20th European Signal Processing Conference (EUSIPCO 2012)
Bucharest, Romania, August 27 - 31, 2012
PERFORMANCE BOUND FOR TIME DELAY AND AMPLITUDE ESTIMATION
FROM LOW RATE SAMPLES OF PULSE TRAINS
Ciprian R. Comsa *,†, Alexander M. Haimovich †
*
Telecommunications Dept, “Gheorghe Asachi” Technical University of Iasi, Romania
†
ECE Dept, New Jersey Institute of Technology, USA
ABSTRACT
The problem considered is estimating the time delays and
amplitudes of a train of pulses of known shape. It was
recently shown that in the absence of noise, if the pulses are
short relative to the interval separating them, the stream of
pulses can be sampled at rates lower than the Nyquist rate
without any loss of information. This paper investigates the
performance of time delay and amplitude estimation from
samples taken at low (sub-Nyquist) rates when the stream of
pulses is affected by additive white Gaussian noise. To this
end, the Cramér-Rao bound (CRB) is developed. For a
particular setup, the CRB for time delay estimation is
inverse proportional to the cube of the sampling rate, while
the CRB for amplitude estimation is inverse proportional to
the sampling rate. Numerical simulations confirm this result.
Index Terms — Cramér-Rao bound, time delay and
amplitude estimation, low rate sampling, pulse train
1. INTRODUCTION
In many applications, signals can be uniquely described by a
small number of parameters. For example, in radar and
wireless communications signal can be constituted of a
stream of short pulses of known shape, stream that is
defined by the time delays and amplitudes of the pulses.
Estimation of these parameters is the subject of this work.
Time delay estimation and amplitude estimation are
conventionally performed from samples taken at rates
higher than the Nyquist rate, which is twice the signal’s
bandwidth. This approach is required when the only
knowledge on the signal is that it is bandlimited. Other
priors on signal structure can lead to more efficient
sampling. An interesting class of structured signals was
considered in [1], where the signals have a finite number of
degrees of freedom per unit time. These signals were termed
to have finite rate of innovation (FRI). A special case of FRI
signals consists of a stream of pulses of known shape per
time interval . For such a stream, any of its segments of
length
is uniquely determined by no more than 2
parameters, i.e., time delays and amplitudes. It is then
© EURASIP, 2012 - ISSN 2076-1465
455
said that the signal’s local rate of innovation is finite and
equals 2 ⁄ .
For streams of pulses of short duration relative to the
reference interval , standard sampling methods require
very high sampling rates. However, the rate of innovation
per interval is much smaller than the number of samples
taken at the Nyquist rate. This observation was exploited to
set the grounds for sampling at rates lower than the Nyquist
rate. To this end, a mechanism to sample at low rates
streams of Diracs can be found in [1], and the references
therein. A scheme for recovering the original stream from
the samples was also proposed. More recent work, [2, 3],
generalizes the approach to sampling at low rates streams of
pulses of known arbitrary shape, rather than just Diracs, by
considering a multichannel sampling setup. Although the
proposed multichannel sampling scheme allows perfect
recovery of the original signal from its noiseless samples, it
was found in [4] that in the presence of noise the
performance of the signal recovery from low rate samples
deteriorates significantly relative to the signal recovery from
samples taken at the Nyquist rate. However, with low rate
sampling, increasing the sampling rate above the rate of
innovation brings substantial performance improvement.
This work considers low rate sampling of streams of
pulses as in [2, 3], but rather than recover the signal itself,
the goal is to estimate specific signal parameters,
specifically, delay and amplitude. The signal recovery in [2,
3] is equivalent to estimating all the unknown signal
parameters, e.g., time delays and corresponding amplitudes.
By contrast, estimation of individual parameters may be
performed independently, e.g., delay estimation does not
require the estimation of amplitudes, and thus its
performance may vary with system parameters differently
than that of signal recovery.
The paper is organized as follows: Section 2 provides the
system model for a multichannel sampling scheme. In
Section 3, CRB expressions are developed for TDE and
amplitude estimation. Discussion of our results vis-à-vis
other work is included. Section 4 contains numerical results,
and conclusions wrap up the paper.
2. SYSTEM MODEL
a).
T
The problem considered
c
is estimation of a vector
th
hat
pparameterizes a FRI segmeent
of length
fro
om
ssamples taken at low rates from
f
a noisy observation
o
.
T
To give practiccal relevance to
o the problem considered,
is viewed as the
t product of
o transmitting
g a signal
∑
,
,
thhrough a multipath channeel
w
where
are the complex valued amplitu
udes and
the
t
tiime delays asssociated with
h each propag
gation path. The
T
consists of a train of
trransmitted sign
nal
o pulses. That is,
is emitted periodically att a
a pulse of kno
own shape
cconstant rate 1⁄ , after being
g modulated by
y some arbitraary,
. Vector
kknown sequencce
contain
ns then the tim
me
ddelays and am
mplitudes as th
he unknowns that
t
parameteriize
thhe signal
. It is assum
med that
≫
, for any
a
∈ 1, … , . Also, the multtipath channel is assumed tim
me
innvariant. Thuss, the amplitud
des
do not change
c
from one
o
1 ,
innterval Ι
to another. Additionally,
A
the
t
rreceived signal is shift invariaant, i.e., the tim
me delays also do
nnot change from
m one intervall Ι to anotherr, with respect to
thhe beginning of
o each intervaal. The received
d signal is then
na
ccomplex valued
d, noisy semi-p
periodic train of
o pulses:
N
K
y  t     ak  n  g  t   k  nT     t  ,
b).
Fig. 1. a). System moodel; b). Filter--bank samplingg block.
main multichannnel sampling methods weree proposed
Two m
recentlyy in the literatture, i.e., the ffilter-bank metthod in [2]
and thee modulator-baank method in [3]. The moduulator-bank
samplinng scheme cann be used to trreat different F
FRI signals
under the assumptioon that the puulse
is compactly
supportted and pulsess do not overlaap boundaries of interval
Ι . In contrast, thee filter-bank sampling sccheme can
accomm
modate arbitrrary pulse sshapes
, including
infinitee-length functiions. Also, thhe filter-bank scheme is
well suuited for sam
mpling long ( → ∞) sem
mi-periodic
signals of form (1). F
For the rest of the paper, the filter-bank
samplinng is assumed,, with the note that similar annalysis can
be carriied out for the modulator-bannk scheme.
The filter-bank schheme is illustrated in Fig. 1bb. On each
branch,, the signal is filtered by ∗
, followedd by actual
⁄ . The signall model is handled easier
samplinng at a rate 1⁄
in the frequency dom
main. For lonng observationn intervals,
→∞
∞, the discreete-time Fourrier transform
m (DTFT),
≜∑ ∈
, cann be applied. With this,
(2) becoomes
(1)
(
n 1 k 1
⁄ . The signal is
an
w
where
nd
.
ddefined by time
t
delays
and by
am
mplitudes
of length is defined by
H
However, any segment of
oonly
time delays and
amplitudes. Moreover, with
w
, the whole observ
kknowledge of the sequence
ved
ssignal
, of length , is defined by tim
me delays and
a
amplitudes . Thus signall
is FRI.
The receiveed signal is affected by complex valu
ued
aadditive white Gaussian noiise (AWGN),
, of pow
wer
sspectral density
y (PSD) Φ . For later use,, let
Φ ⁄
ddenote the pow
wer of the noisee after passing through an ideal
loow pass filter of
o cut-off frequ
uency 1⁄2 .
The system model
m
for the problem
p
considered is depictted
inn Fig. 1a. The
T
continuous-time signal
is passsed
thhrough a Sam
mpling block to
o obtain the set of samples .
B
Based on samp
ples , an estim
mate of is obtained
o
with the
t
E
Estimation blo
ock. For the Sampling
S
block
k a multichann
nel
sstructure is con
nsidered. With a multichanneel sampling settup
is convolveed with
thhe signal
diffferent functio
ons
∗
, …, ∗
, and th
he output of each channel is
ssampled at a rate 1⁄ , [2, 3]. The set of sam
mples is given by
b


∗
,
1,
1 …, .
(3)
where
, and
and
,
,
, noise
,
denote the Foourier transforrms of
, aand sampling ffilters ∗
, respectively. Note that,
∑
based oon (1),
, where
iss the Fourier trransform of
.
Focuusing, as propposed in [2], oon sampling ffilters with
finite support in thee frequency doomain, e.g., coontained in
⁄ ,
⁄ , (33) becomes
the rangge
∗
P
∗
,
m 


(2)
(
∗
,
(4)
q1
⁄ .
T
The system is said
s to have a total sampling rate
r
where
456
2
1
⁄2 ⁄ .
Note
thatt
all
the
expressions in the frequency domain are 2 ⁄ periodic and
is restricted to the interval 0, 2 ⁄ .
from all the
If is a column vector collecting
sampling channels,
,
where
is a
matrix of elements
3. CRAMER-RAO BOUND
The performance of an estimate of a vector parameter
from a set of measurements is typically measured by its
mean squared error (MSE). A lower bound on the MSE is
often used instead to gain insight into the factors affecting
the MSE. The Cramér-Rao bound (CRB) is a lower bound
on the MSE of any unbiased estimate of . For the problem
at hand, the available measurements are the samples ,
obtained as (5) by a filter-bank low rate sampling scheme
in (1) is
applied to the FRI signal
of (1). Since
assumed known, the unknown parameter vector is
is the
,…, , ,…, , ,…,
, where
Re
real part of
and
is the imaginary part of . The CRB
on the MSE of an element of is given by the - element
of a matrix
, i.e., CRB
, . By definition,
the matrix
is the inverse of the Fisher Information
Matrix (FIM), whose elements are given by [8],
(5)
∗
,
,…
,
is a
diag
matrix of elements
,
,
diag
,…
,…
,
is the DTFT of
, is a column vector collecting elements
,
and is a column vector collecting elements
.
Although it not explicitly evident in the notation, note the
of the all the arrays in (5) except
.
dependence on
Also note that the unknown time delays
are embedded in
and , while the unknown amplitudes
are in the
amplitudes
. Matrices and are known.
We briefly review the operations performed within the
Estimation block, as proposed in. First, (5) is normalized by
, with
. Thus, let
. By denoting
and
(dependence
on
continues to be implicit) and taking the inverse
DTFT,
.
,
(8)
where
; is the probability density function (pdf) of the
sample vector , parameterized by .
The CRB for the signal reconstruction from low rate
samples was discussed in [4]. In contrast, we are concerned
here with parameter estimation, rather than estimation of the
signal itself. Insight into the performance is developed from
closed-form results for a particular setting, as stipulated in
Theorem 1 below.
We make the following two assumptions:
(A1) The pulse shape shape
is ideal, in the sense
that
1 for
∈ 2
, 2
and
0
everywhere else.
(A2) Φ
is the DTFT
, where
is assumed constant and
of the sequence
, and Φ
known within the frequency range of interest. With this, we
⁄ . Note
define the signal to noise ratio (SNR) as Φ
that if
is a long bipolar sequence, where the 1
and 1 symbols are equiprobable, it can be further shown
, [9].
that Φ
(6)
Matrix
has a Vandermonde structure and thus
super-resolution techniques, [5, 6], traditionally used in
spectral estimation, can be employed with (6) to estimate the
time delays , where the number of multipaths is a priori
known, [2]. Once is known, the vector can be found
using the linear relation
, where is the
Moore-Penrose pseudo-inverse of
. Further, with
knowledge of the sequence
, the amplitudes
can
be determined.
In the absence of noise , the filter-bank sampling
scheme supports recovery of the signal
only if the
number of samples
per interval
is at least 2 . The
number of samples cannot exceed the number at Nyquist
rate, 2
, where
is the single side bandwidth of the
pulse shape (if band-limited), [2, 7]. Combined, these lead
to condition 2
2
for the filter-bank scheme to
work.
Also note that the values of
are obtained in (6) by
taking the inverse DTFT of
. Thus, the matrix
needs to be stable invertible. One example of sampling
filters satisfying this condition is
,
2 ⁄
, for ∈
0, otherwise. ;
ln
,
Theorem 1. Let in (5) consist of the low rate samples of a
semi-periodic stream of pulses in AWGN noise, of form (1).
Under assumptions (A1) and (A2), with the choice (7) of the
sampling filters, and for a multipath free environment, i.e.,
1, the CRBs for delay and amplitude estimation from
low rate samples are:
3
2
CRB
CRB
(7)
1
| |
1 1
,
.
Proof: See Appendix for a sketch of the proof.
457
(9)
(10)
dB
CRB
MSE
MSE
⁄
⁄
dB
Sim
CRB
dB
,
20
Fig. 4. Delay estimattion errors as ffunction of the number of
samplinng filters.
dB
dB
F
Fig. 2. Accuraccy of delay esttimation versuss SNR.
Sim
MSE
MSE
CRB
CRB
dB
,
2
20
Fig. 5. Amplitude estimation errrors as functiion of the
numberr of sampling ffilters.
F
Fig. 3. Accuraccy of amplitud
de estimation veersus SNR.
D
Discussion. The CRB exp
pression (9) shows that the
t
pperformance of delay estimaation improvess with the SN
NR,
i.e., the CRB decreases as the
t SNR increeases. The sam
me
oobservation hollds for the CRB
B of amplitudee estimation (10).
IInterestingly, the CRB of deelay estimation
n decreases with
w
thhe cube of the
t
number of
o sampling filters,
f
i.e., it is
pproportional to
. This meeans that increaasing the numb
ber
oof sampling fiilters quickly improves the performance of
ddelay estimatiion. In contrrast, the CRB
B of amplitu
ude
eestimation is proportional only to
. If we deno
ote
⁄ , and
d employ the assumption
a
thaat | |
1, the
t
1⁄
C
CRB can be written as 3⁄2
for dellay
eestimation and 1⁄
for
f amplitude estimation. With
W
thhese definition
ns, the expresssion for the CR
RB has the sam
me
fform as the CR
RB for delay esstimation in co
ontinuous sign
nals
ddeveloped in [1
10], where the term
is referred to as
ppostintegration SNR. This iss justified by the fact that the
t
pproduct
is the processing gain du
ue to the tim
me
bbandwidth prod
duct of the sig
gnal. Thus we observe that the
t
vvariation with
of the CR
RB of delay esstimation can be
ssplit into the effect of ban
ndwidth
an
nd the effect of
pprocessing gain
n P. In the casse of amplitud
de estimation, the
t
C
CRB is influenced only by thee SNR gain.
1 , and
10 . Results shhown are averraged over
1000 ruuns in which the noise havee been chosenn randomly
(sourcee location was kept fixed). IIt may be noticed that at
SNR, the M
high S
MSE from simulations aapproaches
asympttotically the CR
RB. This conffirms that the C
CRB given
by (9) is a tight loweer bound. Withh the logarithm
mic scaling
used inn Fig. 2, the CR
RB decreases llinearly with sslope -3. In
contrasst, at low SN
NR, maximum
m likelihood estimation
experieences a threshhold effect, ii.e., the MSE
E increases
steeply . This is beecause, as thhe SNR decrreases, the
estimattion enters a soo called “large--errors” region, where the
noise vvalues become dominant overr the signal. Foor the given
FRI prooblem addressed in this worrk, time delay values are
boundeed by andso are the errorss, resulting in tthe plateau
region evident at the low end of thee SNR scale. IIn Fig. 3, it
is obserrved that for tthe same setupp as delay estim
mation, the
MSE off amplitude esttimation variess linearly with slope -1.
Bas ed on (9) aand (10), the performance of delay
varies frrom 2 to
estimattion improves with
when
2
, while the amplitude eestimation peerformance
improvves only with
. When
2
, the estimation
perform
mance of the ffilter-bank scheme equals thhe one of a
single ffilter scheme w
working at Nyqquist rate [1]. T
This can be
observeed in Fig. 4, annd Fig. 5, wheere for low ratee sampling
the sim
mulated MSE annd CRB are pllotted against tthe number
of samppling filters . The CRB forr Nyquist rate for
10is thhe same as thhe CRB for llow rate samppling with
20
0.
4.
4 NUMERICA
AL EXAMPLES
T
The MSE off delay estim
mation (CRB and maximu
um
liikelihood simu
ulations) as a function of SNR
S
is shown in
F
Fig. 2 for
100 symbolls,
10 sam
mpling channeels,
458
everywhere else, determines that matrix
is an
identity matrix. This leads to further simplification of
(13) to (14).
,…, , ,…, , ,…,
In (14) we use
∑
and
to determine
the elements of the FIM. Particularizing it for
1,
5. CONCLUSIONS
Performance of time delay and amplitude estimation from
low rate samples of pulse trains has been addressed. In
particular, the CRB expression for the case of using a filterbank sampling scheme was developed. With some
simplifying assumptions, closed form expressions were
determined for multipath free signal. The estimation
performance in noise was shown to deteriorate significantly
if the number of samples taken is the one indicated by the
rate of innovation rather than by the Nyquist rate.
Specifically, with the simplifying assumptions considered,
the CRB is inverse proportional to the cube of the number of
sampling filters for delay estimation and to the number of
sampling filters for amplitude estimation.
 P 2 2 1 2 3T 2 0 0 


0
1 0 .


0
0 1


With (15), the equations of Theorem 1 can be easily derived.
Moreover, it can be shown that for
2, for well
separated multipaths, the CRB for the time delay and
amplitude of any th component is also given by Theorem 1.
6. APPENDIX
7. REFFERENCES
Due to space considerations, we can provide in this
appendix only a sketch of the proof of Theorem 1. We start
by expressing the pdf
;
used in (8) in terms of the
signal model (1)
1
;
,
det
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(11)
1 ,…,
,
1 ,…,
,
where
and the observation interval is limited to 0,
. The
denotes the
covariance matrix of the
matrix
noise vector
1 ,…,
. Equation (15.52) from
[8] is applied to (11), together with the observation that the
covariance matrix
does not depend on parameter
[11], to reduce (8) to
2Re
,
(12)
⁄
Re
d
(13)
⁄
⁄
1
∗
d .
Re
(14)
⁄
Term (13) is obtained from (12) by replacing the data vector
by the vector of its Fourier coefficients (obtained by
applying the DTFT to , when → ∞) and following a
is a
reasoning similar to that in [12]. Matrix
given
by
the
DTFT
of the
matrix of elements
,
cross-correlation
1 ,…,
of
sequences
(15)
1 ,…,
and
. With the choice (7) of sampling filters,
⁄
| ,…|
diag |
is
Furthermore, the choice of an ideal pulse shape
sense that
1 for
∈
,
and
| .
, in the
0
459