frequency index $n$, and let $f_i(t)=n\Delta F$ to obtain a time-frequency (TF) two-dimensional (2D) multiplexing waveform \begin{equation}\label{xt} x(t)=\sum_{m=0}^{M-1}\sum_{n=-N/2}^{N/2-1} X[m,n]\overbrace{e^{j2\pi n\Delta F (t-m\Delta T)}g(t-m\Delta T)}^{g_{m,n}(t)}, \end{equation} which is popularly known as multicarrier (MC) modulation, where $N$ and $M$ are the numbers of complex sinusoids and MC symbols, respectively. Since $e^{j2\pi n\Delta F t}$ is often called subcarrier or tone, the transmit pulses $g_{m,n}(t)$ in (\ref{xt}) are truncated or pulse-shaped (PS) subcarriers, corresponding to rectangular or non-rectangular $g(t)$, respectively. Then, the channel bandwidth is matched by \begin{equation} B_x = (N-1)\times \Delta F+B_g\le \mathbb B \label{B_x}, \end{equation} where $B_x$ and $B_g$ are the bandwidths of $x(t)$ and $g(t)$, respectively. Note that the bandwidth of a finite duration signal can be reasonably defined by disregarding its negligibly small high-frequency components.

The symbol interval $\Delta T$ and the fundamental frequency (a.k.a. subcarrier spacing) $\Delta F$ form a TF grid structure in the TF domain, where the TF resolution $(\Delta T, \Delta F)$ is meant to represent the minimum time and frequency "distances" among the pulses - not their actual duration and bandwidth. For such a TF domain MC (TFMC) waveform, we have a well-known design framework based on the Weyl-Heisenberg (WH) frame theory [3], because $g_{m,n}(t)=g(t-m\Delta T)e^{j2\pi n\Delta F (t-m\Delta T)}$ in (\ref{xt}) is exactly the WH or Gabor function widely adopted in TF signal analysis [4].

    A WH set $\left\{\Delta T, \Delta F, g(t)\right \}$ is said to be orthogonal given the inner product $\langle g_{m,n}(t), g_{\dot m, \dot n}(t)\rangle=\delta (m-\dot m)\delta (n-\dot n)$. Let another WH function $r_{m,n}(t)=r(t-m\Delta T)e^{j2\pi n\Delta F (t-m\Delta T)}$ be the corresponding receive pulse. A pair of WH sets $\left\{\Delta T, \Delta F, g(t)\right \}$ and $\left\{\Delta T, \Delta F, r(t)\right \}$ is said to be biorthogonal when $\langle g_{m,n}(t), r_{\dot m, \dot n}(t)\rangle=\delta (m-\dot m)\delta (n-\dot n)$. According to the WH frame theory, when the joint TF resolution $\Delta R=\Delta T \Delta F$ is not less than $1$, we can find (bi)orthogonal WH sets to assemble the MC waveform. Then, the eigenfunction-based transmission for LTI channels can be realized [2] to have $Y[n,m]\approx H(n\Delta F) X[n,m]$, where $H(n\Delta F)$ is the corresponding channel frequency response, and subsequently significantly simplify the channel equalizer.

    Let $\Pi_{\mathcal T}(t)$ stand for the rectangular pulse of length $\mathcal T$, while $T_g$ and $T_{cds}$ denote the duration of $g(t)$ and the channel delay spread, respectively. The most well-known of such MC schemes is orthogonal frequency division multiplexing (OFDM) [2] [5], where $\Delta T =T_g\ge T_{sp}+T_{cds}, g(t)=\Pi_{T_g}(t), r(t)=\Pi_{T_{sp}}(t)$, and $T_{sp}=\frac{1}{\Delta F}$ is known as symbol period. Meanwhile, the WH frame theory also shows that there is no (bi)orthogonal WH set when $\Delta R<1$.

On the other hand, a classic measurement of TF occupancy of $g(t)$ is its TF area (TFA) given by $A_g=\alpha T_g \alpha B_g$, where $\alpha T_g$ and $\alpha B_g$ are the standard deviations of $g(t)$ and its Fourier transform $G(f)$, respectively. Since $g(t)$ and $G(f)$ are related via the Fourier transform, the TF domain for pulse design is an interdependent 2D domain, and is fundamentally different from that for assembling the MC waveform, which has a TF grid structure to translate a prototype pulse already available. According to the uncertainty principle, the TFA obeys a lower bound $A_g \ge \frac{1}{4\pi}$ called the Gabor limit [6]. Therefore, there is no 2D impulse that is "narrow" in time and frequency simultaneously (TF "narrow"), to be confined in a TF grid with small $\Delta T$ and $\Delta F$.

The WH frame theory and the uncertainty principle imply that an MC modulation having a fine TF resolution may not be possible because of the lack of pulse.

For linear time-varying (LTV) channels, similarly, the characteristics for pulse design must first be determined. Unlike LTI channels having elegant and common complex sinusoids eigenfunctions to facilitate pulse design, the underspead LTV channels at best have a structured set of approximate eigenfunctions, which are even channel-dependent [7]. It is therefore impractical to realize eigenfunction-based transmission over LTV channels.

     On the other hand, in addition to the bandwidth, another primary characteristic of the channels is the available time $\mathbb T$, corresponding to the overall duration of the waveform. For LTI channels, we used to not pay attention to $\mathbb T$, because the channel is time-invariant, thus independent of $\mathbb T$. However, $\mathbb T$ is as important as the bandwidth $\mathbb B$ for time-dependent LTV channels. In fact, during an appropriate $\mathbb T$, an LTV channel can be represented by a deterministic delay-Doppler (DD) spread function [8]. Given a practical waveform bounded by bandwidth $\mathbb B$ and duration $\mathbb T$, the waveform received after receiving (or anti-aliasing) filtering can be sampled at an appropriate rate $B\le \mathbb B$ over a period of time $T \le \mathbb T$. The waveform may then be considered to have experienced an equivalent sampled DD (ESDD) channel associated with a delay resolution $\frac{1}{B}$ and a Doppler resolution $\frac{1}{T}$ [8]. As a result, the waveform design for LTV channels may be replaced by that for ESDD channels, which have common characteristics of $B$ and $T$ or equivalently the DD resolution $(\frac{1}{B}, \frac{1}{T})$.

Note that physical units of delay and Doppler are time and frequency, respectively. The DD resolution is exactly a TF resolution and therefore the waveform considering the DD resolution $(\frac{1}{B}, \frac{1}{T})$ or the so-called DD domain modulation is naturally an MC modulation. However, due to the joint TF resolution $\frac{1}{BT}\ll 1$ in practice, such DD domain MC (DDMC) modulation does not exist within the conventional TFMC waveform design framework mentioned before.

As a novel DDMC modulation beyond the conventional TFMC waveform design framework, ODDM is based on a special DD domain orthogonal pulse (DDOP) [9]-[12]. In particular, given a design parameter $T_0$, we set $\Delta T =\frac{T_0}{M}$ and $\Delta F=\frac{1}{NT_0}$ in (\ref{xt}) for ODDM with $B =\frac{M}{T_0}$ and $T=NT_0$. Then, the ODDM waveform without cyclic prefix can be represented as [9]-[12] \begin{align}\label{xtoddm} \check{x}(t)=\sum_{m=0}^{M-1}\sum_{n=-N/2}^{N/2-1}X[m,n]u(t-m\frac{T_0}{M})e^{j2\pi\frac{n}{NT_0} (t-m\frac{T_0}{M})}, \end{align} where the DDOP $u(t)$ depicted in Fig. 1 is a pulse-train defined as \begin{align} u(t)=\sum_{\dot n=0}^{N-1}a(t-\dot nT_0), \end{align} whose elementary pulse $a(t)$ is a root Nyquist pulse parameterized by its Nyquist interval $\frac{T_0}{M}$ and duration $2Q\frac{T_0}{M}$. It has been proved in [9]-[11] that when $2Q\ll M$, although $\frac{1}{BT}=\frac{1}{MN}\ll 1$, $u(t)$ satisfies the orthogonality property of \begin{equation} \mathcal A_{u,u}\left(m\frac{T_0}{M}, n\frac{1}{NT_0}\right)= \delta(m)\delta(n),\label{local_bio} \end{equation} for $|m|\le M-1$ and $|n| \le N-1$, where $\mathcal A_{u,u}(\cdot)$ is the ambiguity function of $u(t)$ given by the inner product$\mathcal A_{u,u}(\tau,\nu) = \langle u(t), u(t-\tau)e^{j2\pi \nu (t-\tau)} \rangle$.

One can see that the orthogonality in (\ref{local_bio}) is subject to a limited number of pulses. On the other hand, the (bi)orthogonality in the conventional TFMC waveform design framework is subject to $\left\{\Delta T, \Delta F, g(t)\right \}$ and $\left\{\Delta T, \Delta F, r(t)\right \}$, which are WH full sets containing unlimited number of pulses corresponding to the entire TF domain. Clearly, such global (bi)orthogonality is not necessary for the design of a practical MC waveform, which contains a limited number of pulses and only occupies a limited region in the TF domain.

    Without loss of practicality, the global (bi)orthogonality for a pair of WH full sets can be replaced by the (bi)orthogonality for a pair of WH subsets $\left\{\Delta T, \Delta F, g(t), M,N\right\}$ and $\left\{\Delta T, \Delta F, r(t), M, N\right \}$. Since we merely need $MN$ pulses to carry $MN$ digital symbols, the WH subsets and the associated local (bi)orthogonality are sufficient to design the MC waveform. Also, because the WH frame theory is applied only to WH full sets, by introducing the local or sufficient (bi)orthogonality for WH subsets, the DDOP-based ODDM can "bypass" the restrictions imposed by the WH frame theory, and therefore goes beyond the conventional TFMC waveform design framework to have fine TF (DD) resolution.

Recall that bandwidth and duration are two primary channel characteristics that determine the delay and Doppler resolutions, respectively. The DDOP considering the DD resolution is actually designed according to the bandwidth and duration of the channel.

    Let $A(f)$ be the Fourier transform of $a(t)$, the frequency domain representation of the DDOP is given by [10] \begin{equation}\label{ufeq} U(f) = \frac{e^{-j2\pi f \widetilde T}}{T_0} A(f) \sum_{m=-\infty}^{\infty} e^{j2\pi \frac{m(N-1)}{2}}\sinc(fNT_0-mN), \end{equation} where $\widetilde T= (2QT_0/M+(N-1)T_0)/2$ and $\sinc(\cdot)$ stands for the normalized $\sinc$ function. As depicted in Fig. 2, $U(f)$ consists of $(\check M+1)$ scaled $\sinc$ functions spaced by $1/T_0$ and centered from $-\check M/(2T_0)$ to $\check M/(2T_0)$ under the envelope of $A(f)$, where $\check M \ge M$ depends on the width of $A(f)$. From (\ref{ufeq}), one can see that the bandwidth of the DDOP is identical to that of the elementary pulse $a(t)$, which is known as a band-limited impulse that occupies the entire bandwidth in the SC modulation. Meanwhile, as seen in Fig. 2, the DDOP also has a duration close to that of the channel. Due to the duality between time and frequency, the DDOP may also be considered as a time-limited frequency domain impulse. These observations imply that the DDOP looks like a 2D impulse.

However, as established by the uncertainty principle, a true 2D impulse does not exist. Does the DDOP violate the uncertainty principle? The answer is no. Recall that the TF domain for designing pulse is an interdependent 2D domain. Due to their interdependence, a signal that is narrow in time tends to be wide in frequency, and vice versa. A pulse that is TF "narrow", like the the non-existent 2D impulse, may also be considered as TF "wide". Although the uncertainty principle sets a lower bound of TFA, there is no upper bound of TFA. In fact, we can have TF "wide" pulses using the pulse-train structure.

    The key properties of the DDOP lie exactly in its pulse-train structure, particularly in the spacing among the elementary pulses. Given the time domain spacing $T_0$, the bandwidth and duration of a pulse-train are determined by the bandwidth and the number of elementary pulses, respectively. They are independent and can be "wide" simultaneously, as long as the pulse-train structure holds. One can see that $U(f)$ is also a frequency domain pulse-train windowed by $A(f)$, where the frequency domain elementary pulses $\sinc(fNT_0-mN)$ are spaced by $1/T_0$.

     Similar to (\ref{B_x}), from (\ref{xt}), we have the duration of $x(t)$ as \begin{eqnarray} T_x = (M-1)\times \Delta T+T_g\le \mathbb T \label{T_x}. \end{eqnarray} In the conventional TFMC waveform design framework, $T_g$ and $B_g$ are comparable to $\Delta T$ and $\Delta F$ as $T_g \ge \Delta T$ in (\ref{T_x}) and $B_g > \Delta F$ in (\ref{B_x}), respectively. It can be seen that (\ref{B_x}) and (\ref{T_x}) are also satisfied by setting $T_g\gg \Delta T$ and $B_g\gg \Delta F$, which require the prototype pulse to be TF "wide" and allows for staggering in both time and frequency domains.

    The DDOP precisely exhibits these characteristics, featuring wide bandwidth, long duration, and the internal TF spaces among its elementary pulses. Without violating the uncertainty principle, the DDOP may be considered as a pseudo 2D impulse, which behaves like the non-existent 2D impulse within a TF region of $(T_0, \frac{1}{T_0})$ w.r.t. the resolution of $(\frac{T_0}{M},\frac{1}{NT_0})$. Thus, the DDOP-based ODDM achieves an pseudo-2D-impulse-based transmission over ESDD channels, where the internal TF spacing of the DDOP results in a staggered signal structure of ODDM in both the time and frequency domains [13].

The DD domain modulation was first considered in form of orthogonal time frequency space (OTFS) modulation [14] [15]. Due to the absence of 2D impulse [16], the OTFS transforms digital symbols from the DD domain to the TF domain via the so-called inverse symplectic finite Fourier transform (ISFFT). The transformed digital symbols are then modulated using the conventional OFDM or TFMC waveform.

    For comparison purposes, let the TF domain grid of OTFS obey $\{\hat nT_0,\hat m\frac{1}{T_0}\}$, then the corresponding DD domain grid is considered to be $\{m\frac{T_0}{M}, n\frac{1}{NT_0}\}$. The CP-free OTFS waveform can be written as [14] \begin{equation}\label{xtotfs} \hat x(t)=\sum_{\hat n=0}^{N-1}\sum_{\hat m=-M/2}^{M/2-1} \mathcal X[\hat n,[\hat m]_M] g(t-\hat nT_0)e^{j2\pi \hat m F_0 (t-\hat n T_0)}, \end{equation} where $[\cdot]_M$ is the mod $M$ operator and the digital symbols \begin{equation}\label{isfft} \mathcal X[\hat n,\hat m] = \frac{1}{\sqrt{MN}} \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} X[m,n]e^{j2\pi(\frac{\hat n n}{N}-\frac{\hat mm}{M})}, \end{equation} are transformed from $X[m,n]$ by the ISFFT. Note that in OTFS, $M$ and $N$ are the number of subcarriers and symbols of the underlying OFDM modulation, respectively. From (\ref{xtotfs}), we can see that the OTFS waveform is designed with $\Delta R=\Delta T \Delta F=1$, because $T_{sp}=T_0$ and $\Delta T= T_{sp}$ in the underlying OFDM. Therefore, OTFS can be considered as a precoded OFDM [17], which is within the conventional TFMC waveform design framework.

    The comparison of ODDM, OFDM, and OTFS is shown in Table I. One can see that ODDM is fundamentally different from OFDM and OTFS, in terms of physical concepts, transmission strategy, waveform design framework, and the resultant orthogonality.

	OFDM	OTFS	ODDM
Physical concepts	TF domain	DD and TF domains are different domains. DD domain modulation is challenging, due to the absence of 2D impulse [16].	A DD domain is a TF domain with fine TF resolution. A piratical DD domain modulation is an MC modulation [9] [11].
Transmission stratery	Eigenfunction-based transmission for LTI channels: OFDM/TFMC based on complex sinusoids pulse-shaped by continuous prototype pulses	Precoded OFDM/TFMC [17]: Use the ISFFT to transfer digital symbols from DD domain to TF domain, then modulate them using OFDM/TFMC waveform	Pseudo-2D-impulse-based transmission for LTV/ESDD channels [12]: DDMC based on complex sinusoids pulse-shaped by a discontinuous prototype pulse (a special pulse-train )
Waveform design framework	WH full set bounded by the WH frame theory [3]	WH full set bounded by the WH frame theory [3]	WH subset beyond the restrictions imposed by the WH frame theory and the uncertainty principle [10]
Orthogonality	Global (bi)orthogonality w.r.t. coarse TF grids $\{m\Delta T, n\Delta F\}$, $\Delta T \ge \frac{1}{\Delta F}$ for unbounded $m$ and $n$ [3]	Global (bi)orthogonality w.r.t. coarse TF grids $\{\hat nT_0, \hat m\frac{1}{T_0}\}$ for unbounded $\hat n$ and $\hat m$ [3]	Local or sufficient (bi)orthogonality w.r.t. fine TF grids $\{m\frac{T_0}{M}, n\frac{1}{NT_0}\}$ for bounded $m$ and $n$ [10]