Wireless Sensor Network, 2010, 2, 462-466
doi:10.4236/wsn.2010.26057 Published Online June 2010 (http://www.SciRP.org/journal/wsn)
Reconstruction of Wireless UWB Pulses by Exponential
Sampling Filter
Juuso T. Olkkonen1, Hannu Olkkonen2
1VTT Technical Research Centre of Finland, Espoo, Finland
2Department of Physics and Mathematics, University of Eastern Finland, Kuopio, Finland
Received November 16, 2009; revised December 25, 2009; accepted January 20, 2010
Abstract
Measurement and reconstruction of wireless pulses is an important scheme in wireless ultra wide band
(UWB) technology. In contrary to the band-limited analog signals, which can be recovered from evenly
spaced samples, the reconstruction of the UWB pulses is a more demanding task. In this work we describe an
exponential sampling filter (ESF) for measurement and reconstruction of UWB pulses. The ESF is con-
structed from parallel filters, which has exponentially descending impulse response. A pole cancellation filter
was used to extract the amplitudes and time locations of the UWB pulses from sequentially measured sam-
ples of the ESF output. We show that the amplitudes and time locations of p sequential UWB pulses can be
recovered from the measurement of at least 2p samples from the ESF output. For perfect reconstruction the
number of parallel filters in ESP should be 2p. We study the robustness of the method against noise and dis-
cuss the applications of the method.
Keywords: Wireless Sensor Networks, UWB, Network Security, Finite Rate of Innovation
1. Introduction
The measurement and reconstruction of some classes of
signals containing discontinuities such as impulses and
edges is difficult [1,2]. Sampling methods are historically
relied on Shannon’s theorem [3]. The perfect reconstruc-
tion (PR) of the continuous signal from the sampled ver-
sion requires that the signal is band limited, i.e., its fre-
quency spectrum has a maximum frequency
M
f
. The PR
is possible only if the sampling frequency 2
s
M
f
f. For
example, signals in optical devices and radiation detec-
tors are not band limited and classical sampling ap-
proaches are not relevant for extracting the information.
Sampling scheme with finite rate of innovation (FRI)
[4,5] has recently got vast interest in signal processing
society, since the FRI methods are not restricted to the
recovery of the band-limited signals. The key idea in FRI
methods is that the signal (e.g., the Dirac impulse stream)
is measured with a sampling filter, which is constructed
using analog circuits. The output of the sampling filter is
measured and the original signal is reconstructed from
the discrete samples. Excellent articles and reviews con-
sider the FRI sampling and reconstruction techniques
[4-9] and many feasible applications have been published.
However, the experimental work on the verification of
the underlying theoretical considerations is lacking, espe-
cially the effect of noise on the reconstruction accuracy.
The information in wireless ultra wideband (UWB)
devices is usually carried out by monocycle Gaussian
pulses. In year 2002 the FCC restricted the allowed fre-
quency band between 3.1-10.6 GHz for unlicensed UWB
transmission [10]. The monocycle Gaussian pulse stream
does not strictly meet this constraint and other pulse
shapes have been introduced, e.g., the family of the or-
thogonal UWB pulse waveforms [11]. In low-range
wireless UWB communication devices, which transmit
sequential pulses, the information is coded to the time
locations of the pulses. The UWB pulse generators are
relatively easy to implement in VLSI [12]. The pulse
stream is designed so that its power spectral density co-
incides with the FCC criteria.
In this work we study the FRI-like method aimed at
sampling and reconstruction of the UWB pulses. As a
sampling device we apply an exponential sampling filter
(ESF), whose output is measured sequentially. We study
the robustness of the method against noise and discuss
the applications of the method.

J. T. OLKKONEN ET AL. 463
2. Theoretical Considerations
2.1. Sampling of the UWB Pulse Train
We consider the UWB pulse train, which is approximated by
1
()( )



p
ii
i
I
tAtt  (1)
where the Dirac distribution ()1 for 0xx

 and
()0 for 0xx

. denotes the amplitude and
time location. The pulse train is fed to the exponential
sampling filter (ESF) consisting of N parallel RC-filters
(Figure 1). The ESF has the causal impulse response
i
Ai t
1
( )exp[]0



N
k
k
htBkt t   (2)
The output signal of the ESF is
11
()()()( )()
exp[ ( )]0
pN
ik i
ik
x
tIthtIht d
ABktt t






 


 

(3)
The key idea is to start the measurement of the ESF
output at time , simultaneously clos-
ing the input of the ESF (Figure 1). We define the dis-
crete time variable n as
0(1, 2,...,) i
tti p
0(0,1,2,... tt nTn)0
nTtt
where T is a sampling period. For the samples of the ESF
output the following is valid [],0,1, 2,...
n
xxnTn
0
11
exp[ ()]



pN
nik i
ik
x
AB ktnTt  (4)
By rearranging and denoting  
io
tt
i
Figure 1. The construction of the ESF from the parallel
exponential filters (EF), which are built using RC-circuits.
The ESF parameter is related as ,
where and are the component values.
kk 1/(R C )αkT 
k
Rk C
11 1
()
 
 

 
i
p NN
kkT nn
nki k
ki k
xBAee b

k

(5)
where
1
and
 


i
p
kkT
kk ik
i
bB Aee

 (6)
2.2. Reconstruction of the Pulse Train
Our task is to recover the amplitudes and time loca-
tions of the pulses from the measured output samples
[],0,1,2,...


n
xxnTn We may write (5) in the form
of matrix/vector equation
01
212
11 1
112
11 1
N
NN N
NN
2
N
x
b
x
b
x
b
 
 
 


 

 

 


 

 

 
 
-1
xλbbλx





(7)
The Vandermonde matrix structure in (7) is non-
singular having rank N and the solution of the vector
requires knowledge of the N measurement values. From
(6) we obtain
λ
b
11
/ =




i
pp
kk
kkk iii
ii
cbB AeAr

(8)
where


i
i
re

(9)
The z transform of the sequence gives
k
c

111
111 0
1
1
1
= ()
(1 )
1
pp
NN
kk k
kiiii
kiik
NN
p
ii i
ii
ZcArzA rz
Arzr z
rz


 



 





(10)
We define the pole cancellation filter (PCF) as
1
11
() 1(1)




p p
n
pc nj
nj
H
zhz rz (11)
where
j
r is defined by (9). Next we consider the prod-
uct filter
()Pz


11
11
() ()
(1)(1)
kpc
p p
NN
ii ij
ij
ji
PzZ cHz
A
rzrzr z




 

k
h
(12)
We may note that the roots of the equal the
roots of the PCF. The impulse response of the is
()Pz
()Pz
0



p
nnk
k
pc (13)
For solution of the roots of the we set ()Pz 0

n
pfor
. This yields the matrix/vector equation 0n

464 J. T. OLKKONEN ET AL.

221221
2122 2312
111
0
ppp p
ppp p
pppp
ccc ch
ccc ch
ccc ch

 

 
 
 

 
 
 
 
 -1 cChh Cc




－
(14)
We may note that the solution of the p coefficients of
the PCF requires the knowledge of the 2p values of the
sequence. The polynomial
k
c12
[1 ]
p
hh h
p
has the
roots , which give

i
i
re

(1, 2,...,)ilo g
i
()
i
r/

and the time locations as tt
0log() / ii
r

.
For solution of the amplitudes we may write (8) in ma-
trix/vector form
1121
22 2
112 2
12
p
p
pp p
pp
crrrA
crr rA
crrrA








 -1
cRaaRc


 
p
p
]
(15)
To summarize, the reconstruction of p sequential pulses
needs the measurement of at least 2p samples from the
ESF output. The impulse response of the ESF must obey
(2), where . It should be pointed out that the
solution of the matrix Equations (7) and (15) require the
inversion of the Vandermonde matrix, which can be ob-
tained by an analytical formula [13]. The inversion
method yields more stable results compared with the
general matrix inversion.
2N
2.3. Reduction of Noise
In practical measurements noise arising in electronic
circuits interferes the results. We apply the SVD based
subspace method for reducing the noise in measurement
signal. Let us construct the Hankel matrix containing the
measurement values [
n
x
xnT , ( 0,...,2 )nMp
1






n
01 /2
12 /2
/2/2 1










 

M
M
MM M
xx x
xx x
xx x
H (16)
where the antidiagonal elements are identical. The sin-
gular value decomposition (SVD) of the matrix is H
T
H = UΣV (17)
where and are unitary matrices. is a diago-
nal matrix consisting of the singular values in descending
order. The decomposition (17) can be separated as
U VΣ
 




T
s
sn
sn
n
TT
sns ssn n
0
Σ
H = UU VV
0Σ
= =
UVU V ΣΣHH
(18)
where contains the smallest singular values.
matrix belongs to noise subspace. The matrix is then
related to the noise free signal subspace [14]. The signal
matrix is not precisely Hankel matrix, but some
variation occurs in the antidiagonal elements. We re-
constructed the noise free Hankel matrix by replacing
the antidiagonal elements by their mean values. This
enables the computation of the noise cancelled xn
n
Σ
s
H
1,...,
n
H
s
H
( 0,n

)
M
sequence. The dimension of the signal
subspace is N, i.e., the number of parallel RC-circuits in
ESF. It should be pointed out that the Hankel matrix (16)
must be a full matrix. Therefore M should be even.
3. Experimental
Extensive simulations were performed to validate the
theoretical results. The T

parameter in (5) varied
between 0.1-0.3. The number of the impulses in one
burst varied between 3-7 and the number N in ESF (2) in
the range 3-15. The amplitudes of the pulses were
randomly distributed between the limits 0.1-0.9. The
simulations warranted the condition that for the recovery
of p pulses at least 2p samples are needed in the de-
scending part of the ESF output. A typical simulation
study is illustrated in Figure 2. In every case the ESF
method recovered the amplitudes and the time locations
of the pulses with a machine precision.
A prototype ESF was constructed with the aid of six
parallel RC-circuits (Figure 1). The output was
measured by a 16 bit analog-to-digital converter (ADC),
which was equipped with a sample and hold amplifier
(S/H). The input of the ESF was closed using the analog
CMOS switch. The ESF was reset by grounding the
output by an analog CMOS switch. Using a commercial
UWB pulse generator two sequential pulses were fed to
the input of the ESF and the descending part was meas-
ured with a 40 MHz sampling frequency. Due to the
noise interference in practical measurements the use of
the SVD based noise suppression method was essential.
The prototype ESF recovered the amplitudes and the
appearance times with a good accuracy. The mean error
(standard deviation divided by the mean value) in ampli-
tudes was 2.7% and in time locations 0.2%.
4. Discussion
The present work proposes a new approach for measuring
of the impulse train using the ESF. The ESF yields expo-
nentially descending pulses, which are sampled sequen-
tially. In the beginning of the measurement period the
input is closed and after measurement period the ESF is
reset. We showed that using 2p parallel RC filters it is
possible to reconstruct p impulses. The required number

J. T. OLKKONEN ET AL. 465
Figure 2. The minimum time interval i1ii must be
for perfect reconstruction using ESF with two
parallel RC circuits and three sequential samples per im-
pulse. The dashed circles denote the measurement values
and the solid line reconstructed ESF output via Equation
(3).
dttt  
2
i
dt T
of the samples was proved to be at least 2p. The simu-
lations using noise free signal indicated an accurate and
precise reconstruction property, in practice the ESF me-
thod showed a high sensitivity to the noise. To obtain a
tolerable reconstruction error, the ESF circuit requires
Faraday gage-type shielding and careful consideration of
grounding and signal cables. The experimental verifica-
tion of the effect of different types of noise sources
(50 Hz + harmonics pick-up, 1/f-noise and random noise)
needs further study.
In this work the noise was cancelled using the SVD
based subspace method (16-18), which eliminates the
noise interference if the number of samples exceeds 2p.
The key idea is to compute the noise free signal as a
mean of the antidiagonal elements in the signal subspace
matrix . The error in the noise cancelled sequence
s
H
0,1,(...,)
n
x
nM attains a minimum in the middle of
the sequence, where M data points are used for computa-
tion of the mean value. Evidently the higher error in both
ends of the sequence is due to the lower number of
points.
The ESF method has a close relationship with the FRI
methods, which are based on the sequential measurement
of the output of the analog circuit network [5]. The re-
construction properties of the Gaussian, sinc-type and
triangle-wave sampling filters have been described [4-6,
8], but to the best of the authors knowledge, ESF-type
circuits have not been previously used for the sampling
and recovery of the UWB pulse sequences. A clear dif-
ference between the FRI sampling filters and the ESF
comes from causality. An exponential impulse response
exp[ ( )]
i
ktt

is causal and can be separated in (5)
only if . On the contrary, FRI sampling filters are
not causal and this restriction is not needed in the
deduction of the reconstruction algorithms. The FRI
reconstruction algorithms usually require the solution of
two matrix/vector equations, which are Yale-Walker and
Vandermonde structures. In our approach three matrix/
vector solutions (7, 14, 15) are required.
i tt
The present ESF measurement scheme can be consid-
ered as a general framework for measurement of the
UWB pulse amplitudes and time locations. The experi-
ments showed that the reconstruction error is mainly due
to the additive random noise affecting on the amplitudes
of the pulses. Hence, as an example of the practical con-
struction of the wireless sensor network the ratio of the
two sequential UWB pulse amplitudes can be related to
the device address. The ratio of the UWB pulse ampli-
tudes is not affected by the attenuation and distance
variations. The time difference between the UWB pulses
is reconstructed more accurately and it can be used to
code the transmitted information.
The ESF method can be applied in many areas of
wireless sensor technology and instrumentation. For
example, in optical instruments the time difference be-
tween laser pulses can be used in testing light transmis-
sion in optical fibres. Many other sensors have pulse-
type outputs.
5. Acknowledgements
This work was supported by the National Technology
Agency of Finland (TEKES).
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