68,967 research outputs found
Transform Domain Analysis of Sequences
In cryptanalysis, security of ciphers vis-a-vis attacks is gauged against
three criteria of complexities, i.e., computations, memory and time. Some
features may not be so apparent in a particular domain, and their analysis in a
transformed domain often reveals interesting patterns. Moreover, the complexity
criteria in different domains are different and performance improvements are
often achieved by transforming the problem in an alternate domain. Owing to the
results of coding theory and signal processing, Discrete Fourier Transform
(DFT) based attacks have proven to be efficient than algebraic attacks in terms
of their computational complexity. Motivated by DFT based attacks, we present a
transform domain analysis of Linear Feedback Shift Register(LFSR) based
sequence generators. The time and frequency domain behavior of non-linear
filter and combiner generators is discussed along with some novel observations
based on the Chinese Remainder Theorem (CRT). CRT is exploited to establish
patterns in LFSR sequences and underlying cyclic structures of finite fields.
Application of DFT spectra attacks on combiner generators is also demonstrated.
Our proposed method saves on the last stage computations of selective DFT
attacks for combiner generators. The proposed approach is demonstrated on some
examples of combiner generators and is scalable to general configuration of
combiner generators.Comment: This is a comprehensive report with over 20 page
Intelligent Reflecting Surface with Discrete Phase Shifts: Channel Estimation and Passive Beamforming
In this paper, we consider an intelligent reflecting surface (IRS)-aided
single-user system where an IRS with discrete phase shifts is deployed to
assist the uplink communication. A practical transmission protocol is proposed
to execute channel estimation and passive beamforming successively. To minimize
the mean square error (MSE) of channel estimation, we first formulate an
optimization problem for designing the IRS reflection pattern in the training
phase under the constraints of unit-modulus, discrete phase, and full rank.
This problem, however, is NP-hard and thus difficult to solve in general. As
such, we propose a low-complexity yet efficient method to solve it
sub-optimally, by constructing a near-orthogonal reflection pattern based on
either discrete Fourier transform (DFT)-matrix quantization or Hadamard-matrix
truncation. Based on the estimated channel, we then formulate an optimization
problem to maximize the achievable rate by designing the discrete-phase passive
beamforming at the IRS with the training overhead and channel estimation error
taken into account. To reduce the computational complexity of exhaustive
search, we further propose a low-complexity successive refinement algorithm
with a properly-designed initialization to obtain a high-quality suboptimal
solution. Numerical results are presented to show the significant rate
improvement of our proposed IRS training reflection pattern and passive
beamforming designs as compared to other benchmark schemes.Comment: Submitted to IEEE conferenc
Efficient function approximation on general bounded domains using wavelets on a cartesian grid
Fourier extension is an approximation method that alleviates the periodicity
requirements of Fourier series and avoids the Gibbs phenomenon when
approximating functions. We describe a similar extension approach using regular
wavelet bases on a hypercube to approximate functions on subsets of that cube.
These subsets may have a general shape. This construction is inherently
associated with redundancy which leads to severe ill-conditioning, but recent
theory shows that nevertheless high accuracy and numerical stability can be
achieved using regularization and oversampling. Regularized least squares
solvers, such as the truncated singular value decomposition, that are suited to
solve the resulting ill-conditioned and skinny linear system generally have
cubic computational cost. We compare several algorithms that improve on this
complexity. The improvements benefit from the sparsity in and the structure of
the discrete wavelet transform. We present a method that requires operations in 1-D and in -D, . We
experimentally show that direct sparse QR solvers appear to be more
time-efficient, but yield larger expansion coefficients
Fast Continuous Haar and Fourier Transforms of Rectilinear Polygons from VLSI Layouts
We develop the pruned continuous Haar transform and the fast continuous
Fourier series, two fast and efficient algorithms for rectilinear polygons.
Rectilinear polygons are used in VLSI processes to describe design and mask
layouts of integrated circuits. The Fourier representation is at the heart of
many of these processes and the Haar transform is expected to play a major role
in techniques envisioned to speed up VLSI design. To ensure correct printing of
the constantly shrinking transistors and simultaneously handle their
increasingly large number, ever more computationally intensive techniques are
needed. Therefore, efficient algorithms for the Haar and Fourier transforms are
vital. We derive the complexity of both algorithms and compare it to that of
discrete transforms traditionally used in VLSI. We find a significant reduction
in complexity when the number of vertices of the polygons is small, as is the
case in VLSI layouts. This analysis is completed by an implementation and a
benchmark of the continuous algorithms and their discrete counterpart. We show
that on tested VLSI layouts the pruned continuous Haar transform is 5 to 25
times faster, while the fast continuous Fourier series is 1.5 to 3 times
faster.Comment: 10 pages, 10 figure
Compression, Restoration, Re-sampling, Compressive Sensing: Fast Transforms in Digital Imaging
Transform image processing methods are methods that work in domains of image
transforms, such as Discrete Fourier, Discrete Cosine, Wavelet and alike. They
are the basic tool in image compression, in image restoration, in image
re-sampling and geometrical transformations and can be traced back to early
1970-ths. The paper presents a review of these methods with emphasis on their
comparison and relationships, from the very first steps of transform image
compression methods to adaptive and local adaptive transform domain filters for
image restoration, to methods of precise image re-sampling and image
reconstruction from sparse samples and up to "compressive sensing" approach
that has gained popularity in last few years. The review has a tutorial
character and purpose.Comment: 41 pages, 16 figure
Discrete Gyrator Transforms: Computational Algorithms and Applications
As an extension of the 2D fractional Fourier transform (FRFT) and a special
case of the 2D linear canonical transform (LCT), the gyrator transform was
introduced to produce rotations in twisted space/spatial-frequency planes. It
is a useful tool in optics, signal processing and image processing. In this
paper, we develop discrete gyrator transforms (DGTs) based on the 2D LCT.
Taking the advantage of the additivity property of the 2D LCT, we propose three
kinds of DGTs, each of which is a cascade of low-complexity operators. These
DGTs have different constraints, characteristics, and properties, and are
realized by different computational algorithms. Besides, we propose a kind of
DGT based on the eigenfunctions of the gyrator transform. This DGT is an
orthonormal transform, and thus its comprehensive properties, especially the
additivity property, make it more useful in many applications. We also develop
an efficient computational algorithm to significantly reduce the complexity of
this DGT. At the end, a brief review of some important applications of the DGTs
is presented, including mode conversion, sampling and reconstruction,
watermarking, and image encryption.Comment: Accepted by IEEE Transactions on Signal Processin
Digital Shearlet Transform
Over the past years, various representation systems which sparsely
approximate functions governed by anisotropic features such as edges in images
have been proposed. We exemplarily mention the systems of contourlets,
curvelets, and shearlets. Alongside the theoretical development of these
systems, algorithmic realizations of the associated transforms were provided.
However, one of the most common shortcomings of these frameworks is the lack of
providing a unified treatment of the continuum and digital world, i.e.,
allowing a digital theory to be a natural digitization of the continuum theory.
In fact, shearlet systems are the only systems so far which satisfy this
property, yet still deliver optimally sparse approximations of cartoon-like
images. In this chapter, we provide an introduction to digital shearlet theory
with a particular focus on a unified treatment of the continuum and digital
realm. In our survey we will present the implementations of two shearlet
transforms, one based on band-limited shearlets and the other based on
compactly supported shearlets. We will moreover discuss various quantitative
measures, which allow an objective comparison with other directional transforms
and an objective tuning of parameters. The codes for both presented transforms
as well as the framework for quantifying performance are provided in the Matlab
toolbox ShearLab.Comment: arXiv admin note: substantial text overlap with arXiv:1106.205
Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition
As a generalization of the two-dimensional Fourier transform (2D FT) and 2D
fractional Fourier transform, the 2D nonseparable linear canonical transform
(2D NsLCT) is useful in optics, signal and image processing. To reduce the
digital implementation complexity of the 2D NsLCT, some previous works
decomposed the 2D NsLCT into several low-complexity operations, including 2D
FT, 2D chirp multiplication (2D CM) and 2D affine transformations. However, 2D
affine transformations will introduce interpolation error. In this paper, we
propose a new decomposition called CM-CC-CM-CC decomposition, which decomposes
the 2D NsLCT into two 2D CMs and two 2D chirp convolutions (2D CCs). No 2D
affine transforms are involved. Simulation results show that the proposed
methods have higher accuracy, lower computational complexity and smaller error
in the additivity property compared with the previous works. Plus, the proposed
methods have perfect reversibility property that one can reconstruct the input
signal/image losslessly from the output.Comment: Accepted by Journal of the Optical Society of America A (JOSA A
Fast evaluation of real and complex exponential sums
Recently, the butterfly approximation scheme and hierarchical approximations
have been proposed for the efficient computation of integral transforms with
oscillatory and with asymptotically smooth kernels. Combining both approaches,
we propose a certain fast Fourier-Laplace transform, which in particular allows
for the fast evaluation of polynomials at nodes in the complex unit disk. All
theoretical results are illustrated by numerical experiments
Fast and Efficient Sparse 2D Discrete Fourier Transform using Sparse-Graph Codes
We present a novel algorithm, named the 2D-FFAST, to compute a sparse
2D-Discrete Fourier Transform (2D-DFT) featuring both low sample complexity and
low computational complexity. The proposed algorithm is based on mixed concepts
from signal processing (sub-sampling and aliasing), coding theory (sparse-graph
codes) and number theory (Chinese-remainder-theorem) and generalizes the
1D-FFAST 2 algorithm recently proposed by Pawar and Ramchandran [1] to the 2D
setting. Concretely, our proposed 2D-FFAST algorithm computes a k-sparse
2D-DFT, with a uniformly random support, of size N = Nx x Ny using O(k)
noiseless spatial-domain measurements in O(k log k) computational time. Our
results are attractive when the sparsity is sub-linear with respect to the
signal dimension, that is, when k -> infinity and k/N -> 0. For the case when
the spatial-domain measurements are corrupted by additive noise, our 2D-FFAST
framework extends to a noise-robust version in sub-linear time of O(k log4 N )
using O(k log3 N ) measurements. Simulation results, on synthetic images as
well as real-world magnetic resonance images, are provided in Section VII and
demonstrate the empirical performance of the proposed 2D-FFAST algorithm
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