Multichannel Sampling of Pulse Streams at the Rate of Innovation

0.34 MB·English

by  Kfir Gedalyahu

1
Multichannel Sampling of Pulse Streams at the Rate
of Innovation
Kfir Gedalyahu*, Ronen Tur* and Yonina C. Eldar, Senior Member, IEEE
Abstract—We consider minimal-rate sampling schemes for standard sampling methods require very high sampling rates.
infinite streams of delayed and weighted versions of a known The main idea is to exploit the fact that the pulse shape is
pulse shape. The minimal sampling rate for these parametric
known, in order to characterize such signals by the time-
signals is referred to as the rate of innovation and is equal
1 delays and amplitudes of the various pulses. Targeting these
to the number of degrees of freedom per unit time. Although
1 samplingofinfinitepulsestreamswastreatedinpreviousworks, parameters allows to reduce the sampling rate way beyond
0 eithertherateofinnovationwasnotachieved,orthepulseshape that dictated by the Shannon-Nyquisttheorem. In fact, it was
2 was limited to Diracs. In this paper we propose a multichannel shown in [14] that the sampling rate can also be reduced for
n architecture for sampling pulse streams with arbitrary shape, short pulses with unknown shape.
a operating at the rate of innovation. Our approach is based
Followingthisparametricpointofview,asamplingscheme
J on modulating the input signal with a set of properly chosen
waveforms,followedbyabankofintegrators.Thisarchitectureis for periodic streams of pulses was developed in [11], [12],
4
motivatedbyrecentworkonsub-Nyquistsamplingofmultiband whichoperatesat the innovationrate. Itrelies onthe observa-

] signals. We show that the pulse stream can be recovered from tionthatthetimedelaysandamplitudescanberecoveredfrom
T the proposed minimal-rate samples using standard tools taken a set of the signal’s Fourier series coefficients. This follows
I from spectral estimation in a stable way even at high rates from the fact that in the frequency domain, the problem
. of innovation. In addition, we address practical implementation
cs issues,suchasreductionofhardwarecomplexityandimmunityto translates into estimating the frequencies and amplitudes of
[ failureinthesamplingchannels.Theresultingschemeisflexible a sum of complex sinusoids (cisoids), a problem which has
and exhibits better noise robustness than previous approaches. been treated extensively in the context of spectral estimation
5
[15].
v
0 I. INTRODUCTION In practical applications finite and infinite streams are usu-
7 ally encountered, rather than periodic streams. For the finite
Digital processing has become ubiquitous, and is the most
0 caseGaussian[11],andpolynomialsorexponentialsreproduc-
common way to process analog signals. Processing analog
5
ing sampling kernels [16], were introduced. The approaches
. signals digitally must be preceded by a sampling stage, care-
4 based on the first two kernels, are unstable for high rates of
fully designed to retain the important features of the analog
0 innovations [13]. An alternative sampling scheme, based on
0 signalrelevantfortheprocessingtaskathand.Thewellknown
a new family of time-limited filters, was presented in [13].
1 Shannon-Nyquist theorem states that in order to perfectly
This approach exhibits better noise robustness than previous
: reconstruct an analog signal from its samples, it must be
v methods,and is stable even for high modelorders.Exploiting
i sampled at the Nyquist rate, i.e., twice its highest frequency.
X the compact support of the sampling kernels in [13], [16],
This assumption is required when the only knowledge on the
bothmethodswereextendedtotheinfinitecase.Unfortunately,
r signalisthatitisbandlimited.Otherpriorsonsignalstructure
a neither techniques achieve the minimal sampling rate, which
[1], [2], which include subspace [3]–[5], sparsity [6]–[8], or
is the rate of innovation, for infinite pulse streams.
smoothness priors [1], [9], [10], can lead to more efficient
All previous methods were composed of a single sampling
sampling.
channel. Multichannel sampling schemes offer additional de-
An interesting class of structured signals was suggested
grees of freedom which can be utilized to achieve the rate
by Vetterli et al. [11], [12], who considered signals with a
of innovation for the infinite setting. In [17] a multichannel
finite number of degrees of freedom per unit time, termed
extension of the method in [16] was presented. This scheme
by the authors as signals with finite rate of innovation (FRI).
allows reduced sampling rate in each channel, but overall
For such models, the goal is to design a sampling scheme
sampling rate similar to [16]. Another multichannel system,
operatingattheinnovationrate,whichistheminimalpossible
composed of two first-order resistor-capacitor (RC) networks,
rate from which perfect recovery is possible. A special case
was proposed in [18]. However, this approach assumed that
that was treated in detail are signals consisting of streams
there is a single pulse per sampling period, an assumption
of short pulses. Pulse streams are prevalent in applications
which limits the method’s time resolution. Two alternative
suchasbio-imaging[13],neuronalactivityandultra-wideband
multichannelmethods,were proposedin [19] and [20]. These
communications.Sincethe pulsesarehighlycompactin time,
approaches, which are based on a chain of integrators [19]
*Bothauthors contributed equally tothiswork. and exponential filters [20], allow only sampling of infinite
Department of Electrical Engineering, Technion—Israel Institute of Tech- streams of Diracs at the rate of innovation. In addition, we
nology, Haifa 32000, Israel. Phone: [phone], fax: [phone],
showinsimulationsthatthesemethodsareunstable,especially
E-mail: {kfirge@techunix,ronentur@techunix,yonina@ee}.technion.ac.il. Y.
Eldariscurrently avisiting ProfessoratStanford, USA. for high rates of innovation. An alternative scheme, proposed
2
in [21], can operate at the rate of innovation for pulses with uppercase letters to matrices. The nth element of a vector a
arbitrary shape. However, this approach constrains the delays iswrittenasa , andA denotestheijthelementofamatrix
n ij
to beconstantin eachperiod.To the bestofourknowledge,a A. Superscripts (·)∗, (·)T and (·)H represent complex conju-
stableminimal-ratesamplingschemeforinfinitepulsestreams, gation,transpositionand conjugatetransposition,respectively.
with arbitrary shape, is still lacking. The Moore-Penrose pseudo-inverse of a matrix A is written
Our first contributiontreats finite pulse streams. We design as A†. We denote by diag(a) a diagonal matrix having the
a multichannel sampling system, based on oscillators, mixers elements of the vector a on its diagonal. The continuous-
and integrators. In each channel the signal is modulated time Fourier transform (CTFT) of a continuous-time signal
by an appropriate waveform, followed by integration over a x(t)∈L is defined by X(ω)= ∞ x(t)e−jωtdt.
2 −∞
compact time interval. We derive conditions which guarantee Consider the finite stream of pulses
R
that the output of each channel is a mixture of the Fourier
L
coefficients of the signal. By properly choosing the mixing x(t)= a h(t−t ), t ∈I ⊂[0,T), a ∈C, (1)
l l l l
parameters, we show that the Fourier coefficients can be
l=1
obtainedfromthesamples.OncethesetofFouriercoefficients X
where h(t) is a known pulse shape, {t ,a }L are the un-
is known, we use standard spectral estimation tools in order l l l=1
known delays and amplitudes, and I is a continuous-time
to recover the unknown times and amplitudes. As we show,
interval in [0,T). The pulse can be arbitrary as long as
the mixingscheme enablessimple and practicalgenerationof
modulating waveforms. Furthermore, mixing the coefficients h(t−t )=0, ∀t∈/ [0,T) l=1...L, (2)
l
allowsrecoveringthe signalevenwhenoneor moresampling
channels fails. i.e., the signal x(t) is confined to the time-window [0,T).
Integrationover a finite intervalenables a simple extension This condition suggests that the support of the pulse has
to the infinite setting. Our infinite sampling approach leads to be finite and smaller than T. In such cases the effective
to perfect reconstruction of the signal, while sampling at the Nyquist rate will be quite large since x(t) will have a large
rate of innovation. In addition, our scheme can accommodate bandwidth.However, the sampling rate can be reduced below
general pulse shapes with finite length support. As we show the Nyquist rate, by noticing that x(t) is uniquely defined
in simulations, our approach exhibits better noise robustness by the delays and amplitudes. Since x(t) has 2L degrees of
compared to previous methods, and allows sampling at high freedom,{tl,al}Ll=1, atleast2Lsamplesare requiredin order
rates of innovation. We also discuss a special case of infinite to represent the signal. Our goal is to design a sampling and
streams of pulses having a shift-invariant (SI) structure, a reconstructionmethodwhichperfectlyrecoversx(t)fromthis
modelpresentedin[21],andcompareourmethodwiththeone minimal number of samples.
in [21]. Finally, we describe how to practically generate the
modulating waveforms and derive conditions on these wave- B. Relation to Model-Based Complex Sinusoids Estimation
forms which guarantee perfect reconstruction of the signal.
Our sampling problem can be related to the well known
Theschemederivedinthisworkfollowsideasofarecently
problemofa model-basedcomplexsinusoids(cisoids)param-
proposedsamplingmethodologyforstructuredanalogsignals,
eter estimation. This approach was originally taken by Hou
termed Xampling [22], [23], [14]. This framework utilizes
and Wu [25], who were the first to show that time delay
the signal model in order to reduce the sampling rate below
estimation can be converted into a frequency estimation of a
the Nyquistrate. A pioneersub-Nyquistsystem formultiband
sum of cisoids [15]. This follows from noticing that delays
signals [7], referred to as the modulated wideband converter
in the time domain are converted into modulations in the
(MWC), was proposed in [24]. Although treating a different
frequency domain. However, their method relied on Nyquist
signal model, our modulation scheme is based on concepts
rate sampling of the signal, and their derivations were only
presented in [24]. Both works share a similar analog front-
approximate. Vetterli et al. [11] addressed this problem from
end, so that the hardware prototypeof the MWC, designed in
an efficient sampling point of view, and derived a low-rate
[22], can also be used to implement our method.
sampling and reconstruction scheme for periodic streams of
The remainder of this paper is organized as follows. In
Diracs.Theirmethodwasbasedonthesamefundamentalrela-
Section II we derive a multichannel scheme for finite pulse
tionbetweenthedelaysintimeandmodulationsinfrequency.
streams. Section III extends our results to the infinite case.
Following a similar path, we show that once a set of Fourier
We discuss the generation of the modulating waveforms in
coefficientsofthesignalareknown,thedelayscanberetrieved
Section IV, and present a practical sampling scheme which
using sinusoidalestimation methods. We then design low-rate
can be implemented in hardware. In Section V we discuss
sampling schemes for obtaining the Fourier coefficients.
in more details the relations of our results to previous work.
Since x(t) is confined to the interval t ∈ [0,T), it can be
Numerical experiments are described in Section VI.
expressed by its Fourier series
x(t)= X[k]ej2Tπkt, t∈[0,T) (3)
II. FINITE STREAMS OF PULSES
k∈Z
A. Problem Formulation X
where
Throughout the paper, we denote matrices and vectors by X[k]= 1 T x(t)e−j2Tπktdt. (4)
bold font, with lowercase letters correspondingto vectorsand T
Z0
3
Substituting (1) into (4) we obtain Each channel consists of modulating x(t) with a complex
exponential,followedbyanintegratoroverthewindow[0,T).
X[k]= 1 L al T h(t−tl)e−j2Tπktdt ThesampletakenbythekthchannelisexactlyX[k],asin(4).
T This direct sampling scheme is straightforward, and may be
l=1 Z0
X implementedusing3basicbuildingblocks:oscillators,mixers
= 1 L ale−j2Tπktl ∞ h(t)e−j2Tπktdt and integrators. However, from a practical point of view this
T
l=1 Z−∞ approachhasthedisadvantagethatitrequiresmanyoscillators,
X
L having frequencies which must be exact multiples of some
= 1H 2πk ale−j2Tπktl, (5) common base frequency.
T T
(cid:18) (cid:19)l=1
X
wherethesecondequalitystemsfromtheconditionin(2),and T1R0T(·)dt Xh−⌊K2⌋i
H(ω) denotes the CTFT of h(t).
Denote by K a set of K consecutive indices for which e+j2Tπ⌊K/2⌋t
H 2πk 6= 0, ∀k ∈ K. We require that such a set exists, x(t)
T
which is usually the case for short time-support pulses h(t).
(cid:0) (cid:1)
Denote by H the K × K diagonal matrix with kth entry
edT1−eHlja2yT(cid:0)πsk2.TπtIlnk,(cid:1)wa,dhadenridteiotbny=dVe{n(tot1,t)e.t.hb.ey,tKaL}t×hiesLltehmnegavtthreic-xLtowrvioethcfttokhrlethwuhnelokesnmeoewltnhnt e−j2Tπ⌊K/2⌋t T1R0T(·)dt Xh+⌊K2⌋i
elementisa ,andbyxthelength-Kvectorwhosekthelement
l Fig. 1. Multichannel direct sampling of the Fourier series coefficients
is X[k]. We may now write (5) in matrix form as X[k],k∈K.
x=HV(t)a. (6)
The matrix H is invertible by construction, and therefore we D. Mixing the Fourier Coefficients
can define y=H−1x, which satisfies
Wenowgeneralizeourframework,towardsamorepractical
y=V(t)a. (7) samplingschemebymixingseveralFouriercoefficients,rather
than limiting ourselves to one coefficient per channel. The
Addressingthe kth elementof the vectory in (7) directly,we
additional degrees of freedom offered by this extension will
obtain
allow the design of waveforms that are easy to implement.
L
yk = ale−j2Tπktl. (8) Our approach is motivated by the hardware reported in [24],
wheresimilarmodulatorsareusedtosamplemultibandsignals
l=1
X
at sub-Nyquist rates.
Giventhevectorx,(7)conformswiththestandardproblem
In addition, in real-life scenarios one or more channels
of finding the frequencies and amplitudes of a sum of L
might fail, due to malfunction or noise corruption, and there-
cisoids. The time-delays can be estimated using nonlinear
fore we lose the information stored in that channel. Unique
techniques,e.g.,theannihilatingfilter[11],matrix-pencil[26],
recoveryof the signal parametersfrom (6), relies on having a
Kumaresan and Tufts method [27] or ESPRIT [28] (see [15]
consecutivesetofFouriercoefficients[12].Hence,whenusing
for a review of this topic), as long as K ≥2L and the time-
thedirectscheme,lossofFouriercoefficients,preventsusfrom
delays are distinct, i.e., t 6= t for all i 6= j. Once the
i j recoveringthesignal.Incontrast,whenmixingthecoefficients
time-delays are known, the linear set of equations (7) may
we distribute the information about each Fourier coefficient
be solved via least-squares for the unknown amplitudes. Due
between several sampling channels. Consequently, when one
to the Vandermondeform of V(t), it is left invertibleas long
or more channels fail, the required Fourier coefficients may
as K ≥L, so that a=V†(t)y.
still be recovered from the remaining operating channels. If
theirnumberisgreaterthan2L,thesignalcanbestillperfectly
C. Direct Multichannel Sampling recovered from the samples. We discuss this feature more
As we have seen, given a vector of K ≥ 2L Fourier thoroughly in Section IV.
series coefficientsx, we may use standard tools from spectral Consider a multichannelsampling scheme with p channels,
analysistodeterminetheset{t ,a }L .Inpractice,thesignal asdepictedinFig. 2. Ineachchannel,we modulatethe signal
l l l=1
is sampled in the time-domain, and therefore we do not have using a weighted sum of cisoids given by
dsairmecptlinacgcsecshsetmoeswamhipclheswoilflaxl.loOwutrogoobatlainnotwheivsetcotodrexsifgrnoma si(t)= sike−j2Tπkt, (9)
time-domain samples. kX∈K
For simplicity, we set K to be an odd number, and choose where the weights sik vary from channel to channel. The
the set K = {−⌊K/2⌋,...,⌊K/2⌋}. However, our results resulting sample of the ith channel is
e|Kxt|en≥d t2oL.anTyheseFtouKrierofcoceofnfisceiceunttisveXi[nkd]icceasn, baes olobntagineads ci = 1 T x(t) sike−j2Tπktdt= sikX[k]. (10)
T
using the multichannel sampling scheme depicted in Fig.1. Z0 k∈K k∈K
X X
4
T1R0T(·)dt c1 is easily verified that this choice yields an invertible matrix
S. The practical advantage of the mixing scheme is already
s1(t)=Xs1ke−j2Tπkt evident, since sine and cosine waves are real valued, whereas
k∈K
x(t) thedirectmultichannelschemerequirescomplexexponentials.
2) Periodic Waveforms: Every periodic waveform can be
expandedinto a Fourier series. Transferringsuch a waveform
T1R0T(·)dt cp through some shaping filter, e.g., a low-pass filter, we can
reject most of the coefficients, leaving only a finite set intact.
sp(t)=Xspke−j2Tπkt Consequently, such a scheme meets the form of (9). In
k∈K Section IV we elaborate on this concept, discuss design con-
Fig.2. MixingtheFouriercoefficients differently ineachchannel. siderations,andshowthatproperlychosenperiodicwaveforms
yield a left invertible matrix S.
Onesimple choiceisperiodicstreamsof rectangularpulses
modulated by ±1 [24]. The strength of the mixing scheme
TorelatethesamplesandtheFouriercoefficients,wedefine over the direct one will be emphasized in Section IV-B. We
the p×K matrix S with s as its ikth element, and by c the show that one periodic stream is sufficient for all channels,
ik
length-psamplevectorwithithelementc .Wemaynowwrite while each channel uses a delayed version of this common
i
(10) in matrix form as waveform. Therefore, the requirement for multiple oscillators
and the need for accurate multiples of the basic frequency,
c=Sx. (11)
are both removed. In addition, periodic streams are easily
As long as S has full column rank, where p ≥ K is a designed and implemented digitally, rather than somewhat
necessary condition, we can recover x from the samples by complicatedanalogdesignofoscillatorscombinedwithanalog
x = S†c. The direct sampling scheme presented earlier is a circuits intended to create exact frequency multiples. Finally,
special case of this more general approach, with p = K and if the period T changes, the analog circuit has to be modified
S = I. In Section IV we exploit the degrees of freedom this substantially, whereas the flexibility of the digital design
general scheme offers, and present sampling schemes which allows simple modifications.
can simplify the hardware design, and are more robust to
malfunctions in the sampling channels. III. INFINITESTREAMS OF PULSES
We summarize this result in the following theorem.
A. General Model
Theorem 1. Consider a finite stream of pulses given by We now consider an infinite stream of pulses defined by
L x(t)= a h(t−t ), t ∈R, a ∈C. (13)
x(t)= a h(t−t ), t ∈I ⊂[0,T), a ∈C, l =1...L, l l l l
l l l l l∈Z
X
l=1
X WeassumethattherearenomorethanLpulsesinanyinterval
where h(t) is a known pulse shape, and condition (2) is I , [(m−1)T,mT], m ∈ Z. We further assume that
m
satisfied. Choose a set K of consecutive indices for which
withineachintervalcondition(2)holds,andconsequently,the
H(2πk/T)6=0, ∀k ∈K.Considerthemultichannelsampling
intervalsareindependentofoneanother.Themaximalnumber
scheme depicted in Fig. 2, for some choice of coefficients
ofdegreesoffreedomperunittime,also knownastherate of
{s } , i=1,...,p.Then,thesignalx(t)canbeperfectly
ik k∈K innovation[11],[12],is2L/T.Wenowpresentamultichannel
reconstructed from the samples {c }p with
i i=1 sampling and reconstruction scheme which operates at the
ci = 1 T x(t) sike−j2Tπktdt, (12) miCniomnasildrearteanpoessxitbelnes,ioi.ne.,otfhethreatseamofpilninngovsacthioenm.e presented
T
Z0 k∈K in Section II-D, where we sample every T seconds. Upon
X
as long as p ≥ |K| ≥ 2L, and the coefficients matrix S in each sample we reset the integrator, so that the mth sample
(11) is left invertible. correspond to an integral over the interval Im. The resulting
samplingschemeisdepictedinFig.3.Sincethemthsampleis
As we discuss in Section V-A, the method in [13] can
influenced by the interval I only, the infinite problem may
m
be viewed as a special case of Fig. 2. Since our work
be reduced into a sequence of finite streams of pulses. The
is a generalization of [13], it benefits from the high noise
resulting samples are given by
robustnessexhibitedby[13],incontrasttopreviouswork[11],
[16]. It should be noted that Theorem 1 holds for a periodic c[m]=Sx[m], (14)
pulse stream as well, since it can be similarly represented by
where the vector x[m] contains the Fourier series coefficients
a Fourier series, and all derivations remain intact.
of the signal within the mth interval, I . As long as S is
We now demonstrateseveralusefulmodulatingwaveforms. m
chosen so that it is left invertible,we can obtain the sequence
1) Cosine and Sine waveforms: First we set p=K. Then,
of Fourier series coefficients by x[m] = S†c[m]. Extending
we choose the first ⌊K/2⌋ waveforms to be cos 2πkt , the
T (7) to the infinite case we obtain:
next ⌊K/2⌋ to be sin 2πkt , and the last to be the constant
function 1. Clearly, theTse waveforms fit the form(cid:0) in ((cid:1)9). It y[m]=H−1x[m]=V(t[m])a[m], (15)
(cid:0) (cid:1)
5
t=mT
T1ZIm(·)dt c1[m] pwehreiored.aAℓ[sns]u∈miℓn2gdceonnodteitsiotnhe(2ℓ)thhpouldlssehaemreplaitsudweelol,n(t1h5e)mcathn
s1(t)= s1ke−j2Tπkt be rewritten as
kX∈K
x(t) y[m]=V(t)a[m], (18)
t=mT sincenowtherelativedelaysineachperiodareconstant.Here,
1 (·)dt cp[m] a[m] denotes the length-L vector with ℓth element aℓ[m].
T ZIm Clearly, the condition for the general model p ≥ 2L is a
sp(t)= spke−j2Tπkt sufficient conditionhere also, howeverthe additional prior on
kX∈K
the signal’s structure can be used to reduce the number of
Fig. 3. Extended sampling scheme using modulating waveforms for an sampling channels. The results obtained in [21], for a similar
infinitepulsestream. setofequations,providethe followingsufficientconditionfor
unique recovery of the delays and vectors a[m] from (18):
where t[m] and a[m] are the times and amplitudes of the K ≥2L−η+1, (19)
pulses in the interval I , respectively, and the matrix V
m
remains as in (6). For each m, (15) represents a sum of where
cisoids problem, and thus may be solved as long as K ≥2L.
η =dim(span({a[m],m∈Z})) (20)
By choosing p = K = 2L we present a sampling scheme
whichoperatesattherateofinnovation,andallowsforperfect denotesthedimensionoftheminimalsubspacecontainingthe
reconstruction of an infinite stream of pulses. vectorset{a[m],m∈Z}.Thisconditionimpliesthatinsome
We state our result in a theorem. cases K, and eventually the number of channels p (since p≥
K),can be reducedbeyondthe lower limit2L for the general
Theorem 2. Consider an infinite stream of pulses given by
model, depending on the value of η.
x(t)= a h(t−t ), t ∈R, a ∈C. Similar to [21], recovery of the delays from (18) can be
l l l l
l∈Z performedusingthe ESPRIT [28] or MUSIC[29] algorithms.
X
These approaches, known as subspace methods, require that
where h(t) is a known pulse shape. Assume that there
η = L. In this case they achieve the lower bound of (19),
are no more than L pulses within any interval I ,
m namely recover the delays using only p ≥ L +1 sampling
[(m−1)T,mT], m∈Z, and that condition (2) holds for all
channels.In caseswhereη <L, anadditionalsmoothing[30]
intervals I , m ∈ Z. Choose a set K of consecutive indices
m stage is required prior to using the subspace methods, and
forwhichH(2πk/T)6=0, ∀k ∈K.Considerthemultichannel
p≥2L sampling channels are needed.
sampling scheme depicted in Fig. 3, for some choice of
To conclude, when the pulse amplitudes vary sufficiently
coefficients {s } , i=1,...,p. Then, the signal x(t) can
ik k∈K from period to period, which is expressed by the condition
be perfectly reconstructed from the samples {c [m]}p with
i i=1 η =L, the common information about the delays can be uti-
ci[m]= 1 x(t) sike−j2Tπktdt, m∈Z, (16) laipzpedrotaochrepdruecseenthteedshaemreplcinagnrimateprtoov(eLth+e1d)e/lTay.sMesotriemoavteior,nthine
T
ZIm kX∈K thepresenceofnoise,comparedtotheoneusedforthegeneral
as long as p ≥ |K| ≥ 2L, and the coefficients matrix S in model, since it uses the mutual information between periods,
(11) is left invertible. rather than recovering the delays for each period separately.
Will demonstrate this improvementin Section VI-B.
To the best of our knowledge Theorem 2 presents the
This result is summarized in the following theorem.
first sampling scheme for pulse streams with arbitrary shape,
operating at the rate of innovation. Furthermore, as we show Theorem 3. Consider the setup of Theorem 2, where now
in simulations, our method is more stable than previous
L
approaches.
x(t)= a [m]h(t−t −mT), t ∈I ⊂[0,T).
ℓ ℓ l
m∈Zℓ=1
X X
B. Stream of Pulses with Shift-InvariantStructure The signal x(t) can be perfectly reconstructed from the sam-
ples {c [m]}p , m ∈ Z as long as the coefficients matrix S
We now focus on a special case of the infinite model (13), i i=1
in (11) is left invertible and
proposed in [21], where the signal has an additional shift-
invariant(SI) structure.This structureis expressedby the fact
≥L+1 when η =L
that in each period T, the delays are constant relative to the p≥|K|
beginning of the period. Such signals can be described as (≥2L when η <L,
L where η = dim(span({a[m],m∈Z})) denotes the dimen-
x(t)= a [m]h(t−t −mT), t ∈I ⊂[0,T), (17) sion of the minimal subspace containing the vector set
ℓ ℓ ℓ
m∈Zℓ=1 {a[m],m∈Z}.
X X
6
C. Channel Synchronization A. General Periodic Waveforms
The sampling scheme of Fig. 3 has two main disadvan- Our aim is to show how to obtain the required modulating
tagesrelativeto singlechannel-basedschemes:each sampling waveforms (9) using a set of p periodic functions, given by
channelrequiresadditionalhardwarecomponents,andprecise pi(t). Such waveforms can be expressed using their Fourier
synchronizationofthechannel’ssamplingtimesisrequired.In series expansion as
tthhiesaspupbrsoeacctihoenswtoeotrveeartcothmeesyitn.chronizationissue, anddiscuss pi(t)= di[k]ej2Tπkt, (25)
k∈Z
One way to synchronize the channels, is on the hardware X
level, for example by using a zero-delay synchronization where the kth Fourier series coefficient of pi(t) is given by
device [24], [31]. Such a device produces accurate trigger
signalsforthesamplersandintegratorsinallthechannels.An di[k]= 1 T pi(t)e−j2Tπktdt. (26)
T
alternative approach is to perform a prior calibration process, Z0
in which the relative delay of each channel is measured. The Thesumin(25)isgenerallyinfinite,incontrasttothefinite
calibration can be performed at the system manufacturing sum in (9). Therefore, we propose filtering p (t) with a filter
i
stageor duringitspower-on,bystimulatingthesystem witha g(t)whichrejectstheunwantedelementsinthesum(25).The
knownsignal.Aswenowshow,oncethetimeoffsetsbetween filtered waveforms at the output of g(t) are given by
the channels are known, they can be compensated.
p˜(t)=p (t)∗g(t), (27)
Suppose that the ith channel has a time offset of ∆ ∈ i i
i
[−∆ ,∆ ] relative to the optimal sampling instants t =
max max and are also periodic. Therefore they can be written as
nT, where ∆ is the maximal possible offset. We assume
max
that for each time interval T p˜i(t)= d˜i[k]ej2Tπkt, (28)
k∈Z
X
h(t−t )=0, ∀t∈/ [∆ ,T −∆ ) l =1...L. (21)
l max max where it can be easily verified that
This condition ensures that the intervals can be processed 2π
d˜[k]=d [k]·G k . (29)
independently using the approach we now propose. i i T
(cid:18) (cid:19)
The effective delay of the lth pulse measured in the ith
Here G(ω) denotesthe CTFT of g(t). From (29), the shaping
channel is t − ∆ . Substituting into (5), the kth Fourier
l i
filter g(t) has to satisfy
coefficient measured in the ith channel satisfies
nonzero ω = 2πk, k ∈K
Xi[k]= T1H 2Tπk L alej2Tπk(tl−∆i) =e−j2Tπk∆iX[k]. G(ω)=0 ω = 2TTπk, k ∈/ K (30)
(cid:18) (cid:19)Xl=1 arbitrary elsewhere,
(22)
so that d˜[k]= 0 for k ∈/ K. This condition is similar to that
i
Therefore, from (10),
obtained in [13] for single channel sampling. Therefore, the
class of filters developed there, can also be used here as a
ci = sike−j2Tπk∆iX[k]= s˜ikX[k], (23) shaping filter.
kX∈K kX∈K Note that (30) implies that the frequency response of g(t)
is specified only on the set of discrete points 2πk, k ∈ Z,
where we defined T
offeringlargefreedomwhendesigningapracticalanalogfilter.
s˜ik =sike−j2Tπk∆i. (24) For instance, when implementing a lowpass filter (LPF) this
allowsasmoothtransitionbandbetweenthepassbandandthe
stopband of the filter, with a width of 2π.
From (24) we conclude that when the time offsets between T
TheresultingschemeisdepictedinFig.4.Thecorrespond-
the channels are known, the effective mixing matrix of the
system is S˜, a matrix whose ikth element is s˜ . The mis-
ik
alignmentbetweenthesamplingchannelscanbecompensated T1R0T(·)dt c1
by inverting S˜. We will further discuss the effects of channel
misalignment, with unknown delays, in Section VI-C. p1(t) g(t)
x(t)
IV. MODULATIONWAVEFORMS
T1R0T(·)dt cp
We now thoroughly treat the example in Section II-D of
generating the modulation waveforms using periodic signals. pp(t) g(t)
We first address the case of general periodic waveforms, and
then focus on the special case of pulse sequences. Fig.4. Proposedsamplingscheme,usingmodulating waveforms.
7
ing elements of the mixing matrix S are given by byproperselectionof the sequencesα [n], where a necessary
i
condition is that p≥K.
S =d˜[−(k−⌊K/2⌋)]. (31)
ik i We summarize our results in the following proposition:
TheinvertibilityofScanbeensuredbyproperselectionofthe
Proposition 1. Consider the system depicted in Fig. 4, where
periodic waveforms p (t). In the next subsection we discuss
i the modulation waveforms are given by (32). If the following
one special case, which allows simple design S.
conditions hold
1) p≥K, N ≥K,
B. Pulse Sequence Modulation 2) The frequency response of the shaping pulse g(t) satis-
We follow practical modulation implementation ideas pre- fies (30),
sented in [22], [24], and consider the set of waveforms 3) The frequency response of the pulse p(t) satisfies
P 2πk 6=0 for k∈K,
N−1 T
p (t)= α [n]p(t−nT/N −mT) (32) 4) The sequences αi[n] are chosen such that the matrix
i i (cid:0) (cid:1)
AW has a full column rank,
m∈Zn=0
X X
then the mixing matrix S in (11) is left invertible.
for i = 1,...,p, where p(t) is some pulse shape and α [n]
i
is a length-N sequence. Our aim is to calculate the mixing We now give two useful configurations, that satisfy the
matrixS,whenusingthefilteredversionof(32)asmodulating conditions of Proposition 1.
waveforms. To this end, we first compute the Fourier series 1) SingleGenerator: Wecreatetheithsequence,bytaking
coefficients di[k] of pi(t) as a cyclic shift of one common sequence α[n] as
di[k]=1 N−1αi[n] T p(t−nT/N −mT)e−j2Tπktdt αi[n]=α[n−i+1 mod N], (37)
T
m∈Zn=0 Z0 where we assume p = N. Clearly, the corresponding wave-
X X
=1 N−1αi[n] −(m−1)T p(t−nT/N)e−j2Tπktdt fwohrmerse tchaenwbaevecfroeramtedatbthyeuistihngchoannnlyeloinseapduellsaeyegdenveerrastioorn,
T
n=0 m∈ZZ−mT of the generator output, delayed by (i−1)T/N time units.
X X
=1 N−1αi[n] ∞ p(t−nT/N)e−j2Tπktdt Twhhiischsurgeqguesirtes,stmhautltiinplecofrnetqrausetntcoytshoeurdcieresc,thesrceheomnleyionneFipgu.ls1e,
T
n=0 Z−∞ generator is required which simplifies the hardware design. It
X
=T1 N−1αi[n]P 2Tπk e−j2Nπkn, (33) imsaetraisxy. Stoucsheea mthaattriwxicthanthbiesdcehcooimcep,oAsedw[i3l3l]baesa circulant
n=0 (cid:18) (cid:19)
X
A=FHdiag(Fα)F, (38)
where P(ω) denotes the CTFT of p(t).
Combining (33) with (31) and (29),
whereFis a N×N unitarydiscreteFouriertransform(DFT)
N−1 matrix,andαisa length-N vectorcontainingtheelementsof
Sik = 1 αi[n]P 2πk′ G 2πk′ e−j2Nπk′n, (34) the sequence α[n]. Therefore, for A to be invertible the DFT
T T T
n=0 (cid:18) (cid:19) (cid:18) (cid:19) of the sequence α[n] can not take on the value zero.
X
where we defined k′ = −(k−⌊K/2⌋). The resulting matrix Wenowgiveanexampleforsuchaselectionofthesystem’s
S can be decomposed as parameters. We set p=N =K, and choose
S=AWΦ, (35) 1 t∈ 0, T
p(t)= N (39)
where A is a p×N matrix with inth element equal to α [n], (0 t∈/ (cid:2)0,NT(cid:3).
i
W isan N×K matrix with nkth elementequalto e−j2Nπk′n, The frequency response of this puls(cid:2)e sati(cid:3)sfies
and Φ is a K×K diagonalmatrix with kth diagonalelement
P(ω)= T e−j2TNω ·sinc T ω . (40)
Φkk = 1P 2πk′ G 2πk′ . (36) N (cid:18)2πN (cid:19)
T T T
(cid:18) (cid:19) (cid:18) (cid:19) Therefore,
Fromthis decompositionit isclear thatΦ hasto beinvertible
2π T k
and the matrix AW has to be left invertible, in order to P k = sinc , (41)
T N N
guarantee the left invertibility of S. We now examine each (cid:12) (cid:18) (cid:19)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
one of these matrices. which is non-ze(cid:12)ro for k ∈(cid:12) K. In addition we choose the
(cid:12) (cid:12)
We start with the matrix Φ. From (30), G 2πk 6= 0 for sequences α [n] as sequences of ±1s, created from cyclic
T i
k ∈ K. Therefore, we only need to require that P 2πk 6= shiftsofonebasic sequence,ina waythatyieldsaninvertible
(cid:0) (cid:1) T
0 for k ∈ K in order for Φ to be invertible. A necessary matrix A. Such rectangular pulses with alternating signs can
(cid:0) (cid:1)
condition for the matrix AW to be left invertible, is that W be easily implemented in hardware [22]. In Figs. 5 and 6,
hasfullcolumnrank.ThematrixWisaVandermondematrix, onemodulatingwaveformis shownin thetime andfrequency
and therefore has full column rank as long as N ≥ K [32]. domains, for p = N = K = 7. The original time-domain
Theleftinvertibilityofthep×K matrixAW canbeensured, waveformiscomprisedofrectangularpulses,whereaslowpass
8
filtering results in a smooth modulating waveform. Switching from the previous discussion, we demonstrate how to reduce
to the frequency domain, the Fourier series coefficients are the number of required generators, for the current setting.
shaped by P(ω), the CTFT of the pulse shape. The shaping For simplicity, we assume p ≥ 2p and that two different
e
filter frequency response, G(ω), is designed to transfer only generatorsareused.Thefirsthalfofthesamplingchannelsuse
the Fourier coefficients whose index is a member of the set delayed versions of the first generator output, and the second
K={−3,...,3}, suppressing all other coefficients. halfuses the secondgenerator.By properselection ofthe two
sequences, the condition mentioned above can be satisfied.
We now give a numerical example for a such choice.
1.5
We assume L = 4 and set N = K = 9 and p = 18
1 sampling channels, which are based on two generators only.
Each generator produces a different sequence of ±1, chosen
0.5 randomly.In Fig. 7 we plot the log of the maximal condition
number of S˜, obtained when going over all possible options
0
for omitting p rows from S. It can be seen that for p ≤ 6
e e
−0.5 a relatively low conditionnumber of the matrix S˜ is achieved
in the worst case, suggesting that its rank is N as required
−1 and S˜† is not ill-conditioned. Therefore, we can overcome
failure in up to 6 sampling channels, using this system. In
−1.5
this case the required K =9 consecutive Fourier coefficients
−2 original modulating waveform can be obtained from the remaining channels, allowing the
filtered modulating waveform perfect recovery of the L = 4 pulses. In contrast, when
−2.5 channels fail in the direct scheme of Fig. 1, a set of K = 9
0 0.2 0.4 0.6 0.8 1
consecutive Fourier coefficients cannot always be obtained.
time [T]
Therefore, perfect recovery of the signal using methods such
as annihilating filter or matrix pencil is not guaranteed.
Fig.5. Modulatingwaveforminthetimedomain,beforeandafterfiltering.
18
r
0.6 be 16
d[k] m
i u 14
0.5 P(ω) n n
G(ω) o 12
diti
0.4 n 10
o
c
al 8
0.3 m
xi 6
a
m
0.2 4
of
g 2
o
0.1 l
0
0 2 4 6 8
0 p
−10 −5 0 5 10 e
frequency [2π/T]
Fig. 7. Robustness to sampling channels failure example, N = K = 9,
p=18,twopulsegenerators.
Fig.6. Modulating waveform inthefrequency domain.
2) Robustness to Sampling Channels Failure: Next we
V. RELATEDWORK
provide a setup which can overcome failures in a given
A. Single-ChannelSampling with the SoS Filter
number of the sampling channels. The identification of the
malfunctioningchannelsis assumed to be performedby some The work in [13] considered single-channel sampling
external hardware. schemesforpulsestreams,basedonafilterwhichiscomprised
We considera maximalnumberp ofmalfunctioningchan- of a Sum of Sincs (SoS) in the frequency domain. This filter
e
nels and assume that p ≥ N +p . In order to ensure unique can be expressed in the time domain as
e
Sr˜e,coovbetaryineodf fxromfroSmbtyheomreimttaininginogf cthheancnoerlrse,spthoendsiunbgmroatwrisx, g(t)=rect t bkej2Tπkt, (42)
T
shouldbeleftinvertible.Sincethishastobesatisfiedforevery (cid:18) (cid:19)kX∈K
possible selection of p rows, we need to design S such that whereK isthechosenindexset,andthecoefficients{b }
e k k∈K
anyp−p rowswillformarank-Kmatrix.Followingourideas have arbitrary nonzero values.
e
9
Toexploretherelationofourmethodto[13],wefirstfocus B. Multichannel Schemes for Shift-Invariant Pulse Streams
on periodic streams of pulses with period T. By sampling a Another related work is [21] which treats the SI signal
periodicstream ofpulsesusingthe schemedepictedin Fig. 8,
model (17) presented in Section III-B. The sampling scheme
the following samples are obtained
proposed in [21] is depicted in Fig. 9. In each channel, the
input signal is filtered by a band-limited sampling kernel
c[n]= bkX[k]ej2TπknTs, n=0,...,p−1, (43) s∗(−t) followed by a uniform sampler operating at a rate
ℓ
kX∈K of 1/T. After sampling, a properly designed digital filter
correction bank, whose frequency response in the DTFT do-
where T =T/p is the sampling period. Using the matrix V
s main is denotedhere by M(eωT), is applied on the sampling
defined in (6) only now with parameter t ={0,T ,...,(p−
s s sequences. The exact form of this filter bank is detailed in
1)T }, and defining the diagonal matrix B with kth diagonal
s [21]. It was shown in [21], that the ESPRIT algorithm can
element b , (43) can be written in matrix form as
k be applied on the corrected samples, in order to recover the
unknown delays.
c=V(−t )Bx. (44)
s
Therefore,this is a special case of our multichannelsampling s∗1(−t) c1[n] d1[n]
t=nT
schemein(11)withmixingmatrixS=V(−t )B.Thematrix
s x(t) ... M ejωT ...
Bisinvertiblebyconstruction,andV(−t )isaVandermonde (cid:16) (cid:17)
s
masaptri≥x wKit.hUdsiisntigncthtitsimcheso,icseoothfaSt,Sthies lpefsta-imnvpelertsibtlaekeans loovnegr s∗p(−t) t=nTcp[n] dp[n]
one period in [13], are equal to the samples at the output of
Fig.9. Proposedsamplingschemein[21].
the p channels in our scheme.
The sampling rate achieved by the method in [21] is
x(t) SoS Filter c[n] generally2L/T,whereforcertainsignalsitcanbereducedto
t=nT (L+1)/T.Suchsignalssatisfydim(span({d[m],m∈Z}))=
s
L,wherethevectorsd[m]containthesamplesattheoutputof
Fig.8. Single-channel samplingschemeusingtheSoSfilter. the scheme depictedin Fig. 9. This conditionis differentthan
(19), which directly depends on the vectors a[m]. Therefore
ExploitingthecompactsupportoftheSoSfilter,themethod the sampling rate, when using the scheme in [21], can be
wasextendedtothefiniteandinfinitesettingsaswell[13].The reduced to (L+1)/T for different signals. This fact is not
extension is based on using an r-fold periodic continuation surprising,sinceeachapproachhasadifferentanalogsampling
of the SoS filter g(t), where the parameter r depends on stage. In both methods, the worst-case minimal sampling rate
the support of the pulse-shape h(t). For the infinite case, the is 2L/T.
The approach in [21] has two main advantages over the
samplesofthetwoschemesstillcoincide.However,since[13]
proposedmethodinthiswork.Thefirstisthatcondition(2)is
is a filtering based scheme, proper separation of at least 1.5T
notrequired.Therefore[21] can also treat pulses with infinite
between periods is required in order to obtain independent
time support, in contrastto our method. Anotheradvantage is
processing of each period. While keeping the same sampling
thatitcansupportsinglechannelconfigurations.Itwasshown
rate of 2L/T, the model in [13] has a rate of innovation of
in[21],[34],thatonesamplingchannelfollowedbyaserialto
2L/(2.5T). Hence the single channel configuration of [13]
parallelconverter,canbeusedin ordertoproducetheparallel
does not achieve the rate of innovation in the infinite case.
sampling sequences in Fig. 9.
Wenowexaminethemodulationwaveformsthatresultfrom
Ontheotherhand,themethoddepictedinFig.3hasseveral
[13]. Since S=V(−t )B,
s
advantagesover[21].First, the equivalentstage forthe digital
si(t)= bke−j2Tπk(t−iT/p). (45) caonrdreSc,tisoinncien [21], is replaced by inversion of the matrices H
k∈K
X
y[m]=(SH)−1c[m]. (47)
It is easily shown that these waveforms can be expressed as
This operation can be viewed as a one-tap digital correction
s (t)=g˜(−(t−iT/p)), (46) filter bank, in contrast to the filter M(eωT), which generally
i
hasalargernumberoftaps.Therefore,theproposedcorrection
where g˜(t) is the periodic continuation of the SoS filter g(t). stage, is much simpler and requires lower computational
Therefore, in each channel the signal is modulated by a complexity, than the one in [21].
delayed version of the periodic SoS filter. The equivalence Anadditionaladvantageofourscheme,isthattheapproach
of the schemes is easy to explain: sampling the convolution of [21] requires collection of an infinite number of samples,
betweentheinputsignalandtheSoSfilterin[13],isequivalent evenwhentheinputsignalcontainsafinitenumberofperiods.
to performinginnerproducts(multiplicationfollowedbyinte- Thisrequirementisduetotheinfinitetimesupportoftheband-
gration)with delayedand reflected versionsof this filter. This limited sampling kernels. Moreover, if one is interested only
relationprovidesanothervalidclassofmodulationwaveforms. in a finite time interval of the signal, the method in [21] does
10
not allow processing it separately. This is in contrast to the VI. SIMULATIONS
proposed scheme, which integrates finite time intervals, and
In this section we provideseveralexperimentsin which we
can collect samples only from the relevant periods. We will
examine various aspects of our method. The simulations are
demonstrate this advantage in Section VI.
divided into 4 parts:
1) Evaluation of the performancein the presence of noise,
C. Modulated Wideband Converter and comparison to other techniques;
2) Demonstrationoftherecoverymethodforpulsestreams
The concept of using modulation waveforms, is based on with SI structure;
ideaswhichwerepresentedin[22]–[24],[35].Wenowbriefly
3) Evaluation of the effects of synchronization errors be-
reviewthesamplingproblemtreatedin[24]anditsrelationto tween the channels;
our setup.As we show the practicalhardwareimplementation 4) Examination of the use of practical shaping filters.
of both systems is similar.
The model in [24] is of multiband signals: signals whose
CTFT is concentrated on N frequency bands, and the A. Performance in the Presence of Noise
bands
width of each band is no greater than B. The location of the We demonstrate the performance of our approach in the
bands is unknown in advance. An example of such a signal presenceof white gaussiannoise,when workingat the rate of
is depicted in Fig. 10. A low rate sampling scheme allowing innovation. We compare our results to those achieved by the
recoveryofsuchsignalsatarateof4BNbands wasproposedin integrators [19] and exponential filters [20] based methods,
[7]. This scheme exploits the sparsity of multiband signals in since these are the only approaches which can work at the
the frequencydomain,to reducethe samplingrate wellbelow same rate, for infinite stream of pulses.
the Nyquist rate. In [22]–[24], this approach was extended to
We examinethree modulationwaveformspresentedin Sec-
tionsII-DandV-A:cosineandsine waveform(tones),filtered
B rectangular alternating pulses (rectangular) and waveforms
··· obtainedfromdelayedversionsoftheSoSfilter(SoS).Forthe
rectangularpulsesscheme,themodulationwaveformsaregen-
0 fmax
erated using a single generator, as discussed in Section IV-B.
Fig.10. Multiband signalmodel. The shaping filter g(t) is an ideal LPF with transition band
of width 2π/T. Following [20], the parameters defining the
a more practical sampling scheme, which uses a modulation impulse response of the exponential filters are chosen as
stage and referred to as the Modulated Wideband Converter α=0.2T and β =0.8T.
(MWC). Ineachchannelofthe MWC, the inputis modulated We focus on one period of the input signal, which consists
with some periodic waveform, and then sampled using a LPF of L = 2 Diracs with t = [0.256T,0.38T]T, and amplitudes
followedbyalowrate uniformsampler.Themainideais that a=[1,0.8]T.Wesetthesystemparametersasp=K =N =
in eachchannel,the spectrumofthe signalis scrambled,such 5. The estimation error of the time-delays versus the SNR is
that a portion of the energy of all bands appears at baseband. depicted in Fig. 11, for the various approaches. Evidently,
Mixingof the frequencybandsin [24] is analogousto mixing our technique outperforms the integrators and exponential
the Fourier coefficients in Fig. 3. filters based methods in terms of noise robustness, for all
We note here some differences between the methods. First, configurations. There is a slight advantage of 2dB for the
following the mixing stage, we use an integrator in contrast schemes based on tones and SoS, over alternating pulses,
to the LPF used in [24]. This difference is a result of the where the first two configurations have similar performance.
differentsignalquantitiesmeasured:Fouriercoefficientsinour Turning to higher order problems, in Fig. 12 we show the
work as opposed to the frequency bands content in [24]. The results for L = 10 Diracs with times chosen in the interval
second difference is in the purpose of the mixing procedure. [0,T) and amplitudes equal one, with N = p = K = 21.
In [24] mixing is performed in order to reduce the sampling The instability of the integratorsand exponentialfilters based
rate relative to the Nyquist rate. In our setting, the mixing is methodsbecomesapparentinthissimulation.Ourapproachin
used in order to simplify the hardware implementation and to contrast achieves good estimation results, demonstrating that
improverobustnesstofailureinoneofthesamplingchannels. our method is stable even for high model orders.
Nonetheless, the hardware considerations in the mixing The performance advantage of the tones and SoS based
stage in both systems is similar. Recently, a prototype of the schemesis now around3.5dB. We concludethatfroma noise
MWC has beenimplementedin hardware[22]. Thisdesign is robustness point of view, using multiple frequency sources or
composed of p = 4 sampling channels, where the repetition SoS waveforms is preferable over a single pulse generator.
rate of the modulatingwaveforms is 1/T ≈20 MHz. In each However,as discussed in Section II-D, pulse sequencesbased
period there are N =108 rectangular pulses. This prototype, schemes can be advantageous from practical implementation
with certain modifications, can be used to implement our considerations, and reduce the hardware complexity. In addi-
samplingscheme as well. These modificationsmainly include tion, the performance degradation is reasonable, and the esti-
adding shaping filters on modulating waveforms lines, and mation erroris still significantly lower than that of competing
reducing the number of rectangular pulses in each period. approaches. Therefore, the flexibility of our architecture, al-