13,359 research outputs found
Sub-Nyquist Sampling: Bridging Theory and Practice
Sampling theory encompasses all aspects related to the conversion of
continuous-time signals to discrete streams of numbers. The famous
Shannon-Nyquist theorem has become a landmark in the development of digital
signal processing. In modern applications, an increasingly number of functions
is being pushed forward to sophisticated software algorithms, leaving only
those delicate finely-tuned tasks for the circuit level.
In this paper, we review sampling strategies which target reduction of the
ADC rate below Nyquist. Our survey covers classic works from the early 50's of
the previous century through recent publications from the past several years.
The prime focus is bridging theory and practice, that is to pinpoint the
potential of sub-Nyquist strategies to emerge from the math to the hardware. In
that spirit, we integrate contemporary theoretical viewpoints, which study
signal modeling in a union of subspaces, together with a taste of practical
aspects, namely how the avant-garde modalities boil down to concrete signal
processing systems. Our hope is that this presentation style will attract the
interest of both researchers and engineers in the hope of promoting the
sub-Nyquist premise into practical applications, and encouraging further
research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin
Time Localization and Capacity of Faster-Than-Nyquist Signaling
In this paper, we consider communication over the bandwidth limited analog
white Gaussian noise channel using non-orthogonal pulses. In particular, we
consider non-orthogonal transmission by signaling samples at a rate higher than
the Nyquist rate. Using the faster-than-Nyquist (FTN) framework, Mazo showed
that one may transmit symbols carried by sinc pulses at a higher rate than that
dictated by Nyquist without loosing bit error rate. However, as we will show in
this paper, such pulses are not necessarily well localized in time. In fact,
assuming that signals in the FTN framework are well localized in time, one can
construct a signaling scheme that violates the Shannon capacity bound. We also
show directly that FTN signals are in general not well localized in time.
Therefore, the results of Mazo do not imply that one can transmit more data per
time unit without degrading performance in terms of error probability.
We also consider FTN signaling in the case of pulses that are different from
the sinc pulses. We show that one can use a precoding scheme of low complexity
to remove the inter-symbol interference. This leads to the possibility of
increasing the number of transmitted samples per time unit and compensate for
spectral inefficiency due to signaling at the Nyquist rate of the non sinc
pulses. We demonstrate the power of the precoding scheme by simulations
Multichannel Sampling of Pulse Streams at the Rate of Innovation
We consider minimal-rate sampling schemes for infinite streams of delayed and
weighted versions of a known pulse shape. The minimal sampling rate for these
parametric signals is referred to as the rate of innovation and is equal to the
number of degrees of freedom per unit time. Although sampling of infinite pulse
streams was treated in previous works, either the rate of innovation was not
achieved, or the pulse shape was limited to Diracs. In this paper we propose a
multichannel architecture for sampling pulse streams with arbitrary shape,
operating at the rate of innovation. Our approach is based on modulating the
input signal with a set of properly chosen waveforms, followed by a bank of
integrators. This architecture is motivated by recent work on sub-Nyquist
sampling of multiband signals. We show that the pulse stream can be recovered
from the proposed minimal-rate samples using standard tools taken from spectral
estimation in a stable way even at high rates of innovation. In addition, we
address practical implementation issues, such as reduction of hardware
complexity and immunity to failure in the sampling channels. The resulting
scheme is flexible and exhibits better noise robustness than previous
approaches
A novel sampling theorem on the rotation group
We develop a novel sampling theorem for functions defined on the
three-dimensional rotation group SO(3) by connecting the rotation group to the
three-torus through a periodic extension. Our sampling theorem requires
samples to capture all of the information content of a signal band-limited at
, reducing the number of required samples by a factor of two compared to
other equiangular sampling theorems. We present fast algorithms to compute the
associated Fourier transform on the rotation group, the so-called Wigner
transform, which scale as , compared to the naive scaling of .
For the common case of a low directional band-limit , complexity is reduced
to . Our fast algorithms will be of direct use in speeding up the
computation of directional wavelet transforms on the sphere. We make our SO3
code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for
publication. Code available at http://www.sothree.or
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