Waveform Optimization for MIMO Joint Communication and Radio Sensing Systems with Training Overhead

A joint communication and (radio) sensing (JCAS, a.k.a., Radar-Communications) system that enables share of hardware and signal processing modules, can achieve efficient spectrum efficiency, enhanced security, and reduced cost, size, and weight [1, 2, 3, 4]. JCAS systems can have many potential applications in intelligent transportation that require both communication links connecting vehicles and active environment sensing functions [5, 6]. For JCAS systems, it is crucial to use a waveform simultaneously performing both communication and sensing function, and help improve the availability of the limited spectrum resources. To this end, one of the main challenges in JCAS systems lies in designing optimal or adequate waveforms that serve both purposes of data transmission and radio sensing.

Mutual information (MI) is an important measure that can be used for studying waveform designs for joint communication and sensing systems. To be specific, for communications the MI between wireless channels and the received communication signals can be employed as the waveform optimization criterion, while for sensing, the conditional MI between sensing channels and the reflected sensing signals can be measured [7, 8]. Despite a significant amount of research effort on waveform design in both communication and sensing systems, existing joint waveform designs for JCAS systems are still limited. It is known that the training sequence for channel estimation has a significant impact on communication capacity, particularly for multiple input multiple output (MIMO) systems [9, 10]. However, there has been no study on the waveform design for JCAS, which takes into consideration the typical signal packet structure containing the training sequence.

Information theory has been used to design radar waveform [7, 11, 8, 12, 13]. Bell [7] was the first to apply information theory to optimize radar waveforms to improve target detection. In [12], the optimal radar waveform was proposed to maximize the detection performance of an extended target in a colored noise environment by using MI as waveform design criteria. Two criteria, namely, the maximization of the conditional MI and the minimization of the minimum mean-square error (MMSE), were studied in [8] to optimize the waveform design for MIMO radars by exploiting the covariance matrix of the extended target impulse response. In [11], the optimal waveform design for MIMO radars in colored noise was also investigated by considering two criteria: maximizing the MI and maximizing the relative entropy between two hypotheses that the target exists or does not exist in the echoes. In [13], a two-stage waveform optimization algorithm was proposed for an adaptive MIMO radar to unify the signal design and selection procedures. The algorithm is based on the constant learning of the radar environment at the receivers and the adaptation of the transmit waveform to dynamic radar scene. In [14], a robust waveform design based on the Cramér-Rao bound was proposed for co-located MIMO radars to improve the worst-case estimation accuracy in the presence of clutters.

For communication and radar co-existing systems that transmit and process respective signals, the MI has also been adopted for waveform design to minimize the interference to each other. In [15, 16], inner bounds on both the radar estimation rate for sensing and the data rate for communication were derived for the co-existing systems. Liu et al. [17] studied transmit beamforming for spectrum sharing between downlink MU-MIMO communication and co-located MIMO radar, to maximize the detection probability for sensing while guaranteeing the transmit power for downlink users. In [18], a minimum-estimation-error-variance waveform design method was proposed to optimize the spectral shape of a unimodular radar waveform and maximize the performance of both the radar and communications In [19], the radar waveform was designed based on a performance bound that is derived from jointly maximizing radar estimation rate and communication data rate.

Only a few studies have investigated the MI for JCAS systems [20, 21, 22]. In [20], considering a JCAS MIMO setup, the expressions for radar mutual information and communication channel capacity were derived. In [21, 22], an integrated waveform design was proposed for OFDM JCAS systems to improve the MI for both communication and sensing by considering extended targets and frequency-selective fading channels.

This paper presents information theoretically optimal waveform designs for a JCAS MIMO downlink system with a signal packet structure, including training sequence and information data symbols. In the JCAS MIMO downlink, a node sends MIMO signals to another node for communications and simultaneously uses the reflected signals for sensing the surrounding environment. We first derive the conditional MI for sensing and communication by taking both the training overhead and channel estimation error (CEE) into consideration, and then provide the optimal waveform designs for several different options of maximizing conditional MI. For sensing, both training and data sequences directly contribute to the MI; while for communications, only the data sequence contributes to the MI in the presence of CEE linked to the training sequence.

The key contributions of the paper are summarized as follows.

1. 1.

   We derive the MI expressions for both sensing and communication. We reveal the significantly different impact of the training and data sequences on the MI of sensing and communications.

2. 2.

   We design the optimal power allocation scheme between the training and data sequences under MMSE estimators for correlated MIMO communication channels.

3. 3.

   We provide the optimal waveform designs for three scenarios, including maximizing the MI only for sensing, and only for communication, and maximizing the weighted MI for joint communication and sensing.

4. 4.

   We conduct extensive simulations to corroborate the effectiveness of our proposed power allocation and waveform design. The results provide important insights into the trade-off of MI between communication and sensing in a JCAS system, and the non-negligible impact of training sequence on the MI.

The rest of this paper is organized as follows. In Section II, the system model is introduced. In Section III, we derive the conditional MI for both communication and sensing in the JCAS system. In Section IV, we derive a lower bound for CEE and develop an optimal power allocation strategy between training and data sequences. In Section V, the optimal waveform design methods for optimal communication, optimal sensing, and JCAS are investigated. Section VI presents simulation results. Section VII concludes the paper.

Notation: Lower-case bold face $(\mathbf{x})$ indicates vector, and upper-case bold face $(\mathbf{X})$ indicates matrix. For a diagonal matrix $\mathbf{X}$, $\mathbf{X}^{a}$ denotes the power $a$ operation to each diagonal element. $\mathbf{I}$ denotes the identity matrix, $\mathbb{E}(\cdot)$ denotes expectation. $(\cdot)^{T}$, $(\cdot)^{H}$, $(\cdot)^{\ast}$, $(\cdot)^{-1}$ and $(\cdot)^{\dagger}$ denote transposition, conjugate transportation, conjugate, inverse and pseudo-inverse, respectively. $\det(\cdot)$ and $\operatorname{Tr}(\cdot)$ denote the determinant and trace of a matrix, respectively.

For communication, the received training and data signals at node B can be respectively given by

where $\mathbf{H}=\left[\mathbf{h}_{1},\cdots,\mathbf{h}_{j},\cdots,\mathbf{h}_{N}\right]\in\mathbb{C}^{N\times N}$ is the channel matrix with $\mathbf{h}_{j}=\left[h_{1,j},h_{2,j},\cdots,h_{N,j}\right]^{T}$ denoting the $j$-th row of $\mathbf{H}$; $\mathbf{N}_{tc}\in\mathbb{C}^{N\times L_{t}}$ and $\mathbf{N}_{dc}\in\mathbb{C}^{N\times L_{d}}$ are both addictive white Gaussian noise (AWGN) with zero mean and element-wise variance $\sigma_{n}^{2}$. It is reasonable to assume that $\mathbf{N}_{tc}$, $\mathbf{N}_{dc}$ and $\mathbf{X}_{d}$ are mutually independent. The signal ${\mathbf{Y}}^{t}_{\rm com}$ is used for channel estimation. We assume that a linear channel estimation based on a minimum mean-square error (MMSE) criterion [24] is applied. In this case, the channel estimate $\hat{\mathbf{H}}$ and the estimation error $\Delta\mathbf{H}$ are uncorrelated [25]. Let $\Delta\mathbf{H}=[\Delta\mathbf{h}_{1},\cdots,\Delta\mathbf{h}_{j},\cdots,\Delta\mathbf{h}_{N}]$, where $\Delta\mathbf{h}_{j}=[\Delta h_{1j},\Delta h_{2j},\cdots,\Delta h_{Nj}]^{T}$ the $j$-th row of $\Delta\mathbf{H}$. The coefficients $\Delta h_{ij}$ are random variables following i.i.d. zero mean circularly symmetric complex Gaussian with variance $\sigma_{e}^{2}$, i.e., $\mathbb{E}\left[\Delta\mathbf{H}\Delta\mathbf{H}^{H}\right]=N\sigma_{e}^{2}\mathbf{I}_{N}$. We will evaluate $\sigma_{e}^{2}$ and link it to $\mathbf{X}_{t}$ and $\mathbf{N}_{tc}$ in Section IV-A.

The matrix $\mathbf{N}^{\prime}_{c}$ combines the CEE and noise, and can be viewed as an equivalent additive noise with zero mean and covariance. The variance $\sigma_{n^{\prime}}^{2}$ can be obtained as

where $\sigma_{n}^{\prime 2}=\frac{P_{d}}{L_{d}}\sigma_{e}^{2}+\sigma_{n}^{2}$.

Let $\mathbf{R}_{H}=\frac{1}{N}\mathbb{E}[\mathbf{H}^{H}\mathbf{H}]$ be the channel covariance matrix, and $\mathbf{R}_{H}$ is a positive semi-definite matrix. We assume that $\mathbf{R}_{H}$ is known to Node A. We can write the random channel matrix as $\mathbf{H}=\mathbf{H}_{0}\mathbf{R}_{H}^{\frac{1}{2}}$, where the entries of $\mathbf{H}_{0}$ are i.i.d. zero mean circularly symmetric complex Gaussian with unit variance.

Node A uses the reflection of the transmitted signal for sensing. The received signal, denoted by $\mathbf{Y}_{\rm rad}$, is given by

where $\mathbf{G}=[\mathbf{g}_{1},\cdots,\mathbf{g}_{N}]$ is the channel matrix to be sensed with its $j$-th column being $\mathbf{g}_{j}=\left[g_{1j},g_{2j},\cdots,g_{Nj}\right]^{T}$, and $\mathbf{g}_{j},\,j=1,\cdots,N$ are independent of each other; $\mathbf{N}_{tr}=[\mathbf{n}_{tr,1},\mathbf{n}_{tr,2},\cdots,\mathbf{n}_{tr,N}]\in\mathbb{C}^{L_{t}\times N}$ and $\mathbf{N}_{dr}=[\mathbf{n}_{dr,1},\mathbf{n}_{dr,2},\cdots,\mathbf{n}_{dr,N}]\in\mathbb{C}^{L_{d}\times N}$ are AWGN with zero mean and covariance matrix $\mathbb{E}\left\{\mathbf{N}_{tr}{\mathbf{N}_{tr}^{H}}\right\}=N\sigma_{n}^{2}\mathbf{I}_{L_{t}}$ and $\mathbb{E}\left\{\mathbf{N}_{dr}{\mathbf{N}_{dr}^{H}}\right\}=N\sigma_{n}^{2}\mathbf{I}_{L_{d}}$. Let $\varSigma_{\mathbf{G}}=\frac{1}{N}\mathbb{E}\{\mathbf{G}\mathbf{G}^{H}\}$ be the spatial correlation matrix. It is assumed to be full-rank and also known to Node A.

For both the communication and sensing channels, we assume that they remain unchanged during the period of a packet. Note that for both communication and sensing, the channel matrices include large-scale path loss and small-scale fading. The path loss of sensing can vary significantly for different multi-path components depending on the number of nearby objects and their locations, and therefore, we consider the mean path loss herein. Our optimization results only depend on the ratio between the mean path losses of communication and sensing.

The MI between the sensing channel matrix $\mathbf{G}$ (or the “target impulse response” matrix in radar) and reflected signals $\mathbf{Y}_{\rm rad}$ given the knowledge of $\mathbf{X}$ can be used to measure the sensing performance [11]. With our model (4), the MI is given by

where $h(\cdot)$ denotes the entropy of a random variable. Provided the noise vector $\mathbf{N}_{r,j}=\begin{bmatrix}\mathbf{n}_{tr,j}\\ \mathbf{n}_{dr,j}\end{bmatrix},\,j=1,\cdots,N$ are independent of each other, the conditional probability density function (PDF) of $\mathbf{Y}_{\rm rad}$ conditioned on $\mathbf{X}$ is given by

where (6b) is obtained based on the PDF of circularly symmetric complex Gaussian distribution, and

where (7b) is conditioned on $\mathbf{X}$, and $\mathbb{E}\{\mathbf{g}_{j}\mathbf{g}_{j}^{H}\}=\frac{1}{N}\mathbb{E}\{\mathbf{G}\mathbf{G}^{H}\}=\varSigma_{\mathbf{G}}$ in (7c) since $\mathbf{g}_{j},\,j=1,\cdots,N$ are independent of each other.

Based on (6), the entropy of $\mathbf{Y}_{\rm rad}$ conditional on $\mathbf{X}$ can be obtained as

where (8c) is based on the Sylvester’s determinant theorem [26], i.e.,

The columns of the noise matrix $\mathbf{N}_{r}$ follow the i.i.d. multivariate complex Gaussian distribution with zero mean and covariance matrix $\sigma^{2}_{n}\mathbf{I}_{N}$, and the entropy of $\mathbf{N}_{r}$ is given by

By substituting (8) and (10) into (5), the MI for sensing can be obtained as

The MI for communication is defined as the mutual dependence between the transmit signals of node A and the received signals of node B, conditional on the estimated channel matrix $\hat{\mathbf{H}}$. With the Gaussian assumption of CEE, the conditional PDF of ${\mathbf{Y}}^{d}_{\rm com}$ on $\hat{\mathbf{H}}$ is given by

where (12a) is under the assumption that the columns of $\mathbf{Y}^{d}_{\rm com}$ (or $\mathbf{X}_{d}$) are i.i.d., and (12b) is from the PDF of circularly symmetric complex Gaussian distribution. The columns of equivalent noise matrix $\mathbf{N}^{\prime}_{c}$ follow the i.i.d. multivariate complex Gaussian distribution with zero mean and covariance matrix $\sigma_{n}^{\prime 2}\mathbf{I}_{N}$. By referring to (8) – (10), the entropy of $\mathbf{N}^{\prime}_{c}$ can be given by

Therefore, the conditional MI between $\mathbf{X}_{d}$ and ${\mathbf{Y}}_{\rm com}^{d}$ is obtained as

Compared to the conventional MI results without consideration of CEE [27], we can see that the CEE here contributes as $\sigma_{n}^{\prime 2}=\frac{P_{d}}{L_{d}}\sigma_{e}^{2}+\sigma_{n}^{2}$.

With an MMSE MIMO channel estimation, the estimated MIMO channel matrix can be expressed as [10]

where, as can be recalled, $\mathbf{X}_{t}$ is an $N\times L_{t}$ training symbol matrix whose elements have the average energy $\sigma_{t}^{2}$. Take the singular value decomposition (SVD) of $\mathbf{R}_{H}$. $\mathbf{R}_{H}=\mathbf{U}_{H}\bm{\Lambda}_{H}\mathbf{U}_{H}^{H}$, where the singular value matrix $\bm{\Lambda}_{H}=\operatorname{diag}(\delta_{1},\delta_{2},\cdots,\delta_{N})$, and $\frac{1}{N}\sum_{i=1}^{N}\delta_{i}=\frac{1}{N}{\operatorname{Tr}}(\mathbf{R}_{H})\triangleq\sigma_{h}^{2}$. Let ${\bm{\Lambda}}_{\rm CRLB}$ be the Cramér-Rao lower bound (CRLB) of the channel matrix estimation [28]. We have

which is due to the fact that $\varSigma_{\mathbf{X}_{t}}=\mathbf{X}_{t}\mathbf{X}_{t}^{H}=L_{t}\sigma_{t}^{2}\mathbf{I}_{N}$. Therefore, a lower bound of MMSE of total channel estimates, denoted by $\mathcal{C}_{t}$, can be represented as

Here, $\mathcal{C}_{t}$ is a function of $\delta_{i},\,i=1,\cdots,N$ with the constraint that $\sum_{i=1}^{N}\delta_{i}={\operatorname{Tr}}(\mathbf{R}_{H})$. Therefore, we can further obtain the lower bound of $\mathcal{C}_{t}$ by applying Lagrange multiplier method. The Lagrangian function can be written as

where $\tau$ is the Lagrange multiplier. By solving $\frac{\partial\mathcal{L}\left({\bm{\Lambda}_{H}}\right)}{\partial\delta_{i}}=0$, we get

which shows that the lower bound is achieved when $\delta_{1}=\cdots=\delta_{i}\cdots=\delta_{N}$ and $\delta_{i}=\frac{1}{N}{\operatorname{Tr}}(\mathbf{R}_{H})=\frac{1}{N}\sum_{i=1}^{N}\delta_{i}{=\sigma_{h}^{2}},\,i=1,\cdots,N$. The lower bound of $\mathcal{C}_{t}$ is then given by

Therefore, for any diagonal element of $\bm{\Lambda}_{\rm CRLB}(i)$, we have

In general, there are some constraints on the maximum and average transmission powers of a transmitter. When such power constraints are applied, there is a motivation for optimizing the power allocation between the training and data symbols, especially for maximizing the MI for communications. Here, we optimize power allocation only by referring to the communication MI, because its impact on communication MI is much stronger than on sensing MI. Larger CEE can cause substantially deteriorate communication performance while sensing MI can only be slightly affected since the training sequence is directly used for sensing.

Since $\hat{\mathbf{H}}=\mathbf{H}-\Delta\mathbf{H}$, we can obtain that $\hat{\mathbf{H}}$ is a random variable with zero mean and variance $\sigma^{2}_{\hat{\mathbf{H}}}=\frac{1}{N^{2}}\mathbb{E}\left[{\operatorname{Tr}}\{\hat{\mathbf{H}}{\hat{\mathbf{H}}}^{H}\}\right]$. According to the orthogonality principle for MMSE [9] and the obtained lower bound of CEE, we have $\sigma^{2}_{\hat{\mathbf{H}}}=\sigma_{h}^{2}-\sigma_{e}^{2}$. Therefore, the estimated channel $\hat{\mathbf{H}}$ can be normalized as $\tilde{\mathbf{H}}=\frac{1}{\sigma_{\hat{\mathbf{H}}}}\hat{\mathbf{H}}$, which has elements following i.i.d. Complex Gaussian distribution $\mathcal{CN}(0,1)$.

The MI in (14) can then be rewritten as

where (21c) is due to the lower bound of CEE, i.e., $\sigma_{e}^{2}\geq\mathcal{C}^{l}_{t}$.

By substituting $\mathcal{C}_{e}$ of (20) into (21) and exploiting the Jensen’s inequality, we can obtain the results for optimal power allocation and minimized CEE as summarized in the following theorem.

Hereafter, we use the optimal power allocation formula (22) for power allocation and let $\mathcal{C}^{l}_{e}=\frac{\mathcal{C}_{t}^{l}}{N}=\frac{\sigma_{n}^{2}{\sigma_{h}^{2}}}{N\sigma_{n}^{2}+\left(1-\kappa_{\rm{op}}\right)P{\sigma_{h}^{2}}}$; unless otherwise stated.

In order to achieve the maximum MI for sensing, or in other words, to make received signals ${\mathbf{Y}}_{\rm rad}$ (including ${\mathbf{Y}}^{t}_{\rm rad}$ and ${\mathbf{Y}}^{d}_{\rm rad}$) containing rich information about $\mathbf{G}$, the transmit signals $\mathbf{X}$ (including the training sequence $\mathbf{X}_{t}$ and data sequence $\mathbf{X}_{d}$) should be designed according to the sensing channel matrix $\mathbf{G}$. Since the training sequence $\mathbf{X}_{t}$ and the data sequence $\mathbf{X}_{d}$ are independent and have different correlations, the optimization problem for maximizing the MI for sensing can be decoupled into two separate optimization problems. As assumed, $\mathbf{X}_{t}$ contains deterministic orthogonal rows and $\mathbf{X}_{t}\mathbf{X}_{t}^{H}=L_{t}\sigma_{t}^{2}\mathbf{I}_{N}$. We only need to consider the optimization problem for the data sequence.

For the data sequence, the spatial correlation matrix can be diagonalized through SVD, that is,

where $\mathbf{U}_{G}$ is a unitary matrix and $\mathbf{\Lambda}_{G}={\operatorname{diag}}\left\{\lambda_{11},\cdots,\lambda_{ii},\cdots,\lambda_{NN}\right\}$ is a diagonal matrix with $\lambda_{ii}$ being the singular values. The mean channel gain $\sigma_{g}^{2}$ for sensing channels is $\sigma_{g}^{2}=\sum_{i=1}\lambda_{ii}$. The MI in (11) can be rewritten as (25),

where (25b) is based on Sylvester’s determinant theorem. Define $\mathbf{Q}^{(t)}=\left(\mathbf{X}^{T}_{t}\mathbf{U}_{G}\right)^{H}\mathbf{X}^{T}_{t}\mathbf{U}_{G}$ and $\mathbf{Q}^{(d)}=\left(\mathbf{X}^{T}_{d}\mathbf{U}_{G}\right)^{H}\mathbf{X}^{T}_{d}\mathbf{U}_{G}$, and their $(i,j)$-th entries are $q^{(t)}_{ij}$ and $q^{(d)}_{ij}$, respectively.

According to Hadamard’s inequality for the determinant and trace of an $N\times N$ positive semi-definite Hermitian matrix, we have the following inequalities:

and

where the equalities are achieved if and only if $\mathbf{Q}_{N\times N}$ is diagonal. As a result, the MI can be rewritten as

Since $\mathbf{U}_{G}$ is unitary, we can find $\operatorname{Tr}\left(\mathbf{Q}^{(d)}\right)=\operatorname{Tr}\left(\left(\mathbf{X}^{T}_{d}\mathbf{U}_{G}\right)^{H}\mathbf{X}^{T}_{d}\mathbf{U}_{G}\right)=\operatorname{Tr}\left(\mathbf{Z}^{H}\mathbf{Z}\right)$, where $\mathbf{Z}=\mathbf{X}^{T}_{d}\mathbf{U}_{G}$ is an $L_{d}\times N$ matrix. Under the constraint that the total transmit power is finite, we have $\operatorname{Tr}\left(\mathbf{Q}^{(d)}\right)\leq P_{d}=\kappa_{\rm{op}}{P}$, where $P_{d}$ is the total power of the transmit data signals. Since $\mathbf{X}_{t}$ satisfies the orthogonality condition, we have $\operatorname{Tr}\left(\mathbf{Q}^{(t)}\right)=\operatorname{Tr}\left(\left(\mathbf{X}^{T}_{t}\mathbf{U}_{G}\right)^{H}\mathbf{X}^{T}_{t}\mathbf{U}_{G}\right)=\operatorname{Tr}\left(\mathbf{X}^{\ast}_{t}\mathbf{X}^{t}_{t}\right)=\operatorname{Tr}\left(\mathbf{X}_{t}\mathbf{X}^{H}_{t}\right)=P_{t}=(1-\kappa_{\rm{op}})P$ and $q^{(t)}_{ii}=\frac{P_{t}}{N}=\frac{(1-\kappa_{\rm{op}})P}{N}$. Therefore, the maximum MI can be obtained by solving the following constrained problem:

We can apply the Lagrange multiplier method to solve (27). The Lagrangian can be written as