44,829 research outputs found
Steerable Discrete Fourier Transform
Directional transforms have recently raised a lot of interest thanks to their
numerous applications in signal compression and analysis. In this letter, we
introduce a generalization of the discrete Fourier transform, called steerable
DFT (SDFT). Since the DFT is used in numerous fields, it may be of interest in
a wide range of applications. Moreover, we also show that the SDFT is highly
related to other well-known transforms, such as the Fourier sine and cosine
transforms and the Hilbert transforms
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
Explicit Hermite-type eigenvectors of the discrete Fourier transform
The search for a canonical set of eigenvectors of the discrete Fourier
transform has been ongoing for more than three decades. The goal is to find an
orthogonal basis of eigenvectors which would approximate Hermite functions --
the eigenfunctions of the continuous Fourier transform. This eigenbasis should
also have some degree of analytical tractability and should allow for efficient
numerical computations. In this paper we provide a partial solution to these
problems. First, we construct an explicit basis of (non-orthogonal)
eigenvectors of the discrete Fourier transform, thus extending the results of
[7]. Applying the Gramm-Schmidt orthogonalization procedure we obtain an
orthogonal eigenbasis of the discrete Fourier transform. We prove that the
first eight eigenvectors converge to the corresponding Hermite functions, and
we conjecture that this convergence result remains true for all eigenvectors.Comment: 21 pages, 4 figures, 1 tabl
Uncertainty Relation for the Discrete Fourier Transform
We derive an uncertainty relation for two unitary operators which obey a
commutation relation of the form UV=exp[i phi] VU. Its most important
application is to constrain how much a quantum state can be localised
simultaneously in two mutually unbiased bases related by a Discrete Fourier
Transform. It provides an uncertainty relation which smoothly interpolates
between the well known cases of the Pauli operators in 2 dimensions and the
continuous variables position and momentum. This work also provides an
uncertainty relation for modular variables, and could find applications in
signal processing. In the finite dimensional case the minimum uncertainty
states, discrete analogues of coherent and squeezed states, are minimum energy
solutions of Harper's equation, a discrete version of the Harmonic oscillator
equation.Comment: Extended Version; 13 pages; In press in Phys. Rev. Let
Discrete Fourier Transform in Nanostructures using Scattering
In this paper we show that the discrete Fourier transform can be performed by
scattering a coherent particle or laser beam off a two-dimensional potential
that has the shape of rings or peaks. After encoding the initial vector into
the two-dimensional potential, the Fourier-transformed vector can be read out
by detectors surrounding the potential. The wavelength of the laser beam
determines the necessary accuracy of the 2D potential, which makes our method
very fault-tolerant.Comment: 6 pages, 5 EPS figures, REVTe
- …