Deep Conditional Generative Adversarial Networks for Efficient Channel Estimation in AmBC Systems

In ambient backscatter communication (AmBC), battery-free devices (tags) harvest energy from ambient radio frequency (RF) signals and communicate with readers. Although reliable channel estimation (CE) is critical, classical pilot-based estimators tend to perform poorly. To address this challenge, we treat CE as a denoising problem using conditional generative adversarial networks (CGANs). A three-dimensional (3D) denoising block leverages spatial and temporal characteristics of pilot signals, considering both real and imaginary components of channel matrices. The proposed CGAN estimator is extensively evaluated against traditional estimators like minimum mean-squared error (MMSE), least squares (LS), convolutional neural network (CNN), CNN-based deep residual learning denoiser (CRLD), and blind estimation. Simulation results show 82% gain of the proposed estimator over CRLD and MMSE estimators at an SNR of 5 dB. Moreover, it has advanced learning capabilities and accurately replicates complex channel characteristics.

SECTION I. Introduction

The recent proliferation of connected devices has led to a substantial surge in spectrum utilization and energy consumption [1]. The Internet of Things (IoT) is one of the driving forces behind this growth. However, logistical and environmental challenges for potentially billions of IoT devices, such as battery replacements and frequent recharging, are formidable. These challenges impede the energy and cost efficiencies necessary for the success and sustainability of this advanced technology [2], [3], [4].

As a solution, ambient backscatter communication (AmBC) has garnered considerable attention. AmBC achieves spectral and energy efficiencies by employing low-power and inexpensive tags that harvest and reflect ambient radio frequency (RF) signals, like Wi-Fi or TV tower signals, rather than generating their own RF signals [5], [6], [7], [8]. Thus, tags eliminate the need for power-intensive components, such as oscillators, mixers, or amplifiers, resulting in minimal power consumption. Consequently, tags achieve power consumption levels 1000 times below that of active mobile devices [2]. These maintenance-free tags offer the advantage of collecting and backscattering data across diverse environmental conditions, from extreme settings like high-pressure or toxic environments to more moderate conditions like farmlands [9]. They find widespread applications in industrial automation, smart logistics, and tracking numerous parcels [3], [10]. Additionally, they are employed in smart agriculture for soil monitoring and livestock tracking, highlighting their versatility. Hence, AmBC represents an energy-efficient future wireless solution.

However, achieving high data rates and reliability requires highly accurate channel estimation (CE) [11]. This problem has different challenges than for CE for conventional wireless links [12], [13], [14], [15], [16], [17]. The first challenge is the existence of two distinct AmBC channel types. Type one is the direct channel from the RF source to the reader, akin to traditional wireless channels. This link is also termed direct link interference (DLI). The reader can eliminate DLI by decoding the symbols transmitted by the RF source. This may be possible in symbiotic radio, where a cooperative receiver merges the reader with the primary user to decode both user and tag signals [18]. The second type involves the cascaded (dyadic) channel, encompassing signal propagation from the RF source to the tag and from the tag to the reader, exhibiting deeper fades compared to one-way wireless links [19], [20], [21]. Furthermore, this channel encounters a double pathloss, significantly reducing received signal power, which impairs the reliability of CE. The CE algorithms will thus achieve different accuracy levels for these two types of channels. Secondly, since a passive tag cannot transmit RF signals independently, the generation of pilot signals for CE requires the cooperation of the RF source [14].

Thus, advanced machine learning algorithms to improve resource allocation and CE for AmBC networks are receiving much attention [22], [23]. Both classical and deep learning-based estimators are available – Table 1. Nevertheless, this problem has not attracted conditional generative adversarial networks (CGANs). CGANs utilize a conditional generation of images by two adversarial networks [24], [25]. The unique absence of CGANs for AmBC CE motivates us to develop and apply them to address this challenge. However, before proceeding, we review existing CE methods (Table 1), including both classical techniques and those grounded in deep learning principles.

TABLE 1 Summary of Existing AmBC Channel Estimators

A. Classical Channel Estimators for AmBC

These estimators can be broadly classified into pilot-based, blind, and semi-blind [12], [13], [14], [15], [16], [17].

1. Pilot-based estimators: These use known pilot signals to estimate channel characteristics, providing structured and predictable analysis [5]. These are widely used, tried, and tested in wireless networks. The most common algorithms are the least squares (LS) and minimum mean-squared error (MMSE) estimators and their variants. Accordingly, the tag transmits pilot symbols, and the reader utilizes those reflected pilot symbols for CE [11], [14], [27]. LS, MMSE, and ML estimators rely on pilot symbols. For instance, the studies in [14] and [27] assign four pilots to the tag frame, which represents 0.25% of the total transmission frame. This allocation ensures that there are still enough information bits for data transmission. Reference [11] introduces a pilot-based space alternating generalized expectation maximization (SAGE) algorithm for joint CE, interference suppression, and data detection.

2. Blind estimators: On the other hand, blind CE techniques rely on the statistical properties of the received signal, eliminating the need for pilot symbols. However, they have poor mean-squared error (MSE) performance [28]. An expectation-maximization (EM)-based estimator is proposed in [16] to obtain the absolute values of the AmBC channel parameters. However, its performance is limited by insufficient knowledge of the RF source. Despite these advancements, a significant performance gap remains when compared to the optimal MMSE and blind estimators, indicating the need for further research and development.

3. Semi-blind estimators: These are a combination of pilot-based and blind estimation techniques, utilizing limited pilot data alongside statistical analysis of received signals for a balanced approach [5]. The semi-blind receiver in [26] features a channel estimator using sample mean or second-order moment estimators.

Moreover, study [15] presents three solutions: a pilot-based maximum likelihood (ML) estimator, a semi-blind estimator using the EM method, and another semi-blind estimator using the decision-directed (DD) approach, balancing between ML and EM.

B. Deep Learning Estimators for AmBC

Deep learning and other machine learning algorithms can uncover patterns in large datasets that traditional methods often miss. By leveraging these hidden patterns, such algorithms can significantly enhance the accuracy of CE. In [29], deep transfer learning (DTL) is introduced in a multi-antenna AmBC system to extract data features using a convolutional neural network (CNN). In [14], the AmBC CE task is modeled as a denoising problem. This work develops the CNN-based deep residual learning denoiser (CRLD) to directly extract channel coefficients from noisy pilot signals.

More specifically, Table 2 highlights the pros and cons of classical and deep learning-based AmBC channel estimators.

TABLE 2 Pros and Cons of Classical Versus Deep Learning-Based CE Methods

C. What are CGANs?

CGAN extends the traditional GAN [24], [25]. A standard GAN uses unsupervised learning where a generator network creates synthetic data, and a discriminator network tries to distinguish between real and generated data. The generator aims to produce indistinguishable data from real data, while the discriminator strives to classify whether the input is real or generated incorrectly. GANs have thus been widely employed in wireless research. For example, [30] introduces an end-to-end system utilizing deep neural networks (DNNs) for encoding, decoding, modulation, and demodulation. A CGAN models channel effects, enhancing the system’s effectiveness across various channel types and addressing the dimensionality challenge in long transmit sequences.

As described by [24] and [25], the GAN training is inherently adversarial, promoting a dynamic wherein enhancements in one component invariably impose challenges upon the other. For instance, a GAN variant is utilized in [31] to estimate extremely large multiple-input multiple-output (MIMO) channels to address the unique challenges of the near-field propagation environment, demonstrating enhanced estimation accuracy. Nonetheless, the intrinsic mechanism of GANs, which involves the conversion of noise into data, is often plagued by instability and randomness. This can lead to notable inconsistencies in output.

In contrast, the generator and the discriminator in a CGAN receive additional information, known as conditioning variables, alongside the random noise or latent vector. This conditioning information guides the generation process, allowing for more controlled and targeted data generation with the desired characteristics of the generated data. Additional information, e.g., class labels, is concurrently fed into the generator and discriminator as an auxiliary input layer, a concept further illustrated by [32] and [33]. The integration of class labels catalyzes image generation’s stability, velocity, and quality during the training phase. It also enables the targeted fabrication of images corresponding to specific categories or labels.

CGANs markedly improve the performance of AmBC CE by effectively simulating complex channel conditions that traditional estimators struggle to replicate. CGAN networks excel at mapping conditional inputs, such as received signals and pilots, onto practical outputs like channel matrices. This capability not only enhances the stability and predictability of the model but also increases the accuracy of CE [30]. CGANs provide a more precise and reliable framework for adapting to AmBC channels by mimicking real-world conditions during the generative process. Our proposed model treats channel covariance matrices as two-dimensional (2D) images, which can be predicted using CGANs.

D. Main Contributions

The main contribution of this paper is to develop an AmBC CE algorithm based on a CGAN. Our proposed CGAN estimator comprises a generator, discriminator, and other components (Fig. 1). Here is a brief description of each component.

1. Dataset Z: The dataset Z includes actual wireless channel measurements, which trains the model to generate channels that closely mimic real-world conditions. The quality of channels produced by the generator during online testing depends on the precision of the data in Z.

2. Generator: It takes input Y (received signal at the reader) and generates a data instance $\hat {\mathbf {Z}}$ (estimated channel). Our generator utilizes the CRLD architecture [14], and it is capable of handling complex channel coefficients (Section IV-A).

3. Discriminator: The discriminator uses a CNN architecture. It takes two sets of inputs: the real data and the generated data. The discriminator is trained to distinguish between real and fake data instances (Section IV-A).

4. CGAN loss: The training process involves a min-max game where the generator tries to minimize this loss by generating a more realistic channel. In contrast, the discriminator tries to maximize its ability to distinguish real from fake channels (Section IV-B).

5. $\mathcal {L}_{2}$ loss: The MSE loss function is used to determine the disparity between generated and real data. It ensures that the generated data not only deceives the discriminator but also closely resembles the real data. By combining the CGAN Loss with the $\mathcal {L}_{2}$ Loss, a composite loss function is formed to enhance the authenticity of the generated data (Section IV-B).

FIGURE 1.

Depiction of the proposed CGAN estimator.

The contributions of this paper are summarized as follows.

• For the first time, a CGAN-based AmBC channel estimator is developed. The CGAN is trained to recognize patterns and variations in actual channel data. Conditioning the GAN on specific channel state information (CSI) generates synthetic channel data that closely resembles real-world conditions. This synthetic data, in turn, facilitates precise estimation of the AmBC channel coefficients.

• The adaptability of our estimator to varying channel conditions is a noteworthy feature, which is necessary when channel conditions are continually changing. For example, at a signal-to-noise ratio (SNR) of 5 dB, the CGAN records a normalized MSE (NMSE) of 0.017 for direct channel link (h), markedly superior to the NMSE of 0.094 demonstrated by both the CRLD and MMSE estimators. Similarly, for the cascade channel link (w), the CGAN achieves an NMSE of 0.013, surpassing the CRLD and MMSE estimators, whose NMSE stands at roughly 0.072.

• Other estimators, e.g., CRLD in [14], often fall short in certain aspects. For instance, it is limited to the real part of channel coefficients, whereas our dataset encapsulates real and imaginary parts. Accordingly, we formulate a unique hybrid training strategy, combining the strengths of classical CE techniques with the generative potential of CGANs. Our approach uniquely models multi-modal channel distributions, distinguishing them from [34], [35], and [36] focusing mainly on tag signal detection, overlooking CSI acquisition.

• The efficacy of the CGAN estimator is assessed against prominent estimators, including LS, MMSE, blind, CRLD, and CNN. Our model demonstrates superior performance in NMSE, even in challenging low SNR environments. Section VI reveals that it consistently outperforms existing CE solutions with gains of 82% over CRLD and MMSE estimators when $\text {SNR}=5$ dB.

Notations: Vectors and matrices are represented by boldface lowercase letters (e.g., a) and uppercase letters (e.g., A), respectively. For a square matrix A, the following notations are used: $\mathbf {A}^{H}$ for the Hermitian conjugate transpose and $\mathbf {A}^{H}$ for the transpose. $\mathbb {R}^{M\times N}$ represents $M\times N$ dimensional real matrices and $\mathbb {C}^{M\times N}$ refers to $M\times N$ dimensional complex matrices. The $(i, j)$ -th element of matrix A is denoted by $[\mathbf {A}]_{ij}$ . Euclidean norms of complex vectors and absolute values of complex scalars are represented by $\|\cdot \|$ and $|\cdot |$ , respectively. The expectation operator is denoted by $\mathbb {E}[\cdot]$ . A circularly symmetric complex Gaussian (CSCG) random vector with mean $\boldsymbol {\mu }$ and covariance matrix C is represented as $\sim \mathcal {C}\mathcal {M}(\boldsymbol {\mu }, \mathbf {C})$ . The symbol $\mathbf {I}_{M}$ represents an identity matrix of dimensions $M \times M$ . Operation $(\cdot)^{-1}$ refers to the inversion of a matrix. Notations $\text {Re}(\cdot)_{i,j}$ and $\text {Im}(\cdot)_{i,j}$ specify real and imaginary parts of the element in the i-th row and the j-th column, respectively. Moreover, $\max (a, b)$ denotes the greatest value of a and b. $\mathbb {1}(\cdot)$ signifies the indicator function. Big-O notation, $\mathcal {O}$ , indicates an upper bound on the growth rate of a function.

This paper is structured as follows. The AmBC system is described in Section II. The CRLD estimator is discussed in Section III, and CGAN as an alternative CE method is introduced in Section IV. Section V reviews several popular traditional CE methods. Comprehensive simulation results are presented in Section VI. The paper concludes with a summary and key insights in Section VII.

SECTION II. AmBC System Model

The system in Fig. 2 comprises an ambient RF source, a tag, and a reader. The source and the tag are equipped with a single antenna. However, the reader is equipped with an M-element antenna array. The ambient RF source transmits signals to both the reader and the tag.

FIGURE 2.

AmBC system model with the multiple-antenna reader and its application in warehouse environments.

A. Tag Description

Upon receiving the ambient RF signal, the tag modulates and reflects it to the reader. The function of the reader is to detect the tag data. The strong direct link interference from the ambient RF source affects the detection process. The tag’s modulation and energy harvesting (EH) processes are described next [21], [37].

1. Backscatter modulation: A tag, a passive device without active electronics, opportunistically reflects ambient RF signals to transmit data. This process is known as load modulation, where the tag adjusts its load impedance based on the bits to be sent. The simplest case is that the tag performs on-off keying (OOK) modulation by switching between two load impedances to represent “0” or “1” by matching or mismatching the antenna impedance, respectively. The reflection coefficient of the tag is expressed as $$\begin{equation*} \Gamma _{i} = \frac {Z_{i}-Z_{a}^{\star }}{Z_{i}+Z_{a}}, \tag {1}\end{equation*}$$ where $Z_{a}$ denotes the antenna impedance of the tag and $Z_{i}$ is the load impedance of state $i = \{1, 2\}$ [5]. In addition, the reflection coefficients of impedance values have a constant magnitude, i.e., $|\Gamma _{i}|^{2}=|\Gamma |^{2}=\xi \in (0, 1$ ], where $\xi$ denotes the power reflection coefficient at the tag satisfying $0 \leq \xi \leq 1$ [12], [13].

2. Energy harvesting: Each tag performs EH and data transmission concurrently [38], [39]. Thus, the tag splits the received RF signal power into two parts based on a power splitting ratio, $\xi$ . To effectively model EH circuits, a linear model is commonly assumed for simplicity. In this model, the harvested direct current (DC) power is a linear function of the input power, represented as $P_{h}=\eta \xi P_{r}$ , where $\eta \in (0,1$ ] denotes the power conversion efficiency, which can be as high as 40% and 30% in photovoltaic and piezoelectric harvesters [40]. However, due to non-linear components like rectifiers or capacitors, the actual output power may behave non-linearly (e.g., −20 dBm for commercially available passive RFID tags [41]), necessitating more complex models to predict the harvested power accurately. For more details, please see [21] and references. While a brief overview of EH is provided for context, the paper primarily focuses on the CE algorithm.

3. Synchronization: The tags are usually synchronized with the incoming RF signal and very small time misalignment [42]. As demonstrated in [43], the tag performs code-word translation on ambient 802.11b packets, embedding its data by altering the phase of each symbol in the packet. Despite potential minor synchronization errors, the backscattered signal retains the 11-bit barker code and can be successfully decoded by a reader. Although synchronization is crucial for low-power and battery-free IoT devices, this paper primarily focuses on CS.

B. Channel Modeling

Backscatter channels exhibit harsher characteristics than conventional communication channels, primarily due to the double-pathloss phenomenon caused by the cascade of channels from the transmitter to the tag and from the tag to the receiver. This phenomenon increases the risk of severe deep fades, potentially resulting in communication outages and higher bit-error rates (BERs) [13], [16], [17]. A rich-scattering propagation model is considered in this study. In this case, channel coefficients can be modeled as zero-mean CSCG random variables. This assumption aligns with the small-scale Rayleigh flat fading model with a predefined coherence time. It is the most commonly used model in AmBC studies for environments without a dominant line-of-sight (LoS) component [44]. Such conditions can be found in settings like factories with obstructions and moving objects causing signal fluctuations or in scenarios involving storage or transportation, where tags may encounter multipath propagation due to reflections.

The considered system comprises three communication channels: the source-to-reader channel ($\mathbf {h}_{sr}$ ), the tag-to-reader channel ($\mathbf {h}_{tr}$ ), and the source-to-tag channel ($h_{st}$ ). The source-to-reader channel is represented by $\mathbf {h}_{sr}=[h^{1}_{sr}, h^{2}_{sr}, {\dots }, h^{M}_{sr}]^{T}$ . Each component of this vector, $h^{m}_{sr} \in \mathbb {C}$ , denotes the channel coefficient between the source and the m-th antenna of the reader. Similarly, the tag-to-reader channel is denoted by $\mathbf {h}_{tr}=[h^{1}_{tr}, h^{2}_{tr}, {\dots }, h^{M}_{tr}]^{T}$ , with each $h^{m}_{tr} \in \mathbb {C}$ representing the channel coefficient between the tag and the m-th antenna of the reader. This channel is responsible for carrying the reflected or backscattered signal from the tag to each antenna of the reader. The source-to-tag channel, $h_{st} \in \mathbb {C}$ , is generally dominated by a strong LoS component [14].

C. Signal Model

Traditionally, in wireless communications, signal models often employ complex Gaussian signals distinguished by their magnitude and phase components. Specifically, these signals are effectively represented as complex Gaussian random variables. Consequently, the signal transmitted by the source is denoted as $s(n) \sim \mathcal {CN}(0, P_{s})$ , adhering to a complex Gaussian distribution with a mean of zero and a power designated as $P_{s}$ . This assumption proves robust across a broad spectrum of communication systems, given that widely adopted modulation schemes, such as orthogonal frequency-division multiplexing (OFDM), tend to exhibit approximately white Gaussian characteristics in the time domain [45]. Additionally, it is customary to assume the independence of source signal samples from each other.

Consequently, the baseband signal received by the tag at the n-th sampling instance can be represented as $x(n) = h_{st}s(n)$ . The tag then reflects $x(n)$ based on its data. Thus, the backscattered signal from the tag can be represented as $x_{b}(n) = \Gamma x(n)$ where $\Gamma$ depends on the data to be sent (see Section II-A). Denote $\mathbf {y}(n) = [y_{1}(n), y_{2}(n), \ldots, y_{M}(n)]^{T}$ as the received signal at the reader, where $n \in \mathcal {N}=\{0, 1, \ldots, N_{T} - 1\}$ . In addition, $N_{T}$ represents the total number of samples within a single frame, and $m\in \mathcal {M}=\{1, 2, \ldots, M\}$ corresponds to the received sample from the m-th antenna element of the reader. Accordingly, the received sampling vector at the reader can be written as $$\begin{equation*} \mathbf {y}(n) = \left ({{\sqrt {P_{r}}\mathbf {h}_{sr} +\sqrt {P_{c}} \Gamma h_{st} \mathbf {h}_{tr}}}\right)s(n) + \mathbf {u}(n), \tag {2}\end{equation*}$$ where $\mathbf {u}(n) \sim \mathcal {CN}(0, \sigma _{u}^{2} \mathbf {I}_{M})$ represents additive white Gaussian noise (AWGN) and follows an independent and identically distributed (i.i.d.) complex Gaussian random vector with zero mean and variance $\sigma _{u}^{2}$ . The average power received directly at the reader is given by $P_{r}$ while $P_{c}$ | denotes the average power from the backscatter link at the reader.

The average received SNRs for both the direct link and the backscatter link can be defined as $\beta _{d} \triangleq \frac {P_{r}}{\sigma ^{2}_{u}}$ and $\beta _{b} \triangleq \frac {P_{c}}{\sigma ^{2}_{u}}$ , respectively [46]. Accordingly, the received power (dBm) at the reader is given by $$\begin{align*} P_{r} = P_{c} \!+\! v_{1} \log (d_{st}) \!+\! v_{2} \log (d_{tr}) \!-\! v_{3} \log (d_{sr}) \!-\! \log (\xi F G_{l}^{2}), \tag {3}\end{align*}$$ where $F = \frac {\lambda ^{2}}{(4\pi)^{2}}$ with $\lambda$ being the wavelength and $G_{l}$ is the tag’s antenna gain. Also, $d_{sr}$ , $d_{st}$ , and $d_{tr}$ denote the distances from the source to the reader, the source to the tag, and the tag to the reader, respectively. The pathloss exponents are given by $v_{i}$ , $i \in \{1,2,3\}$ . By defining the relative SNR as $\zeta \triangleq \frac {\beta _{b}}{\beta _{d}} = \frac {P_{c}}{P_{r}}$ , the received signal at the reader can be represented as $$\begin{equation*} y(n) = \left ({{\mathbf {h}_{sr}+ \sqrt {\zeta } \Gamma h_{st} \mathbf {h}_{tr}}}\right)s(n) + \mathbf {u}(n). \tag {4}\end{equation*}$$ A higher value of $\zeta$ indicates that the backscattered signal is stronger than the direct signal, which can improve the CE performance. Conversely, a higher value suggests that the direct signal is stronger, leading to increased interference and more challenging CE scenarios.

Also, note that the signal received in (2) can be alternatively expressed as $$\begin{align*} \mathbf {y}(n) = \begin{cases} \mathbf {w} s(n) + \mathbf {u}(n) & \text {if} \: \: \Gamma = 1, \\ \mathbf {h} s(n) + \mathbf {u}(n) & \: \text {if} \: \: \Gamma = 0, \end{cases} \tag {5}\end{align*}$$ where $\mathbf {w}= \mathbf {h}_{sr} + \Gamma h_{st} \mathbf {h}_{tr}$ and $\mathbf {h}= \mathbf {h}_{sr}$ . The CE goal is to estimate the vectors of channel coefficients: w for $\Gamma = 1$ and h for $\Gamma = 0$ .

As a result, we propose a simple yet effective AmBC CE protocol. We assume $T\gt 1$ frames, each of which shares an identical structural pattern. For a better understanding, a representative frame denoted as $t \in \mathcal {T}=\{1, 2, {\dots }, T\}$ , is illustrated in Fig. 3. It is divided into three phases: A, B, and C [14], [17]. We next describe these phases and their functionalities.

1. Estimation of h: The tag remains in a non-reflective state for a series of $N_{a}$ consecutive sampling periods, during which the reader estimates the value of h based on the $N_{a}$ pilot bits.

2. Estimation of w: The tag holds a reflective state for $N_{b}$ consecutive sampling periods, allowing the reader to estimate w utilizing the $N_{b}$ pilot bits.

3. Data transmission: During this phase, the tag transmits $N_{d}$ data by either reflecting or absorbing the RF signal.

FIGURE 3.

The frame architecture for the considered AmBC system.

Remark 1:

The chosen frame structure (Fig. 3) is designed for slow-fading channels, and this structure is prevalent in AmBC studies [29], [47]. However, fast-fading channels present a distinct challenge, with the neural network potentially encountering misalignment between its training and real-world operation, adversely affecting accurate CE. To address this, it may be necessary to redesign the frame structure and revise the network architecture. This could include implementing recurrent neural networks (RNNs) or attention mechanisms that excel at capturing the temporal dynamics of the channel [48], [49]. Such architectural improvements offer a robust response to the rapid variations characteristic of fast-fading environments and are a promising avenue for future exploration.

Our protocol uses phases A and B to generate pilot bits for CNN-based CRLD networks or CGANs. We assume ambient RF signals as known pilot sequences, consistently set to one to simplify the estimation process, as done in [14] and [50]. Indeed, the RF source is considered as an imaginary tag assigned with an all-1 pilot sequence, with $N_{a}, N_{b} \leq N_{d}$ for sufficient data transmission [50]. By setting all pilot signals to one [14], [29], the CE problem is converted into a denoising task. Thus, the main objective is retrieving z from a noisy observation. Thus, the signal model becomes $$\begin{equation*} \mathbf {y}(n) = \mathbf {z}+ \mathbf {u}(n), \tag {6}\end{equation*}$$ where z is either w or h.1 Although (6) resembles conventional communications, dyadic channels complicate CE.

Remark 2:

In AmBC, the pilot sequences are required for CE and the critical differentiation of weaker backscatter signals amidst stronger direct path interference, as demonstrated in systems like long-term evolution (LTE) [52]. The employment of structured encoding methods, such as Manchester encoding and Barker codes, and the adoption of orthogonal codes like Hadamard channelization codes, further reinforce this premise, addressing synchronization and demodulation challenges [53]. Also, the integration with symbiotic radio (SR) involves a symbiotic relationship between a tag and a primary transmission. This SR framework, in which the primary system concurrently supports and decodes its own and the tag’s transmissions, highlights the need for pilot sequences to improve synchronization and CE. This underscores the justification for incorporating pilot sequences in AmBC systems, even when dealing with unknown RF sources [54].

SECTION III. CE via CNN-Based CRLD

The CRLD approach is developed in [14]. It is designed to learn residual noise and restore channel coefficients from noisy pilot signals. This method is included here to serve several purposes. Firstly, understanding CRLD provides a benchmark for developing and comparing our primary CGAN-based strategy. Secondly, while [14] presents a basic real-valued model, we enhance it by introducing a complex-valued model. Lastly, our CGAN’s generator model is built upon the foundation of the CNN-based CRLD model in [14].

The CRLD leverages a CNN to enhance feature learning. Building on this foundation, it does not simply map a noisy channel matrix to a denoised channel matrix; instead, CRLD introduces a three-dimensional (3D) denoising block. This block is specifically designed to learn the residual noise both temporally and spatially from the noisy channel matrices, thereby enhancing the denoising process. We analyze CRLD and CGAN to highlight their strengths and potential limitations for CE.

The following sections will discuss the CRLD architecture and algorithmic specifics before its utilization within our CGAN-based approach.

A. CRLD Architecture

As per Fig. 4, the CRLD consists of an input layer, B denoising blocks, one convolutional (Conv) layer, and one output layer. The hyperparameters are summarized in Table 3. Each layer of CRLD is also discussed next.

TABLE 3 Hyperparameters of the CRLD Network

FIGURE 4.

The network architecture of the CRLD for CE in the AmBC system.

1) Input Layer

Consider a scenario of P observed pilot bits represented as $\{\mathbf {y}(0), \mathbf {y}(1), {\dots }, \mathbf {y}(P-1)\}$ , where P is either $N_{a}$ or $N_{b}$ (Fig. 3). To delve deeper into the intricacies of spatial and temporal correlations, an initial step is to reshape each pilot bit $\mathbf {y}(p)$ , for $p\in \mathcal {P}=\{0, 1, {\dots }, P-1\}$ , into a spatial 2D matrix. This matrix is denoted as $\mathbf {Y}(p)\in \mathbb {C}^{M_{a} \times M_{b}}$ . Note that the values of $M_{a}$ and $M_{b}$ are restricted such that $1 \leq M_{a}, M_{b} \leq M$ and their product equals M. Accordingly, the individual 2D matrices are aggregated into a singular 2D matrix, represented as $$\begin{equation*} \tilde {\mathbf {Y}} = [\mathbf {Y}(0), \mathbf {Y}(1), {\dots }, \mathbf {Y}(P - 1)] \in \mathbb {C}^{M_{a} \times PM_{b}}. \tag {7}\end{equation*}$$ Recognizing that $\tilde {\mathbf {Y}}$ is a complex-valued matrix is crucial. To accurately represent its structure, two separate input channels are needed, representing the real and imaginary parts.2 Consider $S_{0}(i, j, \beta)$ , an element of real number space $\mathbb {C}^{M_{a} \times PM_{b}}$ . It represents the i-th row and the j-th column of the $\beta$ -th channel in the $S_{0}$ layer. With this, the real and imaginary components of Y can be expressed as $$\begin{equation*} S_{0}(i, j, 1) = (\text {Re}(\tilde {\mathbf {Y}}))_{i,j},\quad S_{0}(i, j, 2) = (\text {Im}(\tilde {\mathbf {Y}}))_{i,j}.\end{equation*}$$ By stacking $\tilde {\mathbf {Y}}\in \mathbb {C}^{M_{a} \times PM_{b}}$ , with its real and imaginary parts into a 3D matrix, we get $$\begin{equation*} {\mathbf {Y}} = \mathcal {F}\left ({{\text {Re}(\tilde {\mathbf {Y}}),\: \text {Im}(\tilde {\mathbf {Y}}) }}\right) \in \mathbb {C}^{M_{a} \times PM_{b}\times 2}, \tag {8}\end{equation*}$$ where $\mathcal {F}(\cdot):\mathbb {C}^{M_{a} \times PM_{b}} \rightarrow \mathbb {C}^{M_{a} \times PM_{b}\times 2}$ is the mapping function for stacking matrices. This approach allows us to comprehensively represent the complex-valued matrix Y structure in a more manageable form. Utilizing this methodology, the input to the network is characterized by the received signals (data), denoted as $\mathbf {Y}\in \mathbb {C}^{M_{a} \times PM_{b} \times 2}$ . Conversely, the output of the network is represented by the channel matrices (labels), $\mathbf {Z}\in \mathbb {C}^{M_{a} \times PM_{b} \times 2}$ , corresponding to the matrix form of either w or h.

2) Denoising Blocks

The CRLD is a deep learning network architecture specifically designed for denoising applications. The structure of the CRLD comprises B denoising blocks, each designed uniformly. This uniformity implies that each block shares the same structure (Fig. 4).

• Conv+BN+ReLu layers: these span from the first layer of the block to the $(L - 1)$ -th layer. They consist of a convolution operation with learnable filters for transforming input data [24], followed by batch normalization (BN) to accelerate training [56]. The rectified linear unit (ReLu) function, $y = \max (0, x)$ , introduces non-linearity, allowing the learning of complex patterns [24].

• Conv layer: this is the final layer in each denoising block. It applies a convolution operation on the previous layer’s output to derive the residual noise. This is then subtracted from the input element-wise, helping to remove noise from the input.

Each L-layer subnetwork in the CRLD architecture can be represented as a non-linear function ${\mathcal {R}}_{\theta _{i}} (\cdot)$ . In particular, $\theta _{i}$ are the parameters learned by the i-th subnetwork during training, where $i\in \mathcal {I}= \{1, 2, {\dots }, B\}$ . The relationship between the input and output of each block can be represented as $$\begin{equation*} \mathbf {Y}_{i} = \mathbf {Y}_{i-1} - \mathbf {S}_{i} = \mathbf {Y}_{i-1} - {\mathcal {R}}_{\theta _{i}}(\mathbf {Y}_{i-1}),\quad \forall i \in \mathcal {I}, \tag {9}\end{equation*}$$ where $\mathbf {Y}_{0}$ is the initial input to the network. In addition, $\mathbf {Y}_{i-1}$ and $\mathbf {Y}_{i}$ represent the input and output of the i-th denoising block, respectively. $\mathbf {S}_{i} = {\mathcal {R}}_{\theta _{i}}(\mathbf {Y}_{i-1})$ is the difference between $\mathbf {Y}_{i-1}$ and $\mathbf {Y}_{i}$ , which is also referred to as the residual noise.

3) Output Layer

This can be defined as the estimated channel matrix, which undergoes a denoising process utilizing the results from B separate denoising blocks. Hence, the result generated from the B-th denoising block is given by $$\begin{equation*} \mathbf {Y}_{B} = \mathbf {Y}_{B-1} - {\mathcal {R}}_{\theta _{B}}(\mathbf {Y}_{B-1}), \tag {10}\end{equation*}$$ which signifies that the output of the B-th denoising block, $\mathbf {Y}_{B}$ , is achieved through the subtraction of the function $R_{\theta _{B}}(\mathbf {Y}_{B-1})$ from the output of the preceding $(B-1)$ -th block, $\mathbf {Y}_{B-1}$ . By integrating the information given in (9) and (10), the final output of the CRLD can be written as $$\begin{equation*} g_{\boldsymbol {\theta }}(\mathbf {Y}) = \mathbf {Y}- \sum _{i=1}^{B} R_{\theta _{i}} (\mathbf {Y}_{i-1}), \tag {11}\end{equation*}$$ where $g_{\boldsymbol {\theta }}(\cdot)$ is a representative of the overall function implemented by the CRLD, with parameters $\boldsymbol {\theta }= [\theta _{1}, \theta _{2}, \cdots, \theta _{B}]$ consisting of the parameters for each of the B denoising blocks. Term $\sum _{i=1}^{B} R_{\theta _{i}} (\mathbf {Y}_{i-1})$ indicates the cumulative residual noise component generated through the denoising process.

B. CRLD-Based Estimation Algorithm

This subsection will discuss the CRLD estimator proposed in [14]. This discussion is necessary because the generator in our CGAN is based on it. The CRLD scheme comprises offline training and online estimation phases for AmBC CE.

1) Offline Training

Consider a given training set represented by $$\begin{equation*} (\mathcal {Y}, \mathcal {Z}) = {(\mathbf {Y}^{(1)}, \mathbf {Z}^{(1)}), {\dots }, (\mathbf {Y}^{(K)}, \mathbf {Z}^{(K)})}, \tag {12}\end{equation*}$$ where $(\mathbf {Y}^{(k)}, \mathbf {Z}^{(k)})$ corresponds to the k-th example in the training set and K is the total number of the training samples. The MSE-based cost function for the offline training phase is given as [57]: $$\begin{equation*} J_{\text {MSE}}(\boldsymbol {\theta }) = \frac {1}{2K}\sum _{k=1}^{K} \left \|{{\mathbf {Z}^{(k)} - f_{\boldsymbol {\theta }}(\mathbf {Y}^{(k)}) }}\right \|^{2}_{F}, \tag {13}\end{equation*}$$ where $f_{\boldsymbol {\theta }}(\mathbf {Y}^{(k)}) = \mathcal {G}(g_{\boldsymbol {\theta }}(\mathbf {Y}^{(k)}))$ and $\mathcal {G}(\cdot): \mathbb {C}^{M_{a} \times PM_{b}\times 2} \rightarrow \mathbb {C}^{M_{a} \times PM_{b}}$ represents a function that constructs the complex-valued matrix based on the real-valued matrices. To train the CRLD effectively, the backpropagation (BP) algorithm is applied to iteratively update the network weights [24]. This iterative process achieves a well-trained CRLD model denoted by $$\begin{equation*} \hat {\mathbf {Z}} = f_{\boldsymbol {\theta }^{*}}(\mathbf {Y}^{(k)}) = \mathcal {G}(g_{\boldsymbol {\theta }^{*}}(\mathbf {Y}^{(k)})), \tag {14}\end{equation*}$$ where Y is the input matrix and $g_{\boldsymbol {\theta }^{*}}(\cdot)$ indicates the functional expression of the well-trained trained CRLD. The parameters of the trained network are denoted as $\boldsymbol {\theta }^{*} = [\theta _{1}^{*}, \theta _{2}^{*}, \ldots, \theta _{B}^{*}]$ .

2) Online Training

Upon acquiring the pilot-based test data $\mathbf {Y}^{\text {test}}$ , the CRLD model outputs the following: $$\begin{equation*} \hat {\mathbf {Z}} = f_{\boldsymbol {\theta }^{*}}(\mathbf {Y}^{\text {test}}) = \mathcal {G}\left ({{\mathbf {Y}^{\text {test}} - \sum _{i=1}^{B} {\mathcal {R}}_{\boldsymbol {\theta }^{*}_{i}}(\mathbf {Y}^{\text {test}}_{i-1})}}\right), \tag {15}\end{equation*}$$ which represents the CRLD estimated channel coefficients.

C. CRLD Algorithm Steps

Algorithm 1 outlines the CRLD channel estimator [14]. In the offline phase, it computes the cost function $J_{\text {MSE}}(\boldsymbol {\theta })$ and updates network weights $\boldsymbol {\theta }$ using BP across all epochs, given training set $(\mathcal {Y}, \mathcal {Z})$ . The output of this phase is a trained function $f_{\boldsymbol {\theta }^{*}}(\cdot)$ . During the online phase, this function is applied to each test batch $\mathbf {Y}^{\text {test}}$ , using a CRLD estimator to generate an estimated output $\hat {\mathbf {Z}}$ , which is reshaped into a $M_{a} \times PM_{b}$ vector. The algorithm finally outputs the channel coefficient, denoted as either $\hat {\mathbf {w}}$ or $\hat {\mathbf {h}}$ .

Algorithm 1 CRLD-Based CE Algorithm

Input: Training set $(\mathcal {Y}, \mathcal {Z})$ , number of epochs E, batch size D, network architecture parameters

1:

Offline Training:

2:

for $e = 1$ to E do

3:

for each mini-batch $\mathcal {D}$ of size D in $(\mathcal {Y}, \mathcal {Z})$ do

4:

Compute the cost function $J_{\text {MSE}}(\boldsymbol {\theta })$ based on (13)

5:

Update the network weights $\boldsymbol {\theta }$ using BP algorithm

6:

end for

7:

end for

8:

return Well-trained function $f_{\boldsymbol {\theta }^{*}}(\cdot)$

9:

Online Training:

10:

for each test batch $\mathbf {Y}^{\text {test}}$ do

11:

Calculate $\hat {\mathbf {Z}}= f_{\boldsymbol {\theta }^{*}}(\mathbf {Y}^{\text {test}})$ using CRLD estimator in (15)

12:

Reshape $\hat {\mathbf {Z}}$ into a $M_{a} \times PM_{b}$ vector

13:

return Channel coefficient $\hat {\mathbf {w}}$ or $\hat {\mathbf {h}}$

14:

end for

SECTION IV. CE via CGAN

Our CGAN is designed with a Conv structure, which employs layers of CNNs to process the matrix form of inputs and outputs. This architecture can handle images’ spatial and hierarchical patterns [32]. The generator, $G(\cdot)$ , is based on CRLD (Table 4) that maps the received signals Y to the channel matrices $\hat {\mathbf {Z}} = G(\mathbf {Y})$ . The generator includes multiple Conv layers with varying kernel sizes and strides to progressively refine the resolution of output $\hat {\mathbf {Z}}$ . Meanwhile, discriminator $D(\cdot)$ , also structured as a deep CNN, is trained to discern between true channel matrices Z and generated ones $\hat {\mathbf {Z}}$ . It outputs a scalar score indicating the authenticity of the input.

TABLE 4 Hyperparameters of the Discriminator Network

This network is optimized with a minimax two-player game strategy commonly used in GANs [24], [25]. The generator aims to fool the discriminator by producing a channel matrix $\hat {\mathbf {Z}}$ that is as close as possible to the true Z. In contrast, the discriminator aims to correctly classify the inputs as real or generated (Fig. 1).

During the training phase, the generator learns to generate more accurate channel matrices, and the discriminator becomes better at differentiating between real and generated ones. Once the training phase is complete, the generator can be utilized independently for CE.

A. CGAN Architecture

This section describes the architecture of the CGAN. The generator is based on the CRLD framework, complemented by a convolutional PatchGAN classifier for the discriminator [14]. The discriminator is designed to assess the structure at the scale of image patches, enhancing its ability to evaluate local image features. A recurrent challenge in the operation of GANs is the phenomenon of “mode collapse”, where the generator starts to produce a limited variety of outputs. This restricts the diversity of the generated samples, as the generator optimizes its parameters to fool the discriminator with a narrow set of data points. Mode collapse has been extensively studied [58], [59], describing its implications on the diversity and quality of generated samples. To mitigate this, a sigmoid activation function is integrated within the generator, normalizing the output values to the range $[{0, 1}]$ . Such normalization not only assists in stabilizing the training process but also helps in maintaining a wider variety of output, reducing the likelihood of generator collapse and improving the robustness of the model.

The discriminator (Fig. 5) uses a standard CNN that consists of a Conv layer, BN, and a LeakyReLu activation function. The patch architecture is implemented and designed to augment the discriminator’s capacity for recovery and improve its proficiency in distinguishing input data [30], [33]. Instead of providing a single scalar output denoting real/generated, like a traditional discriminator, this architecture transforms the input into a receptive field, with each element representing the authenticity of a corresponding input segment. The hyperparameters of the discriminator are summarized in Table 4.

FIGURE 5.

Architecture of the discriminator.

B. CGAN-Based Estimation Algorithm

As mentioned, GANs concurrently train two interlinked neural networks, the generator and the discriminator [25]. The cost function for this process can be represented as $$\begin{align*} {\mathcal {L}}_{\text {CGAN}}(G_{\boldsymbol {\psi }}, D_{\boldsymbol {\theta }}, \mathbf {Y}, \mathbf {Z}) & = \mathbb {E}\left [{{\log D_{\boldsymbol {\theta }}(\mathbf {Z})}}\right ] \\ & \hspace {10mm} + \mathbb {E}\left [{{\log (1 - D_{\boldsymbol {\theta }}(G_{\boldsymbol {\psi }}(\mathbf {Y}))}}\right ], \tag {16}\end{align*}$$ where $G_{\boldsymbol {\psi }}$ stands for the generator characterized by parameters $\boldsymbol {\psi }$ and $D_{\boldsymbol {\theta }}$ denotes the discriminator parameterized by $\boldsymbol {\theta }$ . The labels “1” and “0” distinguish real and generated data, respectively. This adversarial strategy enables generator $G_{\boldsymbol {\psi }}$ to minimize the GAN loss while discriminator $D_{\boldsymbol {\theta }}$ aims to maximize it. The optimization problem is represented by $$\begin{equation*} \min _{\boldsymbol {\psi }} \max _{\boldsymbol {\theta }} \: {\mathcal {L}}_{\text {CGAN}}(G_{\boldsymbol {\psi }}, D_{\boldsymbol {\theta }}, \mathbf {Y}, \mathbf {Z}). \tag {17}\end{equation*}$$ As mentioned, CGANs further bolster the efficacy of GANs [30]. The generator and the discriminator in a CGAN are conditioned on additional information Y. The principal objective is to train the generator to construct a highly authentic channel matrix that can fool the discriminator. The generator is optimized via the $\mathcal {L}_{2}$ loss function (Fig. 1) defined as $$\begin{equation*} \mathcal {L}_{2} = \mathbb {E}\left [{{\left \|{{\mathbf {Z}- G_{\boldsymbol {\psi }}(\mathbf {Y})}}\right \|^{2}}}\right ]. \tag {18}\end{equation*}$$ Consequently, the composite objective function, encompassing both the GAN loss and the $\mathcal {L}_{2}$ loss, is formulated as below: $$\begin{equation*} \min _{\boldsymbol {\psi }} \max _{\boldsymbol {\theta }} \: {\mathcal {L}}_{\text {CGAN}}(G_{\boldsymbol {\psi }}, D_{\boldsymbol {\theta }}, \mathbf {Y}, \mathbf {Z}) + \mathcal {L}_{2}. \tag {19}\end{equation*}$$ The optimization of $G_{\boldsymbol {\psi }}$ seeks to minimize this combined loss, whereas $D_{\boldsymbol {\theta }}$ strives to maximize its ability to discern genuine data from synthetic. The integration of $\mathcal {L}_{2}$ enhances the precision of $G_{\boldsymbol {\psi }}$ by penalizing output variance from the actual data distribution, promoting the authenticity and accuracy of the generated channel matrix. Thus, (19) is designed to achieve an equilibrium between the authenticity of the generated channel matrix and the precision of the generator’s output. This goal is to ensure that the data produced closely mirrors real data, enhancing the efficacy of the CGAN architecture in CE tasks.3

C. CGAN Algorithm Steps

Algorithm 2 utilizes a CGAN-based process for CE. It takes a training set $(\mathcal {Y}, \mathcal {Z})$ and initializes the generator $G_{\boldsymbol {\psi }}$ and discriminator $D_{\boldsymbol {\theta }}$ . The generator creates fake data, and the discriminator computes loss based on (16) in each epoch. Simultaneously, the generator loss is computed according to (19). The weights of both entities are updated to maximize and minimize respective losses. A practical stopping criterion for training is to set a maximum number of epochs. This approach primarily prevents overfitting and reduces the excessive use of computational resources [58], [60]. By capping the number of epochs, we ensure the training does not continue indefinitely, especially when loss values show no signs of convergence.

Algorithm 2 CGAN-Based CE Algorithm

Input: Training set $(\mathcal {Y}, \mathcal {Z})$ , number of epochs E, batch size D

1:

Offline Training:

2:

Initialize the generator $G_{\boldsymbol {\psi }}$ and discriminator $D_{\boldsymbol {\theta }}$ networks with random weights

3:

for $e = 1$ to E do

4:

for each mini-batch $\mathcal {D}$ of size D in $(\mathcal {Y}, \mathcal {Z})$ do

5:

Generate fake data $\hat {\mathbf {Z}} = G_{\boldsymbol {\psi }}(\mathbf {Y})$

6:

Compute the discriminator loss based on (16)

7:

Update the discriminator weights $\boldsymbol {\theta }$ to maximize the discriminator loss

8:

Compute the generator loss based on (19)

9:

Update the generator weights $\boldsymbol {\psi }$ to minimize the generator loss

10:

end for

11:

end for

12:

Online Training:

13:

for each test batch $\mathbf {Y}^{\text {test}}$ do

14:

Calculate $\hat {\mathbf {Z}}= G_{\boldsymbol {\psi }}(\mathbf {Y}^{\text {test}})$

15:

Reshape $\hat {\mathbf {Z}}$ into a $M_{a} \times PM_{b}$ vector

16:

return Channel coefficient $\hat {\mathbf {w}}$ or $\hat {\mathbf {h}}$

17:

end for

SECTION V. Classical CE Methods

To provide background and context, this section briefly outlines traditional CE techniques: LS, MMSE, and blind estimator, including theoretical analysis and description.

A. LS and MMSE Estimators

The LS and MMSE estimators are based on the examination of CRDL layers theoretically [14]. To understand how to use CRLD as the generator network of CGAN, we analyze the CRLD output and characterize its properties theoretically. The CRLD closely approximates the theoretically optimal MMSE estimator in large datasets. Thus, the CRLD method is used to derive the MMSE estimator [14]. Using CRLD, the convolution operation can be formulated as a product of two matrices [24]. For the sake of simplicity, assume the input is based on (7). Hence, for a well-trained subnetwork, represented as the i-th layer in a total of B layers (with $i \in \{1, 2, {\dots }, B\}$ ), we have: $$\begin{equation*} {\mathcal {R}}_{\theta _{i}}(\tilde {\mathbf {Y}}_{i-1}) = \tilde {\mathbf {Y}}_{i-1} \mathbf {W}_{i}^{*}, \tag {20}\end{equation*}$$ where $\mathbf {W}_{i}^{*}$ are the network weights, $\tilde {\mathbf {Y}}_{0} = \tilde {\mathbf {Y}}$ , and $\tilde {\mathbf {Y}}_{i} = \prod _{j=1}^{i} \tilde {\mathbf {Y}} (\mathbf {I}_{PM_{b}} - \mathbf {W}_{j}^{*})$ . Thus, the output is rewritten as $$\begin{equation*} \tilde {\mathbf {Y}}_{B} = \tilde {\mathbf {Y}}(\mathbf {I}_{PM_{b}} - \mathbf {W}^{*}), \tag {21}\end{equation*}$$ where $\mathbf {W}^{*} = \mathbf {W}_{1}^{*} + \mathbb {1}_{A}(B)\sum _{i=2}^{B} \left ({{ \prod _{j=1}^{i-1} (\mathbf {I}_{PM_{b}} - \mathbf {W}_{j}^{*}) \mathbf {W}_{i}^{*} }}\right)$ with $A = \{2, 3, {\dots }\}$ . The output of the CRLD can be expressed as $\hat {\mathbf {Z}} = \tilde {\mathbf {Y}}(\mathbf {I}_{PM_{b}} - \mathbf {W}^{*})\mathbf {W}_{B+1}^{*}$ , where $\mathbf {W}^{*}_{B+1}$ is the weights of the Conv layer $f_{\theta _{B+1}}(\cdot)$ . Then, the optimal MMSE estimator can be written as $$\begin{equation*} \hspace {-2mm} \mathbf {Z}_{\text {MMSE}} = \tilde {\mathbf {Y}} (\mathbf {I}_{PM_{b}} - \lambda \mathbf {S}^{H} (\lambda \mathbf {S} \mathbf {S}^{H} + \mathbf {R}_{\mathbf {Z}}^{-1})^{-1} \mathbf {S}) \lambda \mathbf {S}^{H} \mathbf {R}_{\mathbf {Z}}, \tag {22}\end{equation*}$$ where $\mathbf {Z} \in \mathbb {C}^{M_{a} \times M_{b}}$ is the channel matrix, $\mathbf {S} = [\mathbf {I}_{M_{b}}, \mathbf {I}_{M_{b}}, {\dots }, \mathbf {I}_{M_{b}}] \in \mathbb {C}^{M_{b} \times PM_{b}}$ , $\mathbf {R}_{\mathbf {Z}} = E(\mathbf {Z}^{H} \mathbf {Z})$ is the statistical correlation matrix, and $\lambda = \frac {1}{M_{a} \sigma _{u}^{2}}$ . Weight matrices $\mathbf {W}^{*}$ and $\mathbf {W}^{*}_{B+1}$ are learned from the training set through offline training using the MMSE-based cost function. For sufficiently large training sets, the CRLD-based estimator can learn and mimic the expression of the optimal estimator under the MMSE criterion. In addition, the LS estimator can also be derived using a similar method. The LS estimator can be written as $\mathbf {Z}_{\text {LS}}= \tilde {\mathbf {Y}} \mathbf {S}^{\dagger }$ , where $\mathbf {S}^{\dagger }$ is the pseudoinverse of the design matrix S.

B. Blind Estimator

This uses the eigenvalue decomposition (EVD) of the covariance matrix of the received signals at the reader [17]. The statistical covariance matrix for the received vector $\mathbf {y}(n)$ can be written as $$\begin{align*} \mathbf {R}_{\mathbf {y}(n)} & = \mathbb {E} \{\mathbf {y}(n)\mathbf {y}(n)^{H}\} =\mathbf {z}\mathbf {z} ^{H} \mathbb {E}\{s^{2}(n)\} + \mathbb {E}\{\mathbf {u}(n)\mathbf {u}^{H}(n)\} \\ & = \mathbf {z}\mathbf {z} ^{H} + \sigma _{u}^{2} \mathbf {I}, \tag {23}\end{align*}$$ where z is either w or h. From (23), $$\begin{equation*} \mathbf {z}\mathbf {z} ^{H} = \mathbf {R}_{\mathbf {y}(n)} - \sigma _{u}^{2} \mathbf {I}, \tag {24}\end{equation*}$$ and $$\begin{equation*} \mathbf {z}^{H} \mathbf {z}= \sum _{i=1}^{M} (\mathbf {z}\mathbf {z} ^{H})_{ii} = \sum _{i=1}^{M} \left ({{\mathbf {R}_{\mathbf {y}(n)} - \sigma _{u}^{2} \mathbf {I}}}\right)_{ii}. \tag {25}\end{equation*}$$ Post-multiplying (23) by z results in $$\begin{equation*} \mathbf {R}_{\mathbf {y}(n)}\mathbf {z}= \mathbf {z}\mathbf {z} ^{H} \mathbf {z}+ \sigma _{u}^{2} \mathbf {z}= (\mathbf {z}^{H} \mathbf {z}+ \sigma _{u}^{2}) \mathbf {z}. \tag {26}\end{equation*}$$ Subsequently, replacing (25) in (26), we have $$\begin{equation*} \mathbf {R}_{\mathbf {y}(n)}\mathbf {z}= \left ({{\sum _{i=1}^{M} \left ({{\mathbf {R}_{\mathbf {y}(n)} - \sigma _{u}^{2} \mathbf {I}}}\right)_{ii} + \sigma _{u}^{2} }}\right)\mathbf {z}, \tag {27}\end{equation*}$$ which serves as the characteristic equation of the covariance matrix $\mathbf {R}_{\mathbf {y}(n)}$ , corresponding to a known eigenvalue $\Lambda$ . Consequently, an estimate $\tilde {\mathbf {z}}$ with unit modulus for z can be acquired by performing EVD on $\mathbf {R}_{\mathbf {y}(n)}$ corresponding to the eigenvalue $\Lambda$ when $\Gamma = 0$ or 1. However, for a given constant $\alpha$ , $$\begin{equation*} \mathbf {R}_{\mathbf {y}(n)}\alpha \tilde {\mathbf {z}} = \left ({{\sum _{i=1}^{M} (\mathbf {R}_{\mathbf {y}(n)} - \sigma _{u}^{2} \mathbf {I})_{ii} + \sigma _{u}^{2} }}\right)\alpha \tilde {\mathbf {z}}. \tag {28}\end{equation*}$$ Therefore, $\alpha \tilde {\mathbf {z}}$ could be an alternative estimate, introducing ambiguity. To resolve this ambiguity, constant $\alpha$ could be determined by using (24) by solving $$\begin{equation*} \alpha ^{2} \tilde {\mathbf {z}}\tilde {\mathbf {z}}^{H} = \mathbf {R}_{\mathbf {y}(n)} - \sigma _{u}^{2} \mathbf {I}. \tag {29}\end{equation*}$$ Furthermore, the value of $\alpha$ can be deduced from $$\begin{equation*} \alpha = \sqrt {\frac {\sum _{i=1}^{M} \left ({{\left ({{\mathbf {R}_{\mathbf {y}(n)} - \sigma _{u}^{2} \mathbf {I}}}\right)\big(\tilde {\mathbf {z}}\tilde {\mathbf {z}}^{H} }}\right)^{-1}\big)_{ii}} {M}}. \tag {30}\end{equation*}$$ Utilizing equation (30), the estimated channels can be obtained as either $\mathbf {z}_{0}= c_{0} \mathbf {h}$ or $\mathbf {z}_{1}= c_{1} \mathbf {w}$ .

In summary, the blind channel estimator initializes and updates the received signal’s covariance matrix, adjusts it by subtracting the noise power, computes the unit eigenvector, and outputs the channel estimates via the EVD. These steps are outlined in Algorithm 3.

Algorithm 3 Blind-Based CE Algorithm

Require: Received signal y, noise power $\sigma _{u}^{2}$ , relative coefficient $\zeta$ , SNR, number of pilots $N_{a}$ and $N_{b}$

1:

for each $\Gamma$ in $\{0, 1\}$ do

2:

Initialize $\hat {\mathbf {R}}_{\mathbf {y}} = 0$

3:

if $\Gamma == 0$ then

4:

Compute and update $\hat {\mathbf {R}}_{\mathbf {y}} = \frac {1}{N_{a}} \sum _{n = 1}^{N_{a}} \mathbf {y}(n)\mathbf {y}^{H}(n)$

5:

else

6:

Compute and update $\hat {\mathbf {R}}_{\mathbf {y}} = \frac {1}{N_{b}} \sum _{n = 1}^{N_{b}} \mathbf {y}(n)\mathbf {y}^{H}(n)$

7:

end if

8:

Compute the adjusted covariance matrix: $\hat {\mathbf {R}}_{\mathbf {y}}-\sigma ^{2}_{u} \mathbf {I}$

9:

Compute unit eigenvector $\tilde {\mathbf {z}}$ of $\hat {\mathbf {R}}_{\mathbf {y}}$ , corresponding to the eigenvalue $\sum _{i=1}^{M} \left ({{\hat {\mathbf {R}}_{\mathbf {y}} - \sigma _{u}^{2} \mathbf {I}}}\right)_{ii} + \sigma _{u}^{2}$ .

10:

Determine $\alpha$ using (29) such that $\alpha ^{2} \tilde {\mathbf {z}}\tilde {\mathbf {z}}^{H} = \hat {\mathbf {R}}_{\mathbf {y}}-\sigma ^{2}_{u} \mathbf {I}$ according to (30)

11:

end for

12:

return Estimated channels: $\hat {\mathbf {z}}_{0} = \alpha _{0}\tilde {\mathbf {h}}$ , $\hat {\mathbf {z}}_{1} = \alpha _{1}\tilde {\mathbf {w}}$

The computational burdens of different algorithms are listed in Table 5. Symbol $s_{l}$ represents the filter size in the l-th Conv layer, while $n_{l-1}$ and $n_{l}$ denote the size of the input and the output of the l-th Conv layer, respectively. Note that the complexity of CNN, CRLD, and CGAN varies with the network size. CGANs incur higher computational costs due to their dual-network adversarial training. The practical deployment must also weigh in model training duration, memory usage, and power demands, especially given the dynamics of AmBC environments. Future work should thus explore optimizing CGANs to balance accuracy and computational efficiency.

TABLE 5 Computational Complexities of Different Estimation Algorithms

SECTION VI. Simulation Results

Simulation results are provided next. The simulation model (Fig. 2) is an AmBC system featuring a multi-antenna reader. Specifically, the ambient RF source is at a distance of $d_{st} = {\mathrm {2.5~\text {m}}}$ from the tag. The tag itself is positioned at $d_{sr} = {\mathrm {4.8~\text {m}}}$ from the source and at $d_{tr} = {\mathrm {0.5~\text {m}}}$ from the reader [46], [51]. These distances reflect realistic system configurations and the fact that tags operate within a short distance (less than 1 m) [4], [51]. In Fig. 11, various relative coefficient values are explored, effectively altering the distances. This examination is limited to this specific set of distances for brevity, and other distance configurations are not reported. The RF ambient source is represented as a Gaussian random signal with a zero mean and a power of $P_{s}=10^{(\text {SNR}/10)}\sigma ^{2}_{u}$ , and a Rayleigh channel model is employed.4 Table 6 summarizes the simulation parameters [44], [51]. The network depth for the generator and discriminator is carefully chosen based on literature and empirical tests, balancing complexity and accuracy to optimize performance without excessive computational demands [14], [29], [46], [61].

TABLE 6 Simulation Parameters

FIGURE 6.

Pseudo-color illustration of the real component of the estimated w using the CGAN approach.

FIGURE 7.

Training loss trends for the proposed CGAN, illustrating the discriminator and generator losses across multiple training iterations.

FIGURE 8.

NMSE versus SNR under $\zeta = -20$ dB and $N_{a} = N_{b} = 2$ .

FIGURE 9.

NMSE versus the number of pilots under $\zeta = -20$ dB and SNR =5 dB.

FIGURE 10.

NMSE versus the number of antenna under $\zeta = -20$ dB and SNR =5 dB.

FIGURE 11.

NMSE versus relative coefficient with SNR =5 dB and $N_{a} = N_{b} = 2$ .

The objective is to assess the performance of the proposed CGAN estimator compared to CRLD, MMSE, LS [14], CNN [62], and blind estimator [17]. This evaluation involves quantifying the performance of these estimators using the NMSE, which represents the disparity between the estimated and actual channels normalized by the actual value. The NMSE is formally defined as follows: $$\begin{equation*} \text { NMSE} = \frac {\mathbb {E}||\mathbf {z} - \hat {\mathbf {z}}||^{2}_{2}}{\mathbb {E}||\mathbf {z}||^{2}_{2}}, \tag {31}\end{equation*}$$ where z and $\hat {\mathbf {z}}$ denote the actual value and the estimated value, respectively. To validate our proposed method, the experiments compare it to LS, MMSE, CRLD, and CNN. The CRLD is trained and tested using datasets of $8,000$ and $2,000$ samples, respectively, ensuring comprehensive model training and validation. The CGAN training and testing datasets consisted of $3,200$ and $1,000$ samples, respectively. The channel matrix elements are normalized between 0 and 1, with 0 representing black and 1 representing white. CRLD utilized the Adam optimizer with a learning rate of 0.001 [14] for optimization. The proposed CGAN model used the Adam optimizer for the generator and the MSProp optimizer for the discriminator [30], [63], with learning rates of $2 \times 10^{-4}$ and $2 \times 10^{-5}$ , respectively.

Fig. 6 visually depicts the CGAN performance gains by representing estimated and actual channels as pseudo-color images (Fig. 6). Images in Fig. 6a and Fig. 6b for the actual or ground truth channel and the CGAN-generated channel bear a striking resemblance. This alludes to the successful replication of intricate details of the channel, offering a powerful visual demonstration of the CGAN effectiveness.

Fig. 7 illustrates the training loss dynamics of the CGAN. The left side of the figure displays the discriminator’s loss, although initially low, increases and fluctuates throughout the training period. Conversely, the right side of the figure shows the generator’s loss, which begins at a considerably higher level, decreases sharply initially, and then tapers off more slowly. The pattern of these loss curves indicates the learning and improvement of both the discriminator and generator as training progresses. This is a characteristic feature of CGANs’ adversarial training process: the generator gradually becomes more proficient at creating data resembling the actual distribution. At the same time, the discriminator improves its ability to differentiate between real and generated data.

Fig. 8 illustrates the NMSE performance as a function of SNR. Obtaining an accurate statistical channel correlation matrix is challenging in real-world scenarios, limiting the effectiveness of the traditional MMSE estimator. The LS estimator approaches the performance of the optimal MMSE estimator at high SNR, with diminishing noise influence. However, a significant performance gap remains between the LS and CGAN methods under low SNR conditions. This disparity arises from the LS method treating channel coefficients as static, while the CGAN and MMSE approaches consider them stochastic variables. MMSE and deep learning methods leverage prior statistical data to enhance CE accuracy. Moreover, the semi-blind estimator outperforms the LS and CNN estimators in high SNR conditions, showcasing its robustness in such environments.

Fig. 8a shows the NMSE of the estimation of the direct link h. The CGAN estimator consistently outperforms MMSE, LS, CRLD, CNN, and blind estimators across a wide range of SNRs. CGAN’s dominance arises from its ability to generate accurate and realistic channel conditions and distinctive architecture that ensures superior performance even in high-noise interference scenarios. Fig. 8b depicts the NMSE for both direct link h and cascade link w. Similarly, the CGAN estimator outperforms its counterparts. Notably, at $\text {SNR}=5$ dB, CGAN achieves a NMSE of 0.017 for h, significantly better than the CRLD and MMSE estimators’ NMSE of 0.094. Likewise, for w, CGAN achieves a NMSE of 0.013, outperforming the CRLD and MMSE with NMSE of approximately 0.072. Thus, CGAN archives approximately 82% gain over CRLD and MMSE estimators.

The CGAN framework leverages the received signal to adeptly model the complex and non-linear characteristics of wireless channels, significantly enhancing performance in fluctuating conditions compared to traditional estimators like MMSE and LS. Its main advantage lies in its adversarial training, where a generator and a discriminator engage in continuous, dynamic competition, refining the model’s predictive accuracy. The generator uses a residual network architecture designed to avoid the vanishing gradient problem common in conventional CNNs by incorporating skip connections that allow gradients to bypass multiple layers directly. This contrasts with traditional CNN architectures, which lack mechanisms to prevent gradient vanishing and tend to underperform in highly non-linear environments. The adversarial process, involving direct feedback from the discriminator to the generator, enables a cycle of continuous improvement markedly different from the straightforward loss function minimization in typical CNN training. Although the CRLD estimator does not match CGAN’s performance, it still improves CE accuracy compared to MMSE and LS. For example, at an SNR of 5 dB, the CGAN achieves a significantly lower NMSE of approximately 93.48% for h, compared to CNN. Similarly, for w, CGAN exhibits a considerable improvement of about 95.22%. These findings underscore the enhanced performance of CGANs in increasing CE accuracy.

Fig. 9 offers a comprehensive examination of the NMSE performance with respect to the number of pilot symbols at $\text {SNR}=5$ dB. It reveals how estimation techniques adapt to more pilot symbols in noisy communication scenarios. Figures 9a and 9b represent the NMSE performance for the channel response vectors, h and w, respectively. A discernible pattern from these subfigures is the decrement in NMSE values for all estimators as the number of pilot symbols increases. Thus, estimation accuracy improves with this trend.

Compared to classical estimators, the CRLD estimator exhibits striking performance attributes. However, its performance mirrors the MMSE and LS estimators in regions with high SNR. This convergence indicates that under high SNR circumstances, the CRLD estimator can generate estimation results with an accuracy that rivals those of the MMSE and LS estimators. This superior performance can be attributed to the unique strengths of the CRLD estimator, which is renowned for exploiting temporal correlations inherent in the pilot signals. By leveraging these correlations, the CRLD estimator can augment the precision of CE even under conditions with low SNR. In addition, the blind estimator shows good performance while it exhibits substantial improvements as the number of pilots increases. It is worth noting that our proposed CGAN outperforms the CRLD estimator in both low and high SNR situations. Both subfigures demonstrate that the CGAN estimator can learn and replicate the intricate characteristics of wireless channels. Consequently, the CGAN estimator delivers superior estimation accuracy, further enhanced by the increasing number of pilot symbols.

Fig. 10 displays the NMSE versus the number of antennas at the reader. From Fig. 10a–10b, the CGAN outperforms the classical and CRLD estimators even with increased antennas at the reader. However, the NMSE value of the CGAN estimator increases slightly when the number of reader antennas changes from 80 to 100. Despite this marginal increment, the performance of the CGAN remains robust and satisfactory, underscoring its adaptability and resilience. The ability of the CGAN estimator to handle complicated and high-dimensional data structures is evident from its remarkable resilience. This characteristic not only allows it to handle increased numbers of reader antennas but also to model and mirror the nuanced attributes of wireless channels precisely. This adaptability of the CGAN to varying numbers of reader antennas further reinforces its potential applicability in a wide array of real-world scenarios when the reader may use different antenna arrays. These observations show that the proposed approach can be used in many practical applications.

Our investigation now turns to the impact of reflection links on CE performance. Fig. 11 explores a realistic range of the relative coefficient $\zeta$ , spanning from −20 to −5. The figure presents a compelling trend: as the relative coefficient $\zeta$ increases, the NMSE for all CE techniques decreases accordingly. This inverse relationship suggests that augmenting the $\zeta$ value can significantly enhance CE performance.

This phenomenon occurs because a stronger reflected path aids in distinguishing tag signals from direct path signals. Consequently, the reader can easily discern tag signals amidst the noise. Additionally, the figure offers insights into how various estimators respond to changes in the relative coefficient $\zeta$ . Techniques such as MMSE, LS, and CRLD all exhibit performance improvements. However, these enhancements are not uniform across all estimators.

Despite these enhancements, the proposed CGAN estimator outperforms other estimators. It estimates wireless channels by accurately capturing intricate features and leveraging stronger reflected paths. This underscores its effectiveness in adapting to changes in the relative strength of the reflected signals, leading to superior CE performance.

SECTION VII. Conclusion

This study uses CGANs for channel state estimation in an AmBc network featuring a multi-antenna reader. Our proposed CGANs perform better than the classical and CRLD estimators, even with limited data availability across various scenarios. For instance, at a realistic SNR of 5 dB, we observed performance gains of 82% over CRLD and MMSE estimators. Additionally, we investigated the influence of variables such as pilot length and the relative coefficient between backscattered and direct paths on NMSE performance. The remarkable performance of the CGAN approach underscores its potential as a compelling alternative to traditional CE methods. Additionally, while initial efforts have concentrated on deploying CGANs within a single-tag system model, supporting multiple backscatter tags presents a promising direction for further investigation.