Integrated Sensing and Communication with Delay Alignment Modulation: Performance Analysis and Beamforming Optimization

For DAM-based sensing, let $N$ be the number of symbol durations for one coherent processing interval (CPI). The transmitted DAM signal in (3) over one CPI can be expressed as

Denote by $\mathbf{F}\triangleq\left[\mathbf{f}_{1},\cdots,\mathbf{f}_{L}\right]\in\mathbb{C}^{M\times L}$ the transmit beamforming matrix, and $\bar{\mathbf{S}}[n]\triangleq\left[\mathbf{s}[n-\kappa_{1}],\cdots,\mathbf{s}[n-\kappa_{L}]\right]^{T}\in\mathbb{C}^{L\times N}$ the transmitted symbols, where

$l=1,\cdots,L$. Based on (3), the transmitted DAM signal over one CPI can be written as

For a sensing target with the channel in (2), the received echo signal over one CPI is

where $\mathbf{d}[\nu]\triangleq[e^{j2\pi\nu(n-(N-1))T_{s}},\cdots,e^{j2\pi\nu nT_{s}}]^{T}\in\mathbb{C}^{N\times 1}$ denotes the phase rotation over one CPI caused by the Doppler frequency $\nu$, and $\mathbf{z}\in\mathbb{C}^{N\times 1}$ is the i.i.d. AWGN vector with $\mathbb{E}[\mathbf{z}\mathbf{z}^{H}]=\sigma^{2}\mathbf{I}_{N}$. Note that for target sensing, the parameters of interest include $\theta$, $\tau$, and $\nu$. As this paper mainly focuses on the delay-Doppler domain sensing, we assume that the target direction $\theta$ in (11) has already been estimated, say via beam scanning in radar searching mode [41].

With the monostatic architecture, the transmitted signal $\bar{\mathbf{X}}[n]$ is known to the ISAC receiver. Therefore, to sense the target in the delay-Doppler domain from the received signal $\bar{\mathbf{y}}_{s}[n]\in\mathbb{C}^{N\times 1}$ in (11), matched filters can be constructed for each delay-Doppler bin parameterized by $(\tau_{p},\nu_{q})$, given by

where $P$ and $Q$ denote the number of bins in the delay and Doppler intervals of interest, with the delay resolution of $T_{s}$ and Doppler frequency resolution of $1/(NT_{s})$, respectively; $\mathbf{d}[\nu_{q}]\triangleq\left[e^{j2\pi\nu_{q}(n-(N-1))T_{s}},\cdots,e^{j2\pi\nu_{q}nT_{s}}\right]^{T}\in\mathbb{C}^{N\times 1}$ is defined as the Doppler frequency estimator, with $\nu_{q}$ denoting the Doppler frequency for the $q$th Doppler bin. After matched filtering (MF), the resulting output is

where $\hat{z}\triangleq\frac{\mathbf{z}^{H}\mathrm{diag}^{H}\left(\mathbf{d}^{T}[\nu_{q}]\right)\bar{\mathbf{S}}^{H}[n-\tau_{p}]\mathbf{F}^{H}\mathbf{a}(\theta)}{\left\|\mathbf{a}^{H}(\theta)\mathbf{F}\bar{\mathbf{S}}[n-\tau_{p}]\mathrm{diag}\left(\mathbf{d}^{T}[\nu_{q}]\right)\right\|}$ is the resulting noise after MF, satisfying $\hat{z}\sim\mathcal{CN}(0,\sigma^{2})$. Therefore, the delay $\tau$ and Doppler frequency $\nu$ in (11) can be estimated as $(\hat{\tau},\hat{\nu})=\arg\max\limits_{\tau_{p},\nu_{q}}\ \left|r(\tau_{p},\nu_{q})\right|$.

Note that the MF output in (13) is random due to the random communication symbols involved. To analyze the DAM sensing performance, let us define a delay-Doppler correlation matrix (DDCM) involved in (13) as

Then (13) can be rewritten as

where

is termed as the discrete equivalent normalized AF of the DAM signal in delay-Doppler domain, w.r.t. the ground-truth delay $\tau$ and Doppler frequency $\nu$. Note that according to (10), the element of the DDCM in (14) at the $i$th row and $j$th column is given by

where $(\cdot)^{\dagger}$ denotes the conjugation operation. Note that (17) involves the summation of $N$ i.i.d. communication symbols $s[\tilde{n}]$ with various delays. When $N$ is large, we have the following result:

Theorem 1: For $N\gg 1$, the DDCM element in (17) approaches to

where $\bm{\Lambda}(d_{\tau},d_{\nu})$ only depends on the delay difference $d_{\tau}\triangleq\tau_{p}-\tau$ and the Doppler difference $d_{\nu}\triangleq\nu-\nu_{q}$. Furthermore, $\psi(d_{\nu})$ in (18) is given by

where $\mathrm{asinc}(d_{\nu},N)\triangleq\frac{\sin(\pi d_{\nu}NT_{s})}{N\sin(\pi d_{\nu}T_{s})}$ denotes the “aliased sinc” function [41].

Note that $\lim\limits_{d_{\nu}\rightarrow 0}\mathrm{asinc}(d_{\nu},N)=1$ and $\lim\limits_{N\rightarrow\infty}\mathrm{asinc}(d_{\nu},N)=\delta(d_{\nu})$, thus we have $\psi(0)=1$. Therefore, according to Theorem 1, when $N\gg 1$, (15) approaches to

with the AF approaching to

which is referred to as the asymptotic AF that is only related to the delay and Doppler differences, i.e., $d_{\tau}$ and $d_{\nu}$. Depending on whether $d_{\tau}=0$ and/or $d_{\nu}=0$ or not, we have the following four cases.

1) $\chi(0,0)$:

In this case, both the considered delay bin $\tau_{p}$ and Doppler bin $\nu_{q}$ perfectly match with the ground-truth $\tau$ and $\nu$, respectively, i.e., $d_{\tau}=0$ and $d_{\nu}=0$. Thus (18) reduces to

Note that since by design that $\kappa_{i}\neq\kappa_{j},\forall i\neq j$ in (3), we have $\left[\bm{\Lambda}(0,0)\right]_{i,j}=0,\forall i\neq j$ and $\left[\bm{\Lambda}(0,0)\right]_{i,j}=1,\forall i=j$, which indicates that the DDCM is simply an identity matrix, i.e.,

By substituting (23) into (21) and (20), we have $\chi(0,0)=1$, and the resulting MF output is

As a result, the maximum output SNR for DAM sensing is

2) $\chi(0,d_{\nu})$:

In this case, the considered delay bin $\tau_{p}$ matches with the ground-truth $\tau$, while the Doppler bin $\nu_{q}$ and the ground-truth $\nu$ are mismatched, i.e., $d_{\tau}=0$ and $d_{\nu}\neq 0$. Thus (18) reduces to

Similar to (22) and (23), the DDCM is a scaled identity matrix, i.e., $\bm{\Lambda}(0,d_{\nu})=\psi(d_{\nu})\mathbf{I}_{L}$, and the delay-cut of $\chi(\tau_{p},\nu_{q};\tau,\nu)$ at $\tau_{p}=\tau$ (i.e., $d_{\tau}=0$) approaches to

which measures the Doppler frequency sensing performance of the DAM signal.

For radar sensing, one important performance metric is the sidelobe level of the AF, which is critically related to the waveform sensibility for multiple targets. Specifically, the high sidelobes of the stronger target echo will mask the MF output of the weaker one [41]. To measure the sidelobe level of $\chi(0,d_{\nu})$, a common metric is the PSR, which is defined as [43]

Note that for the asinc function, the peak sidelobe is approximately $13.2$ dB below the central peak amplitude, as illustrated in Fig. 2(b), where $N$ is only related to the frequency resolution but has no effect on the sidelobe level [41], thus we have $\Phi_{PSR}\approx-13.2$ dB. In practice, to achieve lower PSR, some window functions can be applied, like Hamming or Taylor windows [41].

3) $\chi(d_{\tau},0)$:

In this case, the considered delay bin $\tau_{p}$ mismatches with the ground-truth $\tau$, while the considered Doppler bin $\nu_{q}$ and the ground-truth $\nu$ are matched, i.e., $d_{\tau}\neq 0$ and $d_{\nu}=0$. Thus (18) can be rewritten as

where $\triangle_{i,j}\triangleq\kappa_{i}-\kappa_{j},1\leq i,j\leq L$, is defined as the delay difference between the delay pre-compensation of the $i$th and $j$th paths. Note that with $\kappa_{l}\neq\kappa_{l^{\prime}}$, $\forall l\neq l^{\prime}$ and $\kappa_{l}=n_{\max}-n_{l}$, we can obtain that the delay difference matrix $\mathbf{\Delta}$ consisting of $\triangle_{i,j}$, $1\leq i,j\leq L$, is a skew-symmetric matrix, where $\triangle_{i,j}\in~{}\left\{\pm 1,\cdots,\pm n_{d}\right\}$, $\forall i,j$, $\triangle_{i,j_{1}}\neq\triangle_{i,j_{2}}$, $\forall j_{1}\neq j_{2}$, and $\triangle_{i_{1},j}\neq\triangle_{i_{2},j}$, $\forall i_{1}\neq i_{2}$. For example, for $L=3$ multi-paths with delays $n_{1}=1,n_{2}=3$, and $n_{3}=5$, we have $\kappa_{1}=4$, $\kappa_{2}=2$, and $\kappa_{3}=0$, and the delay difference matrix is

Therefore, according to (29), when $|d_{\tau}|>n_{d}$, we have $\left[\bm{\Lambda}(d_{\tau},0)\right]_{i,j}=0$, $\forall i,j$, i.e., $\bm{\Lambda}(d_{\tau},0)$ is a zero matrix, which means that DAM can ensure perfect delay resolution at least for targets separated by time difference greater than the delay spread of the communication channel. On the other hand, for $0<|d_{\tau}|\leq n_{d}$, i.e., $d_{\tau}=\pm 1,\cdots,\pm n_{d}$, we can obtain that $\left[\bm{\Lambda}(d_{\tau},0)\right]_{i,j}=1$ if and only if $d_{\tau}=\triangle_{i,j}$.

To derive $\bm{\Lambda}(d_{\tau},0)$ for $0<\left|d_{\tau}\right|\leq n_{d}$, we first put all pairs of $(i,j)$ whose corresponding delay difference $\triangle_{i,j}$ satisfying $d_{\tau}=\triangle_{i,j}$ into a set

Taking (30) as an example, we have $\triangle_{1,2}=\triangle_{2,3}=2$. Based on (29), for $d_{\tau}=2$, $\left[\bm{\Lambda}(2,0)\right]_{1,2}=\left[\bm{\Lambda}(2,0)\right]_{2,3}=1$, while all other elements of $\bm{\Lambda}(2,0)$ equal to zero. Thus, we put the pairs of $(1,2)$ and $(2,3)$ into a set $\mathcal{S}(2)=\left\{(1,2),(2,3)\right\}$, and the DDCM when $d_{\tau}=2$ for $L=3$ is given by

Therefore, according to (31), when $0<|d_{\tau}|<n_{d}$, (29) can be further written as

Furthermore, based on (31) and (33), it can be obtained that the Doppler-cut of $\chi(\tau_{p},\nu_{q};\tau,\nu)$ at $\nu_{q}=\nu$ (i.e., $d_{\nu}=0$) approaches to

which measures the delay sensing performance of the DAM signal. To measure the sidelobe level of $\chi(d_{\tau},0)$, in addition to PSR, another effective metric is the ISR, which is particulary useful for measuring the susceptibility to distributed scattering such as clutters [43]. The discrete form ISR of $\chi(d_{\tau},0)$ can be expressed as

In general, the ISR should be no greater than a predetermined threshold to ensure the sensing performance, which can be achieved by applying the appropriate DAM path-based beamforming $\left\{\mathbf{f}_{l}\right\}_{l=1}^{L}$ as discussed in Section III-C.

4) $\chi(d_{\tau},d_{\nu})$:

In this case, neither the delay nor the Doppler bins $(\tau_{p},\nu_{q})$ under consideration match with the ground-truth $(\tau,\nu)$, i.e., $d_{\tau}\neq 0$ and $d_{\nu}\neq 0$. Therefore, according to (18), similar from (29) to (34), the asymptotic AF is

which means that $\chi(d_{\tau},d_{\nu})$ in delay-Doppler domain can be decoupled as the product of $\chi(d_{\tau},0)$ and $\chi(0,d_{\nu})$ in delay and Doppler domains, respectively.

In summary, the asymptotic AF of the DAM signal with $N\gg 1$ can be written as

Note that different from the delay cut AF $\chi(0,d_{\nu})$ in (27), the Doppler-cut AF $\chi(d_{\tau},0)$ in (34) is critically dependent on the DAM beamforming vectors $\left\{\mathbf{f}_{l}\right\}_{l=1}^{L}$. Thus, the Doppler-cut AF can be improved via appropriate beamforming for better sensing performance. In particular, with the transmit power constraint in (4), to maximize the sensing SNR in (25) while minimize the ISR of $\chi(d_{\tau},0)$ in (35) without considering the communication performance, it only needs to retain one beam aligned with the sensing target steering vector $\mathbf{a}(\theta)$ and let other beams to zero, i.e., $\mathbf{f}_{1}=\sqrt{\frac{P_{t}}{M}}\mathbf{a}(\theta)$ and $\mathbf{f}_{l}=\mathbf{0}_{M\times 1},\forall l\neq 1$, which is termed as single-path beamforming. The resulting maximum sensing SNR is $\gamma_{\max}=\frac{\left|\alpha\right|^{2}NMP_{t}}{\sigma^{2}}$, while $\chi(d_{\tau},0)=0$ and $\Phi_{ISR}=0$. Note that for such a single-path beamforming scheme, (37) reduces to

which is equivalent to the asymptotic AF of the conventional SC waveform for sensing.

In Fig. 2, we plot the normalized AF $\chi(\tau_{p},\nu_{q};\tau,\nu)$ and the asymptotic AF $\chi(d_{\tau},d_{\nu})$ for DAM signal with the single-path beamforming. The ground-truth delay and Doppler frequency are set as $\tau=0$ and $\nu=0$. Thus we have $d_{\tau}=\tau_{p}$ and $d_{\nu}=\nu_{q}$. The signals $s[n]$ are set as the $64$ quadrature amplitude modulation (QAM) symbols and the system bandwidth is $B=100$ MHz. The number of the DAM symbols within one CPI are $N=5000,10000,100000$, which correspond to the CPI duration of $T_{d}=NT_{s}=0.05,0.1,1$ millisecond (ms). It is revealed that $\chi(\tau_{p},\nu_{q};\tau,\nu)$ approaches to $\chi(d_{\tau},d_{\nu})$ when $N$ is large, which verifies our theoretical result in Theorem 1. Specifically, as shown in Fig. 2(a), for the Doppler-cut AF of $\chi(\tau_{p},\nu_{p};\tau,\nu)$ at $\nu_{p}=\nu=0$, i.e., $\chi(\tau_{p},0;0,0)$, the PSR decreases as the CPI $T_{d}$ increases, and $\chi(\tau_{p},0;0,0)$ approaches to $\chi(d_{\tau},0)$ when $T_{d}$ is large. On the other hand, the delay-cut AF of $\chi(\tau_{p},\nu_{p};\tau,\nu)$ at $\tau_{p}=\tau=0$, i.e., $\chi(0,\nu_{p};0,0)$, approaches to $\chi(0,d_{\nu})$ for $T_{d}=0.05,0.1$ and $1$ ms. Note that the PSR of $\chi(0,\nu_{p};0,0)$ or $\chi(0,d_{\nu})$ is always about $-13.2$ dB, while increasing the CPI $T_{d}$ can only improve the Doppler frequency resolution but not the PSR.

In this subsection, the beamforming optimization problem for DAM-ISAC is formulated to maximize the communication SNR in (8), subject to the ISI zero-forcing (ZF) condition $\mathbf{h}_{l^{\prime}}^{H}\mathbf{f}_{l}=0,\forall l\neq l^{\prime}$ and the transmit power constraint in (4), while guaranteeing that the sensing output SNR in (25) is no smaller than a threshold $\gamma_{th}$ and the ISR in (35) is no greater than a threshold $\phi_{th}$. By discarding the constant term $\sigma^{2}$ in (8), the problem can be formulated as

For notational convenience, (39c) can be rewritten as

with $\tilde{\gamma}_{th}\triangleq\frac{\gamma_{th}\sigma^{2}}{\left|\alpha\right|^{2}N}$. Furthermore, to simplify the problem, we consider a more stringent ISR constraint than (39d), given by

where $\tilde{\phi}_{th}\triangleq\phi_{th}\tilde{\gamma}_{th}^{2}$. Note that due to the constraint (40), as long as (41) holds, (39d) must also hold. Therefore, problem $\mathrm{(P1)}$ can be recast as