Designing Gabor windows using convex optimization

Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time-frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found

Approximation of dual Gabor frames, window decay, and wireless communications

We consider three problems for Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in $L_2(R)$ by finite-dimensional methods. Utilizing Wexler-Raz type duality relations we derive a method to approximate the dual Gabor frame, that is much simpler than previously proposed techniques. Furthermore it enables us to give estimates for the approximation rate when the dimension of the finite model approaches infinity. The second problem concerns the relation between the decay of the window function $g$ and its dual $\gamma$. Based on results on commutative Banach algebras and Laurent operators we derive a general condition under which the dual $\gamma$ inherits the decay properties of $g$. The third problem concerns the design of pulse shapes for orthogonal frequency division multiplex (OFDM) systems for time- and frequency dispersive channels. In particular, we provide a theoretical foundation for a recently proposed algorithm to construct orthogonal transmission functions that are well localized in the time-frequency plane

Multipliers for Continuous Frames in Hilbert Spaces

In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calder\'on- Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting

A review of RFI mitigation techniques in microwave radiometry

Radio frequency interference (RFI) is a well-known problem in microwave radiometry (MWR). Any undesired signal overlapping the MWR protected frequency bands introduces a bias in the measurements, which can corrupt the retrieved geophysical parameters. This paper presents a literature review of RFI detection and mitigation techniques for microwave radiometry from space. The reviewed techniques are divided between real aperture and aperture synthesis. A discussion and assessment of the application of RFI mitigation techniques is presented for each type of radiometer.Peer ReviewedPostprint (published version

A survey of uncertainty principles and some signal processing applications

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, emphasize their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Space/time/frequency methods in adaptive radar

Radar systems may be processed with various space, time and frequency techniques. Advanced radar systems are required to detect targets in the presence of jamming and clutter. This work studies the application of two types of radar systems. It is well known that targets moving along-track within a Synthetic Aperture Radar field of view are imaged as defocused objects. The SAR stripmap mode is tuned to stationary ground targets and the mismatch between the SAR processing parameters and the target motion parameters causes the energy to spill over to adjacent image pixels, thus hindering target feature extraction and reducing the probability of detection. The problem can be remedied by generating the image using a filter matched to the actual target motion parameters, effectively focusing the SAR image on the target. For a fixed rate of motion the target velocity can be estimated from the slope of the Doppler frequency characteristic. The problem is similar to the classical problem of estimating the instantaneous frequency of a linear FM signal (chirp). The Wigner-Ville distribution, the Gabor expansion, the Short-Time Fourier transform and the Continuous Wavelet Transform are compared with respect to their performance in noisy SAR data to estimate the instantaneous Doppler frequency of range compressed SAR data. It is shown that these methods exhibit sharp signal-to-noise threshold effects. The space-time radar problem is well suited to the application of techniques that take advantage of the low-rank property of the space-time covariance matrix. It is shown that reduced-rank methods outperform full-rank space-time adaptive processing when the space-time covariance matrix is estimated from a dataset with limited support. The utility of reduced-rank methods is demonstrated by theoretical analysis, simulations and analysis of real data. It is shown that reduced-rank processing has two effects on the performance: increased statistical stability which tends to improve performance, and introduction of a bias which lowers the signal-to-noise ratio. A method for evaluating the theoretical conditioned SNR for fixed reduced-rank transforms is also presented