From Torch to Projector: Fundamental Tradeoff of Integrated Sensing and Communications

As the name suggests, the capacity-distortion theory investigates the tradeoff between communication capacity and sensing distortion. Originally proposed by Shannon in the context of rate-distortion theory for lossy data compression [17], distortion refers to a wide range of functions taking the form of $d(\bm{\seta},\hat{\bm{\seta}})$, whose inputs are the true value of some quantity (in sensing problems, the sensing parameter) $\bm{\seta}$ and its estimate $\hat{\bm{\seta}}$. Due to the randomness of the communication message, the ISAC waveform (codeword) $\bm{\mathsfbr{X}}^{N}$ is also random. To reflect the sensing performance over a relatively long period of time, a common practice is to use the expectation of the distortion, instead of its instantaneous values, as the performance metric. For example, in estimation tasks, the squared Euclidean distance $d(\bm{\seta},\hat{\bm{\seta}})=\|\bm{\seta}-\hat{\bm{\seta}}\|^{2}$ is a widely applied distortion function, whose expectation is the mean-squared error (MSE). In binary (e.g. target presence) detection problems in which the task is to determine the value of a binary variable $\seta\in\{0,1\}$, a valid distortion function is the Hamming distance given by $d(\seta,\hat{\seta})=\seta\oplus\hat{\seta}$, whose expectation is related to commonly used detection metrics, including the detection probability $P_{\rm D}$ and the false alarm rate $P_{\rm FA}$, as follows:

Note that for constant false-alarm rate (CFAR) detectors based on the Neyman-Pearson criterion with fixed $P_{\rm FA}$ [18], minimizing the Hamming distance is equivalent to maximizing the detection probability $P_{\rm D}$.

Besides capacity and distortion, there is yet another important ingredient in the capacity-distortion theory, namely the transmission cost. To elaborate, not all ISAC waveforms (or codewords) cost equally in terms of wireless resources. For example, the points in quadrature amplitude modulation (QAM) constellations having different amplitudes would yield different power consumptions. Apparently, as the overall resource budget increases, S&C performances can be simultaneously enhanced. Therefore, one has to discuss the capacity-distortion tradeoff under a specific resource budget. Similar to the sensing distortion, in order to take into account the randomness of the codewords, we typically use the expectation of the resource cost over all possible codewords as the measure of transmission cost.

For the simplicity of discussion, let us consider memoryless channels,²²2It is possible that ISAC channels have memory. However, the capacity-distortion problem of ISAC channels with memory remains open in general, except for some special cases [19]. exemplified by (2), and illustrated in Fig. 4. The effect of such channels can be expressed as

In particular, the channel $p_{\mathsfbr{W},\bm{\mathsfbr{Y}}_{\rm c}^{N},\bm{\mathsfbr{Y}}_{\rm s}^{N},\bm{\mathsfbr{X}}^{N},\bm{\seta}^{N}}(W,\bm{Y}_{\rm s},\bm{Y}_{\rm c},\bm{X},\bm{\eta})$ produces a communication output $\bm{\mathsfbr{Y}}_{\rm c}$ and a sensing output $\bm{\mathsfbr{Y}}_{\rm s}$, whereas their relationship with the channel input $\bm{\mathsfbr{X}}$ can be different from the linear model (2). Moreover, the way that the sensing parameter $\bm{\seta}$ couples with the channel can also be different. For such channels, it is natural to consider per-block resource budgets and distortion metrics, given by

The conditions (8) can then be concretized as

Remarkably, the result (13) suggests that the expected sensing distortion may be alternatively viewed as a kind of “sensing-induced communication cost”. This understanding enables us to extend even further the range that the capacity-distortion theory is applicable, to scenarios in which the sensing performance cannot be described by a proper distortion function. For example, the Cramér-Rao bound (CRB) widely used as the performance metric of estimation problems is not dependent on any specific estimator $\hat{\bm{\seta}}$. Sometimes it does not even depend on the true value of the parameter $\bm{\seta}$. Therefore, CRB is not a distortion function, but it is related to the transmitted codeword $\bm{\mathsfbr{X}}$ (as will be discussed later), and hence the capacity-distortion theory can still be applied in the generalized sense.

In this aforementioned scenario, we did not consider feedback. One may wonder whether designing the ISAC codebook by relying on feedback can further enhance the S&C performance. To this end, the state-dependent memoryless channel with delayed feedback (SDMC-DF) model has been investigated [10]. The effect of an SDMC-DF can be expressed as [8]

where both sensing and communication rely on the same channel, which produces the output $\bm{\mathsfbr{Y}}^{N}$. For sensing tasks, the channel input $\bm{\mathsfbr{X}}_{i}$ at the $i$-th channel use cannot be designed based on the real-time channel output $\bm{\mathsfbr{Y}}_{i}$ or the parameter value $\bm{\seta}_{i}$, since the design process has to be causal. Rather, one can only rely on a delayed feedback $\bm{\mathsfbr{Z}}_{i-1}$, which may be a function of the channel output $\bm{\mathsfbr{Y}}_{i}$.

From the aforementioned discussion, we may conclude that a key condition for (13) to hold is the memorylessness of channels. Indeed, for channels with memory, even the capacity itself remains open. Furthermore, for such channels, online and offline estimators may have very different performances. These problems deserve further investigation.

From a pure theorist’s perspective, the result (13) is a complete information characterization (as opposed to the operational definition (11)) of the capacity-distortion region. But wait! Remember that our expectation on the capacity-distortion theory in the first place was to help understand the nature of the S&C tradeoff, hopefully beyond the ST. However, we cannot even dig ST itself out of (13), not to mention any further insight.

To serve our purpose, a possible approach is to compute explicitly the capacity-distortion functions for some representative scenarios, in the hope of obtaining useful intuitions. It turns out that the renowned Blahut-Arimoto (B-A) algorithm [21] originally proposed for computing the unconstrained capacity can be applied to compute capacity-distortion functions with some modifications. To elaborate, given an initialization of the trail distribution $r(\bm{X})$ for $p_{\bm{\mathsfbr{X}}}(\bm{X})$, the original B-A algorithm solving $C=\max_{p_{\bm{\mathsfbr{X}}}(\bm{X})}\leavevmode\nobreak\ I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}})$ is a fixed-point iteration repeating the following two steps in each round:

1. 1.

   Update the trail distribution $q(\bm{X}|\bm{Y})$ for the a posteriori distribution $p_{\bm{\mathsfbr{X}}|\bm{\mathsfbr{Y}}}(\bm{X}|\bm{Y})$ according to

   $$q(\bm{X}|\bm{Y})\leftarrow\frac{p_{\bm{\mathsfbr{X}}}(\bm{X})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})}{\int r(\bm{X})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X}){\rm d}\bm{X}};$$

   (16)

2. 2.

   Update $r(\bm{X})$ by

   $$r(\bm{X})\leftarrow\frac{e^{\int p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})\log q(\bm{X}|\bm{Y}){\rm d}\bm{Y}}}{\int e^{\int p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})\log q(\bm{X}|\bm{Y}){\rm d}\bm{Y}}{\rm d}\bm{X}},$$

   (17)

   which is equivalent to solving the following constrained entropy maximization problem:

   	$\displaystyle r(\bm{X})\leftarrow$	$\displaystyle\mathop{\arg\max}_{p(\bm{X})}-\int p(\bm{X})\log p(\bm{X}){\rm d}\bm{X},$
   		$\displaystyle{\rm s.t.}\leavevmode\nobreak\ p(\bm{X})\geqslant 0\leavevmode\nobreak\ \forall\bm{X},\leavevmode\nobreak\ \int p(\bm{X}){\rm d}\bm{X}=1,$
   		$\displaystyle\mathbb{E}_{p(\bm{X})}\Big{\{}\int p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})\log q(\bm{X}|\bm{Y}){\rm d}\bm{Y}\Big{\}}\geqslant t$

   for some constant $t$.

Now that our objective function is the conditional mutual information $I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}|\bm{\seta})$, we shall replace (16) with

In addition, since two extra constraints (13b) and (13c) are now enforced, we should replace (17) with

where $\lambda$ and $\mu$ are the Lagrangian multipliers corresponding to the constraints (13b) and (13c), respectively.

From the above discussions, we may get a vague sense that the sensing distortion requirements render the resulting ISAC signal $\bm{\mathsfbr{X}}$ “less random”, as they impose constraints on the entropy maximization problem. Of course, this is not yet a rigorous statement with a well-defined meaning. To validate our intuition, let us consider the toy example of a real-valued, single-input single-output (SISO) Gaussian SDMC-DF channel model (with Rayleigh fading)

where $\seta_{i}$ and $\mathsfbr{N}_{i}$ are mutually independent zero-mean Gaussian variables with unit variance, while $\mathsfbr{X}_{i}$ is the ISAC waveform satisfying the power constraint $\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\{|\mathsfbr{X}_{i}|^{2}\}\leqslant B=10$dB. We consider the perfect feedback $\mathsfbr{Z}_{i}=\mathsfbr{Y}_{i}$ and the quadratic sensing distortion $d(\eta,\hat{\eta})=(\eta-\hat{\eta})^{2}$. In this scenario, the sensing-optimal estimator is the letter-wise minimum mean squared error (MMSE) estimator given by $\hat{\seta}_{i,{\rm MMSE}}=\mathbb{E}\{\seta_{i}|\mathsfbr{X}_{i},\mathsfbr{Y}_{i}\}$, whose MSE can be calculated as

where $\mathsfbr{X}$ is the single-letter representation of the channel input following the probability distribution $p_{\mathsfbr{X}}(X)$.

Under the aforementioned formulation, the capacity-distortion boundary may be numerically computed using the modified B-A algorithm, as portrayed in Fig. 5. At the communication-optimal point $\footnotesize1⃝$, the input distribution is Gaussian, and we have

By contrast, at the sensing-optimal point $\footnotesize4⃝$, the input distribution corresponds to binary phase-shift keying (BPSK) modulation, for which we have

As we move along the capacity-distortion boundary from the communication-optimal point $\footnotesize1⃝$ to the sensing-optimal point $\footnotesize4⃝$, the corresponding input distribution $p_{\mathsfbr{X}}(X)$ exhibits a smooth transition from the Gaussian distribution to the BPSK modulation, which agrees with our intuition that the ISAC signal becomes less random as the sensing distortion requirement becomes more stringent.

Note that there is not enough room to accommodate ST in the SISO scenario. Therefore, the S&C tradeoff in the aforementioned example follows solely from the difference in signal preference between communication and sensing tasks. Intuitively, communication tasks would favor signals with a higher degree of randomness, in order to pack more information into the signal. For example, the capacity for AWGN channels is achieved by using Gaussian-distributed signals, where the Gaussian distribution has the maximum entropy under power constraint. By contrast, sensing tasks favor signals that are deterministic in some sense, in order to better distinguish the received signals (echoes) coupled with sensing parameters taking different values. Such a tradeoff is termed as deterministic-random tradeoff (DRT) [11].

Naturally, we would then wonder whether the DRT exists in more general scenarios, what effect it would exhibit, and how it would interact with the ST when it exists. Unfortunately, although the modified B-A algorithm is universal in principle, it is hardly applicable to the general settings in a practical sense, due to its enormous computational complexity. To elaborate, the modified B-A algorithm relies on numerical integration, which could be computationally prohibitive when the dimensionality of the signals or the sensing parameters is high. For example, if the number of samples per dimension is $K$, the total number of samples $N_{\rm MC}$ that the widely used Monte Carlo integration method would require will be on the order of $N_{\rm MC}=O\big{(}K^{N_{\bm{\mathsfbr{Y}}}+N_{\bm{\mathsfbr{X}}}+N_{\bm{\seta}}}\big{)}$, where $N_{\bm{\mathsfbr{Y}}}$, $N_{\bm{\mathsfbr{X}}}$ and $N_{\bm{\seta}}$ are the dimensionalities of $\bm{\mathsfbr{Y}}$, $\bm{\mathsfbr{X}}$, and $\bm{\seta}$, respectively. The exponentially increasing sample complexity would easily surpass the capability of most computing devices, even for small-scale MIMO systems. Furthermore, the modified B-A algorithm cannot provide analytical solutions, which can be useful for practical system design.

Considering these difficulties, to push further our understanding of the S&C tradeoff, we might need to sacrifice the generality of the capacity-distortion theory to some degree. For example, we may try and find specific distortion functions (or non-distortion sensing-induced communication costs, as implied by (13)), that are both easy to analyze and sufficiently general to reflect ST and DRT. In what follows, we shall consider one such example in detail.

We commence by re-examining the model in (2), which may be extended to a more generic form as

where the sensing channel $\bm{\mathsfbr{H}}_{\rm s}\in\mathbb{C}^{{N_{\rm s}}\times M}$ is now defined as a deterministic, possibly nonlinear function of the sensing parameter $\bm{\seta}\in\mathbb{R}^{K}$, e.g., angular MIMO radar channel [22]. If not otherwise specified, the transmitted codeword $\bm{\mathsfbr{X}}\in\mathbb{C}^{M\times T}$ will be referred to as an ISAC signal matrix in this section. This model may be viewed as a special case of the generic model shown in Fig. 4. Following the preceding memoryless channel assumption, both the sensing parameter $\bm{\seta}$ and communication channel $\bm{\mathsfbr{H}}_{\rm c}\in\mathbb{C}^{{N_{\rm c}}\times M}$ vary every $T$ symbols in an i.i.d. manner. The discussion on channels with memory (e.g., Markov channels) are designated as our future works. For convenience, we also assume that the ISAC Tx has perfect knowledge on $\bm{\mathsfbr{H}}_{\rm c}$. Finally, $\bm{\mathsfbr{Z}}_{{\rm c}}$ and $\bm{\mathsfbr{Z}}_{{\rm s}}$ are zero-mean white Gaussian noise matrices with variances of $\sigma_{{\rm c}}^{2}$ and $\sigma_{{\rm s}}^{2}$, respectively.

At this point, it would be worthwhile to concretize the separate channel models in (18) by illustrating the specific scenarios concerned in this section. As shown in Fig. 6, the ISAC Tx emits a dual-functional signal $\bm{\mathsfbr{X}}$ to simultaneously communicate with a single communication Rx and sense one or more targets. The sensing Rx is either collocated with the Tx (monostatic sensing), or connected with the Tx with a wired link (bistatic sensing). In both cases, both the Tx and sensing Rx have perfect knowledge about the ISAC signal $\bm{\mathsfbr{X}}$. Nevertheless, $\bm{\mathsfbr{X}}$ is unknown to the communication Rx as it conveys useful information intended for the communication user. We are thus temped to model $\bm{\mathsfbr{X}}\sim p_{\bm{\mathsfbr{X}}}(\bm{X})$ as a random matrix, whose realization is perfectly known at both the ISAC Tx and sensing Rx, but is unknown at the communication Rx.

Since the communication performance of (18a) can be directly measured by the mutual information $I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}_{\rm c}|\bm{\mathsfbr{H}}_{\rm c})$, we are now in a position to rethink the sensing performance evaluation in ISAC systems. Indeed, in conventional radar systems, probing signals are typically deterministic, well-designed with good ambiguity properties. In ISAC systems, on the other hand, the transmit signal randomly varies from block to block given the communication data embedded in the signal. This imposes a unique challenge in defining the CRB, which now becomes a function of the random signal $\bm{\mathsfbr{X}}$.

By noting (19), it turns out that the Miller-Chang CRB is nothing but an equivalent “expected sensing distortion” discussed in Sec. III-A (though it is not a real distortion measure), and may hence be viewed as a sensing-induced cost imposed on signaling resources. Although with high dimensionality, one may still deduce useful results on the CRB-rate tradeoff by exploiting the affine structure of the BFIM, as discussed in the sequel.

The CRB-rate tradeoff can be characterized as the following Pareto optimization problem

where $\alpha\in\left[0,1\right]$ is a weighting factor controlling the priority of S&C performance. We highlight here that by moving the CRB from (23a) to (23b), (23) may be equivalently recast as a constrained capacity characterization problem with two cost functions as in (13).

Needless to say, fully depicting the Pareto boundary of the CRB-rate region would incur unaffordable computational overheads, as one has to numerically seek for optimal $p_{\bm{\mathsfbr{X}}}(\bm{X})$ by leveraging the modified B-A algorithm discussed in Sec. III-B. To reveal fundamental insights into the CRB-rate tradeoff, we are interested in the two corner points over the Pareto frontier as shown in Fig. 7, namely, $P_{\rm CS}$, the minimum achievable sensing CRB constrained by the maximum communication capacity, and $P_{\rm SC}$, the maximum achievable rate constrained by the minimized CRB. It is obvious that the line segment linked to two points forms a time-sharing inner-bound. In what follows, we briefly characterize the S&C performance at the two points.

Observe immediately that when $T\rightarrow\infty$, the communication DoFs are lossless, since even a Gaussian matrix would have asymptotically orthogonal rows with the increase of $T$, making it asymptotically equivalent to a semi-unitary matrix.

The above results clearly demonstrate the effect of both ST and DRT in an ISAC system. By comparing the left-most parts of $\bm{\mathsfbr{X}}^{\rm CS}$ and $\bm{\mathsfbr{X}}^{\rm SC}$, we see that the communication- and sensing-optimal ISAC signals should be aligned to $\bm{U}_{\rm c}$ and $\bm{U}_{\rm s}$, respectively, which may be regarded as the orthogonal bases of communication and sensing subspaces. Then the ST is nothing but to flow the signal power from $\operatorname{span}(\bm{U}_{\rm c})$ to $\operatorname{span}(\bm{U}_{\rm s})$, perfectly fitting the picture of the “ISAC torch” metaphor. More interestingly, by comparing the right-most parts of $\bm{\mathsfbr{X}}^{\rm CS}$ and $\bm{\mathsfbr{X}}^{\rm SC}$, e.g., $\bm{\mathsfbr{D}}$ and $\bm{\mathsfbr{Q}}$, we see that communication- and sensing-optimal signals adopt Gaussian and semi-unitary codebooks, respectively, which again reflects the DRT. That is, when the ISAC system moves along the Pareto frontier from $P_{\rm CS}$ to $P_{\rm SC}$, $p_{\bm{\mathsfbr{X}}}(\bm{X})$ gradually changes from Gaussian to less random distribution, and eventually becomes uniform distribution over semi-unitary matrices. In that sense, the DRT considered in Fig. 5 is simply a one-dimensional special case, since the semi-unitary matrix reduces to constant-modulus signals in its scalar form, namely, the BPSK modulation in Fig. 5(e).

In addition to the tradeoff between input distributions, the DRT may also be observed in the attainable communication and sensing DoF at the two corner points. At $P_{\rm CS}$, the communication subsystem apparently acquires the full DoF of the Gaussian channel, namely, $\mathsfbr{M}_{\rm CS}$, which is the rank of $\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS}$. As shown in (28), due to the Gaussian signaling, the sensing subsystem suffers from the DoF loss, or, equivalently, the reduction in the number of individual observations of the target, which is up to $\min\{K,\mathsfbr{M}_{\rm CS}\}$. At $P_{\rm SC}$, the sensing subsystem attains the full DoF of $T$ thanks to the deterministic signal sample covariance matrix. In contrast, the semi-unitary signaling incurs a communication DoF loss of $\frac{\mathsfbr{M}_{\rm SC}^{2}}{2T}$, as indicated by (34).

To achieve the S&C performance tradeoff, it is critical to determine the steering direction of the ISAC signal. More importantly, what kind of codebook is transmitted through that direction also matters. With the above understanding, we may now correct the “ISAC torch” metaphor with a more comprehensive picture, i.e., the “ISAC projector”. As shown in Fig. 8: A child (the ISAC Tx) holds a projector, and wishes to simultaneously illuminate a target (sensing) while sending an image to a receiver (communication). To form an image, the brightness of each pixel may be used to convey information. Nevertheless, those dark pixels result in imperfect illumination of the target.

The lessons learned from the DRT inspired us to rethink the practical design philosophy of ISAC systems. In particular, one has to take the randomness of communication data into account while conceiving a sensing strategy, which is a unique challenge emerged in the context of ISAC. In this subsection, we investigate a novel precoding design for sensing with random signals.

Let us consider again the sensing model (18b), with the sensing parameters being the entries of the channel matrix $\bm{\mathsfbr{H}}_{\rm s}$, i.e., $\bm{\seta}=\operatorname{vec}(\bm{\mathsfbr{H}}_{\rm s})$. For a given instance of $\bm{\mathsfbr{X}}$, the linear minimum MSE (LMMSE) estimator reads

where $\bm{R}_{\bm{\mathsfbr{H}}}=\mathbb{E}(\bm{\mathsfbr{H}}_{\rm s}^{\rm H}\bm{\mathsfbr{H}}_{\rm s})$ represents the channel correlation matrix. The resulting estimation error is given as

Once again, the estimation error relies on the instantaneous realization of the random ISAC signal $\bm{\mathsfbr{X}}$. To depict the average sensing performance, we take the expectation of (37) over $\bm{\mathsfbr{X}}$, yielding

We refer the term (38) to as the ergodic LMMSE (E-LMMSE) [25], as it may be understood as a time average over different realizations of $\bm{\mathsfbr{X}}$. Obviously, it is lower-bounded by the Miller-Chang CRB in (19).

We now investigate a specific form of the ISAC signal by letting $\bm{\mathsfbr{X}}=\bm{W}\bm{\mathsfbr{S}}$, where $\bm{W}\in\mathbb{C}^{M\times M}$ is a precoding matrix, and $\bm{\mathsfbr{S}}=\left[{\bm{\mathsfbr{s}}_{1}},{\bm{\mathsfbr{s}}_{2}},\ldots,{\bm{\mathsfbr{s}}_{T}}\right]\in\mathbb{C}^{M\times T}$ contains column-wise i.i.d. data symbols, satisfying $\mathbb{E}(\bm{\mathsfbr{s}}_{i})=\bm{0}$ and $\mathbb{E}({\bm{\mathsfbr{s}}_{i}}\bm{\mathsfbr{s}}_{i}^{\rm H})=\bm{I}$. A fundamental question to ask is: What is the optimal precoder $\bm{W}$ that minimizes the E-LMMSE? In classical MIMO radar waveform design, strictly orthogonal signals are typically employed, namely, $\frac{1}{T}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}=\bm{I}$, where $\bm{\mathsfbr{S}}$ is a semi-unitary matrix, corresponding to the $P_{\rm SC}$ point discussed in Sec. IV. In such a case, it is known that the LMMSE-optimal precoder has a water-filling structure given by [26]