This paper presents a comprehensive study of virtual 3D object manipulation along 4DoF on real surfaces in mixed reality (MR), using hand-based and tangible interactions. A custom cylindrical tangible proxy leverages affordances of physical knobs and tabletop support for stable input. We evaluate both modalities across isolated tasks (2DoF translation, 1DoF rotation/scaling), semi-combined (3DoF translation+rotation), and full 4DoF compound manipulation. We offer analyses of hand interactions, tangible interactions, and their comparison in MR tasks. For hand interactions, compound tasks required repetitive corrections, increasing completion times—yet surprisingly, rotation errors were smaller in compound tasks than in rotation-only tasks. Tangible interactions exhibited significantly larger errors in translation, rotation, and scaling during compound tasks compared to isolated tasks. Crucially, tangible interactions outperformed hand interactions in precision, likely due to tabletop support and constrained 4DoF design. These findings inform designers opting for hand-only interaction (highlighting trade-offs in compound tasks) and those leveraging tangibles (emphasizing precision gains despite compound-task challenges).

Rate Splitting (RS) has emerged as a flexible multiple-access scheme for Multi-User Multiple-Input Multiple-Output (MU-MIMO) systems, generalizing more conventional approaches such as linear precoding or Non-Orthogonal Multiple Access (NOMA), and providing additional robustness against imperfect channel state information at the transmitter. Actual achievable rates cannot be pre-calculated when facing imperfect and/or outdated channel state information at the transmitter (CSIT), so practical real-time mechanisms in the form of link adaptation (LA) are required. The available CSIT is used to design the RS precoders, which can accommodate some degree of robustness, whereas the physical layer feedback reports the outcome of the decoding of the codewords, in the form of acknowledgement (ACK) or negative ACK (NACK), providing the additional meta-robustness needed when the statistical model of CSIT errors is not accurate, and/or the precoders cannot fully track the channel dynamics. We characterize the outage rate region of RS and analyze the convergence of the complete LA-RS system under partial CSIT (non-accurate and non-permanent), providing relevant insights for addressing the adaptation of private and common rates, designing robust precoders with throughput as relevant metric, and handling the time-variant CSIT quality.

A novel coding strategy is proposed for a broadcast setting with two transmitter (TX) antennas and two single-antenna receivers (RX). The strategy consists of using space-time block coding to send a common message (to be decoded by both RXs) across the two TX antennas, while each TX antenna also sends a private message to one of the RXs. The relative weight of the private and common messages from each TX antenna is tuned to maximize the instantaneous achievable sum-rate of the channel. Closed-form expressions for the optimal weight factors are derived. In terms of the generalized degrees of freedom (GDoF) metric, the new scheme is able to achieve the sum-GDoF with finite precision channel state information at the transmitter (CSIT) of the two user broadcast channel. Moreover, as opposed to the existing rate-splitting schemes, the proposed scheme yields instantaneous achievable rates that are independent of the channel phases. This property is instrumental for link adaptation when only magnitude CSIT is available. Our numerical results indeed demonstrate the superiority of the scheme for the 2-user setting in case of magnitude CSIT. Extension to a more general K-user scenario is briefly discussed.

A beam-free approach to channel allocation in a multi-beam four-color satellite coverage area is taken. Non-Orthogonal Multiple Access (NOMA) and Orthogonal Multiple Access (OMA) are compared as methods to serve users non-necessarily located on the reference beam. A proportional fairness policy is employed for the user scheduling. The naturally occurring SNR imbalances in the user terminal population are exploited in such a way that NOMA outperforms OMA, partly due to the blurring of the boundaries of the satellite beams, in such a way that a non-conventional approach to user pairing can reap more benefits from a system perspective.

Fuzzy-Rough Cognitive Networks (FRCNs) are neural networks that use rough information granules with soft boundaries to perform the classification process. Unlike other neural systems, FRCNs are lazy learners in the sense that we can build the whole model when classifying a new instance. This is possible because the weight matrix connecting the neurons is prescriptively programmed. Similar to other lazy learners, the processing time of FRCNs notably increases with the number of instances in the training set, while their performance deteriorates in noisy environments. Aiming at coping with these issues, this paper presents a new FRCN-based algorithm termed Fast k-Fuzzy-Rough Cognitive Network. This variant employs a multi-thread approach for building the information granules as computed by k-fuzzy-rough sets. Numerical simulations on 35 classification datasets show a notable reduction on FRCNs' processing time, while also delivering competitive results when compared to other lazy learners in noisy environments.

Non-Coherent Rate-Splitting (NCRS) was recently proposed as a practical multiuser coding and decoding scheme to increase the spectral efficiency of multibeam satellite communication systems. In this paper, we further study the practical realization of NCRS. We propose a modified coding scheme (NCRS*) that is robust to a nonzero time offset among beams. In NCRS*, as opposed to NCRS, the beams send independently channel encoded and modulated waveforms. We assess the performance of NCRS* in terms of the achievable rate region. It is shown that NCRS* performs worse than NCRS, but better than or comparable to other competing schemes, which, as opposed to NCRS*, require flexible bandwidth allocation or perfect synchronization at the transmitter. We also propose a new N-MAP algorithm for the practical implementation of NCRS* receivers. Similar to the existing UMAP algorithm, N-MAP takes into account the modulation used by, and the time offset between, the signals received from the different beams. In most cases, however, N-MAP has a significantly lower complexity than U-MAP.

Index Modulations, in the form of Spatial Modulation or Polarized Modulation, are gaining traction for both satellite and terrestrial next generation communication systems. Adaptive Spatial Modulation based links are needed to fully exploit the transmission capacity of time-variant channels. The adaptation of code and/or modulation requires a real-time evaluation of the channel achievable rates. Some existing results in the literature present a computational complexity which scales quadratically with the number of transmit antennas and the constellation order. Moreover, the accuracy of these approximations is low and it can lead to wrong Modulation and Coding Scheme selection. In this work we apply a Multilayer Feedforward Neural Network to compute the achievable rate of a generic Index Modulation link. The case of two antennas/polarizations is analyzed in depth, showing not only a one-hundred fold decrement of the Mean Square Error in the estimation of the capacity as compared with existing analytical approximations, but also a fifty times reduction of the computational complexity. Moreover, the extension to an arbitrary number of antennas is explained and supported with simulations. More generally, neural networks can be considered as promising candidates for the practical estimation of complex metrics in communication related settings.

We propose a novel mode of operation for amplify-and-forward relays in which the spectra of the relay input and output signals partially overlap. This partial-duplex relaying mode encompasses half- and full-duplex as particular cases. By viewing the partial-duplex relay as a bandwidth-preserving linear periodically time-varying system, a spectral efficiency analysis under self-interference is developed. In contrast with previous works, self-interference is regarded as a useful information-bearing component rather than simply assimilated to noise. This approach reveals that previous results regarding the impact of self-interference on (full-duplex) relay performance are overly pessimistic. Based on a frequency-domain interpretation of the effect of self-interference, a number of suboptimal decoding architectures at the destination node are also discussed. It is found that the partial-duplex relaying mode may provide an attractive tradeoff between spectral efficiency and receiver complexity.

Mutual Information is the metric that is used to perform link adaptation, which allows to achieve rates near capacity. The computation of adaptive transmission modes is achieved by employing the mapping between the Signal to Noise Ratio and the Mutual Information. Due to the high complexity of the computation of the Mutual Information, this process is performed off-line via Monte Carlo simulations, whose results are stored in look-up tables. However, in Index Modulations, such as Spatial Modulation or Polarized Modulation, this is not feasible since the constellation and the Mutual Information are channel dependent and it would require to compute this metric at each time instant if the channel is time varying. In this paper, we propose different approximations in order to obtain a simple closed-form expression that allows to compute the Mutual Information at each time instant and thus, making feasible the link adaptation.

This paper deals with the problem of beamformer design in a multibeam satellite, which is shared by several clusters of terminals, each served by an earth station, or gateway. Each gateway precodes the symbols addressed to its respective users; the design follows an MMSE criterion, and a judiciously chosen regularization factor accounts for the presence of mutually interfering clusters, extending more classical results applicable to the case of a single centralized station. More importantly, channel statistics can be used instead of instantaneous channel state information, avoiding the exchange of information among gateways through backhaul links. The on-board satellite beamforming weights are designed to exploit the degrees of freedom of the satellite antennas to minimize the noise impact and the interference to some specific users. On-ground beamforming results are provided as a reference to compare the joint performance of MMSE precoders and on-board beamforming network. A non-adaptive design complements the results and makes them more amenable to practical use by designing a coarse beamforming network.

We consider the problem of estimating the frame error rate (FER) of a given memoryless binary symmetric channel by observing the success or failure of transmitted packets. Whereas FER estimation is relatively straightforward if all observations correspond to packets with equal length, the problem becomes considerably more complex when this is not the case. We develop FER estimators when transmissions of different lengths are observed, together with the Cramer-Rao lower bound (CRLB). Although the main focus is on maximum likelihood (ML) estimation, we also obtain low-complexity schemes performing close to optimal in some scenarios. In a second stage, we consider the case in which FER estimation is performed at a node different from the receiver, and incorporate the impairment of unreliable observations by considering noisy ACK/NAK feedback links. The impact of unreliable feedback is analyzed by means of the corresponding CRLB. In this setting, the ML estimator is obtained by applying the expectation-maximization algorithm to jointly estimate the error probabilities of the data and feedback links. Simulation results illustrate the benefits of the proposed estimators.

A large enough separation in the frequency domain between signals going in opposite directions is instrumental to facilitate the implementation of transceivers able to isolate the receiver from the high power transmitted signal. A more flexible allocation of spectrum requires the application of interference cancellation countermeasures. In this paper we address the digital processing involved in the realization of echo cancelling filters in the analog domain, designed to avoid the saturation of the receiver front-end due to the coupling of the transmitted signal and its out-of-band content. The design goal will be the reduction of the sampling rates, so large bandwidth interference waveforms can be attenuated working at sub-Nyquist rates.

Broadband multibeam satellites may require very high feeder link bandwidths to relay all the signals to a unique gateway. This is the case if they enforce an aggressive spectrum reuse across all the user beams and the overall capacity increases accordingly. If the processing is split among different gateways, then the inter-gateway interference must be counteracted to provide acceptable quality of service. In this paper, we propose to blindly estimate the covariance matrix of the received signals at each gateway, and then perform minimum mean-squared error combining based on that estimate and on the subset of channel responses that the gateway can get to learn. Results show that, even with no collaboration among the gateways, significant throughput increases are achieved with respect to a traditional scenario with partial frequency reuse, at the cost of degraded availability levels.

Frame error rate (FER) prediction in wireless communication systems is an important tool with applications to system level simulations and link adaptation, among others. Although in realistic communication scenarios it is expected to have codewords of different lengths, previous work on FER prediction marginally treated the dependency of the FER on the codeword length. In this paper, we present a method to estimate the FER using codewords of different length. We derive a low complexity FER estimator for frames of different length transmitted over a binary symmetric channel of unknown error probability. We extend this technique to coded systems by the use of effective SNR FER predictors. The proposed estimation scheme is shown to outperform other simpler estimation methods.

Rain attenuation is among the major impairments for satellite systems operating in the K-band and above. In this paper, we investigate the impact of spatially correlated rain attenuation on the performance of a multibeam satellite return link. For a comprehensive assessment, an analytical model for the antenna pattern that generates the beams is also proposed. We focus on theoutage capacityof the link and obtain analytical approximations at high and low signal-to-noise ratio. The derived approximations provide insights into the effect of key system parameters such as the interuser distance, the satellite beam radius, or the rain intensity, and simulation results show that it fits tightly with the Monte Carlo results. Additionally, the derived expressions can be easily particularized for the single-user case, providing some novel insights.

Link adaptation in mobile satellite channels is difficult because of the large propagation delay, the frequent signal blockage and the variation of the channel statistics. In this paper, we propose a cross-layer link adaptation strategy that exploits statistical, long-term CSI to increase the throughput when some retransmissions are available; this increase is obtained while meeting a target outage probability constraint. Our strategy takes as inputs the estimated packet error rates of the available modulation and coding schemes (MCS) and their rates; as an output, it returns the optimum sequence of MCS to be used. Results will also show that simple information acquisition strategies can still provide very good results.

Although full-duplex relaying schemes are appealing in order to improve spectral efficiency, simultaneous reception and transmission in the same frequency results in self-interference, distorting the retransmitted signal and making the relay prone to oscillation. Current feedback cancellation techniques by means of adaptive filters are hampered by the fact that the useful and interference signals are highly correlated. We present a new adaptive algorithm which effectively and blindly restores the spectral shape of the desired signal. In contrast with previous schemes, the novel adaptive feedback canceller has low complexity, does not introduce additional delay in the relay station, and partly compensates for multipath propagation.

The insertion of a secondary transmitter in a multicarrier broadcast single frequency network is studied. The secondary information is overlaid on top of the primary waveform, which is also reinforced by the secondary transmitter. The degradation of the primary service due to the presence of echoes in a strong line of sight environment is taken into account, and mitigated with an appropriate filtering at the secondary transmitter. The transmit rate of the secondary system is maximized while keeping the original primary coverage area, defined as a function of a BER bound. The analytical results are verified by means of software simulations and hardware tests, where the importance of the proposed filtering is clearly shown.

Signal processing in satellite applications is usually performed either on-ground or on-board, i.e. at the gateway station or in the payload. Within the framework of the European Space Agency (ESA) SatNEx III study, a hybrid approach has been considered by splitting the processing between the satellite and the gateway, aiming to strike a better balance between performance and payload complexity. The design of a high-capacity multi-beam system has been carried out, to assess the potential applicability of a hybrid space-ground processing architecture (Digisat) for satellite broadband systems; this is achieved via hybrid space-ground beamforming, MIMO and MIMO-MUD, Precoding, as well as Digital Feeder link techniques.

Next generation high throughput satellite systems are expected to rely on the deployment of a high number of beams in the user link, and therefore operate in an interference limited regime. This paper builds on the combination of advanced interference mitigation techniques and ground based beamforming to cope with this increased level of interference among users. Focussing on the forward link of a multi-beam broadband satellite system, we consider the joint design of linear precoding and ground based beamforming at the gateway. This joint design is modeled, and different linear precoding techniques are considered for comparison. The provided simulation results quantify the performance gain generated by this joint design with respect to considering precoding for a fixed on-board beamforming. The robustness of the considered scheme to channel estimation errors is also analyzed.

In Cognitive Radio scenarios channelization information from primary network may be available to the spectral monitor. Under this assumption we propose a spectral estimation algorithm from compressed measurements of a multichannel wideband signal. The analysis of the Cramer-Rao Lower Bound (CRLB) for this estimation problem shows the importance of detecting the underlaying sparsity pattern of the signal. To this end we describe a Bayesian based iterative algorithm that discovers the set of active signals conforming the band and simultaneously reconstructs the spectrum. This iterative spectral estimator is shown to perform close to a Genie-Aided CRLB that includes full knowledge about the sparsity pattern of the channels.

Dynamic Spectrum Leasing (DSL) is a new paradigm for efficient spectrum sharing in cognitive radio networks that was recently proposed in. Unlike in a hierarchical dynamic spectrum access (DSA) network, DSL allows active participation of both the primary and the secondary users in the spectrum sharing process. In this paper we further generalize the DSL game introduced in by allowing for linear multiuser detectors, in particular the matched filter (MF) and linear minimum mean squared error (LMMSE) receivers, at the secondary base stations. We establish the conditions so that the DSL game has desired equilibrium properties. Performance of the proposed system at the equilibrium is compared through simulations under different linear receivers at the secondary base station.

Detection of resampling traces for digital image blind authentication has been addressed recently by A. C. Gallagher and later extended by B. Mahdian and S. Saic. On the other side, it is well known from the synchronization area in communications that prefiltering is an appropriate tool to improve the performance of those schemes exploiting the underlying cyclostationarity of communication signals. Thus, the detection of resampling manipulations improves significantly when the derivative of the interpolated signal is used for covariance computation. This work focuses on the role of prefiltering as a way of boosting resampling traces and, in particular, on the use of derivation.

In this work, we study the presence of almost cyclostationary fields in images for the detection and estimation of digital forgeries. The almost periodically correlated fields in the two-dimensional space are introduced by the necessary interpolation operation associated with the applied spatial transformation. In this theoretical context, we extend a statistical time-domain test for presence of cyclostationarity to the two-dimensional space. The proposed method allows us to estimate the scaling factor and the rotation angle of resized and rotated images, respectively. Examples of the output of our method are shown and comparative results are presented to evaluate the performance of the two-dimensional extension.

We consider the problem of joint non-data-aided (NDA) estimation of carrier frequency and phase offsets, together with that of signal and noise powers, obtained from the baud-rate samples of a linearly modulated signal distorted by Gaussian noise. The Crame¿r-Rao lower bound (CRLB) is derived, for quadrature-symmetric constellations showing that the carrier parameters are decoupled from signal and noise powers. Furthermore, a joint NDA maximum-likelihood estimator is developed, based on the application of the expectation-maximization algorithm, which exhibits a jitter performance close to the CRLB for a wide SNR range.

Starting from local observations, iterative consensus algorithms attempt to drive a sensor network to a common estimate in a decentralized, incremental manner. When additive noise perturbs the sensor exchanges, a decreasing stepsize guarantees convergence under certain conditions, although the design of such stepsize sequence for fastest convergence is an unsettled issue. We present a greedy approach to stepsize sequence design, which minimizes the mean squared error at each iteration. This design requires knowledge of the network topology; in order to overcome this drawback, a modified design based only on average descriptors of the network is also developed.

In this paper, we propose the concept of centralized dynamic spectrum leasing (C-DSL), in which multiple primary users belonging to the same primary system participate in the spectrum leasing process with secondary users (potential bidder for spectrum) under centralized control. We develop a new game-theoretic user interaction model suitable for C-DSL in a cognitive radio network. Dynamic Spectrum Leasing (DSL), proposed in allows active participation of both primary and secondary users in the spectrum sharing process. Motivated by network spectrum utilization considerations, we propose generalizations to the primary system utility function defined in and a new utility function for the secondary users. We also generalize the proposed non-cooperative C-DSL game to allow for linear multiuser detectors at the secondary base stations. We formulate the conditions on the primary system and the secondary user utility functions so that the proposed C-DSL game has desired equilibrium properties. We prove that the proposed C-DSL game has unique Nash equilibria (NE) under both matched filter (MF) and linear minimum mean-squared error (LMMSE) receivers. Equilibrium performance of the system and robustness of the proposed game theoretic adaptive implementation are investigated through simulations.

Dynamic spectrum leasing (DSL) is one of the schemes proposed for dynamic spectrum sharing (DSS) in cognitive radio networks. In DSL, spectrum owners, denoted as primary users, dynamically adjust the amount of secondary interference they are willing to tolerate in response to the demand from secondary transmitters. In this correspondence we investigate how much can be gained by primary users if this limited interaction with secondary system is allowed, compared to a scheme in which the interference cap allowed by primary users is fixed {a priori} by a regulatory authority. To that end, we define performance metrics for both primary and secondary systems based on the theoretically achievable multiuser sum-rate of the secondary system and analyze both schemes' behavior with respect to different system parameters. This analysis shows that (i) in dynamic environments DSL based schemes may present an important advantage over other schemes with fixed interference constraints, and (ii) DSL schemes are robust against inaccurate a priori information that may degrade system performance.

Signal-to-noise ratio (SNR) estimation for linearly modulated signals is addressed in this letter, focusing on envelope-based estimators, which are robust to carrier offsets and phase jitter, and on the challenging case of nonconstant modulus constellations. For comparison purposes, the true Cramer-Rao lower bound is numerically evaluated, obtaining an analytical expression in closed form for the asymptotic case of high SNR values, which quantifies the performance loss with respect to coherent estimation. As the maximum-likelihood algorithm is too complex for practical implementation, an expectation-maximization (EM) approach is proposed, achieving a good tradeoff between complexity and performance for medium-to-high SNRs. Finally, a hybrid scheme based on EM and moments-based estimates is suggested, which performs close to the theoretical limit over a wide SNR range.

The problem of distributed sequential estimation of a nonrandom parameter over noisy communication links is considered, with observations that are correlated spatially across the sensor field. A recursive algorithm for updating the sequential estimator is derived assuming wide-sense stationary observations. It is shown that the performance of the sequential estimator is a trade-off between the quality of node observations and communication channels. A sufficient condition for the convergence of the estimator variance is derived, and asymptotic expressions for variance when this condition is met are obtained for both iid and correlated observations.

Quantization-based methods, such as dither modulation (DM), have gained wide acceptance due to their host rejection capabilities which afford significant performance gains over spread-spectrum-based methods in additive white Gaussian channels. Unfortunately, existing quantization-based schemes are not robust against simple linear-time-invariant (LTI) filtering, which is a common operation with multimedia signals. We propose a new algorithm, named discrete Fourier transform-rational dither modulation (DFT-RDM) which is robust against LTI filtering and yet does not assume any prior knowledge of the filter at either the embedder or the detector. DFT-RDM basically combines a DFT operation with a quantization-based scheme robust to amplitude scaling. Two easily implementable improvements over the basic DFT-RDM are proposed: windowing and spreading. In particular, the latter leads to performance gains that are much larger than those achieved with spreading in regular DM. We also provide a thorough analysis of our scheme which leads to both accurate predictions and bounds on the per-DFT-channel bit-error rate, for the basic DFT-RDM and its combination with spreading and windowing. These tools let the designer choose the main embedding parameters without actually requiring any simulation. The results of several simulations for practical filters validating our analysis are presented as well. The benefits of combining DFT-RDM with windowing, spreading, and Reed-Solomon channel coding are illustrated with an example.

A novel quantization-based data-hiding method, named rational dither modulation (RDM), is presented. This method amounts to simple modifications of the well-known dither modulation (DM) scheme, which is largely vulnerable to scaling attacks. With such modifications, RDM becomes invariant to those attacks. Since RDM does not work by trying to estimate the step-size of the quantizers, it does not need any pilot-sequence. Moreover, RDM is suitable for a scalar operation, thus avoiding the cumbersome constructions of spherical codes. It is also shown that RDM approaches the performance of DM asymptotically with the size of the memory needed for the method to operate. Simulation results show the accuracy of our theoretical analysis and the superiority of RDM compared to the improved spread spectrum method.

Linear periodically time-varying systems are the basis of many practical systems, ranging from electrical networks with periodically time-varying components to filters which incorporate modulators in the signal path. Despite the pervasive use of digital systems, the study of analog time-varying systems can be considered an important branch of applied mathematics and control, with applications to' signal processing and communications. Given the well-known modulators filter bank description of a linear periodically time-varying system the spectral transformation between the input and the output would follow readily. This paper shows a simple way to find such a description for a continuous-time linear periodically time-varying system within a feedback loop, motivated by applications such as gated phase-locked loops.

A novel car speed estimation technique from a pair of omnidirectional microphones is presented. This approach is based on the maximum likelihood principle and it directly estimates the speed without any assumptions on the acoustic signal emitted by the vehicle. This has the advantages of bypassing troublesome intermediate delay estimation steps as well as eliminating the need for an accurate yet general enough acoustic model. An analysis of the estimate for the particular case of a narrowband source is provided. The estimation algorithm is well suited to DSP implementation and performs well with preliminary field data.

This paper studies the performance of the clock transfer scheme for burst-mode communication systems for which data are received during short, equally spaced intervals. Its main focus is on satellite-based time-division multiple-access (TDMA) communication systems with data regeneration and switching onboard the satellite, although the results apply to other TDMA systems as well. The system reference clock is generated onboard from an incoming, very stable ground source, based on a burst-mode demodulator that extracts the clock from a discontinuous modulated carrier due to the bursty nature of TDMA signals. If good enough, this onboard regenerated clock avoids the use of bulky and expensive clocks in the satellite payload and can act as the master clock of the TDMA system.

This work considers the implementation of recursive identification algorithms based on hyperstability concepts with polyphase structures. It is shown that the strictly positive real (SPR) condition required for convergence of these schemes can always be met by using a sufficiently high polyphase expansion factor M. For a given M, the degree of persistent excitation required for parameter convergence is obtained. When a priori knowledge about the unknown system is available, a compensating filter can be designed to avoid the need for a high M.

This paper is concerned with systems in which the output error of an adaptive IIR filter is subsequently filtered, e.g., an active noise control system where a transfer function models the path between the injection point of the cancellation noise and the point where the residual error is measured. We propose a family of algorithms suited to this type of scenarios, deriving conditions for their deterministic convergence. The analysis of the convergence is particularized to the filtered-U recursive LMS algorithm, a popular scheme whose global convergence has never been proved formally. Finally, some results based on real measurements are also presented.

The paper addresses the properties of the noise in gridding reconstruction, an algorithm for reconstruction from nonuniform samples. Sequences with time-varying gradients, such as spiral or projection reconstruction (PR) techniques, are being increasingly used in magnetic resonance imaging (MRI). Since these techniques sample k-space nonuniformly, some kind of algorithm is needed to map the data onto a Cartesian frame to allow an inverse Fourier transform through an FFT. The authors present an analytical characterization of the image noise after gridding and inverse Fourier transform for the most popular sampling techniques used in MRI.