Communication-Assisted Sensing Systems: Fundamental Limits and ISAC Waveform Design
Fuwang Dong Fan Liu Yifeng Xiong Yuanhao Cui Wei Wang Shi Jin

	
 



Communication-Assisted Sensing Systems: Fundamental Limits and ISAC Waveform Design
    Fuwang Dong, Member, IEEE, Fan Liu, Senior Member, IEEE, Yifeng Xiong, Member, IEEE, 
 Yuanhao Cui, Member, IEEE, Wei Wang, Senior Member, IEEE, Shi Jin, Fellow, IEEE 

(Corresponding author: Fan Liu.)
Part of this paper was presented at IEEE International Symposium on Information Theory (ISIT), 2024 <cit.>.

Fuwang Dong, and Fan Liu are with the School of System Design and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen 518055, China. (email: {dongfw, liuf6}@sustech.edu.cn).

Yifeng Xiong, and Yuanhao Cui are with the School of Information and Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China. (email: {yifengxiong, cuiyuanhao}@bupt.edu.cn).

Wei Wang is with the College of Intelligent System Science and Engineering, Harbin Engineering University, Harbin 150001, China. (email: wangwei407@hrbeu.edu.cn).

Shi Jin is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China. (e-mail: jinshi@seu.edu.cn). 
    September 9, 2024
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§ ABSTRACT 
The communication-assisted sensing (CAS) systems are expected to endow the users with beyond-line-of-sight sensing capabilities without the aid of additional sensors. In this paper, we study the dual-functional signaling strategy, focusing on three primary aspects, namely, the information-theoretic framework, the optimal distribution of channel input, and the optimal waveform design for Gaussian signals. First, we establish the information-theoretic framework and develop a modified source-channel separation theorem (MSST) tailored for CAS systems. The proposed MSST elucidates the relationship between achievable distortion, coding rate, and communication channel capacity in cases where the distortion metric is separable for sensing and communication (S&C) processes. Second, we present an optimal channel input design for dual-functional signaling, which aims to minimize total distortion under the constraints of the MSST and resource budget. We then conceive a two-step Blahut-Arimoto (BA)-based optimal search algorithm to numerically solve the functional optimization problem. Third, in light of the current signaling strategy, we further propose an optimal waveform design for Gaussian signaling in multi-input multi-output (MIMO) CAS systems. The associated covariance matrix optimization problem is addressed using a successive convex approximation (SCA)-based waveform design algorithm. Finally, we provide numerical simulation results to demonstrate the effectiveness of the proposed algorithms and to show the unique performance tradeoff between S&C processes.             
                       



Integrated sensing and communication, communication assisted sensing, rate-distortion theory, source-channel separation theorem, Blahut-Arimoto algorithm.  




§ INTRODUCTION

Integrated sensing and communication (ISAC) systems are widely acknowledged for their potential to enhance snc performance by sharing hardware and spectrum resources <cit.>. Over the past few decades, substantial research efforts have focused on improving resource efficiency and mitigating mutual interference through waveform design <cit.>, beamforming <cit.>, and dedicated signal processing techniques <cit.>. To further explore the coordination gains between snc, the authors of <cit.> proposed a sensing-assisted communication scheme, which leverages sensing capability of the transmitting ISAC signal, such as beam tracking and prediction, to establish user links thereby reducing the communication overhead. On the other hand, sensing-as-a-service is expected to become a key feature of upcoming 6G perceptive networks <cit.>. This necessitates the exploration of how the sensing quality can be enhanced by leveraging communication functionality, namely, the cas techniques.
          




 §.§ Communication-Assisted Sensing Systems

The effective sensing range may be significantly extended by leveraging the data transmission capabilities of communication systems. This capability, referred to as beyond-line-of-sight (BLoS) sensing, allows for detecting obstructed or exceptionally distant targets. In <cit.>, BLoS sensing was investigated by sharing measurements collected by vehicle-mounted sensors among users. Recently, a novel cas system has been proposed in <cit.>, which enables BLoS sensing as an intrinsic network capability. Unlike the techniques proposed in <cit.>, the CAS system eliminates the need for additional sensors.

As shown in Fig. <ref>, the base station (BS) or roadside unit (RSU) with favorable visibility illuminates targets and captures observations during the sensing process while simultaneously transmitting the acquired sensory information to the end-users during the communication process. Therefore, the snc processes are coupled at the BS and concurrently determine the sensing quality attained by the user. The authors of <cit.> have proposed two signaling strategies, i.e., the separated snc and the dual-functional signaling, to minimize the total distortion between the ground truth and the corresponding reconstruction at end-users.      

 

In the dual-functional signaling strategy, the BS transmits a sophisticatedly tailored ISAC waveform to simultaneously perform snc tasks. Namely, the ISAC waveform is used to sense the current targets' states and to convey the previously acquired state information to the end-users. However, in this strategy, where there is strong snc coupling, two critical problems still remain unclear. 1) The problem formulation in <cit.> is established at an information theoretical level, whose operational meaning is not proved from a coding theory perspective. 2) The ISAC waveform is assumed to be a Gaussian distribution, which may not be necessarily optimal for the considered CAS scenario. For instance, Gaussian distribution is communication-optimal under Gaussian channels, whereas 2-ary pulse amplitude modulation is sensing-optimal <cit.>. This motivates us to delve into the fundamental limits and the resulting snc tradeoff of the dual-functional signaling strategy for CAS systems.    



        


 §.§ Related Works


Exploring the fundamental limits plays an important role in the ISAC systems <cit.>. The groundbreaking studies <cit.> have considered a scenario where the transmitter (Tx) communicates with a user through a memoryless state-dependent channel while simultaneously estimating the state from generalized feedback. The capacity-distortion-cost tradeoff of this channel is characterized to illustrate the optimal achievable rate for reliable communication while maintaining a preset state estimation distortion. Concurrently, the work of <cit.> has addressed a general scenario where the Tx senses arbitrary targets rather than the specified communication channel state. This study reveals the deterministic-random tradeoff between snc within the dual-functional signaling strategy by characterizing the Cramér-Rao bound (CRB)-communication rate region. Unfortunately, the fundamental limits of CAS systems, whose setups significantly differ from the existing literature, remain widely unexplored. 

The information-theoretic framework of the dual-functional signaling strategy for CAS systems is depicted in Fig <ref>. The ISAC channel input X is employed both to acquire the observation Z of the original state S from the sensing channel, and to carry the estimate S̃ to the communication channel. Although this framework shares some similarities with conventional remote source coding and remote estimation problems, it exhibits a unique challenge in ISAC channel input design. In Remote source coding <cit.>, the encoder cannot access the original source information S but only its noisy observation Z. In this context, S↔Z↔X↔Y↔Ŝ forms a Markov chain, where the observation Z does not depend on the communication channel input X. However, in our scheme, the sensing channel output Z is determined by the ISAC channel input X due to the dual-functional signaling strategy. This implies that the process is no longer a Markov chain in CAS systems, complicating the characterization of the rate-distortion relationship using remote source coding theory.   

The remote estimation problem involves a sensor measuring the state of a linear system and transmitting its observations to a remote estimator over a wireless fading channel <cit.>. A typical application of the remote estimation problem in the ISAC field is vehicular networks, such as the sensory data-sharing scheme considered in <cit.>. This widely investigated problem differs from the proposed CAS system in two ways. First, remote estimation requires additional sensors, whereas sensing capability is an intrinsic function in CAS system. Second, similar to the remote source coding problem, the observations and channel inputs are not coupling. Consequently, the snc performance tradeoff leads to a significantly different working pipeline and unique challenges in CAS system design.         

It would be worthwhile to remark that, in most works on remote source coding and remote estimation, the encoder directly transmits the observation Z, which is estimated at the user-end. This strategy is referred to as the compress-and-estimation. In the proposed CAS systems, however, we focus on a scenario where the BS transmits the estimate S̃ instead of the observation. This estimate-and-compress strategy achieves an optimal tradeoff between rate and distortion, as discussed in <cit.>. We will leave the compress-and-estimate strategy for future research. 



 §.§ Our contributions
 
To fill the aforementioned research gap, we aim to bring new insights into the dual-functional signaling strategy for CAS systems, from the fundamental limits to the specific waveform design. Compared to the conference version <cit.>, which only discussed the corresponding information-theoretic framework, we further elaborate on the the optimal distribution of channel input and propose a novel method for Gaussian waveform design in MIMO CAS systems. For clarity, the main contributions of this work are summarized as follows. 


	
  * First, we establish the information-theoretic framework for CAS systems to illustrate the relationship between achievable distortion, coding rate, and communication channel capacity. We develop a modified source-channel separation theorem (MSST) specific to the cases of separable distortion metric for snc processes. Compared to the existing works <cit.>, we provide a rigorous proof of the operational meaning for the MSST.      

	
  * Second, we formulate a unified optimization problem for the input distribution design of the ISAC channel, where the total distortion is minimized while adhering to the MSST and resource constraints. To cope with the functional optimization problem, we conceive a two-step Blahut-Arimoto (BA)-based optimal search algorithm in an effort to tackle the challenges of lacking explicit expressions for the rate-distortion bound and channel capacity.    
	
	
  * Third, we propose an optimal waveform design scheme for Gaussian signaling in MIMO CAS systems. The associated covariance matrix optimization problem is solved using an successive convex approximation (SCA)-based waveform design algorithm.
		
	
  * Finally, we provide numerical simulation results to show the effectiveness of the proposed algorithms, while demonstrating the unique performance tradeoff between snc in the considered CAS systems. 
	
 


This paper is structured as follows. We commence with establishing the information-theoretic framework in Section <ref>, including the definitions of distortions, rate-distortion function, and constrained channel capacity. In Section <ref>, we prove the achievability and converse of the proposed MSST. Then, we formulate the optimization problem for ISAC channel input design and develop a two-step BA-based optimal search algorithm in Section <ref>. In Section <ref>, we present the Gaussian ISAC waveform design method for the MIMO CAS systems. Finally, we provide the simulation results and conclude this paper in Section <ref> and <ref>, respectively.
 
The notations used in this paper are as follows. The uppercase normal letter A, lowercase italic letter a, and fraktur letter 𝒜 denote a random variable, its realization and a set, respectively. Aa represent a probability distribution function and A|Ba|b specifies to channel transition probability. Uppercase and lowercase bold letters 𝐀 and 𝐚 denote the matrix and column vector. (·)^T, (·)^*, and (·)^H represent the transpose, conjugate, and complex conjugate transpose operations, respectively. · is the statistical expectation, and · is the trace of a matrix.
  



§ INFORMATION-THEORETIC FRAMEWORK OF CAS
 



 §.§ Sensing and Communication Processes


As shown in Fig. <ref>, X∈𝒳 serves as the input to both the snc channel. The random variables of the target's state S, the sensing channel output Z, and the communication channel output Y take values in the sets 𝒮, 𝒵, and 𝒴, respectively. Here, the state sequence {S_i}_i ≥ 1 is iid subject to a prior distribution Ss. In general, the cas system may be described as the following snc processes.

∙ Sensing Process: The sensing channel output, Z_i, at a given time i is generated based on the sensing channel law Z|XS·|x_i,s_i given the ith channel input X_i=x_i and the state realization S_i=s_i. We assume that the sensing channel output Z_i is independent of past inputs, outputs and state signals. Let 𝒮̃ denote the estimate alphabet. The state estimator is a map from the acquired observations 𝒵^n to 𝒮̃^n. Thus, the expected average per-block estimation distortion in the sensing process can be defined by

    Δ_s^(n):= d(S^n,S̃^n)=1/n∑_i=1^n d(S_i,S̃_i),
    
where d(·, ·) is the distortion function bounded by d_max.  

∙ Communication Process: The BS encodes the estimate S̃_i-1 into the communication channel input X_i. The user receives the communication channel output Y_i in accordance with channel law Y|Xy|x. The decoder is a map from 𝒴^n to 𝒮̂^n, where 𝒮̂ denotes the reconstruction alphabet. Therefore, the expected average per-block estimation distortion in the communication process can be defined by  

    Δ_c^(n):= d(S̃^n,Ŝ^n) =1/n∑_i=1^n d(S̃_i,Ŝ_i).
 
Additionally, the performance of the CAS systems may be evaluated by the total distortion between the ground truth and its reconstruction at the user,

    Δ^(n):= d(S^n,Ŝ^n) = 1/n∑_i=1^n d(S_i,Ŝ_i).
  

By denoting R as the bit rate, a (2^nR, n) coding scheme for the CAS systems consists of 

1) A state parameter estimator h: 𝒳^n ×𝒵^n →𝒮̃^n [We will show that the optimal estimator h can be achieved by a symbol-by-symbol estimation later in Lemma 1. Namely, the estimate S̃_i only depends on X_i and Z_i but independent of past inputs and outputs. Here, we define the mapping in block form to maintain generality.];

2) A message set (also the estimate set) 𝒮̃^n = [1:2^nR];

3) An encoder ϕ: 𝒮̃^n →𝒳^n;

4) A decoder ψ: 𝒴^n →𝒮̂^n.

In practical systems, the snc channel input X may be restricted by limited system resources. Let us define the cost-function b(x):𝒳→ℝ^+ as the channel cost, then a rate-distortion-cost tuple (R,D,B) is said achievable if there exists a sequence of (2^nR,n) codes that satisfy

    _n →∞Δ^(n)≤ D,  5pt _n →∞𝔼[b(X^n)] ≤ B,
 
where D and B represent the total distortion and resource budget, respectively.
    


 §.§ Optimal Sensing Estimator

The estimate S̃, as well as the communication source, is determined by the choice of sensing estimator at the BS. The optimal estimator is given through the following lemma.

Lemma 1 <cit.>: By recalling that S↔XZ↔S̃ forms a Markov chain, the sensing distortion Δ_s^(n) is minimized by the deterministic estimator

    h^⋆(x^n,z^n):= (s̃^⋆(x_1,z_1),s̃^⋆(x_2,z_2),⋯,s̃^⋆(x_n,z_n)),
   
where

    s̃^⋆(x,z):= min_s'∈𝒮̃∑_s ∈𝒮S|XZs|x,zd(s,s'),

with posterior transition probability being

    S|XZs|x,z = SsZ|XSz|x,s/∑_s' ∈𝒮Ss'Z|XSz|x,s' ,

which is independent to the choice of encoder and decoder.

The detailed proof can be found in <cit.>. By applying the estimator (<ref>), the estimation-cost function wrt the channel input realization can be defined by

    e(x) = d(S,s̃^⋆(X,Z))|X=x.
 
Thus, a given estimation (or sensing) distortion D_s satisfies 

    _n →∞Δ_s^(n) = _n →∞1/n∑_i=1^n e(X)≤ D_s.






 §.§ Constrained Communication Channel Capacity


The snc processes are strongly coupled through the common snc channel input, leading to a performance tradeoff between the sensing distortion and the achievable communication channel capacity. To proceed, let us define the channel capacity and the rate-distortion function under the CAS regime.  

Definition 1: The information-theoretic channel capacity constrained by the desired sensing distortion D_s and resource cost B can be defined by  

    C^IT(D_s, B) = Xx∈𝒫_D_s∩𝒫_B I(X;Y),
    
where I(X;Y) denotes the mutual information (MI) between the communication channel input and output, 𝒫_D_s is the sensing feasible probability set whose element satisfies

    𝒫_D_s= {Xx|e(X)≤ D_s },
  
and 𝒫_B is the resource feasible probability set described by

    𝒫_B={Xx|b(X)≤ B }.
    

Note that the sequence {X_i}_i ≥ 1 is iid with distribution Xx for achieving the capacity-distortion region. By recalling the iid state sequence {S_i}_i ≥ 1, the sensing channel output {Z_i}_i ≥ 1 is also iid subject to the distribution Zz=∑_s ∈𝒮∑_x ∈𝒳Z|XSz|x,sSsXx. Furthermore, according to the symbol-by-symbol optimal sensing estimator in Lemma 1, we have the fact that the estimate {S̃_i}_i ≥ 1 is iid with the following distribution

    S̃s̃ = ∑_z ∈𝒵∑_x ∈𝒳∑_s ∈𝒮Z|XSz|x,sSsXx𝕀{s̃^⋆(x,z)=s̃},

with 𝕀(·) being the indicator function. Therefore, for the iid source S̃, the relationship between communication distortion D_c and bit rate R can be characterized by the rate-distortion theory.  

Definition 2: The information-theoretic rate distortion function can be defined by 

    R^IT(D_c) = Ŝ|S̃ŝ|s̃: d(S̃,Ŝ)≤ D_c I(S̃;Ŝ).

Here, we have

    I(S̃;Ŝ) =^(a) I(X,Z;Ŝ) =^(b) I(X;Ŝ),

where (a) is due to the deterministic sensing estimator (<ref>) in Lemma 1, and (b) follows the fact that Ŝ is conditionally independent to Z with a given X.  




§ MODIFIED SOURCE-CHANNEL SEPARATION THEOREM
 

In this section, we develop the modified source-channel separation theorem tailored for the dual-functional signaling CAS systems, aiming to elucidate the relationships between total distortion, coding rate and communication channel capacity. The MSST retains a form analogous to the conventional source-channel separation theorem with distortion in lossy data transmission <cit.>.    

Theorem 1: Under the condition that the distortion metric d(·, ·) is separable for the snc processes, i.e.,

    d(S,Ŝ) = d(S,S̃) + d(S̃,Ŝ),
    
the total distortion D can be achieved by the sum of snc distortions, if and only if 

    R^IT(D_c) ≤ C^IT(D_s, B),

where C^IT(D_s, B) and R^IT(D_c) are the constrained channel capacity (<ref>) and the rate-distortion function (<ref>), respectively. 

The above MSST is proposed mathematically within the framework of information theory. Next, we elucidate the operational meaning of the MSST by proving that there must exist a practical coding scheme satisfying it.   



 §.§ Converse

We start with a converse to show that any achievable coding scheme must satisfy (<ref>). Consider a (2^nR,n) coding scheme defined by the encoding and decoding functions ϕ and ψ. Let S̃^n = h^⋆(X^n,Z^n) be the estimate sequence as given in (<ref>) and Ŝ^n = ψ(ϕ(S̃^n)) be the reconstruction sequence corresponding to S̃^n. 

Let us focus on the communication process S̃↔X↔Y↔Ŝ. By recalling from the proof of the converse in lossy source coding, we have

    R    ≥1/n∑_i=1^n I(S̃_i;Ŝ_i) ≥^(a)1/n∑_i=1^n R^IT( d(S̃_i,Ŝ_i)) 
       ≥^(b)  R^IT(1/n∑_i=1^nd(S̃_i,Ŝ_i)) ≥^(c) R^IT(D_c),

where (a) follows the Definition 2 that R^IT is the minimum required MI, (b) and (c) are due to the convexity and non-increasing properties of the rate distortion function. 

On the other hand, by recalling from the proof of the converse in channel coding, we have

    R    ≤1/n∑_i=1^n I(X_i;Y_i) 
       ≤^(d)1/n∑_i=1^n C^IT( ∑_x∈𝒳X_ixc(x), ∑_x∈𝒳X_ixb(x)) 
       ≤^(e)  C^IT(1/n∑_i=1^n∑_x∈𝒳X_ixc(x), 1/n∑_i=1^n∑_x∈𝒳X_ixb(x)) 
       ≤^(f) C^IT(D_s,B),
  
where (d) follows the Definition 1 that the channel capacity is the maximum MI, (e) and (f) are due to the concavity and non-decreasing properties of the capacity constrained by estimation and resource costs <cit.>. By combing the inequalities (<ref>), (<ref>) and the data processing inequality I(S̃;Ŝ) = I(X;Ŝ) ≤ I(X;Y) benefit from the Markov chain X↔Y↔Ŝ, we complete the proof of the converse.  



 §.§ Achievability

Let S^n be drawn iid ∼Ss, we will show that there exists a coding scheme for a sufficiently large n and rate R, the distortion Δ^n can be achieved by D if (<ref>) holds. The core idea follows the famous random coding argument and source channel separation theorem with distortion.  

1) Codebook Generation: In source coding with rate distortion code, randomly generate a codebook 𝒞_s consisting of 2^nR sequences Ŝ^n which is drawn iid ∼Ŝŝ. The probability distribution is calculated by Ŝŝ = ∑_s̃S̃s̃Ŝ|S̃ŝ|s̃, where S̃s̃ is defined in (<ref>) and Ŝ|S̃ŝ|s̃ achieves the equality in (<ref>). In channel coding, randomly generate a codebook 𝒞_c consisting of 2^nR sequences X^n which is drawn iid ∼Xx. The Xx is chosen by satisfying the constrained capacity with estimation- and resource-cost in (<ref>). Index the codeword S̃^n and X^n by w ∈{1,2,⋯, 2^nR}. 

2) Encoding: Encode the S̃^n by w such that

    (S̃^n, Ŝ^n(w)) ∈𝒯^(n)_d,ϵ_s (S̃Ŝs̃,ŝ),

where 𝒯^(n)_d,ϵ_s (S̃Ŝs̃,ŝ) represents the distortion typical set <cit.> with joint probability distribution S̃Ŝs̃,ŝ = S̃s̃Ŝ|S̃ŝ|s̃. To send the message w, the encoder transmits x^n(w). 

3) Decoding: The decoder observes the communication channel output Y^n=y^n and look for the index ŵ such that 

    (x^n(ŵ), y^n) ∈𝒯^(n)_ϵ_c(XYx,y),
 
where 𝒯^(n)_ϵ_c(XYx,y) represents the typical set with joint probability distribution XYx,y=XxY|Xy|x.  If there exists such ŵ, it declares Ŝ^n = ŝ^n(ŵ). Otherwise, it declares an error. 

4) Estimation: The encoder observes the channel output Z^n = z^n, and computes the estimate sequence with the knowledge of channel input x^n by using the estimator s̃^n = h^⋆(x^n,z^n) given in (<ref>). 

5) Distortion Analysis: We start by analyzing the expected communication distortion (averaged over the random codebooks, state and channel noise). In lossy source coding, for a fixed codebook 𝒞_s and choice of ϵ_s > 0, the sequence s̃^n ∈𝒮̃^n can be divided into two categories:

∙ (s̃^n, ŝ^n(w)) ∈𝒯^(n)_d,ϵ_s, we have d(s̃^n, ŝ^n(w)) < D_c + ϵ_s; 

∙ (s̃^n, ŝ^n(w)) ∉𝒯^(n)_d,ϵ_s, we denote P_e_s as the total probability of these sequences. Thus, these sequence contribute at most P_e_sd_max to the expected distortion since the distortion for any individual sequence is bounded by d_max.    

According to the achievability of lossy source coding <cit.>, we have P_e_s tends to zero for sufficiently large n whenever R ≥ R^IT(D_c). 

In channel coding, the decoder declares an error when the following events occur:

∙  (x^n (w), y^n) ∉𝒯^(n)_ϵ_c; 

∙  (x^n (w'), y^n) ∈𝒯^(n)_ϵ_c, for some w'  w, we denote P_e_c as the probability of the error occurred in decoder. The error decoding contribute at most P_e_cd_max to the expected distortion. Similarly, we have P_e_c→ 0 for n →∞ whenever R ≤ C^IT(D_s, B) according to channel coding theorem <cit.>. 

On the other hand, the expected estimation distortion can be upper bounded by

    Δ_s^(n) = 1/n∑_i=1^n d(S_i,S̃_i)|Ŵ wPr(Ŵ w) 
        30pt + 1/n∑_i=1^n d(S_i,S̃_i)|Ŵ = wPr(Ŵ = w) 
       ≤ P_e_cd_max + 1/n∑_i=1^n d(S_i,S̃_i)|Ŵ = w (1-P_e_c).


Note that (s^n, x^n(w), s̃^n) ∈𝒯^(n)_ϵ_e (SXS̃s,x,s̃) where SXS̃s,x,s̃ denotes the joint marginal distribution of SXZS̃s,x,z,s̃ = SsXxZ|SXz|s,x𝕀{s̃=s̃^⋆(x,z)}, we have 

    _n →∞1/n∑_i=1^n d(S_i,S̃_i)|Ŵ = w≤ (1+ϵ_e) d(S,S̃),
    
according to the typical average lemma <cit.>. In summary, the total distortion can be attained by

    Δ^(n)   =^(a)Δ_s^(n) + Δ_c^(n)
       ≤^(b) D_c + (P_e_s+ 2P_e_c)d_max + (1+ϵ_e)(1-P_e_c)D_s,
 
where (a) follows the condition of the separable distortion metric in MSST and we omit the terms containing the product of P_e_s and P_e_c in step (b). Consequently, taking n →∞ and P_e_s, P_e_c, ϵ_c,ϵ_e,ϵ_s → 0, we can conclude that the expected total distortion (averaged over the random codebooks, state and channel noise) tends to be D → D_c+D_s whenever R^IT(D_c) ≤ R ≤ C^IT(D_s, B).  

This completes the proof of the proposed MSST. ▪ 

Remark: The sensing process determines the accuracy of the state information acquisition at the BS, whereas the communication process governs the quantity of information transmitted to the user. Overemphasis on either process can result in substantial performance degradation. For instance, achieving optimal sensing performance alone may not ensure the accurate reconstruction at the end-user's side due to limitations in the communication channel capacity. Conversely, excessive communication capacity may be wasted if the sensory data lacks sufficient accuracy. This underscores the need to investigate optimal channel input as well as the waveform design that minimize the total distortion. 



§ CHANNEL INPUT DISTRIBUTION OPTIMIZATION FOR THE CAS SYSTEMS
 
In this section, we explore the optimal input distribution for the CAS system under scalar channels, aiming to minimize total distortion under the MMST constraint. It should be noted that the MSST applies only when the distortion metric is separable. Fortunately, we demonstrate that the mse or quadratic distortion, a widely used distortion metric in parameter estimation, satisfies the separability condition. Specifically, we have

    D       = S-Ŝ
       = ^(a)𝔼[𝖲- 𝖲̃_2^2 ]+𝔼[𝖲̃ - 𝖲̂_2^2 ]  = ^Δ D_s+D_c,
     
where (a) holds from the properties of the conditional expectation <cit.>

    𝔼[(𝖲- 𝖲̃)^T(𝖲̃ - 𝖲̂)]= 𝔼[(𝖲-𝔼[ 𝖲|𝖹, 𝖷])^Tf(𝖹, 𝖷)]=0,

with 𝖲̃=𝔼[ 𝖲|𝖹, 𝖷] being the optimal estimator in (<ref>) [We highlight that the (<ref>) is exactly the minimum MSE (MMSE) estimator. An arbitrary non-optimal estimator except MMSE, which cannot apply the conditional expectation properties, may not satisfy the separability condition. ]. 

Hereinafter, we adopt the mse as the distortion metric to guarantee the MSST. Meanwhile, we drop the superscript “IT” in the MSST.      
   


 §.§ Problem Formulation


The ISAC channel input design can be formulated by the following optimization problem [Here, we use the symbol I(D_s,B) instead of C(D_s,B) to emphasize that the optimal distribution does not necessarily achieve the channel capacity.]

    𝒫_0  { 2pt
    min_P_𝖷(x) 5pt     D = D_c + D_s   
    subject to 5pt     R(D_c) ≤ I(D_s, B). 
    .
 
By substituting the associated expressions into (<ref>), problem 𝒫_0 can be reformulated by

    𝒫_1  { 2pt
    	min_P_𝖷(x) 5pt    d(S,S̃) + d(S̃,Ŝ)
    subject to 5pt     I(𝖲̃;Ŝ) ≤ I(X;Y), 
     5pt    b(X)≤ B.
    	.

It is challenging to directly solve the functional optimization problem 𝒫_1 due to the difficulty in obtaining explicit expressions of distortions and MI for arbitrary distributions. Typically, the BA algorithm can be employed to solve problem 𝒫_1 numerically. Inspired by this idea, we expand the expectation operation and the MI wrt the variable Xx as follows.

∙ Sensing distortion:

    d(S,S̃)   = d(𝖲,s̃(𝖷,𝖹))|𝖷,𝖹
       = ∑_x,z P_𝖷𝖹(x,z) ∑_s Q_𝖲|𝖷𝖹(s|x,z) d(s,s̃(x,z)) 
       ≜∑_x P_𝖷(x) e(x),

where the sensing cost e(x) defined in (<ref>) can be expressed by  

    e(x)=∑_z Q_𝖹|𝖷(z|x) ∑_s Q_𝖲|𝖷𝖹(s|x,z) d(s,s̃(x,z)).


∙ Channel MI:

    I(𝖷;𝖸) =  ∑_x,y P_𝖷(x) Q_𝖸|𝖷(y|x) logQ_𝖷|𝖸(x|y)/P_𝖷(x).


∙ Resource budget:

    b(X) = ∑_x P_𝖷(x)b(x).


     
∙ Communication distortion:

    𝔼[d(𝖲̃,Ŝ)] = ∑_s̃ P_𝖲̃(s̃) ∑_ŝ Q_Ŝ|𝖲̃(ŝ|s̃)d(s̃,ŝ).
    

∙ Rate distortion function:

    I(𝖲̃;Ŝ) = ∑_s̃,ŝ P_𝖲̃(s̃) Q_Ŝ|𝖲̃(ŝ|s̃) logQ_𝖲̃|Ŝ(s̃|ŝ)/P_𝖲̃(s̃).


Here, we emphasize the unique challenge in solving problem 𝒫_1. In the conventional BA algorithm, the primary approach involves constructing the Lagrangian function wrt the variable and obtaining the optimal solution based on the first-order necessary condition. However, the probability distribution S̃s̃ in (<ref>) and (<ref>) is indeed a function of the variable Xx as describe in (<ref>). Consequently, deriving a closed-form expression of Xx from first-order necessary condition is challenging, which imposes a significant obstacle to solving problem 𝒫_1 by using the conventional BA algorithm. To address this issue, we propose a two-step BA-based optimal search method to seek for a sub-optimal solution in the following subsection.      



 §.§ Two-step BA-based Optimal Search Algorithm


We divide the original problem 𝒫_1 into two sub-problems.   

1) Sub-problem 1: We determine the constrained communication channel capacity I(X,Y) for a given sensing distortion D_s and identify the optimal distribution P_𝖷(x).  

2) Sub-problem 2: We calculate the minimum communication distortion D_c for the given channel input distribution P_𝖷(x) and determine the communication distortion D_c, thereby obtaining the total distortion D.

By varying the preset sensing distortions, one may generate a set 𝒟 collecting the total distortions calculated through the above two steps. Thus, the minimum total distortion and its corresponding distribution P_𝖷(x) can be identified by the minimum value search. The detailed procedure is as follows.

∙ Constrained communication channel Capacity for a given sensing distortion.

In this sub-problem, I(𝖷;𝖸) is maximized under the constraints of sensing distortion D_s and resource budget B, which can be expressed by   


    max_P_𝖷(x) 10pt     I(𝖷;𝖸) 
    subject to    5pt ∑_x P_𝖷(x) e(x) ≤ D_s,  5pt  ∑_x P_𝖷(x)b(x) ≤ B.

Inspired by <cit.>, we reformulate (<ref>) by incorporating the sensing distortion cost as a penalty term in the objective function, i.e.,

    max_P_𝖷(x) 5pt     I(P_𝖷(x), Q_𝖸|𝖷(y|x)) - μ∑_x P_𝖷(x) e(x)  
    subject to    5pt  ∑_x P_𝖷(x)b(x) ≤ B,
      
where μ is the penalty factor to balance the weight of snc performance. For a given μ, the Lagrangian function of (<ref>) can be written by 

    ℒ(P_𝖷(x), λ)     = I(P_𝖷(x), Q_𝖸|𝖷(y|x)) 
       - μ∑_x P_𝖷(x) e(x) + λ∑_x P_𝖷(x)b(x).
 
By setting the derivative ∂ℒ(P_𝖷(x), λ) / ∂ P_𝖷(x) to zero and taking the fact that ∑_x P_𝖷(x)=1 into account, we have   

    P_𝖷(x) = e^h(x)/∑_x' e^h(x'),

with

    h(x) = ∑_y Q_𝖸|𝖷(y|x) log Q_𝖷|𝖸(x|y)-μ e(x)-λ b(x).


In formula (<ref>), the Lagrangian multiplier λ and the posterior distribution Q_𝖷|𝖸(x|y) are unknown. The former may be determined by a bisection search method such that ∑_x P_𝖷(x)b(x) → B. For the latter, we have the optimal posterior distribution which maximizes the MI with given P_𝖷(x) and Q_𝖸|𝖷(y|x), expressed by <cit.> 

    Q_𝖷|𝖸(x|y) =  P_𝖷(x)Q_𝖸|𝖷(y|x)/∑_x'P_𝖷(x')Q_𝖸|𝖷(y|x').

After setting an initial distribution X_0x, problem (<ref>) can be solved by updating (<ref>) and (<ref>) iteratively. The detailed procedure is summarized in Algorithm <ref>, whose convergence can be guaranteed <cit.>. 



∙ Source distribution S̃s̃.

The tradeoff between sensing distortion and constrained communication capacity can be adjusted by varying the penalty factor μ. Let us denote the optimal channel input distribution, sensing distortion, and channel MI obtained by Algorithm <ref> as P_𝖷^(μ)(x), D_s^(μ), and I^(μ)(X;Y), respectively, to highlight their dependence on the factor μ. 

Subsequently, given a specific channel input distribution P_𝖷^(μ)(x), our objective is to compute the estimate distribution P_𝖲̃^(μ)(s̃), which is essential for determining the rate-distortion function. However, deriving the explicit expression of the estimate distribution for an arbitrary state distribution Ss is a challenging task. Instead, we can compute the estimate S̃ and its distribution P_𝖲̃^(μ)(s̃) using formulas (<ref>) and (<ref>), respectively, by performing a sufficiently large number of random trials. For analytical convenience, we consider a Gaussian state distribution and a linear sensing model, which allows for explicit expressions to be derived.  

Assume that the target's state follows Gaussian distribution with 𝒞𝒩(0,ν_s^2). Let us consider the following linear sensing model                

    𝖹=𝖷𝖲+𝖭,
 
where 𝖭∼𝒞𝒩(0,1) represents Gaussian channel noise. In such a Gaussian linear model, the estimator (<ref>) is indeed an MMSE estimator for each channel realization x. Therefore, the estimate S̃ and the corresponding conditional probability distribution of P_𝖲̃|𝖷(s̃|x) can be expressed by  


    𝖲̃ = x^2ν_s^4/(1+x^2ν_s^2)^2Z, 5pt P_𝖲̃|𝖷(s̃|x) ∼𝒞𝒩(0,x^2ν_s^4/1+x^2ν_s^2).
     
Moreover, sensing cost is the MSE with the given realization x, which can be written by 

    e(x)= ν_s^2/(1+x^2ν_s^2).

Consequently, the source distribution can be given by    

    P_𝖲̃^(μ)(s̃) = P_𝖷^(μ)(x)P_𝖲̃|𝖷(s̃|x).





∙ Communication and total distortions.

In this sub-problem, we aim to evaluate the minimum communication distortion D_c^(μ) achieved under the source distribution P_𝖲̃^(μ)(s̃) and the channel capacity I^(μ)(X;Y). This scenario corresponds to a typical rate-distortion problem in lossy data transmission. The distortion-rate curve can be obtained by the optimization problem <cit.>   

    min_Q_Ŝ|𝖲̃(ŝ|s̃) 5pt I(𝖲̃;Ŝ) - λ_s D_c

where λ_s represents the slope of the distortion-rate curve. The conventional BA algorithm can be leveraged to solve problem 
(<ref>). By following a similar procedure in sub-problem 1, the optimal solution can be calculated by

    Q_Ŝ|𝖲̃(ŝ|s̃) = P_Ŝ(ŝ)e^λ_s d(s̃,ŝ)/∑_ŝP_Ŝ(ŝ')e^λ_s d(s̃,ŝ').
   
The unknown nuisance term Ŝŝ can be updated by

    P_Ŝ(ŝ) = ∑_s̃ P_𝖲̃(s̃)Q_Ŝ|𝖲̃(ŝ|s̃).
 
By setting an initial distribution Ŝ|S̃ŝ|s̃, the optimal solution of (<ref>) can be obtained by updating (<ref>) and (<ref>) iteratively. The detailed derivations can be found in <cit.>. 

For a given slop λ_s, the tangent point ( D^(λ_s)_c, I^(λ_s)(𝖲̃;Ŝ) ) of the distortion-rate curve can be obtained by solving problem (<ref>). We may collect a series of distortion-rate data points to fit a distortion-rate function by varying the slop λ_s. Thus, the communication distortion constrained by a given channel capacity is attained by   

    D_c^(u) = D(I^(u)(X;Y)).
    
We summarize the communication distortion computation algorithm into Algorithm <ref>. 

In the outer loop, we may collect a set 𝒟 of total distortion D^(μ) and the associated channel input distribution P_𝖷^(μ)(x) by repeating the above steps with various penalty factor μ. Therefore, the optimal channel distribution that minimizes the total distortion can be found by     

    D^(μ^⋆) = min_D ∈𝒟 2pt D,  5pt  P_𝖷^(⋆)(x) =  P_𝖷^(μ^⋆)(x).
     
The proposed two-step BA-based optimal search algorithm is summarized in Algorithm <ref>. 





§ WAVEFORM DESIGN FOR GAUSSIAN CHANNEL INPUT
 

The previous section illustrates that using Gaussian channel input may not be optimal for CAS systems. Despite this limitation, the assumption of Gaussian signals is prevalent in communications research literature. In this section, we introduce an ISAC waveform design scheme specifically developed for multi-input multi-output (MIMO) systems, while adhering to the framework of Gaussian signaling. 
      


 §.§ System Model

The widely employed signal models of the snc processes for the MIMO systems are expressed by     

    𝐙=𝐇_s𝐗+𝐍_s,  5pt  𝐘=𝐇_c𝐗+𝐍_c,
  
where 𝐗∈ℂ^N_t × T, 𝐙∈ℂ^M_s × T, and 𝐘∈ℂ^M_c × T represent the transmitting ISAC signal, the received signals at the BS and the end-user, respectively; 𝐇_s and 𝐇_c denote the snc channel state information (CSI) matrices; 𝐍_s and 𝐍_c are the channel noises whose entries follow the complex Gaussian distribution with 𝒞𝒩(0,σ_s^2) and 𝒞𝒩(0,σ_c^2); Finally, N_t, M_s, M_c, and T denote the numbers of Tx antennas, snc receiver (Rx) antennas, and the transmitting symbols, respectively.

We focus on the sensing task of trm estimation, where the to-be-estimate parameters can be written by 𝐬 = vec(𝐇_s). After vectorization operations, the sensing signal observed at the BS can be recast by         

    𝐳=(𝐈_M_s⊗𝐗^H )𝐬+𝐧_s,
  
with 𝐳=vec(𝐙^H), and 𝐧_s = vec(𝐍_s^H). In the CAS process, the BS estimates trm 𝐬̃ from the observations 𝐳, then transmits it to the user through communication channel 𝐇_c. Subsequently, the user can reconstruct 𝐬̂ from the communication received data 𝐘. Here, we adopt the quadratic distortion metric (mse), which meets the separable condition as shown in (<ref>). Before proceeding the ISAC waveform design, we make the following assumptions.       

Assumption 1: The ISAC waveform 𝐗 follows complex Gaussian distribution with 𝒞𝒩(0, 𝐑_x), where 𝐑_x ∈ℂ^N_t × N_t is the covariance matrix.          

Assumption 2: The trm vector 𝐬 follows complex Gaussian distribution 𝒞𝒩(0,𝐈_M_s⊗Σ_s), where Σ_s ∈ℂ^N_t × N_t denotes the covariance matrix of each column of 𝐇_s.[This assumption corresponds to the scenario that the Rx antennas for sensing are sufficiently separated so that the correlations among the rows of 𝐇_s can be ignored <cit.>. Here, we specify this Kronecker structure in TRM covariance matrix to simplify the expression of sensing distortion <cit.>. To avoid the deviation of our core contribution in this paper, we will leave the general TRM covariance matrix cases for future research.] 

The above assumptions enable us to derive the explicit expressions of the snc distortions, the estimate distribution 𝐬̃, and the communication channel capacity wrt the ISAC waveform matrix 𝐗.  
   
1) Sensing distortion: For the Gaussian linear model (<ref>), the optimal estimator (<ref>), i.e., the well-known mmse estimator, is leveraged to obtain the estimate 𝐬̃ by 

    𝐬̃ = (𝐈_M_s⊗(Σ_s𝐗(𝐗^H Σ_s 𝐗 + σ^2_s𝐈_T)^-1))z,

Furthermore, the sensing distortion is attained by <cit.>  

    D_s (𝐑_x) = 𝔼[𝐬-𝐬̃^2] = M_s ( 1/σ^2_s𝐑_x + Σ_s^-1)^-1  .


2) Communication distortion: The estimate 𝐬̃ in (<ref>) follows the complex Gaussian distribution with 𝒞𝒩(0,𝐈_M_s⊗𝐑_s̃), with the covariance matrix of <cit.> 

    𝐑_𝐬̃ = Σ_s - ( 1/σ^2_s𝐑_x + Σ_s^-1)^-1.
  
Let us temporarily omit the Kronecker product. For a Gaussian source 𝒞𝒩(0, 𝐑_s̃), the rate-distortion function can be characterized by the following optimization problem <cit.>     

    R(D_c) = 𝐃∈𝕊_N_t^+   log( (𝐑_s̃)/(𝐃)) 
    subject to    5pt 𝐃≼𝐑_s̃,  2pt 𝐃≤ D_c,

where 𝕊_N_t^+ denotes the set of all N_t × N_t positive definite matrices. More details on (<ref>) can be found in the proof of <cit.>. 

3) Channel Capacity: The MIMO Gaussian channel capacity can be expressed by

    C(𝐑_x) = I(𝐘;𝐗|𝐇_c) = 
    log(1/σ^2_c𝐇_c𝐑_x𝐇_c^H+𝐈_M_c).
      





 §.§ ISAC Waveform Design

Observe that the expressions (<ref>) to (<ref>) are all related to the covariance matrix 𝐑_x rather than the specific waveform matrix 𝐗. Consequently, our ISAC waveform design problem can be transformed into determining the optimal covariance matrix that minimizes the total distortion. Regarding the system resource constraint, we only consider the transmit power at the BS. By substituting the expressions into the original problem 𝒫_1 and introducing an auxiliary variable of positive definite matrix 𝐃, the ISAC waveform design problem can be reformulated by  

    𝒫_2 {𝐑_x,𝐃∈𝕊_N_t^+    D_s(𝐑_x) + 𝐃
    subject to    5pt  C(𝐑_x) - M_s log( (𝐑_s̃)/(𝐃)) ≥ 0,   
        5pt   𝐃≼𝐑_s̃,  2pt 𝐑_x ≽ 0,  2pt 𝐑_x≤ P_T,
    .
   
where P_T is the power budget. 𝒫_2 is non-convex due to the nonlinear intermediate variables log (𝐑_s̃) and 𝐑_s̃. To relax 𝒫_2 into a convex problem, we employ an SCA technique based on Taylor series expansion. To be specified, for any given point 𝐑_0, the matrix 𝐑_s̃ can be approximated by       

    𝐑_s̃≃Σ_s - 𝐏 + 1/σ_s𝐏 (𝐑_x - 𝐑_0) 𝐏≜𝐑_s̃,

with the constant matrix defined by

    𝐏= ( 1/σ^2_s𝐑_0 + Σ_s^-1)^-1.
 
Furthermore, we have 

    log (𝐑_s̃) ≃log( Σ_s - 𝐏) 
        10 pt + 1/σ_s^2 (Σ_s - 𝐏)^-1𝐏 (𝐑_x - 𝐑_0) 𝐏≜ f(𝐑_x).

Note that 𝐑_s̃ and f(𝐑_x) are both the linear functions wrt variable 𝐑_x with a given point 𝐑_0. By substituting (<ref>) and (<ref>) into 𝒫_2, the problem can be relaxed into

    𝒫_3 {𝐑_x,𝐃∈𝕊_N_t^+    D_s(𝐑_x) + 𝐃
    subject to    5pt  C(𝐑_x) - M_s ( f(𝐑_x) - log (𝐃) )  ≥ 0,   
        5pt   𝐑_s̃ - 𝐃≽ 0,  2pt 𝐑_x ≽ 0,  2pt 𝐑_x≤ P_T.
    	.

For a given point 𝐑_0, 𝒫_3 is convex since both the objective and constraints are either convex or linear. Therefore, it can be efficiently solved by using the off-the-shelf CVX toolbox <cit.>. For an initial covariance matrix, e.g., 𝐑_0 = P_T/N_t𝐈, the non-convex problem 𝒫_2 can be addressed by iteratively computing the optimal solution 𝐑^⋆_x of 𝒫_3 and updating 𝐑_0 = 𝐑^⋆_x until convergence. The procedure is outlined in Algorithm <ref>.        





§ SIMULATION RESULTS
 

The simulation results are divided into two sections. The first section evaluates the optimal distribution for the ISAC channel input in scalar cases, demonstrating the effectiveness of the proposed MSST and the two-step BA-based optimal search algorithm. Moreover, the unique distortion-capacity (D-C) curves of the CAS system are provided to illustrate the performance tradeoff between snc processes. The second section presents the results of the optimal waveform design for the Gaussian MIMO channel, showing its superiority over the snc-optimal schemes.      

       


 §.§ Optimal Distribution for ISAC Channel Input


In this subsection, we aim to reveal the unique performance tradeoff between snc processes. The simulation parameters are set as follows. The prior distribution of the target's sate is assumed to be Ss∼𝒞𝒩(0,1), i.e, ν_s^2=1. We define the normalized signal-to-noise ratios (SNRs) for the snc channels as 10log(1/σ^2) dB and set the power budget to be B=5. Unless otherwise specified, the snc channel SNRs are set to be SNR_s=SNR_c=0 dB in this subsection. To improve the computational efficiency, the weighted factor μ takes values from the sets [0,1], [1,5], and [5,30] with non-uniform spacings of 0.1, 0.5, and 5, respectively.    
  








  §.§.§ The D-C curves and the optimal distribution of ISAC channel input

Fig. <ref> presents the D-C curves for both snc processes. As expected, sensing distortion D_s increases as the values of μ decrease, implying that the estimation performance at the BS becomes less significant. Conversely, the communication channel capacity increases accordingly, leading to a reduction in communication distortion D_c. Consequently, these opposing trends in snc distortions result in an uncertain total distortion. In Fig. <ref>, we show the resultant D-C curve for the CAS systems with different sensing SNRs. A CAS-optimal point that minimizes the total distortion is evident, highlighting a unique phenomenon in CAS systems compared to the existing studies <cit.>.


Fig. <ref> depicts the specific channel input distributions corresponding to three special points in Fig.<ref>, namely, the snc-optimal and CAS-optimal points. The channel capacity is maximized without the sensing constraint when μ = 0. It is well-known that Gaussian channel input achieves the maximum MI for the Gaussian channels. By contrast, at the sensing-optimal point, as μ becomes sufficiently large, the channel input exhibits a 2-ary pulse amplitude modulation. These results are consistent with the findings in <cit.>. However, the unique CAS-optimal distribution, presenting a compromise between the snc-optimal distributions, is attained at μ = 2.5.                        







  §.§.§ The impact of the target's prior variance

Subsequently, we investigate the impact of the target's state variance by selecting the values of ν_s^2 ∈{0.1,0.2,0.5,5}. Due to the significant differences in the order of magnitude of the values for MI and sensing distortions in (<ref>), we adjust the interval of μ for each state variance rather than using a fixed interval. The essential purpose of the CAS systems is to reduce the uncertainty of the target's prior information, specifically the variance of the state's prior distribution. 

In Fig. <ref>, each point represents a lower total distortion compared to the prior variance, even at the snc-optimal points. This reduction is attributed to the resource multiplexing gain achieved through the dual-functional signaling strategy. Additionally, we observe that the CAS-optimal point gradually shifts from sensing-optimal to communication-optimal as variance increases. A small variance indicates relatively accurate prior information about the target's state, necessitating an ISAC channel input with strong sensing capabilities for better accuracy performance. Conversely, as variance increases, greater emphasis on communication performance is required to convey more information.     

We provide the detailed distributions of the CAS-optimal channel input for different state variances. For a small variance (e.g., ν_s^2=0.1), the sensing distortion becomes small accordingly. It is at least less than the prior variance; otherwise, the user can `guess' the state information in terms of the prior distribution. Therefore, a large weighted factor is required, e.g., μ = 690, to balance the values of MI and sensing distortion in (<ref>). In this case, the CAS-optimal distribution aligns with the sensing-optimal distribution, specifically a standard 2-ary pulse amplitude modulation. As anticipated, the optimal μ values decrease with increasing state variance, and the CAS-optimal distribution changes accordingly.    
   
 


 §.§ Optimal Waveform Design for Gaussian Signal


In this subsection, we demonstrate the effectiveness of the proposed ISAC waveform design scheme for Gaussian signaling. The general system setups are as follows. The communication channel is modeled by Rayleigh fading, where each entry of 𝐇_c obeys the standard complex Gaussian distribution. All results are obtained by the average of 100 Monte Carlo trials and the average distortion in each Monte Carlo trial is calculated by D/(M_sN_t). 
          




  §.§.§ Waveform design for a 2-D special case

We begin with a two-dimensional (2-D) special case where the state covariance matrix, Σ_s = diag{a_1,a_2}, is assumed to be diagonal. In this scenario, the global optimal solution of problem 𝒫_2 can be attained through a 2-D exhaustive search, as outlined in <cit.>. Thus, this optimal solution [The optimal eigenspace of 𝐑_x must align with the communication eigenspace of 𝐇_c^H𝐇_c as the sensing eigenspace is an arbitrary unitary matrix. Thus, the original problem can be reduced to finding two optimal eigenvalues over 2-D grids, with the sum of each 2-D point equal to P_T.] can serve as a benchmark for comparing the superiority of our method. Accordingly, the numbers of the Tx and snc Rx antennas are set by N_t=M_c=M_s=2. 


Fig. <ref> illustrates the impact of sensing channel quality on the achievable average distortion. The sensing SNRs vary from 0 dB to 20 dB with a spcing 2 dB. Besides the global optimal benchmark, the snc optimal waveform schemes refer to the solutions that minimize the MMSE in (<ref>) and maximize the channel capacity in (<ref>), respectively. At a glance, we observe that the achievable average distortion decreases with increasing sensing channel SNRs. Furthermore, the sensing-optimal scheme is preferable for acquiring accurate target information when the communication channel is sufficiently good at SNR_c=5 dB. However, the sensing-optimal scheme may cause significant performance degradation as the communication SNR decreases. Particularly, the sensing-optimal curve even tends to be flat at SNR_c=-5 dB, implying that a poor communication channel severely limits performance improvement. This clearly highlights the unique tadeoff between snc processes in CAS systems. Finally, it is worth noting that the proposed algorithm achieves satisfactory performance compared to the global optimal solution.                

 

In Fig. <ref>, we investigate the impact of state variances on the achievable average distortion in the MIMO systems. The state variances are scaled by a factor ranging from 2 to 10. Two interesting observations emerge. 1) For small state variances, the global optimal scheme coincides with the sensing-optimal scheme at SNR_c = 10 dB. However, both the snc-optimal schemes experience performance degradation as state variances increase. 2) When the communication channel quality is insufficient at SNR_c = -5 dB, the global optimal scheme tends towards the communication-optimal scheme to enhance communication capabilities. As state variances increase, the performance degradation of the sensing-optimal scheme becomes more significant. Nevertheless, our proposed algorithm can effectively balance the snc processes, resulting in improved average distortion.                   





  §.§.§ Waveform design for general MIMO cases

Subsequently, we extend the results to high-dimensional matrix scenarios, where the number of Tx antennas is N_t = 10, and the numbers of snc Rx antennas are M_s = 2 and M_c = 5, respectively. The state covariance matrix Σ_s is randomly generated as an N_t × N_t Hermitian matrix. It is worth noting that obtaining the global optimal solution is challenging due to the highly coupled snc processes. For comparison purposes, we provide the results of the Heuristic algorithm proposed in <cit.>, whose formulation is given by

    𝐑_x    5pt β I(𝐘;𝐗|𝐇_c) + (1-β) I(𝐙;𝐇_s|𝐗)  
    subject to    5pt  𝐑_x≤ P_T,
 
where I(𝐙;𝐇_s|𝐗) = log(1/σ^2_sΣ_s𝐑_x+𝐈_N_t) is defined by sensing MI and I(𝐘;𝐗|𝐇_c) is defined in (<ref>). The weighted factor β takes values over the interval [0,1] with L grid points. In our simulations, L is set to be 11.   
       
Fig. <ref> illustrates the average distortions versus sensing channel SNRs for various waveform design schemes. It can be observed that the proposed algorithm outperforms its counterparts. Similar trends to those shown in Fig. <ref> for the 2-D case are also noticeable. Besides, we can observe that the proposed ISAC waveform design achieves significantly greater performance gains in high-dimensional scenarios.  





Fig. <ref> and Fig. <ref> demonstrate the influence of snc Rx antennas on average distortion across various waveform design schemes. Fig. <ref> exhibits a linear increase in average distortion with the addition of sensing Rx antennas. The linearity feature consistent with the Kronecker product form in assumption 2. Moreover, regarding the TRM estimation, an increase in sensing Rx antennas necessitates more channel information to be estimated, leading to a larger average distortion.

In contrast, Fig. <ref> shows a declining trend in average distortion as the number of communication Rx antennas increases. This performance improvement is attributed to enhanced communication channel capacity facilitated by multiple Rx antenna gain, thereby improving overall system performance. Additionally, under the communication-optimal scheme, there is a gradual performance degradation as channel capacity increases. This trend arises because improving sensing performance becomes critical in scenarios with excessive communication channel capacity. Once again, the proposed ISAC waveform design method is superior to the other counterparts in all scenarios. 
        
          


§ CONCLUSION
 

In this paper, we delve into the dual-functional signaling strategy for communication-assisted sensing (CAS) systems across three primary aspects. Firstly, we develop a modified source-channel separation theorem (MSST) tailored for CAS systems under the condition of separable distortion metric. We elucidate the operational meaning of the proposed MSST from the perspective of coding theory. Secondly, we develop an input distribution optimization scheme for the CAS system under scalar channels, which minimizes the total distortion while adhering to the MSST and resource constraints. We determine the optimal distribution for CAS systems, balancing between the conventional communication-optimal (Gaussian distribution) and sensing-optimal (2-ary pulse amplitude modulation) schemes. Thirdly, we propose an ISAC waveform design method for Gaussian signaling in MIMO CAS systems, where a successive convex approximation (SCA) algorithm is conceived to solve for the optimal covariance matrix. Simulation results show the unique performance tradeoff between the sensing and communication processes.  















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