BiLiMO: Bit-Limited MIMO Radar via Task-Based Quantization

We consider a colocated MIMO radar consisting of two linear antenna arrays with $N$ receive antennas and $M$ transmit antennas. The locations of the $N$ receive antennas and $M$ transmit antennas are denoted by $\zeta_{0}\lambda,\cdots,\zeta_{N-1}\lambda$ and $\xi_{0}\lambda,\cdots,\xi_{M-1}\lambda$, respectively. Here, $\lambda$ is the wavelength of the carrier signal. Without loss of generality, we assume that $\zeta_{0}=\xi_{0}=0$. The MIMO radar uses two uniform linear arrays (ULAs) as the receive antennas and transmit antennas, located at $\zeta_{n}=n/2$ and $\xi_{m}=Nm/2$ for $0\leq n\leq N-1$ and $0\leq m\leq M-1$. As such, the resulting $MN$ channels can generate a virtual ULA array with length $MN\lambda/2$ [1].

Each transmit antenna sends out $Q$ pulses, such that the $m$th transmitted signal is given by

$$s_{m}(t)=\sum_{q=0}^{Q-1}h_{m}(t-qT_{0})e^{j2\pi f_{c}t},\quad 0\leq t\leq QT_{0},$$

(1)

where $h_{m}(t)$, $0\leq m\leq M-1$, are narrowband pulses with bandwidth $B_{h}$, modulated with carrier frequency $f_{c}$, and $T_{0}$ denotes the pulse repetition interval (PRI). For simplicity, we only consider one PRI, i.e., $Q=1$. However, our analysis can be generalized to the case of multiple pulses, namely, $Q>1$.

MIMO radar architectures commonly utilize orthogonal waveforms for radar probing [1]. Here, we consider orthogonality achieved using frequency division multiple access (FDMA) signaling. In FDMA, the transmitted baseband waveform $h_{m}(t)$ can be expressed as $h_{m}(t)=h_{0}(t)e^{j2\pi f_{m}t}$, where $h_{0}(t)$ is the lowpass waveform with spectral support $[-\frac{B_{h}}{2},\frac{B_{h}}{2}]$, each $f_{m}$ is chosen in $[-\frac{MB_{h}}{2},\frac{MB_{h}}{2}]$ so that the intervals $[f_{m}-\frac{B_{h}}{2},f_{m}+\frac{B_{h}}{2}]$ do not overlap. In such setups, different waveforms lie in distinct spectral bands.

The targets are represented as non-fluctuating point reflectors in the far field, and we let $K$ be the number of targets. Each target is characterized by the following parameters: its reflection coefficient $\tilde{\alpha}_{k}$, its distance from the array origin $R_{k}$, and the azimuth angle relative to the array $\theta_{k}$. We assume the targets lie in the radar unambiguous time-frequency region.

Let $\tau_{k}=\frac{2R_{k}}{c}$ and $\vartheta_{k}=\sin(\theta_{k})$ be the delay and azimuth sine of the $k$th target, respectively. The received signal $\tilde{x}_{n}(t)$ at the $n$th antenna in one PRI can then be written as:

$$\tilde{x}_{n}(t)\!=\!\sum_{m=0}^{M-1}\sum_{k=1}^{K}\tilde{\alpha}_{k}e^{j2\pi f_{c}(\xi_{m}\!+\!\zeta_{n})\vartheta_{k}}h_{m}(t\!-\!\tau_{k})e^{j2\pi f_{c}(t\!-\!\tau_{k})}\!+\!\tilde{w}_{n}(t),$$

where $\tilde{w}_{n}(t)$ is the interference plus noise signal. By defining $x_{n}(t)=\tilde{x}_{n}(t)e^{-j2\pi f_{c}t}$ as the baseband component, we have

$$x_{n}(t)\triangleq\sum_{m=0}^{M-1}\sum_{k=1}^{K}\alpha_{k}e^{j2\pi(\xi_{m}+\zeta_{n})\vartheta_{k}}h_{m}(t-\tau_{k})+w_{n}(t),$$

(2)

with $\alpha_{k}=\tilde{\alpha}_{k}e^{-j2\pi f_{c}\tau_{k}}$ and $w_{n}(t)=\tilde{w}_{n}(t)e^{-j2\pi f_{c}t}$. In Fig. 1 we illustrate our MIMO radar model for $K=2$.

The goal of MIMO radar receiver processing is to identify the targets based on the received echos. In particular, the receiver is required to resolve the $K$ delay-azimuth pairs $\{\tau_{k},\vartheta_{k}\}_{k=1}^{K}$ from the received signals $\{x_{n}(t)\}_{n=0}^{N-1}$. In classic MIMO radar, after demodulation, the baseband components of the received signals $\{x_{n}(t)\}_{n=0}^{N-1}$ are converted from analog signals to digital representations using ADCs which sample above the Nyquist rate and utilize high-resolution quantizers. The outputs of the ADCs are then processed in the digital domain in order to estimate the delays and azimuth sines of the $K$ targets from the received signals [2]. The usage of high-resolution ADCs, which assign a relatively large number of bits to represent each sample, induces minimal distortion [41], and thus the effect of quantization on radar signal processing is usually ignored. Nonetheless, the fact that the power consumption of ADC devices grows exponentially with the number of bits assigned to each sample [6], dramatically affects the power and cost of MIMO radar systems operating at high frequencies with a large number of receive antennas.

Among the leading design approaches to reduce the cost and power usage of such MIMO radar systems are $(i)$ utilize low-resolution ADCs; and $(ii)$ reduce the number of RF chains and ADCs by operating in a HAD manner. The usage of low-resolution ADCs implies that each ADC can output up to $b$ different levels, e.g., $b=4$ for two-bit ADCs. HAD architectures introduce pre-acquisition analog processing, combining the $N$ analog signals $\{x_{n}(t)\}_{n=0}^{N-1}$ into $P$ outputs $\{y_{p}(t)\}_{p=0}^{P-1}$, which are then acquired by the ADCs. Setting $P<N$ implies that HAD systems reduce the number of costly RF chains and ADCs compared to conventional MIMO radar receivers. An illustration of such a HAD receiver is depicted in Fig. 2.

Analog combining prior to analog-to-digital conversion is commonly studied in the MIMO communication literature [39, 38], typically as a mean to reduce the number of costly RF chains. HAD MIMO receivers were also shown to facilitate operation with low resolution quantizers for communication tasks [32, 35]. This is achieved using task-based quantization methods [31, 32, 33, 34, 35], which tune the acquisition mapping in light of the overall system task, allowing to accurately recover the desired information under limited bit budgets. This motivates the design of HAD receivers for the task of recovering the target parameters as a form of task-based quantization.

In order to design such HAD radar receivers, one must first introduce some constraints on the feasible mappings of the components of the system in Fig. 2. The motivation for imposing such constraints is two-fold: First, they enforce the resulting system to correspond to architectures which are feasible in terms of hardware. For instance, while in principle the analog processing in Fig. 2 can be any mapping, in practice it is likely to be implemented using analog hardware based on filters and multiplexers. The second motivation for introducing these constraints is to obtain an analytically tractable design problem, which is very challenging due to the complex relationship between the target parameters $\{\tau_{k},\vartheta_{k}\}_{k=1}^{K}$ and the received signals $\{x_{n}(t)\}_{n=0}^{N-1}$. In the following section we introduce the considered constrained HAD radar receiver architecture, which we refer to as BiLiMO. We then tackle the challenge in designing the receiver to recover the targets by formulating a relaxed problem, as shown in the sequel.

The analog processing consists of three stages: analog combining, analog mixing, and filtering. First, the $N$ received signals are combined to output $P$ ($P\leq MN$) channels. Then each of the $P$ channels are separately mixed with a mixing signal and low-pass filtered before being fed to the ADCs. The motivation for using this architecture stems from the fact that it equivalently implements channel separation, which is typically the first step in MIMO radar processing required to achieve its desired virtual array capabilities, followed by a controllable analog combiner. However, while the direct implementation of such analog hardware requires $MN$ bandpass filters (for channel separation) followed by $MNP$ controllable filters, each applied to a different bandpass component, the architecture illustrated as Analog processing in Fig. 3 can be shown to implement the same mapping while utilizing merely $NP$ filters applied directly to the full-band received signals, followed by $P$ identical low-pass filters.

Let $b_{p,n}(t)$ be the $(p,n)$th analog filter. The $p$th output of the analog combining can be expressed as

	$\displaystyle\tilde{y}_{p}(t)=\sum_{n=0}^{N-1}b_{p,n}(t)\ast x_{n}(t)$
	$\displaystyle=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}\sum_{k=1}^{K}\alpha_{k}e^{j2\pi(\xi_{m}+\zeta_{n})\vartheta_{k}}[b_{p,n}(t)\ast h_{m}(t-\tau_{k})].$		(3)

Since $\tilde{y}_{p}(t)$ is limited to $t\in[0,T_{0}]$, it can be equivalently expressed by its Fourier series

$$\tilde{y}_{p}(t)=\sum_{i=\lceil-MT_{0}B_{h}/2\rceil}^{\lfloor MT_{0}B_{h}/2\rfloor}\tilde{c}_{p}[i]e^{j2\pi it/T_{0}},\quad t\in[0,T_{0}],$$

(4)

where

	$\displaystyle\tilde{c}_{p}[i]=\frac{1}{T_{0}}\int_{0}^{T_{0}}\tilde{y}_{p}(t)e^{-j2\pi it/T_{0}}dt$
	$\displaystyle\!=\!\!\sum_{n=0}^{N-1}\!\sum_{m=0}^{M-1}\hat{h}_{m}\!\left(\!\frac{2\pi i}{T_{0}}\!\right)\!\hat{b}_{p,n}\!\left(\!\frac{2\pi i}{T_{0}}\!\right)\!\sum_{k=1}^{K}\frac{\alpha_{k}}{T_{0}}e^{j2\pi\left((\xi_{m}\!+\!\zeta_{n})\vartheta_{k}\!-\frac{i\tau_{k}}{T_{0}}\right)},$

where $\hat{b}_{p,n}(\omega)$ and $\hat{h}_{m}(\omega)$ denote the Fourier transform of $b_{p,n}(t)$ and $h_{m}(t)$, respectively.

Each of the $P$ output channels is mixed with the signal $q(t)=\sum_{m=0}^{M-1}e^{-j2\pi f_{m}t}$ and filtered by a lowpass filter with passband $[-\pi{B_{h}},\pi{B_{h}}]$. The motivation of using such mixing signals is to combine the spectrum of the $M$ transmitted signals, such that a portion of energy from each band appears in baseband. Its combination with low-pass filtering results in equivalent outputs to those which would have been produced by first applying channel separation based on $MN$ matched filters, as shown in the sequel, without having to implement these channel separation filters in analog hardware. This structure is similar to that used in Xampling [16], which has been shown to be conveniently implemented in hardware. Using (4), the output of the $p$th channel is

$$y_{p}(t)=\sum_{m=0}^{M-1}\sum_{i=\lceil-T_{0}B_{h}/2\rceil}^{\lfloor T_{0}B_{h}/2\rfloor}\tilde{c}_{p}[i+f_{m}T_{0}]e^{j2\pi it/T_{0}}.$$

(5)

Let us define

$$\hat{b}_{p,n}^{m}(\omega)\triangleq\begin{cases}\frac{1}{T_{0}}\hat{h}_{0}(\!\omega)\hat{b}_{p,n}(\!\omega\!+\!2\pi\!f_{m})&\!\omega\!\in\![\!-\!\pi\!B_{h},\!\pi\!B_{h}\!]\\ 0&\text{else}.\\ \end{cases}$$

(6)

Then, for $\lceil-T_{0}B_{h}/2\rceil\leq i\leq\lfloor T_{0}B_{h}/2\rfloor$, $\tilde{c}_{p}[i+f_{m}T_{0}]$ can be expressed as

$$\tilde{c}_{p}[i+f_{m}T_{0}]=\sum_{n=0}^{N-1}c_{m,n}[i]b_{p,mN+n}[i],$$

(7)

where

$$c_{m,n}[i]\triangleq\sum_{k=1}^{K}\alpha_{k}e^{j2\pi\left((\xi_{m}\!+\!\zeta_{n})\vartheta_{k}\!-\frac{i\tau_{k}}{T_{0}}-f_{m}\tau_{k}\right)},$$

(8)

and $b_{p,mN+n}[i]\triangleq\hat{b}_{p,n}^{m}\left(\frac{2\pi i}{T_{0}}\right)$. Define $L\triangleq B_{h}T_{0}$, assumed to be an odd integer in the following discussion for convenience. Then the output in (5) can be written as

$$y_{p}(t)=\sum_{i=-\frac{L-1}{2}}^{\frac{L-1}{2}}\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}b_{p,mN+n}[i]c_{m,n}[i]e^{j2\pi it/T_{0}},$$

(9)

which is equivalent to the outputs achieved by applying channel separation followed by an analog combiner comprised of $MNP$ individual filters with the frequency responses in (6).

We can now write the output of the analog processing component in multivariate form by defining $\mathbf{y}(t)\triangleq[y_{0}(t),\cdots,y_{P-1}(t)]^{T}$, and obtaining from (9) that

$$\mathbf{y}(t)=\sum_{i=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{j2\pi it/T_{0}}\mathbf{B}_{i}\mathbf{c}_{i},$$

(10)

where $\mathbf{B}_{i}\triangleq[b_{p,\tilde{n}}[i]]\in\mathbb{C}^{P\times MN}$ with $0\leq\tilde{n}<MN$, and $\mathbf{c}_{i}\triangleq[\mathbf{c}_{0}^{T}[i],\cdots,\mathbf{c}_{M-1}^{T}[i]]^{T}$ with $\mathbf{c}_{m}[i]=[c_{m,0}[i],\cdots,c_{m,N-1}[i]]^{T}$ for each $0\leq m<M$. By using the above analog processing, rather than classic matched filtering, the number of channels to be processed is reduced from $MN$ to $P$, facilitating the sampling and quantization operations. The design of the analog processing is equivalent to designing the $NP$ analog filters $b_{p,n}(t)$, which can be constructed from the desired value of $\mathbf{B}_{i}$. Specifically, the frequency response of $b_{p,n}(t)$ becomes $\hat{b}_{p,n}(2\pi(\frac{i}{T_{0}}+f_{m}))=T_{0}b_{p,mN+n}[i]\hat{h}_{0}^{\ast}(\frac{2\pi i}{T_{0}})/|\hat{h}_{0}(\frac{2\pi i}{T_{0}})|^{2}$ for $-\frac{L-1}{2}\leq i\leq\frac{L-1}{2}$ by (6). After analog processing, the $P$ signals represented by $\mathbf{y}(t)$ are forwarded to the ADCs as detailed in the following.

The output of the analog processing $\mathbf{y}(t)$ is converted into a set of digital streams via sampling and quantization. This conversion is carried out using $P$ identical pairs of ADCs which independently discretize the real and imaginary parts of each analog input signal. We focus here on the low-resolution quantization aspect of analog-to-digital conversion, assuming that Nyquist sampling is applied, and leave the analysis and joint design of such systems with sub-Nyquist sampling and low resolution quantization for future work.

To formulate the ADC operation, we let $\mathbf{y}_{i}\triangleq\mathbf{y}(\frac{i}{B_{h}})$, $i=0,\cdots,L-1$, be the Nyquist samples of $\mathbf{y}(t)$. Define the $PL\times 1$ samples vector $\mathbf{y}\triangleq[\mathbf{y}_{0}^{T},\cdots,\mathbf{y}_{L-1}^{T}]^{T}$, and the $MNL\times 1$ Fourier coefficients vector as $\mathbf{c}\triangleq[\mathbf{c}_{-(L-1)/{2}}^{T},\cdots,\mathbf{c}_{(L-1)/{2}}^{T}]$. Using these definitions, the samples of the outputs of the analog combining filters can be expressed as

$$\mathbf{y}=\bar{\mathbf{F}}\bar{\mathbf{B}}\mathbf{c},$$

(11)

where $\bar{\mathbf{B}}\triangleq\operatorname{blkdiag}(\mathbf{B}_{-(L-1)/2},\cdots,\mathbf{B}_{(L-1)/2})\in\mathbb{C}^{PL\times MNL}$, $\bar{\mathbf{F}}\triangleq\mathbf{F}_{L}^{H}\otimes\mathbf{I}_{P}$, and $\mathbf{F}_{L}$ is the $L\times L$ discrete Fourier transform (DFT) matrix.

Using a similar derivation, we represent the samples of the interference and noise signal as

$$\mathbf{n}=\bar{\mathbf{F}}\bar{\mathbf{B}}\mathbf{w},$$

(12)

where $\mathbf{w}\in\mathbb{C}^{MNL}$ is the frequency-domain representation of the interference and noise signal observed at the $N$ antennas.

The sampled signal $\mathbf{y}$ is converted into a digital representation using uniform complex-valued quantizers with $b$ decision regions and support $\gamma>0$. The resulting quantization mapping is given by $\mathcal{Q}_{C}^{\gamma,b}(\cdot)=\mathcal{Q}^{\gamma,b}(\Re\{\cdot\})+j\mathcal{Q}^{\gamma,b}(\Im\{\cdot\})$, where $\mathcal{Q}^{\gamma,b}(\cdot)$ is the real-valued quantization operator applied element-wise to any real vector or matrix, and is given by

$$\mathcal{Q}^{\gamma,b}(x)=\begin{cases}-\gamma+\frac{2\gamma}{b}(l+\frac{1}{2})&\begin{array}[]{l}x-l\frac{2\gamma}{b}+\gamma\in[0,\frac{2\gamma}{b}],\\ l\in\{0,1,\cdots,b-1\},\end{array}\\ \operatorname{sign}(x)(\gamma-\frac{\gamma}{b})&|x|>\gamma.\end{cases}$$

The output of the ADCs is the vector $\mathbf{z}\in\mathbb{C}^{PL}$, given by

$$\mathbf{z}=\mathcal{Q}_{C}^{\gamma,b}(\mathbf{y}+\mathbf{n})=\mathcal{Q}_{C}^{\gamma,b}(\bar{\mathbf{F}}\bar{\mathbf{B}}\mathbf{c}+\bar{\mathbf{F}}\bar{\mathbf{B}}\mathbf{w}).$$

(13)

In our design of the BiLiMO receiver in Section IV we model the ADCs as implementing non-subtractive dithered quantization [42]. This model facilitates the design and analysis of bit-constrained HAD systems [31], while being a faithful approximation of conventional uniform quantization operation under various statistical models [43]. The number of bits used for representing $\mathbf{z}$ is thus $2P\lceil\log b\rceil$. Processing of $\mathbf{z}$ in the digital domain is detailed in the following subsection.

The digital representation $\mathbf{z}$ is used to recover the target parameters. However, the relationship between $\mathbf{z}$ (13) and the target parameters $\{\tau_{k},\vartheta_{k}\}_{k=1}^{K}$ is quite complex, making the joint design of the analog processing and digital mapping very difficult. Therefore, in the following we partition the digital processing into two stages, as illustrated in Fig. 3. The first part is comprised of a digital filter, which is jointly designed with the analog combiner and ADC support based on a relaxed problem of recovering a vector $\mathbf{s}$ in the sense of minimal MSE. The relaxed task vector $\mathbf{s}$ is selected such that the target parameters can be recovered from it using conventional linear sparse recovery algorithms in the second part of the digital processing. The BiLiMO receiver thus builds upon the ability to recast target identification as a sparse recovery problem [44]. Therefore, to formulate the problem of designing the BiLiMO receiver, we first rewrite our parameter estimation task as sparse recovery, after which we present the resulting digital processing structure, which is designed based on the relaxed objective detailed in Section IV.

As in classic MIMO radar, we now assume the parameters $\tau_{k}$ and $\vartheta_{k}$ are located on the Nyquist grid, i.e., $\tau_{k}\in\{\frac{T_{0}l}{ML}\}_{l=0}^{ML-1}$ and $\vartheta_{k}\in\{-1+\frac{2l}{MN}\}_{l=0}^{MN-1}$. It follows from (8) that the vector $\tilde{\mathbf{c}}_{m}\triangleq[(\mathbf{c}_{m}[-\frac{L-1}{2}])^{T},\cdots,(\mathbf{c}_{m}[\frac{L-1}{2}])^{T}]\in\mathbb{C}^{NL}$ obeys the following sparse representation

$\displaystyle\tilde{\mathbf{c}}_{m}$

$\displaystyle=\operatorname{vec}(\mathbf{U}_{m}\mathbf{A}\mathbf{V}_{m}^{T})=(\mathbf{V}_{m}\otimes\mathbf{U}_{m})\mathbf{a},$

(14)

where $\mathbf{U}_{m}\in\mathbb{C}^{N\times MN}$ with $(\mathbf{U}_{m})_{n,l}=e^{j2\pi(\xi_{m}+\zeta_{n})(-1+\frac{2l}{MN})}$, $\mathbf{V}_{m}\in\mathbb{C}^{L\times MN}$ with $(\mathbf{V}_{m})_{i,l}=e^{-j2\pi(i/T_{0}+f_{m})\frac{T_{0}l}{ML}}$, $\mathbf{A}\in\mathbb{C}^{MN\times ML}$ is a sparse matrix that contains $K$ nonzero elements, and $\mathbf{a}=\operatorname{vec}(\mathbf{A})\in\mathbb{C}^{M^{2}NL}$ is thus a $K$-sparse vector. The sparsity pattern of $\mathbf{a}$ encapsulates the values of the unknown delays and angles, i.e., if the $k$th non-zero element of $\mathbf{a}$ is located at its index $(l_{1}-1)MN+l_{2}$ where $0\leq l_{1}<ML$ and $0\leq l_{2}<MN$, then $\tau_{k}=\frac{T_{0}l_{1}}{ML}$ and $\vartheta_{k}=-1+\frac{2l_{2}}{MN}$.

Stacking $\{\tilde{\mathbf{c}}_{m}\}_{m=1}^{M-1}$ into the $MNL\times 1$ vector $\tilde{\mathbf{c}}$, we obtain

$$\tilde{\mathbf{c}}=\mathbf{\Phi}\mathbf{a},$$

(15)

where $\mathbf{\Phi}\in\mathbb{C}^{MNL\times M^{2}NL}$ is defined as

$$\mathbf{\Phi}\triangleq[(\mathbf{V}_{0}^{T}\!\otimes\!\mathbf{U}^{T}_{0})),(\mathbf{V}^{T}_{1}\!\otimes\!\mathbf{U}^{T}_{1}),\cdots,(\mathbf{V}^{T}_{M\!-\!1}\!\otimes\!\mathbf{U}^{T}_{M\!-\!1})]^{T}.$$

(16)

Using the sparse representation (15), the quantized $\mathbf{z}$ (13) becomes

$$\mathbf{z}=\mathcal{Q}_{C}^{\gamma,b}(\bar{\mathbf{F}}\bar{\mathbf{B}}(\mathbf{P}\mathbf{\Phi}\mathbf{a}+\mathbf{w})),$$

(17)

where $\mathbf{P}\in\mathbb{R}^{MNL\times MNL}$ is a permutation matrix which aligns the elements between $\mathbf{c}$ and $\tilde{\mathbf{c}}$, i.e., $\mathbf{c}=\mathbf{P}\tilde{\mathbf{c}}$.

The MIMO radar task is thus equivalent to recovering the $K$-sparse vector $\mathbf{a}$ from the quantized $\mathbf{z}$. Therefore, to facilitate the joint design of BiLiMO as a HAD receiver, we set its digital processing to first recover a $J\times 1$ vector $\mathbf{s}$, with $J\leq PL$, which can be written as a linear compressed representation of $\mathbf{a}$. This is achieved by first applying a digital filter $\mathbf{D}\in\mathbb{C}^{J\times PL}$, which can be designed using task-based quantization methods, as we show in the following section, such that its output $\hat{\mathbf{s}}=\mathbf{D}\mathbf{z}$ is an accurate estimate of $\mathbf{s}$. Then, the fact that $\hat{\mathbf{s}}$ can be written as a linear compressed representation of $\mathbf{a}$ with some additive estimation error term is exploited in the subsequent digital processing, which resolves the targets from $\hat{\mathbf{s}}$ using conventional algorithms for recovering sparse vectors from noisy linear compressed measurements. The formulation of the relaxed problem and the resulting design of BiLiMO are detailed in the following section.

Task-based quantization is a framework for designing HAD acquisition systems operating under bit constraints. Such methods aim to facilitate the recovery of information embedded in the observed analog signals, rather than preserving sufficiency of the digital representation with respect to the signal itself [33]. In particular, the work [31] jointly designed the components of an acquisition systems including analog pre-quantization filtering and digital linear processing of a similar structure as those in BiLiMO for the task of recovering a linear function of the measurements. However, unlike the setup in [31], the task of the BiLiMO receiver is to recover the parameters of the targets, which do not obey a linear form. To encompass this challenge in utilizing task-based quantization tools for MIMO radar, we next present a relaxed problem formulation, which decomposes target identification into a linear recovery problem followed by a linear sparse recovery task. This relaxation allows the former to be treated using existing results in task-based quantization theory, and the latter be tackled using conventional CS methods.

In the proposed BiLiMO receiver, rather than recovering the $K$-sparse vector $\mathbf{a}$ in (15) directly, a digital filter is first applied to estimate a compressed vector $\mathbf{s}\in\mathbb{C}^{J}$ ($J\leq PL$) from the quantized data $\mathbf{z}$. In particular, we set $\mathbf{s}=\mathbf{M}\mathbf{\Phi}\mathbf{a}$, i.e., $\mathbf{s}$ is a linear compressed representation of the desired $\mathbf{a}$, where $\mathbf{M}\in\mathbb{C}^{J\times MNL}$ is a pre-defined compressive measurement matrix. Then, the $K$-sparse vector $\mathbf{a}$ is recovered from the estimate of $\mathbf{s}$ by applying sparse recovery techniques.

Although we refer to $\mathbf{s}$ as the task vector when applying task-based quantization tools, the true task of the system is to recover $\mathbf{a}$, and the estimation of $\mathbf{s}$ is an intermediate step in that aim. Therefore, the setting of $\mathbf{s}$ can be treated as part of the design procedure. Specifically, the $J$-dimensional vector $\mathbf{s}$ is related to the $MNL$-dimensional vector $\mathbf{\Phi}\mathbf{a}$ via the compressive measurement matrix $\mathbf{M}$. As a result, $\mathbf{M}$ should be selected such that the desired $\mathbf{a}$ is still recoverable, while allowing BiLiMO to obtain an accurate estimate of $\mathbf{s}$ at the first stage of its digital processing. The additional dimenionality reduction induced by $\mathbf{M}$ can be translated into improved accuracy when jointly designing the HAD system including the digital filter $\mathbf{D}$ to estimate $\mathbf{s}$ via task-based quantization. In particular, the accuracy of task-based quantization typically improves when the task dimensionality is reduced, as the same number of bits can be utilized to recover less quantities via HAD processing [31]. Thus, our relaxed problem formulation considers the estimation of the further compressed $\mathbf{s}$, from which the targets are still recoverable via sparse recovery, rather than $\mathbf{\Phi}\mathbf{a}$. In the following we formulate our problem for a given $\mathbf{M}$, providing guidelines for its setting and numerically evaluating different selections in Subsection IV-D and Section V, respectively.

The relaxed objective of the jointly designed hybrid acquisition system is therefore to minimize the MSE between the compressed vector $\mathbf{s}$ and the digital filter output, given by

$$\hat{\mathbf{s}}=\mathbf{D}\mathcal{Q}_{C}^{\gamma,b}(\bar{\mathbf{F}}\bar{\mathbf{B}}(\mathbf{P}\mathbf{\Phi}\mathbf{a}+\mathbf{w})).\vspace{-0.1cm}$$

(18)

The resulting analysis characterizes the corresponding HAD processing and low-bit quantizers which achieve this MSE. In particular, we assume that the BiLiMO receiver has knowledge of: 1) the statistical model of $\mathbf{a}$ and $\mathbf{w}$; 2) the compressive measurement matrix $\mathbf{M}$, which allows recovery of $\mathbf{a}$ from $\mathbf{s}$.

Quantizers are typically designed to operate within their dynamic range, namely, that their input lies within the support $[-\gamma,\gamma]$, to avoid inducing additional distortion due to saturation [41]. Our derivation is thus carried out assuming that non-overloaded quantizers, i.e., the magnitudes of the real and imaginary parts of $\mathbf{z}$ are not larger than $\gamma$ with sufficiently large probability. To guarantee this, we fix $\gamma$ to be some multiple $\eta$ of the maximal standard deviation of the inputs:

$$\gamma^{2}=\eta^{2}\max_{l=1,2,\cdots,P}\mathbb{E}\big{\{}\big{|}(\mathbf{y}+\mathbf{n})_{l}\big{|}^{2}\big{\}}.\vspace{-0.1cm}$$

(19)

For instance, for proper-complex Gaussian inputs, setting $\eta\geq\sqrt{2}$ yields overload probability smaller than $6\%$ [32]. For arbitrary inputs, one can set $\eta$ to obtain a desired overload probability bound via Chebyshev’s inequality [45, Pg. 64].

Following the framework of task-based quantization in [31], we aim to jointly design the analog combining matrix $\bar{\mathbf{B}}$, the digital processing matrix $\mathbf{D}$, and the support of the quantizer $\gamma$ via (19), such that $\hat{\mathbf{s}}$ approaches the linear minimal MSE (LMMSE) estimator of $\mathbf{s}$ from $\mathbf{P}\mathbf{\Phi}\mathbf{a}+\mathbf{w}$, denoted $\tilde{\mathbf{s}}$. The motivation for this formulation stems from the fact that the output of the digital filter can be treated as an estimate of $\mathbf{s}$ from $\mathbf{P}\mathbf{\Phi}\mathbf{a}+\mathbf{w}$ by (18). Let ${\boldsymbol{\Gamma}}$ be the LMMSE transformation, i.e., $\tilde{\mathbf{s}}={\boldsymbol{\Gamma}}(\mathbf{P}\mathbf{\Phi}\mathbf{a}+\mathbf{w})$, and let $\mathbf{R}_{c}$ and $\mathbf{R}_{w}$ be the covariance matrices of $\mathbf{c}=\mathbf{P}\mathbf{\Phi}\mathbf{a}$ and $\mathbf{w}$, respectively. As $\tilde{\mathbf{s}}$ is the LMMSE estimate of $\mathbf{s}=\mathbf{M}\mathbf{\Phi}\mathbf{a}$, it holds that ${\boldsymbol{\Gamma}}$ is given as

$${\boldsymbol{\Gamma}}=\mathbf{M}\mathbf{P}^{T}\mathbf{R}_{c}{\boldsymbol{\Sigma}}^{-1},\vspace{-0.1cm}$$

(20)

where ${\boldsymbol{\Sigma}}=\mathbf{R}_{c}+\mathbf{R}_{w}$. Accordingly, the LMMSE $\operatorname{\epsilon_{M}}\triangleq\mathbb{E}\{\|\tilde{\mathbf{s}}\!-\!\mathbf{s}\|^{2}\}$ is

$$\operatorname{\epsilon_{M}}=\operatorname{Tr}\left[\mathbf{M}\mathbf{P}^{T}\mathbf{R}_{c}\mathbf{P}\mathbf{M}^{H}\!-\!\mathbf{M}\mathbf{P}^{T}\mathbf{R}_{c}{\boldsymbol{\Sigma}}^{-1}\mathbf{R}_{c}\mathbf{P}\mathbf{M}^{H}\right].$$

(21)

To summarize, our goal is to optimize the components of the BiLiMO receiver to minimize the excess MSE (EMSE):

$$\min_{\bar{\mathbf{B}},\mathbf{D},\gamma}\mathbb{E}\{\|\tilde{\mathbf{s}}-\hat{\mathbf{s}}\|^{2}\}.$$

(22)

Our derivation of the jointly designed BiLiMO receiver is presented in the sequel, where we first focus on the special case of monotone waveforms, i.e., $L=1$, after which we show how these results extend to multitone waveforms with $L>1$.

We now characterize the BiLiMO receiver design based on the objective (22). In particular, we derive the analog combining matrix, digital processing matrix, and the support $\gamma$, for the scenario in which $L=1$, which means that monotone waveforms are transmitted. In this case, the matrix $\bar{\mathbf{F}}$ in (11) is the identity matrix. As a result, the signal samples and noise samples can now be written as $\mathbf{y}=\mathbf{B}\mathbf{P}\mathbf{\Phi}\mathbf{a}$ and $\mathbf{n}=\mathbf{B}\mathbf{w}$, respectively, where $\mathbf{B}=\mathbf{B}_{0}$ is the $P\times MN$ analog combining matrix. Thus, the quantized output $\mathbf{z}$ is given by

$$\mathbf{z}=\mathcal{Q}_{C}^{\gamma,b}(\mathbf{B}(\mathbf{P}\mathbf{\Phi}\mathbf{a}+\mathbf{w})).$$

(23)

We begin by characterizing the digital processing matrix which minimizes the MSE for a fixed analog combining matrix $\mathbf{B}$. By applying [31, Lemma 1], we obtain the follow result:

Using Lemma 1, we optimize the analog filter matrix $\mathbf{B}$, which also dictates the support of the quantizers via (19). We do so by designing $\mathbf{B}$ to minimize the EMSE $\operatorname{\epsilon}(\mathbf{B})$ in (25), yielding the filter $\mathbf{B}^{o}$ given in the following theorem.

The unitary matrix $\mathbf{U}^{o}$ in Theorem 1 can be obtained via [46, Alg. 2.2]. With the analog combining matrix $\mathbf{B}^{o}$, we can derive the EMSE and the resulting support of the quantizers.

The characterization of the BiLiMO receiver configuration in Theorem 1 and the corresponding accuracy in Corollary 1 are obtained by expressing the problem of recovering the desired compressed representation as a task-based quantization setup [31]. This follows since for monotone waveforms, i.e., $L=1$, the effect of the analog filters $\{b_{p,n}(t)\}$ in (9) can be expressed as the matrix $\mathbf{B}$, without imposing any structure constraints on the equivalent analog combining matrix. However, radar applications commonly utilize multitone signals, resulting in $L>1$, which yields an equivalent formulation in which the analog combiner is constrained to take a structured form. Therefore, in the following we characterize the configuration of BiLiMO under such structure constraints.

For $L>1$, the analog combining matrix, which represents the analog filters $\{b_{p,n}(t)\}$, is expressed as the product of $\bar{\mathbf{F}}$ and a block diagonal matrix $\bar{\mathbf{B}}$ as shown in Subsection III-A. By repeating the derivation in Lemma 1 with fixed analog processing, the digital filter which minimizes the MSE for a fixed analog combiner $\bar{\mathbf{B}}$ is stated in the following lemma.

The derivation of the digital processing for a given analog filter is invariant to whether the waveforms are monotone or multitone. However, when optimizing the analog combining matrix in light of (30), one must account for its block-diagonal structure induced when $L>1$. In particular, we design the matrix $\bar{\mathbf{B}}$ which accounts for the aforementioned constraint by formulating the following objective:

$$\min_{\mathbf{B}_{1},\cdots,\mathbf{B}_{L}\in\mathbb{C}^{P\times MN}}\operatorname{\epsilon}(\bar{\mathbf{B}}=\operatorname{blkdiag}\{\mathbf{B}_{1},\cdots,\mathbf{B}_{L}\}).$$

(31)

In order to tackle (31), we can choose an appropriate matrix $\mathbf{M}$ such that the matrix $\mathbf{M}\mathbf{P}^{H}$ is also block diagonal. Let $\mathbf{M}_{i}$ be the $i$th block of $\mathbf{M}\mathbf{P}^{H}$ with its dimension $J_{i}\times MN$, where $\sum_{i=1}^{L}J_{i}=J$. Then the compressed vector $\mathbf{s}$ can be separated into $L$ sub-vectors, each related to a different vector in the set $\{\mathbf{c}_{i}\}$, i.e., $\mathbf{s}_{i}=\mathbf{M}_{i}\mathbf{c}_{i}$. Furthermore, we introduce the following assumption on the underlying statistical model:

• A1

  The covariance matrices of $\mathbf{c}$ and $\mathbf{w}$, i.e., $\mathbf{R}_{c}$ and $\mathbf{R}_{w}$, are block diagonal, with their $i$-th blocks being the $MN\times MN$ matrices $\mathbf{R}_{c_{i}}=\mathbb{E}\{\mathbf{c}_{i}\mathbf{c}_{i}^{H}\}$ and $\mathbf{R}_{w_{i}}$, respectively.

Assumption A1 means that we only consider the correlation within the vector $\mathbf{c}_{i}$ and ignore the correlation between the different vectors $\mathbf{c}_{i}$ and $\mathbf{c}_{i^{\prime}}$ for $i\neq i^{\prime}$. Since each $\mathbf{c}_{i}$ represents the samples of the received echos after channel separation at a given tone index $i$ by (8), A1 implies that echos observed at different frequency bins are uncorrelated.The validity of this assumption clearly depends on the distribution of $\{\alpha_{k}\}$, $\{\tau_{k}\}$, and $\{\varphi_{k}\}$. As shown in [47], if $\alpha_{k}$, $\tau_{k}$, and $\varphi_{k}$ are mutually independent, with $\{\alpha_{k}\}$ being zero-mean i.i.d. and $\tau_{k}$ and $\varphi_{k}$ being uniformly distributed on $[0,T_{0}]$ and $[-1,1]$, respectively, then $\{c_{m,n}[i]\}$ are uncorrelated. In this case, the covariance matrix $\mathbf{R}_{c}$ is diagonal, satisfying assumption A1. With this assumption, ${\boldsymbol{\Sigma}}$ also becomes a block diagonal matrix, with its $i$-th block being the $MN\times MN$ matrix ${\boldsymbol{\Sigma}}_{i}=\mathbf{R}_{c_{i}}+\mathbf{R}_{w_{i}}$.

Under assumption A1, we characterize the block diagonal matrix $\bar{\mathbf{B}}^{o}$ which solves (31) in the following theorem: