1MBZUAI, UAE 2Institute of Space Technology, Pakistan 3IBM Research

# FANet: Feature Amplification Network for Semantic Segmentation in Cluttered
Background

###### Abstract

Existing deep learning approaches leave out the semantic cues that are crucial
in semantic segmentation present in complex scenarios including cluttered
backgrounds and translucent objects, etc. To handle these challenges, we
propose a feature amplification network (FANet) as a backbone network that
incorporates semantic information using a novel feature enhancement module at
multi-stages. To achieve this, we propose an adaptive feature enhancement
(AFE) block that benefits from both a spatial context module (SCM) and a
feature refinement module (FRM) in a parallel fashion. SCM aims to exploit
larger kernel leverages for the increased receptive field to handle scale
variations in the scene. Whereas our novel FRM is responsible for generating
semantic cues that can capture both low-frequency and high-frequency regions
for better segmentation tasks. We perform experiments over challenging real-
world ZeroWaste-f [1] dataset which contains background-cluttered and
translucent objects. Our experimental results demonstrate the state-of-the-art
performance compared to existing methods. The source code can be found at
https://github.com/techmn/fanet.

Index Terms—  Semantic segmentation, image sharpening, waste segmentation,
convolution neural network, feature enhancement

##  1 Introduction

Semantic segmentation is a fundamental computer vision task having objective
to obtain pixel-level predictions and is critical for various practical
applications such as autonomous driving, scene understanding, robot sensing,
and image editing. Traditional convolutional neural networks (CNNs)-based
methods rely on local short-range structure to capture the semantics of the
image [2, 3]. However, due to intrinsic fixed geometric structures, they are
limited to short-range contextual information. Various efforts have been made
to tackle this issue including dilated convolution [4, 5, 6] and channel or
spatial attention models [7, 8].

Fig. 1: The main challenges in semantic segmentation, e.g., translucent
objects, background clutter, and scale variations. The first row indicates the
input image while the bottom row shows the input image overlay with the ground
truth.

Since the advent of vision transformers [9] which make use of the self-
attention mechanism, semantic segmentation has been considerably
revolutionized. For instance, [10, 11] effectively utilizes the global
contextual relationships of the transformers to perform reasonably on semantic
segmentation tasks. However, they pay less attention to capturing the
semantic-level contextual information. In addition, the dominance of global
receptive fields and long-range dependencies hinder its ability to capture the
fine details. Moreover, attention-based models are data-hungry and require a
large amount of data for optimal convergence due to the lack of inductive bias
and the struggle to capture local dependencies due to dominant global
representations which restrict their practicality. To this end, several hybrid
attention models [12, 13, 14, 15, 16] are introduced to improve the
performance by utilizing convolutions and self-attention mechanisms. However,
these models still face difficulties to perform better when the segmentation
objects lie in a cluttered background. Furthermore, the segmentation of
translucent objects is another challenging task that needs special
consideration. As shown in Fig. 1, these models may suffer from three
challenges. (i)- The model may encounter notable difficulties when dealing
with translucent objects due to their intrinsic nature of opacity and unclear
boundaries between object and background. (ii)- Background clutter makes
appearance representations more ambiguous. (iii)- Diverse scale variations
increase the difficulty of capturing subtle objects.

Contributions: In this work, to handle the aforementioned challenges for the
semantic segmentation task, we propose a feature amplification network (FANet)
as shown in Fig. 3-(a). We introduce FANet as a backbone to capture the
enhanced features and generate multi-stage features using our novel adaptive
feature enhancement (AFE) block, as shown in Fig. 3-(b). Our plug-and-play AFE
block aims to benefit from a larger kernel and semantic cues in a parallel
manner, which can extract more comprehensive features to preserve the coarse-
to-fine details so that the boundaries of the objects are easily
distinguishable. The focus of the design is to encode the intrinsic properties
of the target objects and segment the objects from the complex scenes,
especially in the presence of cluttered backgrounds. Our block is responsible
for explicitly exposing the spatial descriptors while simultaneously
performing feature enhancement to preserve both high-frequency and low-
frequency components in an image. To excavate the spatial context, we propose
a spatial context module (SCM) that uses a large kernel to handle the scales
of the objects in complex scenes. Meanwhile, to inscribe the semantic cues, we
introduce a novel feature refinement module (FRM) which aims to capture low-
frequency components as well as highlight the fine details of the objects.
Experimental results indicate a considerable improvement in the complex
ZeroWaste-f [1] dataset compared to the existing state-of-the-art methods.

Fig. 2:  Given an image $I$ (a) and its smoothed version $I_{b}$ (b), element-
wise multiplication of $I$ and $I_{b}$ emphasize the color information and
preserve the blob regions (c). Whereas (d) fine details can be highlighted by
subtracting the smoothed image $I_{b}$ from the original image $I$. Motivated
by this, we introduce our feature refinement module (FRM).  Fig. 3: The (a) is
the overall illustration of our proposed feature amplification network
(FANet), as a backbone network, for background-cluttered semantic
segmentation. The input is passed to a backbone network (FANet) to produce the
multi-stage features ($S1,S2,S3,$ and $S4$). These multi-stage features are
input to a UperNet decoder [17] as a segmentation head for prediction. The (b)
shows our novel adaptive feature enhancement (AFE) block. Our (b) AFE block is
designed to capture the rich information. It comprises convolutional
embeddings (CE), spatial context module (SCM), feature refinement module
(FRM), and ConvMLP. Our AFF block adaptively aggregates the large kernel
information using SCM which increases the receptive field and FRM which
refines the features in the spatial dimension.

##  2 Background

Typically, image sharpening and contrast enhancement are the two fundamental
concepts that are applied in the spatial domain to improve the quality of the
image, as shown in Fig. 2. The sharpening is performed to emphasize the high-
frequency or fine details in a given visual feature. To do so, the visual
feature is subtracted from the Laplacian function to obtain a sharpened
result. Suppose, $f(x,y)$ is the given input feature then the resultant
sharpened feature $g(x,y)$ is obtained using Laplacian follows:

$g(x,y)=f(x,y)-c[\Delta^{2}f(x,y)],$ (1)

where $c$ is the Laplacian mask center coefficient which is empirically
determined.

Contrast enhancement is another image enhancement technique that improves the
contrast in a feature by stretching the range of intensity values to span a
desired range of values to highlight the low-frequency regions in the feature
map. The contrast-enhanced feature can be obtained as:

$q(x,y)=f(x,y)\odot m(x,y),$ (2)

where $q(x,y)$ and $\odot$ represent the output feature and Hadamard product
operation, respectively. The $m(.)$ is designed to stretch the contrast and
enhance features and can be defined as follows:

$m(x,y)=\gamma(\frac{1}{1+e^{-\alpha((x,y)-\beta}})-0.5),$ (3)

where $\gamma$ is a scaling factor used to control the strength of the
enhancement. The $\alpha$ and $\beta$ control the contrast of the enhancement
function.

Finally, the outputs from Eq. 1 and Eq. 2 can be combined to get the enriched
features in the spatial domain that can capture both high-frequency and low-
frequency regions. Motivated by this, we leverage our feature amplification
module described in Sec. 3.3 for token-mixer design.

##  3 Method

###  3.1 Overall Architecture

Fig. 3-(a) shows the overall architecture of our proposed feature
amplification network (FANet) for the semantic segmentation task. The focus of
our design is a backbone network that can capture the intrinsic properties of
the object and segment it from the cluttered background, using our novel
adaptive feature enhancement (AFE) block. Specifically, our novel FANet can
capture the enriched features and generate multi-stage features ($S1,S2,S3,$
and $S4$). To obtain the multi-stage features, the input
$x\in\mathbb{R}^{H\times W\times 3}$ is input to non-overlapping convolution
stem layers (kernel size = 5×5, stride = 4) to generate the tokens of size
$\mathbb{R}^{H/4\times W/4}$. Following hierarchical design [18, 19, 20], our
FANet comprises four stages to obtain hierarchical feature representations.
There exists a down-sampling convolution layer (kernel size = 3×3, stride = 2)
between two stages to reduce the spatial resolution of the features. The
multi-stage features are input to a UperNet [17] decoder to get the final
segmentation mask.

###  3.2 Adaptive Feature Enhancement (AFE) Block

The proposed AFE block comprises four key components: convolutional embeddings
(CE), spatial context module (SCM), feature refinement module (FRM), and
convolutional multi-layer perceptron (ConvMLP). The focus of this design is to
adaptively capture the enriched features of cluttered backgrounds for semantic
segmentation. The input features are passed through a LayerNorm and a CE to
learn the generalization and discriminative ability [12]. The output of CE is
passed to a 1x1 convolution layer that squeezes the channels to half. The
channel squeeze helps to reduce the computational overhead and encourages the
model to perform feature mixing. The squeezed features are passed to the SCM
module which comprises group-wise convolution with a larger kernel (kernel
size = 7x7). The objective of SCM is to increase the receptive field which can
capture the spatial context over a broader range to handle scale variations.
In parallel, the squeezed features from the CE layer are also fed to the
feature refinement module (FRM) to refine the features (see details in Sec.
3.3). The outputs from SCM and FRM are fused and projected through a 1x1
convolution layer and a ConvMLP to further enhance the representations.

###  3.3 Feature Refinement Module (FRM)

Inspired by the image sharpening and contrast enhancement concept, we
introduce a novel feature refinement module (FRM) to capture the low-frequency
context and emphasize the blob regions while simultaneously highlighting the
high-frequency details. The block diagram for FRM is depicted in Fig. 4.
Suppose $F\in\mathbb{R}^{C\times H\times W}$ be the input of FRM. We pass it
from a depthwise convolution layer to obtain down-sampled feature maps
$P\in\mathbb{R}^{C\times H/2\times W/2}$. In the image processing domain, high
frequencies are highlighted by taking the difference between the image and its
blurred version. We adopt a similar procedure by up-sampling the smoothed
feature maps $P$ to the same spatial resolution of $F$ to obtain $Q$ features.
Later, the difference between $F$ and $Q$ highlights the fine details
resulting in $R$ refined feature embeddings (as shown in Fig. 4).

The second branch (bottom) in FRM strives to capture the low-frequency regions
in the feature maps. Given a normalized image, the blob features can be
emphasized by element-wise multiplication of the image with its blurred
version. To do so, we perform element-wise multiplication operations between
the $F$ and $Q$ features to obtain $S$ which capture the low-frequency
components. The idea is to highlight the low frequencies by focusing on the
blobs in the feature maps. Our FRM emphasizes the low and high-frequency
regions and concatenates them after depthwise convolution in the channel
dimension (to obtain $T$). Finally, these features are realized with a
projection layer to obtain the final enriched/amplified features $\bar{F}$.

##  4 Experimentation

Fig. 4: The illustration of our novel feature amplification module. The input
features $F$ are downsampled using depthwise convolution (DWConv) and
upsampled to get $Q$ features. The input features $F$ are subtracted from $Q$
features to get $R$ features that highlight the fine details. Similarly, the
input features are multiplied with $Q$ features to obtain $S$ features which
highlight the low-frequency components in the spatial dimension. Later, these
low-frequency and high-frequency features are aggregated after DWConv to
obtain enhanced features. Finally, the aggregated features are input to the
projection layer to obtain the final $\bar{F}$ features. Table 1: Comparison
of ours FANet with state-of-the-art methods for semantic segmentation on the
test set of the ZeroWaste-f dataset. We report the results in terms of mIoU
and pixel accuracy (Pix. Acc.). Here, the FocalNet-B provides improved
performance compared to the DeepLabv3+. Our FANet when utilized as a backbone
network improves mIoU performance by 1.63% over the FocalNet-B. The best
results are in bold.

Methods | mIoU | Pix. Acc.
---|---|---
CCT [21] | 29.32 | 85.91
ReCo [22] | 52.28 | 89.33
DeepLabv3+ [23] | 52.13 | 91.38
FocalNet-B [18] | 53.26 | 91.28
FANet (ours) | 54.89 | 91.41

Fig. 5: Qualitative comparison on ZeroWaste-f dataset. Our method better
segments the objects from cluttered backgrounds compared to existing state-of-
the-art methods.

###  4.1 Dataset and Evaluation Protocol

We have chosen to focus on the ZeroWaste-f dataset [1] which is introduced to
advance the field of waste management and segmentation, particularly in
environments with extreme clutter, while also boosting innovations in
recycling applications. It encompasses a diverse range of waste types,
including various paper, cardboard, plastics, metals, and organic materials,
often in cluttered and overlapping states. Our motivation in selecting this
dataset is twofold: firstly, to push the boundaries of what’s possible in the
realm of automated waste segmentation, and develop a model that can accurately
differentiate and classify a wide array of waste materials. Secondly, we aim
to contribute to environmental sustainability efforts. By improving the
precision of waste segmentation, we can enhance recycling processes and waste
management practices, ultimately contributing to a more sustainable and
environmentally conscious approach to waste disposal. This dataset comprised
3002 training, 572 validation, and 929 test sets. Following [1], we use the
mean intersection over union (mIoU) and pixel accuracy to measure the
performance of the models. The mIoU is a crucial metric for assessing
segmentation accuracy by quantifying the overlap between prediction and ground
truth.

###  4.2 Implementation Details

We implemented our method in PyTorch and trained over NVIDIA V100 GPU. For
this purpose, we leverage the MMSegmentation open-source toolbox [24]. The
training of our model is conducted using the UperNet architecture, with the
backbone initialized using weights pre-trained on the ImageNet-1K [25]
dataset. This model undergoes a training regimen of 40k iterations on the
ZeroWaste dataset. We employ the AdamW optimizer [26] for this process,
starting with an initial learning rate of $9e^{-5}$. In line with standard
research methodologies, our training approach includes a learning rate decay,
specifically utilizing a polynomial decay strategy with a power exponent of
1.0. We configure our training setup with a crop size of 512x512 and a batch
size of 2. Following [20, 27], we first trained the backbone on the
ImageNet-1K [25] dataset for 300 epochs with batch size 80 using 16 NVIDIA
V100 GPUs and a learning rate of 1e-3. We adopt the AdamW optimizer [28] for
training and set weight decay of $0.05$. After pertaining, we finetuned the
model on the semantic segmentation dataset.

Fig. 6: Illustration of the impact of feature refinement module (FRM). (a) is
the input to the model, (b) shows the output feature maps of the third stage
and input to the FRM, (c) and (d) show the highlighted features in the FRM as
$R$ and $S$, respectively. The (e) shows the enhanced features as output
provided by FRM, and (f) highlights the resulting segmentation map provided by
the model overlay with the input image. The (g) is the ground truth overlay
with the input image. Table 2: Ablation study of our FANet pretrained over
ImageNet-1K for 300 epochs. First, we use the depthwise convolutional
embeddings using a skip connection called CE and ConvMLP as our baseline. Row
2 shows that introducing SCM into the baseline achieves a performance gain.
Similarly, integrating FRM (rows 3, 4, and 5) into the baseline leads to a
consistent gain in the performance. Our final approach FANet (last row), which
comprises both SCM and FRM, achieves a significant improvement in performance
over the baseline. The best results are in bold.

Exp. No. | SCM | FRM | mIoU | Pix. Acc.
---|---|---|---|---
High Frequency Context | Low Frequency Context
1\. CE+ConvMLP (baseline) |  |  |  | 38.15 | 89.26
2\. baseline | ✓ |  |  | 47.66 | 91.07
3\. baseline |  | ✓ |  | 49.03 | 90.98
4\. baseline |  |  | ✓ | 43.99 | 90.65
5\. baseline |  | ✓ | ✓ | 50.45 | 90.95
6\. FANet (Ours) | ✓ | ✓ | ✓ | 54.89 | 91.41

###  4.3 Quantitative Comparison

Tab. 1 presents the comparison over ZeroWaste-f [1] of our method with other
state-of-the-art methods including CCT [21], ReCo [22], DeepLabV3+ [23], and
FocalNet-B [18]. The DeepLabV3+ [23] achieves a leading IoU score of 52.13% on
the ZeroWaste-f dataset. In addition, the FocalNet-B [18] is one of the best-
performing models for semantic segmentation, and also obtained mIoU of 53.26%.
We notice that our model surpasses these compared methods and obtains mIoU and
pixel accuracy of 54.89% and 91.41%, respectively. Such improvements are
indicative of our novel adaptive feature enhancement module’s efficiency. Our
method sets a new state-of-the-art performance with this significant gain
obtained in the challenging mIoU metric.

Table 3: Comparison of FANet with FocalNet on test set. Methods | Params | mIoU | Pix. Acc.
---|---|---|---
FocalNet-T [18] | 28.6 M | 51.71 | 91.03
FocalNet-B [18] | 88.7 M | 54.26 | 91.28
FANet (ours) | 36.7 M | 54.89 | 91.41
Table 4: Comparison on the validation set of ImageNet-1K. Methods | Params (M) | Flops (G) | Top1 Acc.
---|---|---|---
FocalNet-T | 28.6 | 4.5 | 82.3
FANet (ours) | 36.7 | 5.8 | 82.5

###  4.4 Qualitative Comparison

Fig. 5 illustrates the qualitative comparison of our method with DeepLabV3+
[23] and FocalNet-B [18]. We observe that our FANet can accurately segment the
object in complex scenarios. For example, in the first row, our method can
segment the translucent object better compared to the others. Similarly, in
the second row, our method can segment objects in the presence of a heavily
cluttered background. These results show our method’s capability to adapt a
variety of image characteristics and correctly classify each item which
validates the robustness of our segmentation method.

###  4.5 Ablation Study

For all the ablation study experiments, the FANet backbone is pre-trained on
ImageNet for 300 epochs.

Effectiveness of SCM and FRM: Tab. 2 showcases the impact of our
contributions. Our baseline consists of a depthwise convolution embedding with
a skip connection, dubbed as CE, and ConvMLP. Introducing the SCM (row-2) into
the baseline which is responsible for capturing the semantic information over
the larger context leads to improved performance gain compared to the
baseline. On the other hand, integrating the FRM which refines the features
highlighting both low-frequency and high-frequency regions also obtains a
significant gain compared to the baseline as shown in rows 3, 4, and 5,
respectively. Finally, the integration of SCM and FRM into the baseline
achieves the best performance which indicates that efficiently captures the
larger context information as well as preserves the low-frequency and high-
frequency components. This integration improves the model’s capacity to better
capture the spatial descriptors as well as coarse-to-fine details, which show
the metrics of our contributions.

Comparison of ours FANet with the FocalNet: We compared our method with
FocalNet-T and FocalNet-B frameworks with our method in Tab. 3. From the
table, it is clearly shown that our method exhibits better capability to
handle the cluttered backgrounds. Although FocalNet-T has less the number of
parameters, it exhibits a significantly reduced mIoU of 3.18% compared to
ours. In addition, our method has significantly reduced the number of
parameters compared to the FocalNet-B frameworks and still achieves better
performance. This reduction in parameters signifies a more efficient model,
both in terms of computational resources and processing speed, without
compromising the quality of segmentation. It underscores the potential of our
approach in setting new standards for segmentation tasks, particularly in
complex and resource-constrained environments.

Feature Visualization of FRM: In Fig. 6, we show the effectiveness of our FRM
using feature visualization. To do so, we take the stage 3 features and show
the input features to our FRM, output features for both $R$ and $S$, and
output of the FRM as (b), (c), (d), and (e), respectively. Whereas, (a), (f),
and (g) show the input image, our prediction overlay with the input image, and
ground truth overlay with the input image, respectively. From these feature
visualizations, we can observe that our FRM can preserve both low-frequency
and high-frequency components in the presence of a heavily cluttered
environment for better semantic segmentation. This ability to preserve
detailed textural information alongside broader contextual elements
significantly contributes to the accuracy.

Comparison of FANet over ImageNet-1K: Finally, in Tab. 4, we compare our FANet
method with FocalNet-T over the validation set of ImageNet-1K. Although our
approach has more parameters, it has improved accuracy in terms of top-1.
Moreover, from Tab. 3, it is notable that our FANet has significantly improved
performance for the target ZeroWaste-f dataset compared to FocalNet-T. This
shows that FocalNet-T has limited ability to tackle the cluttered background
challenge for the semantic segmentation task.

##  5 Conclusion

In this work, we propose a feature amplification network (FANet) to capture
the semantic context information and generate multi-stage features for better
semantic segmentation in complex scenarios especially in heavily cluttered
objects. Particularly, we propose an adaptive feature enhancement (AFE) block
that exploits both larger contexts using the spatial context module (SCM) and
semantic cues in the feature refinement module (FRM) in a parallel fashion.
Our FRM, inspired by image sharpening and contrast enhancement, exploits low-
frequency and high-frequency regions in the latent space to perform feature
refinement. Our extensive experimental study reveals the effectiveness of our
method.

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# Beamforming Design for Integrated Sensing and Communications Using Uplink-
Downlink Duality

Kareem M. Attiah and Wei Yu  Electrical and Computer Engineering Department
University of Toronto, Canada
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper presents a novel optimization framework for beamforming design in
integrated sensing and communication systems where a base station seeks to
minimize the Bayesian Cramér-Rao bound of a sensing problem while satisfying
quality of service constraints for the communication users. Prior approaches
formulate the design problem as a semidefinite program for which acquiring a
beamforming solution is computationally expensive. In this work, we show that
the computational burden can be considerably alleviated. To achieve this, we
transform the design problem to a tractable form that not only provides a new
understanding of Cramér-Rao bound optimization, but also allows for an uplink-
downlink duality relation to be developed. Such a duality result gives rise to
an efficient algorithm that enables the beamforming design problem to be
solved at a much lower complexity as compared to the-state-of-the-art methods.

##  I Introduction

Next-generation wireless systems and their emerging applications are expected
to offer high-throughput communication services as well as accurate sensing
functionalities [1, 2]. Integrated sensing and communication (ISAC) is a
promising strategy which is anticipated to play a central role in meeting
those demands. In contrast to traditional systems in which the sensing and
communication operations are performed separately, ISAC systems offer better
utilization of resources and network infrastructure. However, these benefits
also come with the challenge of addressing more complicated signal processing
tasks and hardware design.

This paper is motivated by the recent interest in beamforming design for the
simultaneous operation of sensing and communications. Such problem is often
posed as that of optimizing beamformers for a sensing-related metric, e.g.,
the Cramér-Rao bound (CRB) of an estimation problem, while guaranteeing
certain quality-of-service, e.g., signal-to-interference-and-noise (SINR)
constraints for the communication problem [3, 4, 5, 6]. This optimization is
challenging not only due to its nonconvexity but also because the objective
function typically has rather complicated forms [5, 4, 6]. The best known
strategy for tackling this problem relies on transforming the problem to the
covariance domain, where the problem is convexified after dropping rank-one
constraints using the semidefinite relaxation (SDR) technique. Despite the
convexity of the transformed problem, solving the SDR problem is typically
inefficient due to the need for lifting the solution space. Moreover, it often
requires a randomization step whenever the SDR solution does not satisfy the
rank-one constraint.

This paper provides a novel framework for solving this challenging problem.
The main idea is to transform the complicated ISAC problem into a simple form
that involves minimizing the beamforming power subject to downlink SINR
constraints. The main benefit of doing so is that the latter problem allows
for an efficient solution based on uplink-downlink (UL-DL) duality [7, 8, 9,
10, 11].

The task of transforming the ISAC problem into the desired form can be
accomplished in two stages. First, we show that the optimization of a CRB
objective can be viewed as that of maximizing the power in certain point of
interest. This observation is related to a result in [6] for a specific angle
estimation problem that we extend herein for an arbitrary vector parameter
using a more general technique. The second step is to leverage Lagrangian
theory in order to transform the resulting problem to that of minimizing the
power subject to SINR constraints. Given the new formulation, we draw upon the
existing theory in [7] to establish an UL-DL duality relation for the ISAC
problem. Such duality result reveals that the same efficient algorithms
developed for the classical communication problem [7, 10, 11] are applicable
here, with a few modifications. In contrast to the SDR approach, the proposed
solution methodology is computationally efficient and does not lift the
solution space. Finally, we present numerical results demonstrating the
effectiveness of the proposed solution.

##  II System Model

###  II-A ISAC Model and Performance Metrics

We consider the multi-user ISAC system illustrated in Figure 1 over a
coherence interval spanning $T$ symbols. In this setup, a base station (BS)
equipped with co-located $N_{\text{T}}$ transmit antenna array and
$N_{\text{R}}$ receive antenna array seeks to learn an unknown vector of real
parameters $\boldsymbol{\eta}\in\mathbb{R}^{L}$. Simultaneously, it aims to
convey information to $K$ single-antenna communication users in the downlink.
During the $t$-th symbol, the baseband transmitted signal
$\mathbf{x}[t]\in\mathbb{C}^{N_{\text{T}}}$ follows a beamforming model given
by

$\mathbf{x}[t]=\mathbf{V}\mathbf{s}[t]=\sum_{k}\mathbf{v}_{k}s_{k}[t],$ (1)

where
$\mathbf{V}\triangleq\left[\mathbf{v}_{1},\ldots,\mathbf{v}_{K}\right]\in\mathbb{C}^{N_{\text{T}}\times
K}$ is a matrix of beamforming vectors satisfying a total power constraint
$\operatorname*{Tr}\left(\mathbf{V}\mathbf{V}^{\textsf{H}}\right)\leq P$ and
$\mathbf{s}[t]=\left[s_{1}[t],\ldots,s_{K}[t]\right]^{\textsf{T}}\in\mathbb{C}^{K}$
is a zero-mean vector of communication symbols satisfying
$\mathbb{E}\left\\{\mathbf{s}[t]\mathbf{s}^{\textsf{H}}[t]\right\\}=\mathbf{I}_{K}$.

Figure 1: The ISAC system considered in this work. The BS seeks to serve $K$
communication users and learn some unknown vector parameter
$\boldsymbol{\eta}$.

Observe that the beamforming model (1) adopted herein is identical to that of
conventional communication systems. Such model is particularly useful in
scenarios where the length of the parameter to be estimated is short compared
to the number of users (i.e., $L\ll K$), in which case $K$ degrees-of-freedom
(DoFs) are enough to estimate the parameter [5, 12]. In contrast, if $L$ is
large, the beamforming model (1) should be augmented with an additional
component dedicated to sensing [3, 4, 13]. The extended model can exhibit
better performance in this case due to the increased DoF, but at the cost of a
more complicated design and increased RF-chain requirements. For simplicity,
we restrict our attention to the model (1).

In the $t$-th symbol, the BS transmits the waveform $\mathbf{x}[t]$ and
subsequently observes the measurement vector
$\mathbf{y}[t]\in\mathbb{C}^{N_{\text{R}}}$. In addition, each user observes a
scalar baseband signal, denoted by $\tilde{y}_{k}[t]$ for the $k$-th user.
Such received signals are given by

$\mathbf{y}[t]=\mathbf{G}^{\left(\boldsymbol{\eta}\right)}\mathbf{x}[t]+\mathbf{n}[t],$
(2)

and

$\tilde{y}_{k}[t]=\mathbf{h}_{k}^{\textsf{H}}\mathbf{x}[t]+\tilde{n}_{k}[t],\quad\forall
k\in\left\\{1,\ldots,K\right\\},$ (3)

where $\mathbf{h}_{k}\in\mathbb{C}^{N_{\text{T}}}$ is the $k$-th user channel
and $\mathbf{n}[t]\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N_{\text{R}}})$ and
$\tilde{n}_{k}[t]\sim\mathcal{CN}\left(0,\sigma^{2}\right)$ are noise terms
whose variances are normalized to be the same without loss of generality. The
matrix $\mathbf{G}^{(\boldsymbol{\eta})}\in\mathbb{C}^{N_{\text{R}}\times
N_{\text{T}}}$ models the “round-trip" channel between the transmit and
receive antenna arrays. Observe that $\boldsymbol{\eta}$ is the set of
underlying parameters of the sensing channel. This model encompasses a wide
variety of practical scenarios (e.g., angle-of-arrival estimation of a radar
target). Finally, we assume that the communication channel
$\mathbf{H}\triangleq\left[\mathbf{h}_{1},\ldots,\mathbf{h}_{K}\right]\in\mathbb{C}^{N_{\text{T}}\times
K}$ is perfectly known at the BS.

We aim to design the beamforming matrix $\mathbf{V}$ so that certain
performance goals are met for communication and sensing. For sensing, we
assume a prior distribution on $\boldsymbol{\eta}$ and adopt the Bayesian CRB
(BCRB) as a measure of performance. Unlike the classical CRB used in previous
works [14, 4, 5, 15, 16] which depends on the actual value of the unknown
parameter $\boldsymbol{\eta}$, the BCRB provides a bound on the mean-squared
error (MSE) averaged over the prior. For an estimator
$\hat{\boldsymbol{\eta}}$ we have
$\mathbb{E}\\{\left(\boldsymbol{\eta}-\hat{\boldsymbol{\eta}}\right)\left(\boldsymbol{\eta}-\hat{\boldsymbol{\eta}}\right)^{\textsf{T}}\\}\succcurlyeq\mathbf{J}^{-1}_{\mathbf{V}}$,
where $\succcurlyeq$ denotes inequality with respect to the positive
semidefinite (PSD) cone and $\mathbf{J}_{\mathbf{V}}\in\mathbb{R}^{L\times L}$
is the Bayesian Fisher information matrix (BFIM). For the additive Gaussian
noise channel model (2), the elements of BFIM can be expressed as follows [17,
18]:

$\left[\mathbf{J}_{\mathbf{V}}\right]_{ij}=\frac{2T}{\sigma^{2}}\Re\left(\operatorname*{Tr}\left(\mathbb{E}\left\\{\dot{\mathbf{G}}_{i}^{\textsf{H}}\dot{\mathbf{G}}_{j}\right\\}\mathbf{V}\mathbf{V}^{\textsf{H}}\right)\right)+[\mathbf{C}]_{ij}$
(4)

where
$\dot{\mathbf{G}}_{i}\triangleq\tfrac{\partial\mathbf{G}^{\left(\boldsymbol{\eta}\right)}}{\partial\eta_{i}}$
and $\mathbf{C}\in\mathbb{R}^{L\times L}$ is a matrix whose elements are given
by $[\mathbf{C}]_{ij}\triangleq-\mathbb{E}\\{\tfrac{\partial^{2}\log
f(\boldsymbol{\eta})}{\partial\eta_{i}\partial\eta_{j}}\\}$. Observe that
$\mathbf{C}$ does not depend on $\mathbf{V}$. Our aim is to minimize the trace
of the inverse of the BFIM

$\mu(\mathbf{V})\triangleq\operatorname*{Tr}\left(\mathbf{J}^{-1}_{\mathbf{V}}\right).$
(5)

The above metric constitutes a lower bound on the sum of average MSEs across
the individual elements of $\boldsymbol{\eta}$. The main advantage of the
metric (5) is that it abstracts the receiver design since the bound holds for
any estimator.

For communications, most existing works on ISAC assume a performance metric of
satisfying certain rate constraints for the users [3, 4, 5]. This paper also
follows the same formulation. Since the rate is a monotone function of the
signal-to-interference-and-noise-ratio (SINR), the rate requirements can be
described as constraints on the SINR, i.e.,

$\text{SINR}^{\text{DL}}_{k}(\mathbf{V})\triangleq\frac{\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{k}\right|^{2}}{\sum_{i\neq
k}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{i}\right|^{2}+\sigma^{2}}\geq\gamma_{k},\quad\forall
k,$

where $\gamma_{k}$ denotes the SINR threshold for the $k$-th user.

###  II-B Problem Formulation and Solution Strategy

Based on the previous considerations, the beamforming design problem can be
mathematically stated as follows:

$\displaystyle\underset{\mathbf{V}}{\mathrm{minimize}}~{}~{}~{}~{}$
$\displaystyle\operatorname*{Tr}\left(\mathbf{J}^{-1}_{\mathbf{V}}\right)$
(6a) $\displaystyle\mathrm{subject\ to}~{}~{}~{}$
$\displaystyle\frac{\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{k}\right|^{2}}{\sum_{i\neq
k}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{i}\right|^{2}+\sigma^{2}}\geq\gamma_{k},\quad\forall
k,$ (6b)
$\displaystyle\operatorname*{Tr}\left(\mathbf{V}\mathbf{V}^{\textsf{H}}\right)\leq
P.$ (6c)

We make the assumption that the constraints of the above problem are feasible.
Problem (6) is challenging to solve for a number of reasons. First, the
objective is a complicated function to compute let alone optimize. Second,
neither the objective nor the feasible set is convex in $\mathbf{V}$.

One possible strategy for solving (6) is SDR. For example, the work [5]
considers a special case of (6) in which the sensing task is that of
estimating the angle-of-arrival for an unknown target. For this simpler case,
the authors of [5] show that an SDR strategy can effectively relax the problem
and transform it into a convex form. Despite the convexity, the SDR approach
suffers from a number of drawbacks. First, the SDR solution is not guaranteed
to satisfy the rank constraint. If the SDR solution is not rank one, we must
employ an additional, and often suboptimal, algorithm to recover a rank-one
solution. Secondly, it lifts the optimization space from ${N}_{\text{T}}K$
complex dimensions to $N_{\text{T}}^{2}K$ complex dimensions. Consequently,
the interior-point method requires $\mathcal{O}(N_{\text{T}}^{6}K^{3})$ per
iteration to compute a search direction [19]. This order of complexity is
infeasible in many practical scenarios.

A key contribution of this paper is to develop an efficient optimization
method that does not require lifting the problem dimensionality. We accomplish
this by establishing a connection between (6) and the following classical
problem

$\displaystyle\underset{\mathbf{V}}{\mathrm{minimize}}~{}~{}~{}~{}~{}$
$\displaystyle\sum_{k}\|{\mathbf{v}}_{k}\|^{2}$ (7a)
$\displaystyle\mathrm{subject\ to}~{}~{}~{}~{}$
$\displaystyle\frac{\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{k}\right|^{2}}{\sum_{i\neq
k}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{i}\right|^{2}+\sigma_{\text{c}}^{2}}\geq\gamma_{k},\quad\forall
k.$ (7b)

Problem (7) arises in the context of downlink MIMO beamforming where the goal
is to ensure reliable communication for the users (by constraining the SINRs)
while minimizing the transmit power at the BS. In [10], a duality notion is
observed that allows the downlink problem (7) to be transformed to a virtual
uplink problem, a result now commonly referred to as UL-DL duality [10, 9, 8,
7]. The key advantage of such transformation is that the uplink problem is
more computationally efficient to solve than its downlink counterpart.

##  III Problem Reformulation as Downlink Power Maximization Problem

We begin by showing that the ISAC problem (6) can be expressed as a simpler
problem that involves maximizing a quadratic term subject to the constraints
(6b) and (6c). In particular, we aim to rewrite problem (6) as a sequence of
maximization problems in the following form

$\displaystyle\underset{\mathbf{V}}{\mathrm{maximize}}\quad$
$\displaystyle\operatorname*{Tr}{\left(\mathbf{Q}\mathbf{V}\mathbf{V}^{\textsf{H}}\right)}$
(8a) $\displaystyle\mathrm{subject\ to}\quad$
$\displaystyle\frac{\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{k}\right|^{2}}{\sum_{i\neq
k}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{v}_{i}\right|^{2}+\sigma_{\text{c}}^{2}}\geq\gamma_{k},\quad\forall
k.$ (8b)
$\displaystyle\operatorname*{Tr}\left(\mathbf{V}\mathbf{V}^{\textsf{H}}\right)\leq
P.$ (8c)

where $\mathbf{Q}\in\mathbb{C}^{N_{\text{T}}\times N_{\text{T}}}$ is a PSD
matrix. Problem (8) has the property that it has zero duality gap. This
follows from the observation that a rank-one solution must exist for problems
of this particular type; see [20]. We now proceed to transform problem (6)
into this desired form.

###### Theorem 1

If problem (6) is feasible, it is equivalent to

$\displaystyle\underset{\boldsymbol{\beta}\in\mathbb{R}^{L\times
L}}{\mathrm{maximize}}~{}~{}\min_{\mathbf{V}\in\mathcal{V}}~{}~{}\sum_{\ell=1}^{L}\left(2\boldsymbol{\boldsymbol{\beta}}_{\ell}^{\textsf{T}}\mathbf{e}_{\ell}-\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{J}_{\mathbf{V}}\boldsymbol{\beta}_{\ell}\right)$
(9)

where
$\boldsymbol{\beta}\triangleq[\boldsymbol{\boldsymbol{\beta}}_{1},\ldots,\boldsymbol{\boldsymbol{\beta}}_{L}]\in\mathbb{R}^{L\times
L}$, $\mathbf{e}_{\ell}$ is the $\ell$-th column of $\mathbf{I}$, and
$\mathcal{V}$ denotes the constraints (8b)-(8c). Further, problem (9) can be
re-written as

$\displaystyle\underset{\boldsymbol{\beta}\in\mathbb{R}^{L\times
L}}{\mathrm{maximize}}~{}\min_{\mathbf{V}\in\mathcal{V}}\left[\left(\sum_{\ell=1}^{L}2\boldsymbol{\boldsymbol{\beta}}_{\ell}^{\textsf{T}}\mathbf{e}_{\ell}-\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{C}\boldsymbol{\beta}_{\ell}\right)-\operatorname*{Tr}{\left(\mathbf{Q}_{\boldsymbol{\beta}}\mathbf{V}\mathbf{V}^{H}\right)}\right]$
(10)

where
$\mathbf{Q}_{\boldsymbol{\beta}}\triangleq\frac{2T}{\sigma^{2}}\sum_{\ell=1}^{L}\mathbb{E}\\{\tilde{\mathbf{G}}_{\boldsymbol{\beta_{\ell}}}^{\textsf{H}}\tilde{\mathbf{G}}_{\boldsymbol{\beta_{\ell}}}\\},$
and
$\tilde{\mathbf{G}}_{\boldsymbol{\beta_{\ell}}}\triangleq\sum_{i}[\boldsymbol{\boldsymbol{\beta}}_{\ell}]_{i}\dot{\mathbf{G}}_{i}$.
In particular, there exists $\boldsymbol{\boldsymbol{\beta}}^{*}$ such that by
setting $\mathbf{Q}=\mathbf{Q}_{\boldsymbol{\beta}^{*}}$ in (8), we can obtain
the optimal beamformers in problem (6).

###### Proof:

The first part of Theorem 1 is related to a similar result in [6] but for a
specific angle estimation problem and using an extra sensing beamformer, so
its formulation is different. Further, the proof technique in [6] relies on a
quadratic transform for fractional programming [21], while we provide an
alternative proof based on Schur complement. For notational simplicity, we
suppress the dependence on $\mathbf{V}$ in $\mathbf{J}$. We start with the
convex relaxation, i.e., SDR of problem (6)

$\displaystyle\underset{\mathbf{R}_{1},\ldots,\mathbf{R}_{K}}{\mathrm{minimize}}~{}~{}$
$\displaystyle\operatorname*{Tr}\left(\mathbf{J}^{-1}\right)$ (11a)
$\displaystyle\mathrm{subject\ to}~{}$
$\displaystyle\frac{1}{\gamma_{k}}\mathbf{h}_{k}^{\textsf{H}}\mathbf{R}_{k}\mathbf{h}_{k}-\sum_{i\neq
k}\mathbf{h}_{k}^{\textsf{H}}\mathbf{R}_{i}\mathbf{h}_{k}\geq\sigma^{2}$ (11b)
$\displaystyle\sum_{k}\operatorname*{Tr}\left(\mathbf{R}_{k}\right)\leq
P,\quad\mathbf{R}_{k}\succcurlyeq 0~{}~{}\forall k,$ (11c)

where $\mathbf{R}_{k}\triangleq\mathbf{v}_{k}\mathbf{v}_{k}^{\textsf{H}}$.
Note that by $\mathbf{J}$ is an affine function of $\mathbf{R}_{k}$ by (4), so
the above is a convex optimization problem.

We use the following trick to express the objective (11a) using Schur
complement [22][18, 23] as follows:

$\displaystyle\operatorname*{Tr}\left(\mathbf{J}^{-1}\right)=\min_{d_{1},\ldots,d_{L}}~{}~{}$
$\displaystyle\sum_{\ell}d_{\ell}$ (12a) $\displaystyle\mathrm{s.t.\
}~{}~{}~{}~{}$ $\displaystyle\left[\begin{matrix}\mathbf{J}&e_{\ell}\\\
e_{\ell}^{\textsf{T}}&d_{\ell}\end{matrix}\right]\succcurlyeq
0,~{}\forall\ell.$ (12b)

Combining (11) with (12), we obtain

$\displaystyle\underset{\mathbf{R}_{1},\ldots,\mathbf{R}_{K},d_{1},\ldots,d_{L}}{\mathrm{minimize}}~{}~{}$
$\displaystyle\sum_{\ell}d_{\ell}$ (13a) $\displaystyle\mathrm{subject\
to}~{}~{}~{}~{}$
$\displaystyle\eqref{eq:sdp_const_1},\eqref{eq:sdp_const_2},\eqref{eq:SDP_const_main_text}$
(13b)

Problem (13) is convex with strong duality. We formulate its dual problem with
respect to (12b). Let
$\tilde{\mathbf{B}}_{1},\ldots,\tilde{\mathbf{B}}_{L}\in\mathbb{R}^{(L+1)\times(L+1)}$
denote the dual variables associated with (12b)

$\tilde{\mathbf{B}}_{\ell}\triangleq\left[\begin{matrix}\mathbf{B}_{\ell}&-\boldsymbol{\beta}_{\ell}\\\
-\boldsymbol{\beta}_{\ell}^{\textsf{T}}&b_{\ell}\end{matrix}\right]\succcurlyeq
0,$ (14)

with $\mathbf{B}_{\ell}\in\mathbb{R}^{L\times L}$,
$\boldsymbol{\beta}_{\ell}\in\mathbb{R}^{L}$, and $b_{\ell}\in\mathbb{R}$. The
dual problem is

$\displaystyle\underset{\tilde{\mathbf{B}}_{1},\ldots,\tilde{\mathbf{B}}_{L}}{\mathrm{maximize}}\min_{\underset{d_{1},\ldots,d_{L}}{(\mathbf{R}_{1},\ldots,\mathbf{R}_{K}})\in\mathcal{R}}\sum_{\ell}d_{\ell}(1-b_{\ell})+2e_{\ell}^{\textsf{T}}\boldsymbol{\beta}_{\ell}-\operatorname*{Tr}(\mathbf{B}_{\ell}\mathbf{J}).$

where $\mathcal{R}$ denotes the constraints (11b)-(11c). By minimizing the
above over $d_{\ell}$, we conclude $b^{*}_{\ell}=1$ for all $\ell$. Then, the
dual problem becomes

$\displaystyle\underset{\mathbf{B}_{\ell}\succcurlyeq\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}},\forall\ell}{\mathrm{maximize}}~{}~{}\min_{(\mathbf{R}_{1},\ldots,\mathbf{R}_{K})\in\mathcal{R}}\sum_{\ell}2e_{\ell}^{\textsf{T}}\boldsymbol{\beta}_{\ell}-\operatorname*{Tr}(\mathbf{B}_{\ell}\mathbf{J}),$
(15)

where the condition
$\mathbf{B}_{\ell}\succcurlyeq\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}$
follows because $\tilde{\mathbf{B}}_{\ell}\succcurlyeq 0$ and $b_{\ell}=1$, so
the Schur complement must satisfy
$\mathbf{B}_{\ell}-\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\succcurlyeq
0$. Now, since strong duality holds, the primal optimal
$\mathbf{R}^{*}_{1},\ldots,\mathbf{R}^{*}_{K}$ and dual optimal
$\mathbf{B}_{1}^{*},\ldots,\mathbf{B}^{*}_{L},\boldsymbol{\beta}^{*}_{1},\ldots,\boldsymbol{\beta}^{*}_{L}$
is a saddle-point solution of the max-min problem. By interchanging $\min$ and
$\max$, we can optimize over $\mathbf{B}_{\ell}$. In particular, for any fixed
$\mathbf{R}_{1},\ldots,\mathbf{R}_{K}$, we have

$\mathbf{B}_{\ell}\succcurlyeq\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\Rightarrow\operatorname*{Tr}(\mathbf{B}_{\ell}\mathbf{J})\geq\operatorname*{Tr}(\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{J}).$
(16)

which holds because $\mathbf{J}$ is positive definite. Thus, the optimal
$\mathbf{B}_{\ell}$ must be
$\mathbf{B}_{\ell}=\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}^{\textsf{T}}_{\ell}$
at the saddle-point. In other words, the saddle-point in (15) can be
alternatively written as

$\displaystyle\underset{\boldsymbol{\beta}\in\mathbb{R}^{L\times
L}}{\mathrm{maximize}}~{}~{}\min_{{(\mathbf{R}_{1},\ldots,\mathbf{R}_{K})\in\mathcal{R}}}\sum_{\ell}\left(2e_{\ell}^{\textsf{T}}\boldsymbol{\beta}_{\ell}-\
\operatorname*{Tr}(\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{J})\right).$
(17)

We now show that there exists a rank-one solution in which
$\operatorname*{rank}(\mathbf{R}_{1}^{*})=\cdots=\operatorname*{rank}(\mathbf{R}_{K}^{*})=1$.
To see this, note that based on (4), we can rewrite

$\sum_{\ell}\operatorname*{Tr}(\boldsymbol{\beta}_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{J})=\sum_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{J}\boldsymbol{\beta}_{\ell}=\sum_{\ell}\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{C}\boldsymbol{\beta}_{\ell}+\sum_{k}\operatorname*{Tr}(\mathbf{Q}_{\boldsymbol{\beta}}\mathbf{R}_{k}),$
(18)

where $\mathbf{Q}_{\boldsymbol{\beta}}$ is as defined in the theorem
statement. In other words, for any fixed $\boldsymbol{\beta}$, the inner
minimization in (17) has the form of the SDR of a problem (8). But the SDR of
(8) is tight by results of [20]. This means that there exists a set of rank-
one solution $\mathbf{R}_{1}^{*},\ldots\mathbf{R}_{K}^{*}$, i.e.,
$\mathbf{R}_{k}^{*}=\mathbf{v}_{k}^{*}{\mathbf{v}_{k}^{*}}^{\textsf{H}}$, for
the inner minimization problem in (17), for any $\boldsymbol{\beta}$ and in
particular for $\boldsymbol{\beta}=\boldsymbol{\beta}^{*}$. This implies that
(11), the SDR of (6), is equivalent to (6) itself, and
$\mathbf{V}^{*}\triangleq[\mathbf{v}_{1}^{*},\ldots,\mathbf{v}_{K}^{*}]$ is
the solution of the original ISAC problem (6). This shows that problem (6) is
equivalent to problems (9) and (10). ∎

Theorem 1 allows us to express the ISAC problem (6) in a more tractable form.
In addition, it offers the following interesting interpretation. Consider the
max-min problem (10). By construction, the matrix
$\mathbf{Q}_{\boldsymbol{\beta}}$ is PSD, so the trace term in (10) can be
interpreted as a sum of beamforming powers measured with respect to the matrix
$\mathbf{Q}_{\boldsymbol{\beta}}$. Thus, for a fixed
$\boldsymbol{\boldsymbol{\beta}}$, the inner problem in (10) can be viewed as
maximizing the beamforming power at certain point of interest. Changing
$\boldsymbol{\boldsymbol{\beta}}$ corresponds to changing the point of
interest. Theorem 1 essentially states that there exists a judicious choice
$\boldsymbol{\boldsymbol{\beta}}^{*}$ for which the problem of minimizing the
CRB is equivalent to that of maximizing the power along a certain direction.

##  IV Uplink-Downlink Duality for ISAC

We now show that the ISAC problem admits an alternative formulation in terms
of an uplink problem. The reason that an UL-DL duality relation exists for
ISAC beamforming is that problems (7) and (8) are closely related. On one
hand, problem (8) (or the ISAC version when
$\mathbf{Q}=\mathbf{Q}_{\boldsymbol{\beta}^{*}}$) seeks to maximize the
transmit power in certain directions under a total power constraint. On the
other hand, problem (7) aims to minimize the transmit power. Consider the
inner problem in (10) for a given $\boldsymbol{\beta}$. We introduce the
Lagrangian function

$\mathcal{L}_{\lambda,\boldsymbol{\beta}}\left(\mathbf{V}\right)=\sum_{k}\mathbf{v}_{k}^{\textsf{H}}\left(\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\right)\mathbf{v}_{k}-\lambda
P+\sum_{\ell=1}^{L}2\boldsymbol{\boldsymbol{\beta}}_{\ell}^{\textsf{T}}\mathbf{e}_{\ell}-\boldsymbol{\beta}_{\ell}^{\textsf{T}}\mathbf{C}\boldsymbol{\beta}_{\ell}$

which gives rise to the following dual problem

$\displaystyle\underset{\boldsymbol{\beta},\lambda\geq
0}{\mathrm{maximize}}~{}~{}~{}\min_{\mathbf{V}}~{}~{}~{}~{}~{}$
$\displaystyle\mathcal{L}_{\lambda,\boldsymbol{\beta}}\left(\mathbf{V}\right)$
(19a) $\displaystyle\mathrm{s.t.\ }~{}~{}~{}~{}~{}$
$\displaystyle\text{SINR}^{\text{DL}}_{k}(\mathbf{V})\geq\gamma_{k},~{}~{}\forall
k.$ (19b)

Because strong duality holds for the inner problem in (10), an optimal
beamforming solution of problem (10) (or problem (6)) must also be a minimizer
of $\mathcal{L}_{\lambda^{*},\boldsymbol{\beta}^{*}}\left(\cdot\right)$.

Now, consider the inner problem in (19) for fixed
$(\lambda,\boldsymbol{\beta})$:

$\displaystyle\underset{\mathbf{V}}{\mathrm{minimize}}~{}~{}~{}~{}$
$\displaystyle\sum_{k}\mathbf{v}_{k}^{\textsf{H}}\left(\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\right)\mathbf{v}_{k}$
(20a) $\displaystyle\mathrm{subject\;to}~{}~{}~{}$
$\displaystyle\text{SINR}^{\text{DL}}_{k}(\mathbf{V})\geq\gamma_{k},~{}~{}\forall
k.$ (20b)

For suitable choice of $\lambda$ and $\boldsymbol{\beta}$, this problem can be
viewed as a downlink communication problem, similar to the classical problem
(7). For example, if
$\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\succcurlyeq 0$, then (20a)
represents a sum of beamforming powers measured with respect to
$\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}$. A natural question is the
following. For what values of $(\lambda,\boldsymbol{\beta})$ does problem (20)
admit a similar interpretation as problem (7)? This motivates the following
definition.

###### Definition 1

The pair $\lambda,\boldsymbol{\beta}$ is said to be admissible if
$\sum_{k}\mathbf{v}_{k}^{\textsf{H}}\left(\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\right)\mathbf{v}_{k}\geq
0$ for all $\mathbf{V}$ satisfying (20b). ∎

Admissibility of $\lambda,\boldsymbol{\beta}$ implies that the matrix
$\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}$ behaves as a “weak" PSD
matrix. It may not be PSD globally but behaves like one over the feasible
region of SINRs. This also implies that the objective function (20a) is a
convex function when restricted to $\mathbf{v}_{k}$’s that satisfy the set of
SINRs, but can be nonconvex outside such set.

Now, let $\mathcal{A}$ be the set of all admissible pairs. The next result
establishes an UL-DL relation for problem (20) if
$(\lambda,\boldsymbol{\beta})\in\mathcal{A}$.

###### Theorem 2

Problem (20) is bounded below if and only if
$(\lambda,\boldsymbol{\beta})\in\mathcal{A}$. In addition, whenever
$(\lambda,\boldsymbol{\beta})\in\mathcal{A}$, the matrix

$\mathbf{V}^{*}_{\lambda,\boldsymbol{\beta}}\triangleq\mathbf{U}^{*}_{\lambda,\boldsymbol{\beta}}\mathbf{p}_{\lambda,\boldsymbol{\beta}}^{*},$
(21)

is an optimal solution of the problem. Here,
$\mathbf{p}^{*}_{\lambda,\boldsymbol{\beta}}\triangleq\sigma^{2}(\mathbf{D}_{\mathbf{U}_{\lambda,\boldsymbol{\beta}}^{*}}-\mathbf{F}_{\mathbf{U}_{\lambda,\boldsymbol{\beta}}^{*}})^{-1}\mathbf{1}\in\mathbb{R}^{K}$
is a vector of downlink powers and
$\mathbf{U}^{*}_{\lambda,\boldsymbol{\beta}}\in\mathbb{C}^{N_{\text{T}}\times
K}$ is a matrix of normalized beamforming directions obtained by solving the
uplink problem

$\displaystyle\underset{\mathbf{U},\;\mathbf{z}}{\mathrm{minimize}}~{}~{}$
$\displaystyle\sigma^{2}\mathbf{1}^{\textsf{T}}\mathbf{z}$ (22a)
$\displaystyle\mathrm{subject\ to}~{}$
$\displaystyle\frac{z_{k}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{u}\right|^{2}}{\sum_{i\neq
k}z_{i}\left|\mathbf{h}_{i}^{\textsf{H}}\mathbf{u}\right|^{2}+\mathbf{u}^{\textsf{H}}\left(\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\right)\mathbf{u}}\geq\gamma_{k},$
(22b)
$\displaystyle\rho_{\text{max}}\left(\mathbf{D}_{\mathbf{U}}^{-1}\mathbf{F}_{\mathbf{U}}\right)<1,$
(22c) $\displaystyle\|\mathbf{u}_{k}\|_{2}=1,\quad\forall k.$ (22d)

In problem (22),
$\mathbf{U}\triangleq\left[\mathbf{u}_{1},\ldots,\mathbf{u}_{K}\right]$
represents a matrix of combining vectors and
$\mathbf{z}\triangleq\left[z_{1},\ldots,z_{K}\right]^{\textsf{T}}\in\mathbb{R}^{K}$
is a vector of uplink powers. Finally, $\rho_{\text{max}}(\cdot)$ denotes the
spectral radius of a matrix,
$\mathbf{D}_{\mathbf{U}}\triangleq\operatorname*{diag}\left(\tfrac{|\mathbf{h}_{1}^{\textsf{H}}\mathbf{u}_{1}|^{2}}{\gamma_{1}},\ldots,\tfrac{|\mathbf{h}_{K}^{\textsf{H}}\mathbf{u}_{K}|^{2}}{\gamma_{K}}\right)\in\mathbb{R}^{K\times
K}$ is a diagonal matrix and $\mathbf{F}_{\mathbf{U}}\in\mathbb{R}^{K\times
K}$ is a matrix defined by

$\displaystyle\left[\mathbf{F}_{\mathbf{U}}\right]_{ij}\triangleq$
$\displaystyle$0$,$ for $i=j$ (23a)
$\displaystyle\left[\mathbf{F}_{\mathbf{U}}\right]_{ij}\triangleq$
$\displaystyle|\mathbf{h}_{i}^{\textsf{H}}\mathbf{u}_{j}|^{2},$ for $i\neq j$
(23b)

###### Proof:

The idea of the proof is rooted in the classic result of [24]. We omit the
details due to space limitations. ∎

Theorem 2 states that whenever the pair
$\left(\lambda,\boldsymbol{\beta}\right)$ is admissible, the downlink solution
of problem (20) can be recovered by solving the uplink problem (22). More
specifically, the optimal combiners for the uplink problem (22) are also
optimal as beamforming directions for the downlink problem (20).

Note that problem (22) differs from the standard uplink formulation, e.g.,
[11], in that it has the additional constraint (22c). This constraint arises
because the matrix $\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}$ does
not need to be PSD in a global sense, and we only need
$(\lambda,\boldsymbol{\beta})\in\mathcal{A}$. If such matrix is already PSD
then a simple proof [24] reveals that
$\rho_{\text{max}}\left(\mathbf{D}_{\mathbf{U}}^{-1}\mathbf{F}_{\mathbf{U}}\right)<1$
is equivalent to $\mathbf{z}\geq 0$. In which case, problem (22) reduces to
the conventional uplink problem.

Observe that $(\lambda^{*},\boldsymbol{\beta}^{*})$ must be admissible since
problem (20) is unbounded below whenever
$(\lambda,\boldsymbol{\beta})\not\in\mathcal{A}$, but the BCRB can never be
negative.

##  V Proposed Algorithm

The previous uplink interpretation is useful in devising efficient algorithms
for solving the ISAC problem (6). Let us first focus on solving the uplink
problem (22). If problem (22) is given by the classical formulation (i.e.,
with constraint (22c) replaced by $\mathbf{z}\geq 0$), the iterative procedure
of [10] would recover an optimal uplink solution. Such iterations alternate
between maximum SINR combining for fixed $\mathbf{z}$

$\mathbf{u}_{k}[n]\in\arg\min_{\mathbf{u}\neq 0}\frac{\sum_{i\neq
k}z_{i}[n]\left|\mathbf{h}_{i}^{\textsf{H}}\mathbf{u}\right|^{2}+\mathbf{u}^{\textsf{H}}\left(\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\right)\mathbf{u}}{\tfrac{1}{\gamma_{k}}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{u}\right|^{2}},$
(24)

and a power control update for fixed $\mathbf{U}$

$~{}z_{k}[n+1]=\frac{\sum_{i\neq
k}z_{i}[n]\left|\mathbf{h}_{i}^{\textsf{H}}\mathbf{u}_{k}[n]\right|^{2}+{\mathbf{u}^{\textsf{H}}_{k}[n]}\left(\lambda\mathbf{I}-\mathbf{Q}_{\boldsymbol{\beta}}\right)\mathbf{u}_{k}[n]}{\tfrac{1}{\gamma_{k}}\left|\mathbf{h}_{k}^{\textsf{H}}\mathbf{u}_{k}[n]\right|^{2}}.$
(25)

It turns out that the same procedure, when initialized properly, can also
recover the optimal solution of problem (22).

###### Theorem 3

Let $\mathbf{z}^{*}_{\lambda,\boldsymbol{\beta}}$ and
$\mathbf{U}^{*}_{\lambda,\boldsymbol{\beta}}$ denote an optimal uplink
solution of problem (22). If $(\lambda,\boldsymbol{\beta})\in\mathcal{A}$,
then starting from
$\mathbf{z}[0]\geq\mathbf{z}^{*}_{\lambda,\boldsymbol{\beta}}$, we have
$\mathbf{z}[n]\rightarrow\mathbf{z}^{*}_{\lambda,\boldsymbol{\beta}}$ and
$\mathbf{U}[n]\rightarrow\mathbf{U}^{*}_{\lambda,\boldsymbol{\beta}}$.

###### Proof:

Omitted due to space limitations. ∎

Note that the iterations converge to an uplink solution if
$\left(\lambda,\boldsymbol{\beta}\right)\in\mathcal{A}$. If the pair is
inadmissible, the decreasing sequence $\\{\mathbf{z}[n]\\}$ would fail to
converge, which can be detected by checking if the sequence falls below zero.

After solving the uplink problem (22) for fixed
$\left(\lambda,\boldsymbol{\beta}\right)$, a downlink solution can be obtained
using (21). The problem of finding the optimal pair
$\left(\lambda^{*},\boldsymbol{\beta}^{*}\right)$ can be solved iteratively as
an outer maximization of problem (19), where each iteration involves solving
an inner subproblem of the form (20) (see [7]), along with a projection
subgradient method onto $\mathcal{A}$ in the outer problem. Unfortunately, a
projection operator on $\mathcal{A}$ cannot be readily obtained. Instead, we
propose a simple scheme that works quite well. The idea is to update the
parameters using the subgradient method (without projection onto
$\mathcal{A}$) to obtain a tentative
$(\tilde{\lambda},\tilde{\boldsymbol{\beta}})$. If the tentative parameters
are inadmissible, we reduce the step size by a factor and repeat the process
until an admissible pair is found. When this simple scheme converges to the
optimal pair, it would find the optimal beamformers
$\mathbf{V}^{*}_{\lambda^{*},\boldsymbol{\beta}^{*}}$ of the ISAC problem (6).

##  VI Simulations

In this section, we examine the performance of the proposed beamforming
solution for the angle of arrival problem in [5] against that of the SDR
scheme. In this case, we set $N_{\text{T}}=N_{\text{R}}=20$ antennas and $K=2$
users with SINR thresholds of $10$ and $12$ dB. We model the communication
channels as line-of-sight with angles $-30^{\circ}$ and $50^{\circ}$. We place
the target at $0^{\circ}$ with unit path gain and choose a Gaussian prior with
mean equal to the true values.

In Figure 2, we plot the array response of the beamformers produced by both
algorithms. We consider two distinct cases for the standard deviation of the
prior distribution of the angle $\sigma=2.5^{\circ}$ and $10^{\circ}$. In both
cases, it is seen that the array response of the beamformer obtained by the
proposed algorithm is identical to its SDR counterpart. However, the proposed
method requires much a lower computational complexity. This highlights the
benefits of the propose solution. We additionally observe that both algorithms
form beams in the direction of communication users and the sensing target, but
with a wider sensing beam when the variance of the prior is larger. Finally,
we also plot the beam pattern for the beamformer that solves the classical
communication problem (7) without considering sensing. We see that such
beamformer can focus the beam towards the communication users but is clearly
oblivious to the sensing target.

Figure 2: Beam pattern of the proposed solution vs SDR for optimizing the
angle CRB. Here, we set $N_{\text{T}}=N_{\text{R}}=20$ and $K=2$.

##  VII Conclusion

This paper develops a novel optimization framework for beamforming design in
ISAC systems. Such framework is a consequence of the observation that a
general optimization of a BCRB objective can be viewed as a maximization of
beamforming power along with a set of auxiliary variables. We leverage this
observation to develop an UL-DL duality for beamforming optimization in ISAC
systems. This duality result allows the problem to be solved in the
beamforming space at much lower computational complexity as compared to the
existing SDR methods.

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