Innovation Rate Sampling of Pulse
Streams with Application to Ultrasound Imaging Ronen Tur, Noam Wagner,Yonina C. Eldar
& Zvi Friedman

Introduction

Signals comprised of a stream of short pulses appear
in many applications including bio-imaging and radar.
The recent finite
rate of innovation framework, has paved the way to low
rate sampling of such pulses by noticing that only a
small number of
parameters per unit time are needed to fully describe
these signals. Unfortunately, for high rates of
innovation, existing
sampling schemes are numerically unstable. In this work
we propose and demonstrate a novel sampling approach
which leads to stable recovery even in the presence of
numerous pulses. We derive a condition on
the sampling kernel which allows perfect reconstruction
from the minimal number of samples.
We then propose a compactly supported class of filters,
satisfying this condition. Our solution is noise robust
and numerically stable even for a large number of pulses.
Finally, we apply our techniques
on ultrasound imaging data, and show
that substantial rate reduction with respect to
traditional ultrasound sampling schemes can be achieved.

Signal Model

Consider a tau-periodic stream
of pulses, defined as

where h(t) and tau are the known
pulse shape and period, respectively, and {t_{l},a_{l}}
are the unknown delays and amplitudes.
The goal is to sample and reconstruct x(t) efficiently. This is done by expanding the signal to its
Fourier series, and noticing that the Fourier series
coefficients are of the form:

The right hand side of this equation is a sum of
exponentials with frequencies {t_{l}},
which can be found using standard spectral analysis
methods as long as we have a set of Fourier coefficients
{X[k]}, with cardinality greater than 2L.
The challenge is to obtain the set of Fourier
coefficients from time domain samples of the signal x(t).

X-ADC: SoS Filters

In order to obtain the set {X[k]}, we propose
uniformly sampling the signal with a any sampling kernel
satisfying:

where К denotes
the index set of the Fourier Coefficients. For this
filter choice, it can be shown that the Fourier
coefficients can be obtained from the samples by simple
matrix inversion, as long as the number of samples is
larger than the cardinality of the index set К.
Following the general condition presented above, we
propose a compactly supported filter which consists of a
sum of sinc functions in the frequency domain:

where b_{k} are free parameters of the
filter. Switching to the time domain it is evident that
the filter is compactly supported:

The SoS filter results in a system which is highly
robust to noise. In addition, the compact support of the
SoS filter class enables extension of our solution to
the infinite setting.

Infinite and Finite Pulse Streams

Consider now the case of an infinite stream of pulses

We assume that the infinite signal has a bursty
character, i.e., the signal has two distinct phases: a)
bursts of maximal duration tau containing at most L pulses, and b) quiet phases between bursts. For the sake
of clarity we begin with the case h(t)=δ(t).

Consider uniformly sampling z(t) with a filter
comprising three periods of g_{3P}(t)
=g(t-tau) + g(t) + g(t+tau). If the minimal spacing between any two consecutive
bursts is 3T/2, then we are guaranteed that
each sample taken during the burst is influenced by one
burst only. Here
we exploited the compact support of the SoS filter.
Once we reduced the infinite problem to a sequence of
finite pulse streams, it can be shown that the samples
obtained by the g_{3P}(t) sampling
kernel form a set of equations which allows to obtain
the set of Fourier coefficients exactly as in the periodic case, for each
burst independently. Therefore the unknown delays and
amplitudes can be determined throughout the infinite
signal z(t).
The extension to arbitrary h(t) is possible as long as
the pulse-shape has finite support, which is a
rather weak condition, since our primary interest is in
very short pulses which have wide, or even infinite,
frequency support. The derivation in this section
obviously holds for finite pulse streams, as a simple
special case.

Note that in the infinite case the SoS approach is not a
minimal rate sampling scheme. In order to achieve the
minimal possible rate, the approach in Multichannel pulse stream acquisition may be chosen,
in the expense of implementation difficulties related to
multichannel schemes.

Ultrasound Imaging Application

An
interesting application of our scheme is ultrasound imaging,
in which the signal received from the tissue under test
comprises a stream of short Gaussian pulses. Applying our
scheme on data recorded with GE Healthcare's Vivid-i system,
we reconstructed the original signal as depicted in the
figure below. The reconstruction is based on 17 samples
only, whereas current ultrasonic imaging systems use for the
same scenario 4000 samples, emphasizing the potential of our
scheme in reducing sampling rate in such systems.

Description:
This simulation demonstrates the proposed sampling and
reconstruction approach, for a stream of Gaussian pulses.
In addition, our method is applied to real ultrasound
imaging data, where the pulse is modeled as a Gaussian.
Since ultrasonic data is quite noisy, we oversample (typically a factor of 2), apply hard-thresholding to the received
signal, and use Cadzow's denoising method.
Analog filtering is replaced by digital processing via a
dense grid mimicking the corresponding analog operations.

Installation:
Unzip all files to a directory of your choice.
Usage:
Follow the instructions in
"SoS_GaussianPulse_Description.doc".

Description:
This simulation analyzes the performance of our method in
the presence of noise, compared to previous approaches. The
signal is chosen to be a stream of Dirac delta functions,
which allows to calculate the resulting samples
analytically. At this stage noise is added to the samples to
give the desired SNR. The MSE of time-delay estimation is
then given for each method as a function of SNR.

Installation:
Unzip all files to a directory of your choice. Usage:
Follow the instructions in "SoS_Noise_Performance.doc".