In cognitive radio networks, compressive sensing has the potential to allow a secondary user to efficiently monitor a wideband spectrum at a sub-Nyquist sampling rate without complex hardware. In general, compressive sensing techniques leverage the assumption of sparsity of the wideband spectrum to recover the spectrum by solving a set of ill-posed linear equations. In this paper, we adopt the framework of a generative adversarial neural network (GAN) in deep learning and propose a deep compressive spectrum sensing GAN (DCSS-GAN), where two neural networks are trained to compete with each other to recover the spectrum from undersampled samples in the time domain. The proposed DCSS-GAN is a data-driven learning approach that does not require a priori statistics about the radio environment. In addition, it is an end-to-end algorithm that directly recovers the information of spectrum occupancy from raw samples and without the need of energy detection. Various simulations show that the proposed DCSS-GAN has a 12.3% to 16.2% performance gain on prediction accuracy at a 1/8th compression ratio compared to the conventional LASSO approach.

We derive new optimal power allocation solutions for a distributed antenna system with orthogonal frequency division multiplexing (DAS-OFDM), in which K remote radio heads (RRHs) allocate power over N subchannels with both in-dividual RRH and total system power constraints. The individual RRH constraints may be necessary due to hardware limitations, while the total power constraint allows the system to limit the overall energy consumption for cost and/or green environmental factors. In order to obtain some insight on the optimal power allocation, we focus on a simple two-RRH case in this paper. The resulting optimal power allocation solution has an interesting sparse feature: Among N subchannels, at most 1 subchannel can be allocated power for joint transmission by the two RRHs, and the rest of the subchannels must be served by a single RRH. A novel subchannel rearrangement scheme is presented to identify the possible joint-transmission subchannel, as well as which RRH the remaining subchannels are associated with. Numerical results are presented to verify our theoretical results.

This paper analyzes the optimal energy beamforming solution for a multiple-input single-output wireless power transfer (WPT) system over frequency-selective fading channels with joint total and individual antenna power constraints. To maximize the total harvested energy, we derive the optimal co-phasing power allocation rule which reveals that all K antennas will participate in energy beamforming with T<; K antennas transmitting with their maximum individual powers due to the total power constraint. We prove that optimally no more than T+1 subchannels are selected for power allocation. We highlight that the optimal power allocation solution can be efficiently obtained based on the corresponding low- complexity dual problem. Our proposed power allocation algorithm generalizes previous solutions considering only total or individual power constraints. Numerical examples verify our theoretical results and show the impact of the joint total and individual power constraints on the average harvested power.

Wireless sensor networks (WSNs) are the foundation of the Internet of Things (IoT), and in the era of the fifth generation of wireless communication networks, they are envisioned to be truly biquitous, reliable, scalable, and energy efficient. To this end, topology control is an important mechanism to realize self-organized WSNs that are capable of adapting to the dynamics of the environment. Topology optimization is combinatorial in nature, and generally is NP-hard to solve. Most existing algorithms leverage heuristic rules to reduce the number of search candidates so as to obtain a suboptimal solution in a certain sense. In this paper, we propose a deep reinforcement learning-based topology optimization algorithm, a unified search framework, for self-organized energy-efficient WSNs. Specifically, the proposed algorithm uses a deep neural network to guide a Monte Carlo tree search to roll out simulations, and the results from the tree search reinforce the learning of the neural network. In addition, the proposed algorithm is an anytime algorithm that keeps improving the solution with an increasing amount of computing resources. Various simulations show that the proposed algorithm achieves better performance as compared to heuristic solutions, and is capable of adapting to environment and network changes without restarting the algorithm from scratch.

We propose relay-energy access points (REAPs) as a novel Internet-of-Things (IoT) network architecture that supports wireless energy harvesting of IoT devices and long-range relay communications to data access points (DAPs). In this paper, we formulate a Stackelberg game for energy trading with the REAPs and a single relay selection (SRS) scheme for the REAP to assist communications with the DAP. We consider a harvest-then-transmit protocol in which REAPs transmit energy-bearing signals to multiple (active) IoT devices for energy harvesting. Using the harvested energy, the IoT devices then send information to the DAP via the REAP. Numerical examples highlight the effectiveness of the proposed game-theoretic solution and improvements achieved by relaying data through REAPs.

Despite promising capacity gains, small cell densifications in ultra-dense HetNets can lead to frequent handovers (HOs), which may cause significant network overheads and decline in user experience. With the aim of modelling handovers in the context of ultra-dense HetNets, we propose a novel approach to model coverages of overlapping small cells. Based on this model, we derive the cumulative distribution function (CDF) of the user equipment's (UE) time-of-stay in small cells using boundary length and chord length distributions of small cell coverage. Our model is comprehensive enough to capture the nature of both inter-tier and intra-tier HOs in small cell networks, the latter of which is a major challenge in ultra-dense HetNets. Our analytical results can provide guidance for optimisations of HO parameters based on user velocity and small cell density to reduce network overheads and improve user experience.

This paper analyzes the optimal transmission strategy and power allocation for a distributed wireless power transfer (WPT) system operating over frequency-selective fading channels. We consider K coordinated energy transmitters (CETs) coherently transmitting energy to a single user over N > K subchannels with individual power constraints. To maximize the total harvested energy, we derive the optimal co-phasing power allocation rule which has the following properties: 1) For any given subchannel, if the optimal power allocation of one CET is zero, then the power allocated by all the other K - 1 CETs to that subchannel is also zero (i.e., the subchannel is inactive); 2) For the non-zero power subchannels, the optimal power allocation obeys a proportionality principle that establishes a relationship between the powers allocated by all K CETs to all active subchannels. Based on this property, we prove that the optimal distributed WPT strategy is for all CETs to select no more than K subchannels. This is in sharp contrast to wireless information transmission where more than K subchannels may be used for capacity maximization. Numerical examples verify our theoretical results and show the performance gains of our proposed scheme compared to two benchmark schemes.

This paper derives the optimal power allocation for a coordinated orthogonal frequency-division multiplexing (OFDM) transmission system in which K coordinated transmission points (CTPs) coherently transmit and allocate power across N subchannels under both total and individual power constraints. In maximizing the system capacity, previous works showed that, under a total power constraint, the optimal transmission strategy is a maximum-ratio transmission (MRT) for CTPs with a waterfilling type of power allocation solution for the subchannels. For CTPs with both total and individual power constraints, we derive a new optimal co-phasing power allocation with the following property: For any given subchannel, if the optimal power allocation of one CTP is zero, then the power allocation of all the other K - 1 CTPs on that subchannel must also be zero; otherwise, the non-zero power allocation on all CTPs must follow a proportional principle which establishes the relationship between the optimal power allocation for all subchannels and all CTPs. This property highlights that the optimal power allocation for CTPs with individual power constraints is different from waterfilling and MRT, as more power is not necessarily allocated to the subchannels with better channel conditions. Numerical results are presented to verify our theoretical findings.

This paper considers the use of full-duplex technology in cognitive radio networks to allow secondary users to sense the presence of primary users and transmit data simultaneously. This is the main advantage over half-duplex radios. In such networks, the so-called sensing-throughput trade-off exists due to the fact that while a higher transmit power results in higher secondary network throughput, sensing performance is degraded by the self-interference at the full-duplex transceiver. This paper presents a novel deep reinforcement learning-based joint spectrum sensing and power control algorithm for downlink communications in a cognitive small cell. The proposed algorithm can adapt to the unknown radio environment to transmit data opportunistically to the secondary users while avoiding interference to the primary network. Simulation results show that our algorithm achieves better performance than the traditional energy detection-based sensing method and performs close to a genie-aided method with the optimal spectrum utilization, especially in the high-SNR regime.

This paper considers an orthogonal frequency division multiplexing (OFDM) coordinated transmission system in which K coordinated transmission points (CTPs) coherently transmit and allocate power across N subchannels under individual power constraints. In maximizing the system capacity, we derive an optimal co-phasing power allocation solution with the following property: For any subchannel, if the power allocation of one CTP is zero, then the power allocation of all the other CTPs must be zero. Otherwise, the non-zero power allocation of all CTPs follows a proportional rule establishing a relationship between the power allocation for all the subchannels and CTPs. This highlights that the optimal power allocation is different from classical waterfilling and maximum ratio transmission (MRT), as more power is not necessarily allocated to the subchannels with better channel conditions. Based on this property and our derived solution, we successfully reduce the constrained optimization problem with NK variables into an unconstrained one with only K variables, which simplifies computation significantly. Numerical results are presented to verify our theoretical findings.

In this paper, we propose a new optimization framework that supports energy harvesting user fairness in a wireless information and power transfer OFDMA network. We consider the downlink of a multiuser network, where non-overlapping subchannels are allocated to the users with power splitting receivers for information transfer and energy harvesting. We propose a two-stage optimization algorithm to maximize the minimum energy harvested by the users while satisfying a minimum required user rate. We apply the proposed algorithm to optimize the subchannel allocation, power allocation, and power splitting ratio for each user. Numerical examples are provided to highlight the fairness and efficiency of our two-stage algorithm.

Cooperative relays have been used in many wireless applications to reduce transmit power as well as adding receiver diversity. However, the energy efficiency trade-offs of relay networks have not been thoroughly investigated using a more complete energy model of the system. In this paper we present an energy model for a cooperative dual-relay system using decode-and-forward (DF) protocol with receiver diversity. Energy consumption is based on models of both analog and digital components for transmission and reception. A quartic transmit power amplifier (PA) energy equation is derived that is a function of the following parameters: relay location, relative transmit power allocation amongst the relays, spectral efficiency, transmission distance, pathloss, and target bit-error rate (BER). The energies of the remaining circuit components are added to form a multi-relay DF system energy equation, which is then numerically evaluated to minimize total energy by finding the optimal relay locations, relay transmit PA power allocations, and spectral efficiency. Results are compared to our previous work that considered the energy of direct transmission and a single-relay system, and the minimum-energy system as a function of source-destination distance is considered.

We propose a new resource allocation algorithm to support user fairness for downlink coordinated multipoint (CoMP) transmission with joint processing in multiuser multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems. The proposed algorithm utilizes successive convex approximation (SCA) to dynamically allocate the transmit powers between multiple CoMP base stations transmitting to multiple co-channel user terminals (UTs). The aim is to minimize the total transmit power subject to per-antenna power and per-UT rate constraints. Simulation results compare our SCA-based algorithm with iterative waterfilling.

In this study, the authors propose a continuous rate and power allocation algorithm for multiuser downlink multiple‐input–multiple‐output orthogonal frequency‐division multiplexing systems with coordinated multi‐point transmission. The optimisation problem is formulated as a weighted sum‐rate maximisation problem subject to per‐antenna power constraints across multiple cooperating base stations (BSs). The practical consideration of the per‐antenna power constraint limits the average transmit antenna power which indirectly controls the inherent issue of high peak powers in OFDM. The proposed algorithm employs a successive convex approximation (SCA) technique to dynamically allocate powers to multiple co‐channel user terminals. They provide a convexity proof of the transformed optimisation problem and they show that the proposed algorithm converges to a unique solution. They compare the proposed SCA algorithm with two alternative approaches: (i) iterative waterfilling (IWF) and (ii) zero‐forcing beamforming (ZFB) with semi‐orthogonal user selection under both per‐antenna and per‐BS power constraint scenarios. Their simulation results highlight that the proposed SCA algorithm outperforms the existing IWF and ZFB in noise‐limited environments under both power constraint scenarios.

In this paper, we propose two efficient and practical resource allocation algorithms to maximize the weighted sum-rate of coordinated multipoint (CoMP) transmission with joint processing in multiuser multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems subject to per-antenna power constraints. We first propose a continuous-rate algorithm that utilizes successive convex approximation (SCA) to dynamically allocate the transmit powers of multiple CoMP base stations (BSs) transmitting to multiple co-channel user terminals (UTs). Next, we propose a discrete-rate algorithm that takes the continuous-rate result as a starting point and redistributes the transmit powers to obtain a discrete solution. Simulation results are provided to benchmark our continuous-rate algorithm with two alternative approaches: iterative waterfilling (IWF), and zero-forcing beamforming (ZFB). Results show that SCA provides significant sum-rate improvements over IWF in medium to high interference scenarios, and outperforms ZFB in low to medium interference scenarios. Moreover, our proposed discrete rate algorithm produces a higher discrete sum-rate with much lower computational complexity compared to existing algorithms.

In this paper, we consider a heterogeneous network where the statistics of fading channels between the Base Station (BS) and various users are different. The network can operate either in a single-user mode (i.e. the BS transmits to one user at a time) or in a multi-user mode via opportunistic beamforming. Given the profile of users' fading channel statistics, we are able to derive the exact downlink capacities associated with different modes assuming a 2-Tx BS, as well as determine the optimal mode that maximizes the throughput. The switching point between transmission modes depends on the signal-to-noise ratio (SNR), the distribution of users and the path loss of the fading channel. Simulations are provided to verify the derived expressions and to illustrate the movement of the switching point. In addition, a method is proposed to address the fairness issue in user scheduling, which allows for a reasonable trade-off between performance and fairness.

This paper describes an efficient approach to discrete rate and power allocation for MIMO-OFDM systems with individual power-constrained antennas. The system is configured with multiple base stations, each with a single transmit antenna, for coordinated multi-point transmissions. The traditional OFDM loading problem can be extended to the MIMO-OFDM case provided that the channel state information between all transmit and receive antennas is known. However, due to unequidistant between coordinated base stations and user terminal, a perantenna average power constraint is more practical than a total average power constraint in order to keep each antenna peak power manageable to avoid low power efficiency of the power amplifier in each antenna. The replacement of the per-antenna power constraint results in a more complex loading optimization problem, and we show it can still be solved efficiently using the dual Lagrange decomposition method. A multidimensional Lagrange search is needed to find the optimal solution and is accomplished via the Ellipsoid Method and the Analytical Center Cutting-Plane Method.

This paper looks at the problem of theoretically characterizing the nonlinear biasing and clipping (BAC) effects on an optical Orthogonal Frequency-Division Multiplexing (OFDM) signal in intensity-modulated, direct-detected (IM/DD) optical systems. Due to the unipolarity of the IM/DD optical channel, a large DC bias and associated nonlinear clipping distortion (NLCD) is inevitable, resulting in a significant performance penalty. This NLCD can be well modelled as a linear deterministic attenuation plus an uncorrelated random additive clipping noise in the time domain. In the frequency domain, the NLCD results in an additive or impulsive noise on the received OFDM constellation. A total effective signal-to-noise ratio (SNR) formula is then presented which is a function of biasing power, modulation constellation and receiver SNR figure. This suggests that rather than eliminating all clippings, the system performance is indeed optimized with some deliberately introduced NLCD as a result of higher power efficiency. Analytical results are in agreement with simulations for various cases which help us to accurately and efficiently evaluate the performance of such systems.

This paper investigates Orthogonal Frequency- Division Multiplexing (OFDM) transmission in intensity modulated, direct-detected (IM/DD) optical systems. Due to the unipolarity of the IM/DD optical channel, DC biasing and associated nonlinear clipping distortion (NLCD) is inevitable, unless an anti-periodic signal is created by limiting bit loading to only half of the available bandwidth. To generalize this idea, we first provide a class of such signals generated with different subchannel combinations. Using the theoretical characterization of NLCD on these signals, we show that by parallel transmission and successive decoding, a set of anti-periodic OFDM signals can be transmitted simultaneously and recovered without a distortion penalty. The proposed method requires minimum change on the existing optical plant, and has the potential to achieve both high power and high spectral efficiencies.

This paper looks at the problem of performance enhancement for orthogonal frequency-division multiplexing (OFDM) transmission in intensity modulated, direct detected (IM/DD) optical systems. Due to the unipolarity of the IM/DD channel, asymmetric clipping and associated biasing is inevitable, which results in a significant signal degradation. ACO-OFDM has been previously proposed to avoid this detrimental distortion with the cost of a lower spectral efficiency. In this paper, with the help of a detailed analysis of such distortion, a simple combining algorithm is proposed. This algorithm extracts the instructive information inside the clipping distortion, and requires no changes to the existing optical plant and no extra transmitting power. By exploiting this spectral diversity gain, the receiver SNR can be significantly improved.

This paper investigates the effects of nonlinear biasing and clipping (BAC) on a random optical orthogonal frequency-division multiplexing (OFDM) signal with direct detection. We look at the problem of theoretical performance evaluation of such a system. Specifically, we derive an approximation for the symbol error rate (SER). The nonlinear BAC process is modeled as a linear deterministic gain plus a random additive clipping noise. The total effective SNR is presented as a function of biasing power, modulation constellation and receiver SNR. Analytical results are in good agreement with simulations in various cases.

We consider the multi-user discrete bit loading problem in OFDMA systems with objective and constraints in terms of ergodic rates and powers. This differs from most existing techniques, which allocate rate and power according to instantaneous constraints and channel conditions. The maximum weighted ergodic sum-rate optimization problem is non-convex in general, but a dual decomposition results in an convex optimization problem where the duality gap vanishes with an increasingly large number of subchannels. We propose an approach tailored for the discrete case based on our previous work that is a flexible, near optimal, and results in an efficient algorithm. Simulation results showing convergence behavior are presented.

We consider the discrete bit loading problem in OFDM systems with objective and constraints in terms of ergodic rates and powers. This differs from almost all existing techniques, which allocate rates and powers using instantaneous channel gains. The resulting optimization problem is still convex in the single user case, and a dual decomposition is employed along with a tailored approach to the discrete rate case used in our previous work. Our proposed approach is a flexible, near optimal, and efficient algorithm, and simulations results are presented.

This paper considers the discrete bit loading problem in OFDMA downlink systems where each user has a minimum-rate constaint. Whereas OFDMA loading problems are generally non-convex and combinatorial, their optimal solutions can be approximated with a dual decomposition. The dual approach in the per-user minimum-rate case has a K-dimensional Lagrange search space for K users, and as a result is a very difficult problem. We consider two cases of the minimum-rate problem: instantaneous and ergodic allocation, and compare approaches and complexity in obtaining their solutions. We propose a subgradient ergodic allocation approach that is both practical and yields near optimal solutions.

The high peak-to-average-power ratio (PAR) of orthogonal frequency division multiplexing (OFDM) transmission systems significantly reduces the power efficiency or performance of such systems, and many techniques have been proposed to deal with this problem. Recently, the active constellation extension (ACE) algorithm has received much attention, where the main idea is to reduce the PAR by introducing a perturbation of the transmit constellation that does not reduce the constellationpsilas minimum-distance. In this paper, a different approach is taken where the constellationpsilas minimum-distance is reduced to lower the PAR but the constellation perturbation is done in an optimal way to improve the system performance. The problem of optimally trading off PAR reduction and constellation distortion is shown to be a convex optimization problem that could be solved very efficiently. The proposed idea has no data-rate loss and does not require the transmission of side-information.

This paper proposes a flexible, efficient approach to discrete rate and power allocation for OFDMA systems. We consider the weighted-sum-rate maximization problem, and utilize the Lagrange dual decomposition (as done in some recent works) along with our previous single-user approach to simplify the non-convex, combinatorial optimization problem. The result is an optimal algorithm (up to a very small worst-case duality gap) with very low complexity. Furthermore, the approach is very flexible and can easily incorporate such factors as various QoS measures, fading channels, imperfect channel knowledge and prediction errors, and error control codes provided that a simple power scalability property is satisfied.

Orthogonal frequency division multiple access (OFDMA) has been widely recognized as a promising solution for broadband wireless networks. Unfortunately, in OFDMA uplink scenarios, high peak-to-average power ratios (PARs) dramatically degrade the power efficiency of the mobile users. This turns out to be one of the most critical problems when implementing OFDMA systems. In this paper, we propose efficient PAR reduction schemes based upon tone reservation. We show by simulation that the PAR can be significantly reduced without incurring multiple- access interference and with a very small loss in bandwidth.

This paper investigates the power minimization problem in OFDMA based wireless multihop systems with the knowledge of channel state information (CSI). The joint allocation problem is divided into two stages: bandwidth allocation and subchannel assignment. Efficient optimal and suboptimal algorithms for both stages are jointed together to provide beneficial and stable improvements by exploiting a form of multiuser diversity via intelligent subchannel partitioning on the frequency-selective fading channel.

This paper looks at the problem of total power minimization in multiuser OFDM systems while maintaining individual fairness (rate) and QoS (BER) requirements. A practical and efficient subchannel, power and bit allocation algorithms is described, where an average-SNR approximation is used to substantially reduce complexity. The proposed algorithm guarantees improvement through each iteration and converges quickly to a stable suboptimal solution. Numerical results and complexity analysis show that the proposed algorithm offers beneficial performance improvement compared to existing approaches.

The implementation of practical adaptive resource allocation scheme remains a key criterion to be satisfied for realising spectrally efficient multitone wireless communications. The ever-increasing demand for spectrally efficient broadband wireless transmission technologies has spurred intensive research leading towards the implementation of adaptive OFDM and adaptive MIMO systems. Efforts in this direction have been frustrated however by the lack of a clear and accurate description of the fading behaviour typically encountered in the broadband wireless transmission environment. This has been partially been overcome by the use of mathematical modelling which captures certain large-scale characteristics of the channel and facilitates theoretical research. The "average" channel parameters gleaned from these processes is typically then used to inform the design and configuration of wireless networking equipment after the broad application of generous safety margins. The resulting solution is therefore quite robust to certain transient channel quality degradation yet the generous safety tolerances render it unable to exploit other transient transmission quality improvements We seek to overcome the problems associated with this approach by applying a theoretically sound novel adaptive resource allocation framework to actual broadband wireless channel development data. The allocation framework is derived from the optimal OFDM allocation scheme for a known channel: the channel development data is obtained from actual measurement of a broadband wireless mobile environment, Prediction techniques are employed to overcome the time lag between channel assessment and symbol transmission. We present the details of the predictive resource allocation scheme used and include a broad characterisation of the transmission environment in terms of the time-varying fading processes observed. We provide some results of the application of this scheme as typical performance levels that may be achieved in an actual transmission environment.

We propose ad framework to optimally allocate transmission resources for a digital-user multichannel transmission link operating a dynamically-fading wireless environment. Beginning with the statistical properties of fading phenomena observed in such channels, we demonstrate how convex optimisation theory may be harnessed for this purpose. Examples are presented to illustrate how service quality criteria may be used to dynamically identify operating points for the allocation algorithm.

The high peak-to-average power ratio (PAR) in orthogonal frequency division multiplexing (OFDM) modulation systems can significantly limit performance. Methods exist which alter or introduce new signal constellations to combat large signal peaks. We present an improved PAR-reduction method that dynamically extends outer constellation points in active (data-carrying) channels, within margin-preserving constraints, in order to minimize the peak magnitude. This scheme simultaneously decreases the bit error rate slightly while substantially reducing the peak magnitude of an OFDM transmit block and requiring no side information. PAR reduction for an approximated analog signal is considered, and a 4.5 dB reduction is obtained for 256-channel QPSK OFDM. The results show great promise for use in commercial systems.

The high peak-to-average power ratio (PAR) in OFDM systems can significantly limit performance. Tone reservation techniques use unavailable or reserved tones to design a peak-cancelling signal that lowers the PAR of a transmit data block. These techniques have been shown to reduce the PAR for real-baseband OFDM signals, such as for ADSL. Our fast-converging real-baseband tone-reservation approach based upon active-set methods is extended to reduce PAR in complex-baseband signals. This new technique converges very quickly toward a minimum-PAR solution at a low cost.

We present an optimal, computationally efficient, integer-bit power allocation algorithm for discrete multitone modulation. Using efficient lookup table searches and a Lagrange-multiplier bisection search, our algorithm converges faster to the optimal solution than existing techniques and can replace the use of suboptimal methods because of its low computational complexity. Fast algorithms are developed for the data rate and performance margin maximization problems.

We present an optimal, efficient power allocation algorithm for discrete multitone modulation (DMT). Using efficient lookup table searches and a Lagrange multiplier bisection search, our algorithm converges much faster to the optimal solution than existing techniques and can replace the use of suboptimal methods because of its low computational complexity. A fast algorithm is developed and a pseudocode is provided.

Interference suppression in spread spectrum communication systems is often essential for achieving maximum system performance. Existing interference suppression methods do not perform well for most types of nonstationary interference. We first consider interference suppression schemes based on adaptive orthogonal time-frequency decompositions, such as wavelet packet and arbitrary dyadic time-frequency tilings. These methods often reduce interference substantially, but their performance can vary dramatically with minor changes in interference characteristics such as the center frequency. To circumvent these drawbacks, we propose a multiple overdetermined tiling (MODT) with an accompanying blind interference excision scheme which appears very promising for mitigating time-frequency-concentrated interference. Simulations with narrowband, impulsive, and simultaneous impulsive and narrowband interference compare the performance of the various methods and illustrate the promise of approaches based on multiple overdetermined tilings.

It is commonly known that wavelet expansions are very sensitive to translation of an input signal due to their dyadic structure. Without proper time alignment of the signal to the decomposition basis vectors, a compact expansion may be unattainable. Recently, work has been done to combat translation sensitivity by determining the best translation-invariant expansion of a signal with respect to a set of bases. However, the dyadic structure of wavelet expansions also results in frequency alignment sensitivity. In this work, we introduce a frequency-shifted wavelet packet library as well as an efficient best basis algorithm for determining the best signal representation among all frequency shifts of a signal.