From OTFS to DD-ISAC: Integrating Sensing and Communications in the Delay Doppler Domain

The main idea of DD communications is to place the information-conveyed data symbols onto a two-dimension (2D) DD plane. Physically, there are not tangible ‘delay’ or ‘Doppler’ resources. These are concepts mathematically derived from time and frequency resources. Given the bandwidth $B$ and time duration $T_{f}$ of the transmitted signal, we can construct a DD frame composed of $M\times N$ bins, where each bin is occupied by one data symbol. The length and width of a DD bin are defined by the delay resolution $1/B$ and Doppler resolution $1/T_{f}$, respectively. Since the radio waves are transmitted in the time domain, the main task of DD communication transceivers is two-folded: at the transmitter side, to convert data symbols from the DD domain to the time domain signal; and, conversely, at the receiver end, to obtain the signal in the DD domain from the received time domain signal.

It is useful to compare DD communications with conventional TF communications, e.g., OFDM modulation. Considering an OFDM system under critical sampling, where the total TF resources are partitioned into $M\times N$ bins with size $T\times 1/T$. Here $1/T$ is the subcarrier spacing, while $T$ is the time slot duration, where $B=M/T$, and $T_{f}=NT$, with $M$ being the number of subcarriers and $N$ being the number of time slots, respectively. Comparing to TF communications, DD communications enable a more dedicated synthesis of TF signals, making it more sensitive to small channel fluctuations in time and frequency, e.g., delay and Doppler. More specifically, the setup of DD communications indicates that a DD frame can support scatterings with maximum delay of $T$ and Doppler shift of $1/T$, which is sufficient for general underspread channels.

DD communications and TF communications are mathematically connected by the symplectic finite Fourier transform (SFFT). In fact, in the early works on DD communications, the signal transmission was built upon the overlay of OFDM. This implementation is commonly referred to as the SFFT-based implementation. In general, SFFT-based DD communications relies on a two-stage conversion process where the data symbols in the DD domain are initially transformed into a TF domain signal via the inverse SFFT (ISFFT). This 2D transformation will spread each DD symbol over the entire TF plane. As a result, DD symbols can experience all channel fluctuations in the TF domain, enabling the potential of obtaining the full channel diversity. Then, the standard OFDM modulation is performed, incorporating a TF domain pulse shaping filter, yielding the transmitted signal in the time domain. After passing through the channel, the received signal undergoes multi-carrier demodulation, resulting in the TF domain symbols. The DD domain received symbols are then obtained by applying the SFFT to the TF domain symbols. Different from TF communications, DD communications enables a channel input-output relation characterized by forms of convolution, e.g., circular convolution, depending on the pulse shapes. An obvious advantage for SFFT-based DD communications is its compatibility with OFDM modulation, which minimizes the need for modifications of existing hardware equipment. Nevertheless, the introduction of additional processing blocks incurs increased implementation complexity and latency.

Recent work has shown that the Zak transform can be used to characterize the modulation and demodulation for general DD communications. By leveraging the Zak transform, we are capable of directly converting the time domain signal to the DD domain, bypassing the TF domain processing. The Zak transform, also known as the Gelfand mapping, is an operation that outputs a two-variable function with an input function of one variable, which is essentially a type of Poisson summation. Particularly, the Zak transform has many unique properties, such as quasi-periodicity, which not only simplifies the DD signal representation (The signal in the fundamental rectangle, i.e., a DD region of $\tau\in\left[{0,T}\right)$ and $\nu\in\left[{0,1/T}\right)$, fully characterizes the signal in the entire DD domain.), but also gives rise to special signal structures in both time and frequency. To place the DD symbols on a carrier, the DD domain basis functions (carriers) can be constructed in a way that aligns with the properties of the Zak transform. Specifically, it was shown in [8] that the DD domain basis function is quasi-periodic globally, while twisted-shifted locally within the fundamental rectangle. Such a basis function turns out to be the shape of a “pulsone” in both time and frequency as shown in Fig. 2. In this figure, the pulsone in time is a train of time-delayed pulses modulated by a signal tone determined by the Doppler. Correspondingly, it can be shown that the pulsone in frequency is a train of frequency-shifted pulses modulated by a signal tone determined by the delay. This symmetrical treatment in time and frequency allows a full exploitation of the channel dynamics thereby improving the communication performance.

In line with the properties of the Zak transform, DD modulation can be implemented digitally using the inverse discrete Zak transform (IDZT), followed by filters and windows for converting it into time-continuous waveform with approximately limited TF resources. Similarly, at the receiver side, we can apply matched filters and windows to discretize the signal, followed by a DZT, to yield the received DD symbols. We shall refer to this type of implementation as the Zak transform-based implementation. In fact, both SFFT-based and Zak-based DD communications will produce similar input-output relations depending on the underlying filters and windows. Nevertheless, the Zak-based DD communication requires only an $N$-point Doppler-time finite Fourier transform with a symbol-wise interleaver, leading to a lower implementation complexity compared to the SFFT-based method. Furthermore, Zak-based DD communications enjoys many practical benefits. For example, the information carrier can be directly designed in the DD domain without leveraging TF domain pulse shaping. Therefore, Zak-based DD communication does not suffer from the pulse discontinuity issue, which in return improves the out-of-band emission. However, the filter and window design for Zak transform-based implementation are of critical importance for achieving orthogonality and preserving the DD communication features. This important topic will be discussed in Section-III of this paper.

Since the establishment of DD communications, its capability for radar sensing has attracted significant interest. Intuitively, DD parameters, which can describe a target’s state, align perfectly with the sensing tasks. To shed light on this, we can consider radar sensing. Typical radar sensing studies revolve around the calculation of the range-Doppler matrix. This computation involves applying the discrete Fourier transform (DFT) to the slow-time axis of the fast-time slow-time matrix. Here, fast time corresponds to the sampling rate, which relies on the available bandwidth of the instantaneous pulse, while slow time represents the sampling interval between successive pulses. After matched-filtering the time domain sensing signal and sampling at an appropriate rate, the samples can be reorganized into a 2D matrix, as shown in Fig. 3. Then, performing a DFT along the slow-time axis produces the range-Doppler matrix, which essentially indexes the matrix elements by delay and Doppler (the range and delay as a one-to-one correspondence related to the signal propagation speed). Recall the earlier discussions on the demodulation of Zak-based DD communications. The time domain sequence after matched-filtering is converted to a matrix in the DD domain via the DZT. In fact, the DZT can be regarded as the combination of a symbol-wise interleaver and a DFT applied along the time axis. Therefore, it is straightforward to conclude that the derivation of range-Doppler matrix is exactly identical to the demodulation process in DD communications [9]. The only difference lies in that radar sensing aims to extract the state parameters from the DD matrix, whereas communications seeks to detect symbols conveying information.

Next, we will show how sensing and communication sub-systems can collaborate with each other to achieve coordination gain. As discussed in [5], the DD and angular parameters extracted from the sensing signals can facilitate inference about the communication channel, which is referred to as ‘sensing-assisted communications’. For instance, suppose a base station (BS) transmits a dual-functional DD-ISAC signal. Assuming a mono-static radar setup, the sensing echo is received at the BS. Through advanced estimation techniques, delay and Doppler shifts associated with moving targets can be obtained, which are adopted for constructing the topology of the mobile network. Subsequently, the network topology in the next few time instants can be predicted, allowing the prediction of the DD communication channels. In scenarios with line-of-sight (LoS) dominated channel model, both the path loss and phase offset of the channel can be accurately predicted and compensated at the BS, leading to a very low cost at the user side. When dealing with the generalized multi-path channel case, although perfect prediction of the communication channel remains elusive, the knowledge of DD shifts can provide a known interference pattern in the DD domain, which can reduce the channel estimation overhead.

Moreover, the DD domain channel intrinsically reveals the underlying wireless propagation environment. Hence, the estimates of the communication channel can serve a dual purpose: not only facilitating communications but also providing the capability of coarse environmental sensing, termed ‘communications-assisted sensing’[10]. Following standard communication processing, channel estimation precedes data detection. The estimated delay and Doppler indices directly mirror the physical attributes linked to in-environment scatterings. To illustrate, consider an environment with a BS and a moving vehicle. At each time instant, the vehicle estimates the downlink DD channel and obtains the delay and Doppler shifts associated with different paths. Through straightforward manipulations, the distance of the BS-scattering-vehicle path can be measured based on the estimated delay. Having obtained the range measurements over several consecutive instants, a standard elliptical localization problem can be formulated where the BS and vehicle are the foci of the ellipses. In addition, the Doppler shift containing angular information can help differentiate measurements with respect to different paths.

Consider an unknown number $P$ of targets in the sensing Field of View (FoV) of the ISAC system. For simplicity, assume that these targets can be modeled as single scatterers. Since the number of targets is initially unknown, a detection step is required to distinguish between targets, clutter, and noise. To achieve this, we formulate the target detection as a standard Neyman-Pearson hypothesis testing problem. Under this framework, at each step we have hypotheses ${\cal H}_{0}$ and ${\cal H}_{1}$ corresponding to the absence or presence of a $p$-th target only. This means that when detecting a target and estimating the relevant parameters, the targets already detected are assumed to have already been canceled from the received signal, whereas the contributions of the remaining targets are modeled as additional noise. The observation under the two hypotheses is given by

where $\mathring{\Theta}_{p}$ denotes the true parameters of the $p$-th target, ${\bf A}$ is the radar channel matrix, containing the channel gain, delay-Doppler, and possibly beamforming effects, and ${\bf w}$ is the noise term. The Neyman-Pearson hypothesis testing problem above, for which the solution that maximizes the detection probability subject to a bound on the false-alarm probability is given by the (Generalized-) Likelihood Ratio Test (LRT) [11]. Here a threshold determines the tradeoff between detection and false-alarm probabilities for target detection. Since the statistics of noise and interference in realistic scenarios are generally not known, an adaptive threshold function according to the Constant False Alarm Rate Detection (CFAR) approach can be used. In particular, the Ordered Statistic (OS) -CFAR method, is known to provide good performance in those scenarios [12]. As an example, Fig. 6 shows the performance of the OS-CFAR technique applied to the DD radar image in a multi-target scenario.

Parameter estimation can be carried out as a side-product of the LRT-based detection above. More precisely, by defining a discrete grid over the parameter search space (e.g. delay, Doppler, angle) and evaluating the likelihood ratio at each grid point, the grid cells passing the defined threshold indicate the target parameters. In multi-target cases, a target located closely may create ‘masking’ effects. This is especially significant when a target is a much stronger reflector than adjacent targets. Hence, a sequential detection and estimation framework of targets was proposed in [12], where after each target is detected and its parameters are estimated, a Successive Interference Cancellation (SIC) routine is carried out. This step effectively cancels out the contribution of a detected target from the received signal and proceeds to the detection of the next target.

In this subsection, we will present numerical results discussing the sensing benefits of DD-ISAC in comparison to OFDM-based ISAC. We argue that DD-ISAC has the potential to outperform OFDM-based ISAC thanks to its advantage in terms of the ambiguity function. Particularly, the ambiguity function of DD-ISAC shows a robust and ‘spike-like’ response for normalized delay and Doppler. This observation clearly indicates good sensing performance when applying target detection and parameter estimation based on a matched-filtering type of receiver. Without loss of generality, both DD and TF waveforms are pulse shaped by the RRC pulse with roll-off factor $\beta=0.3$ via the DMT structure.

We present the numerical results of the target detection probability based on matched-filtering in Fig. 7, where we consider $M=N=16$ and the number of targets is set to be $1$. As can be observed from the figure, DD-ISAC shows a better detection probability than the OFDM-based ISAC using the CFAR approach. This can be attributed to the ‘less noisy’ radar image in the DD domain produced by the DD waveform, as observed from Fig. 4 and Fig. 5. Specifically, both DD and TF waveforms start to work when the SNR is higher than $-10$ dB, while the detection probability achieves almost one with SNRs higher than $0$ dB.

In Fig. 8, the MSE performance of parameter estimation is shown, where $M=N=16$ and a single target is considered. Specifically, we calculate the associated MSE based on the detected target. Furthermore, if the CFAR fails to detect the target, we simply take the corresponding delay and Doppler with the highest response after matched-filtering as the estimator output. From the figure, we notice that both DD and TF waveforms provide similar MSE performance in general. This is because the considered frame length is sufficiently long to ensure a satisfactory auto-correlation performance of the transmitted symbols, such that the parameter estimation based on matched-filtering is good enough.

We highlight that we only adopted a single target ISAC in the above examples because it is the simplest and most common scenario for sensing. However, in practice, the wireless channel for sensing can be quite complex with many unintended scatterers. In this case, the DD waveform may provide further performance improvement for sensing in comparison to the TF waveforms, thanks to its ‘spike-like’ ambiguity function.

Some preliminary results show that a single symbol placed in the DD domain might be sensing-optimal from the AF perspective. Nevertheless, rigorous mathematical validation is needed. Starting from this point, a very interesting topic is DD-ISAC waveform design. Since communication systems lean towards randomized signals while sensing favors deterministic ones, there is an inherent deterministic-random trade-off for ISAC waveform design. When our aim is to improve sensing performance, a possible solution is to optimize the transmitted DD sequence that closely resembles the optimal sensing sequence in the expectation sense. In contrast, if we take the communication performance as the primary goal, a water-filling type of DD precoding design should be adopted to achieve channel capacity, which however inevitably affects the correlation of the DD sequences and degrade the sensing performance. Hence, ‘optimal’ DD-ISAC waveforms vary based on specific applications, drawing considerable attention nowadays.

Existing works study the performance of ISAC using channel coding for communications and using either rate-distortion theory or the Cramér-Rao bound for sensing. The communication task is very well understood while the sensing task, especially the performance trade-off between communication and sensing is less understood. Based on Rényi’s pioneering work [13], the ideas from 20 questions estimation [14] might be adopted, which converts the sensing and communication tasks to continuous and discrete random variable estimation problems. Specifically, the sensing task of DD-ISAC, which focuses on the estimation delay, Doppler shifts, and reflection coefficients, can be converted to a 20 questions estimation problem of multiple targets in multi-dimensions, as studied in [15]. As a result, the theoretical results can be adapted to yield bounds on the estimation error for targets. Using the connection of the sensing task of ISAC to 20 questions estimation, the performance bounds on the estimation error could be obtained, which depend on the sensing channel via the mutual information term and an additional dispersion term that characterizes the variance of the channel. Correspondingly, the transmission performance, defined as the maximal transmission rate subject to constraints on transmit power and error probability is a function of the communication channel via similar mutual information and dispersion terms. In case of quasi-static fading, both performance criteria for sensing and communication essentially reduce to mutual information over the corresponding channels.

Although we can potentially use 20 questions estimation to obtain theoretical bounds on the sensing performance, another concern is the complexity of algorithms. Some preliminary results rely on random coding with maximum likelihood decoding, which has exponential complexity with the signal length. However, no practical systems could support such high complexity due to power and latency constraints. Thus, another challenge is how to design low complexity algorithms for time and resource efficient target sensing.

To solve this problem, we can resort to the construction of low-complexity query procedures for 20 questions estimation. Specifically, since any channel code can be used to construct a query procedure for 20 questions estimation, we can use practical low-complexity codes such as LDPC or polar codes to construct query procedures and adapt the query procedure to design the signal and the sensing decoder.

The reconfigurable intelligent surface (RIS) technique aims at modifying the phase of the transmitted signal to enhance the communication performance and save on hardware costs and energy consumption. Relying on the DD-ISAC framework, one cannot only improve the communication performance but also can attain a better sensing performance as well. In particular, the adoption of of RIS technology can directly modify the DD response of the physical channel, which in return improves the sensing performance.

As a further comment, near-field communication has emerged as a topic of interest in the community recently. In contrast to traditional far-field communications, the range distance information will be exploited for beamforming and multiple access design in near-field communication, which actually reflects the delay parameter. Therefore, incorporating DD signaling into near-field communications is interesting and deserves further investigation.