Resumen de Atomic norm decomposition for sparse model reconstruction applied to positioning and wireless communications
Sparse models have very limited degrees of freedom compared to the dimension of available data and are present in many science and engineering fields. The sparsity of those models can be due to factors such as data gathering by means of sensor networks where size and cost of the sensors yield sparsity, magnetic resonance imaging (MRI) where time for data acquisition is constrained, or wireless communications in millimeterwave (mmWave) channels where the number of scatterers and thus propagation echoes is limited.
These sparse models are formulated by a linear combination of a set of so called atoms belonging to a known atom set. In this thesis we focus on a specific set with infinite cardinality whose atoms are a Fourier transform fully characterized by a d-dimensional parameter. The linear combination is done over a small number of atoms if compared to the dimension of the available data. Due to the sparsity of the model, its measured observations need fewer samples than required by the Nyquist-Shannon sampling theorem to recover the data, which is known as compressed sensing.
Existing methods for parameter extraction via compressed sensing can be classified in two main groups: on-grid methods and gridless methods. The former group searches for the parameters in a discrete space, forcing the estimates to lay on the grid. Thus, even with a very intensive sampling of the continuous space, there will always be an error between the estimates and the real values. Conversely, the latter operates directly on the continuous space, and therefore, avoids any errors caused by the grid. Gridless methods are usually more computationally expensive.
In this thesis we aim to a gridless solution that allows the recovery of the d-dimensional parameters that characterize the atoms in the continuous domain in which they are defined. The objective of this thesis is to give some new results on the gridless recovery of multidimensional parameters in wireless communication and radar scenarios. The contributions of the thesis are: • Validation via numerical analysis of the available theoretical results in the literature.
• Application of gridless methodology to the design of a low complexity precoder in mmWave channels. We limit as much as possible the number of radio-frequency (RF) chains needed in a hybrid precoder, enabling up to 3-dimensional MIMO propagation, whereas in the literature, hybrid precoders are mostly studied in the linear case (1D).
• Application to positioning by estimation of parameters such as direction, distance, and velocity. Compared to state-of-the-art techniques based on grid, our gridless methods outperform them for equal computational complexity.
• Application to full channel estimation, including relevant parameter extraction such as angles of departure and arrival of propagation echoes. Propagation is studied in MIMO mmWave sparse channels.
• A machine learning based methodology to estimate the number of composite atoms in the sparse model. We leverage the prior knowledge of the system to enable the possibility of training a model offline and use its predictions online to have an estimation of the number of atoms. We explore and test several models.
1 Gridless methodology based on Atomic Norm for sparse model recovery As introduced earlier, in this thesis we focus on a continuous atom set. The atoms in the set are fully characterized by two vectors: vector J containing the number of elements per dimension and the d-dimensional parameter f, which is defined in the continuous interval [0, 1). Each atom in the atom set defines, then, a d-dimensional Fourier transform of that parameter and will be denoted as a(f). The observations are measurements of linear combinations of K atoms. That measurement will be formulated by means of a matrix B and it is known and fully dependant on the application. Thence, the observation can be modeled as y = Bx where x = [a(f1), . . . , a(fK)]u, being u a vector modelling the linear combination of atoms. In the applications studied in the thesis we will have additive Gaussian noise in the observations denoted as w which can be added to the observation formula as y = Bx + w.
Thus, in the thesis we focus on the following problem: knowing the observed noisy vector y and knowing the matrix that maps the linear measurements B, the objective is to robustly recover the signal data x, the atoms a(fk) alongside with the recovery of the parameter fk for k = 1, . . . ,K.
For doing that, we define a set of optimization problems based on atomic norm. However, it is still needed to address how to solve atomic norms so, we define a rank minimization problem and its convex relaxation nuclear minimization problem to solve them. The solution to those optimization problem is a matrix which we can decompose following a Vandermonde decomposition to extract the d-dimensional parameter fk for k = 1, . . . ,K.
2 Sparsity in mmWave channels Fifth (5G) and sixth (6G) generations of mobile communications demand a huge increase in the systems capacity as well as other requirements such as multidimensional space awareness or ubiquitous connectivity. This has led to the development of a series of technologies to meet those needs, among which we can find Multiple-Input Multiple-Output (MIMO), carrier aggregation, advanced channel coding and interference coordination. Large antenna arrays have been widely explored and are usually deployed embedded in already existing spaces that have other additional usages, such as surfaces or smart surfaces. However, despite the importance of the aforementioned technologies, the current spectrum is saturated which has led to the exploration of alternative and underused bands.
The GSM Association identifies three frequency ranges that meet the needs imposed by the novel paradigms in wireless communications. The first proposed range is the band below 1 GHz, which is aimed to give wide coverage and also low rate applications such as IoT; 1 to 6 GHz, intended to give service to emerging utilities such as intelligent transport including self-driving cars; and the range above 6 GHz and especially the mmWave range, which goes from 30 GHz to 300 GHz, will support ultra-high broadband services. The latter range is the one that meets the requirements and will be studied in this work.
Nevertheless, the nature of mmWave band presents severe propagation drawbacks. First, mmWave channels experience extremely high free-space path loss due to 10× increase in carrier frequency. Fortunately, it makes multidimensional antenna arrays achievable due to the consequential decrease in wavelength which allows the deployment of small sized arrays that can mitigate the inherent path loss in mmWave channels via MIMO multiplexing and diversity gains.
The path loss experienced in mmWave band systems is worsened by any obstacle the signal may encounter and also by other phenomena such as atmospheric absorption resulting in a small number of propagation paths in the system which leads to sparse channel models. Thus, a sparse MIMO mmWave channel can be modeled following the general definition stated in section 1 where the number of dimensions d is referred to the dimensions of the transmit and receive antennas arrays, the number of atoms K is the number of propagation paths and the d-dimensional parameter f can contain relevant information about the communication system: angular, spread, range, etc.
3 Machine Learning approach for estimation of number of composite atoms Ideally, in the noiseless case, the rank (r) of the data signal covariance matrix, which results from the optimization problems stated in section 1, should be equal either to the number of composite atoms K if K < J, being J the total number of elements in all dimensions, or equal to J in the other case. Furthermore, given the resolvability conditions of the optimization problems mentioned in 1 we seek that the matrix is rank deficient having r = K < J. With that we would ensure unique signal and frequency recovery. However, there are several scenarios where this rank condition would not hold and we end with a full-rank matrix having r = J. Fortunately, in these scenarios, the first K eigenvalues are dominant and the rest are close to 0 and can be considered as spurious.
A Machine Learning approach is proposed to identify the relevant eigenvalues and then estimating the number of composite atoms. The problem consists in estimating the number of dominant eigenvalues from the set of eigenvalues by training a model whose output is an estimate of the number of composite atoms.
In order to achieve that we generate a synthetic dataset with hundreds of thousands of examples where we simulate several distinct noise and frequency realization scenarios. Once the dataset is gathered we train a set of models, both regression (linear regression, xgboost, fully-connected neural network) and classification (logistic regression, naive bayes, xgboost, fully-connected neural network), and compare their performance. The best model results are obtained by the fully-connected neural network classifier with 3 hidden layers of sizes 200, 50 and 10.
4 Applications 4.1 Super-resolution in automotive pulse radar mmWave radars are one of the key technologies for advanced driver assistant systems (ADASs) and eventually for future self–driving cars. This adoption is made possible given recent hardware developments, which allow building cost-effective wideband transceivers equipped with a large number of antennas and powerful digital processing units. Automotive radars are already employed in adaptive cruise control with stop-and-go, automatic emergency brake, lane change and park assistance, cross traffic alert, blind spot detection, and collision warning. Differently from other optical and ultrasound sensors, mmWave radars can operate under adverse weather and light conditions and enable the simultaneous measurement of the delay, radial velocity, and azimuth and elevation angles of arrival (AoA) of prospective targets/obstacles within a single Coherent Processing Interval (CPI), thus allowing the construction of a four dimensional image of the inspected region.
Classical approaches for delay and Doppler estimation rely on the correlation properties of the probing signal and achieve a resolution approximately equal to the inverse of the signal bandwidth and duration, respectively, also known as Rayleigh limit. More advanced algorithms may obtain a resolution beyond the Rayleigh limit (usually referred to as super-resolution) for a sufficiently large SNR at the price of an increased complexity. Unfortunately, well-established solutions based on the use of the sample covariance matrix, such as the multiple signal classification (MUSIC), the estimation of signal parameters via rotational invariance techniques (ESPRIT), and their subsequent extensions, may hardly be employed in automotive applications. Indeed, the surrounding environment is rapidly changing over time following the vehicle speed and the dynamics of the vehicle surroundings, limiting the number of valid snapshots that can be processed since in most cases the scenario will change even at every new snapshot. Recently, an adaptive matched filter with Adaptive Matched Filter with Iterative Interference Cancellation (IIC-AMF) has been derived, which extracts the prospective echoes one-by-one, after removing the interference caused by the previously-detected (stronger) signal components. The major limitation of this technique is the need of a search-grid, whereby not only we will face off-grid losses, but more importantly its implementation may become unaffordable in a multi-dimensional parameter space due to the exponential increase of required grid points.
To overcome the limitations of the above methods, we propose to address gridless techniques as the ones mentioned in section 1.
For this application, we consider a wideband mmWave pulse radar and study the problem of super-resolving the echoes generated by multiple prospective targets. The signal model can be represented as a sparse linear combination of atoms. The model unveils the four dimensional environment image with delay and Doppler dimensions as well as a non-uniform 3-D receive array for azimuth and elevation measurement. Furthermore, since this problem has a complexity scaling up with the dimension of the observed data vector, we propose and study a reduced-complexity approximation. Numerical analysis show that the proposed solution outperforms the state-of-the-art IIC-AMF technique.
4.2 Low complexity hybrid precoding strategies for MIMO in sparse mmWave channels An important challenge of mmWave and very relevant for this application, relates to mmWave hardware design, a subject of significant research since the 1970s. While the first mmWave system implementations were based on gallium arsenide (GaAs), the improvement that CMOS technology brought to traditional microwave systems has recently led to the development of CMOS subsystems for mmWave systems. Nonetheless, mmWave system design based on CMOS technology is still under ongoing investigation. A last challenge to take into consideration is the cost of mmWave hardware, mainly due to its reduced size and low power consumption requirements, which creates the need for the design of low complexity MIMO communication solutions in these bands. With MIMO becoming a key technology for 5G due to its improved spectral efficiency and diversity gains, it is important to address the specific challenges of implementing MIMO, or even massive MIMO, in the mmWave band.
For this application, we focus on the design of a low complexity precoder in a multiuser MIMO (MU-MIMO) downlink (DL) scenario. Under conventional microwave propagation, MIMO precoding can be easily implemented digitally at baseband (BB), requiring dedicated Radio Frequency (RF) hardware, i.e. an RF chain for each antenna element. This approach is named fully-digital (FD) implementation. Hybrid (HB) precoding divides the precoding implementation into digital BB processing, bandpass modulation and analog processing, allowing a potential reduction on the number of RF chains and thus enabling a viable precoding architecture for mmWave systems.
By reformulating the hybrid precoder design as a matrix factorization problem, and adopting the atomic norm minimization approach, we propose a new hybrid precoding algorithm that takes advantage of the sparse nature of the mmWave channel and that it is able to closely approach the performance of the optimal fully-digital precoder. We compare, via numerical analysis, the proposed gridless method with state-of-the-art techniques such as OMP and MUSIC showing that the method proposed by this thesis outperforms both state-of-the-art techniques, achieving the same error (distance to optimal precoder) with less RF chains.
4.3 Full channel estimation in MIMO mmWave channels using atomic norm As stated in previous sections, the available bandwidth in the mmWave range is promising, being a very good candidate to meet the specified requirements in wireless communications speed, especially when combined with MIMO technology, which allows to mitigate the severe propagation path loss in this band. Channel estimation in MIMO scenarios is more challenging than in single-antenna transmissions given that the overhead grows with the number of antennas in the system. Thus, leveraging the sparse nature of propagation in the mmWave range to reduce the overhead in the transmission is of high interest. Several studies in the literature have explored compressed sensing techniques to recover the relevant information from the received signal which allows to reduce the number of necessary pilots and therefore, reducing the overhead in the communication. Other works, inspired by recent studies on deep neural networks for sparse signal recovery have applied convolutional neural networks (CNN) for sparse channel estimation in mmWave MIMO channels.
A sparse model as the one studied in the thesis is fully characterized by the structures of both transmitter and receiver, the angular information (departure and arrival) and the channel fading coefficients for every propagation path. We aim to apply the gridless methodology mentioned in section 1 to extract the angular information simultaneously, and with that, reconstruct the channel to provide a full channel estimation. We also compare via numerical analysis the proposed gridless method with state-of-the-art techniques such as OMP and MUSIC showing that the method proposed by this thesis outperforms both state-of-the-art techniques, needing less pilots to achieve the same error in channel estimation.
5 Conclusion We delivered numerical analysis for several applications in this thesis that show how the proposed solution outperforms some of the state-of-the-art techniques for frequency and parameter extraction.
5.1 Automotive radar in mmWave frequency range We have considered a mmWave pulse radar and using the signal measurements at the antenna elements of a 3D receive array we have studied the super-resolution of multiple target echoes. We compare our proposed solution with the state-of-the-art technique IIC-AMF via numerical analysis showing that the proposed technique outperforms the state-of-the-art one.
5.2 Hybrid precoder design The precoder takes advantage of the sparse nature of mmWave channels to reduce the number or RF chains, therefore scaling down the total hardware cost, and yielding a feasible transmission strategy in this millimeter bands. We compared ourselves with some of the state-of-the-art algorithms for hybrid precoding, such as MUSIC and OMP. For a given error in terms of distance to the optimal fully-digital precoder, the proposed ANbased precoder needs less RF chains than the state-of-the-art precoders. Thus, mmWave spectrum can be exploited in a feasible way by the use of MIMO and hybrid precoding.
5.3 Full channel estimation In this application we leverage the sparse nature of mmWave propagation to estimate the channel. This is done by extracting the K frequency parameters of the channel containing information on the angles of departure and arrival. We compare the proposed solution with some of the state-of-the-art methods such as OMP and MUSIC.We see via numerical analysis that our method outperforms both alternative algorithms.
5.4 Future work Next we outline some points for further research: • Explore the theoretical conditions under which robust parameter recovery is achieved for noisy measurements.
• Study the feasibility of a different data gathering and training approach for the Machine Learning model to estimate the number of composite atoms that allow us to do the predictions without having to solve the AN-based optimizations mentioned in section 1.
• Extend the applications to other science and engineering fields where the studied sparse models fit, such as sensor networks, magnetic resonance imaging, computer vision, etc.
