Wireless Sensor Network, 2010, 2, 850-853
doi:10.4236/wsn.2010.211102 Published Online November 2010 (http://www.SciRP.org/journal/wsn)
Design of Orthogonal UWB Pulse Waveform for
Wireless Multi-Sensor Applications*
Hannu Olkkonen1, Juuso T. Olkkonen2
1Department of Physics and Ma thematics, University of Eastern Finland, Kuopio, Finlan d
2VTT Technical Research Centre of Finland, 02044 VTT, Finland
Received August 13, 2010; revised September 21, 2010; accepted Octo be r 28, 2010
Abstract
In this work we propose an orthogonal pulse waveform for wireless ultra wideband (UWB) transmission.
The design is based on an ideal low-pass prototype filter having a windowed sinc impulse response. The
frequency response of the prototype filter is transferred to the high frequency region using a specific sign
modulator. The UWB pulse waveform comprises of the weighted summation of the left singular vectors of
the impulse response matrix. The power spectral density of the pulse waveform fulfils the FCC constraint
(allowed frequency band 3.1-10.6 GHz) for unlicensed UWB transmission. Applications of the UWB pulse
waveform in multi-channel wireless sensor networks are considered.
Keywords: UWB, Wireless Sensor Networks
1. Introduction
The ultra wideband (UWB) technology is becoming a
widely used wireless data transmission method in a vari-
ety of systems. The spectral density of the UWB pulses
is restricted by the FCC constraint (allowed frequency
band 3.1-10.6 GHz) for unlicensed UWB transmission
[1]. Compared with the conventional impulse radio
method, which uses the Gaussian monocycle pulses, this
criterion brings new challenges to the design of the UWB
pulse waveforms. In multi-channel wireless systems, the
minimal crosstalk between the neighbouring channels is
an essential requirement. One design method is based on
the M-channel FIR filter bank and the UWB pulses are
the impulse responses of the narrow band-pass FIR fil-
ters. The filter bank matching with the FCC criterion is
designed with the Parks-MacClellan algorithm. A disad-
vantage of the FIR filter bank method is the slowly de-
caying impulse response, which prolongs the throughput
rate of the transmission system. Recently, a different
approach has been presented, where the UWB pulse
()pt is constructed as a linear combination of the sh ifted
orthogonal function()t

[2]
0
()( )
p
k
k
ptBt k 


 (1)
where ()t

obeys the orthogonal constraint
1
()() 0
f
or kl
tk tldt
f
or kl








 (2)
The information is coded to the k
Bsequence. The re-
construction is based on th e correlation
0
() ()()) ()
p
km
k
pttmdtBt kt mdtB 


 
 

 (3)
where 0,1,...,mp

. The pulse generators and the cor-
relators are usually analog circuits and k B is a binary
sequence [0,1]
k
B

. A disadvantage of the correlation
method is the long integration time required to obtain an
adequate signal-to-noise ratio.
In this work, we study a method the UWB pulse wav e-
form consists of the weighted summation of orthogonal
vectors k u,1,2,...,kM

having the dot product condi-
tion 1

T
kl
uu for kl

and 0
f
or kl
T
kl
uu . In
Section 2 we describe the design of the prototype filter,
the UWB pulse waveform and the reconstruction method.
Section 3 describes experimental results. In section 4 we
discuss the application of the orthogonal UWB pulse
waveform in wireless multi-sensor environments.
2. Theoretical Considerations
2.1. Filter Design
The prototype filter consists of the ideal low-pass filter,
which has the impulse response
*This work was supported by the National Technology Agency o
f
Finland (TEKES).

H. OLKKONEN ET AL. 851
sin( )
()sinc()
cc
c
t
ht AAt
t



 (4)
where c

is the cut-off frequency. The frequency re-
sponse for
() 0 for
c
c
A
Hj









 (5)
The discrete-time is defined as ,
n
tnTnand the
discrete impulse response of the low-pass filter as

sinc( )
nc
hA nT  .The impulse response should be
symmetrical and compactly supported. The compact
support is achieved by weighting the discrete impulse
response n
h by a windown w, which eliminates the side
lobes due to the truncated impulse response. We have for

sinc()
nnc
hAw nT  ,...,nNN . The design of the
window functions (Hanning, Hamming, Blackman etc.)
is well described in signal processing literature [3]. We
applied the cosine window
(1)cos[() /]
n
wnNN

 ,,...,nNN (6)
where 0.5

 for Hanning window and 0.54


for
Hamming window.
The power spectral density of the prototype filter is
concentrated on the frequency range [0, c

]. To validate
the FCC constraint (allowed band 3.1 -10.6 GHz) for
unlicensed UWB transmission [1] the key idea in this
work is that the discrete impulse response of the low-
pass filter is sign modulated as

(-1)nn
n
hh (7)
In continuous-time domain this corresponds to the
modulation of the signal bycos( )t

. The sign modulator
has been used in VLSI wavelet transform chips to sim-
plify the construction of the analysis and synthesis filters
[4]. In z transform domain this equals to the replacement
of the z variable by –z. In frequency domain the fre-
quency spectrum becomes a mirror image with respect to
/2

  (half of the Nyquist frequency). The original
low-pass filter has the frequency components up to the
cut-off frequency. The sign modulated impulse response
has the mirror image spectrum, which is concentrated in
the rangemin

 . The upper limit frequency is defined
by the Nyquist frequency, which should be selected un-
der 10.6 GHz. Then the cut-off frequency c

of the
low-pass filter so that the lowest frequency component of
the sign modulated sequence matches to 3.1 GHz.
2.2. UWB Pulse Waveform
The impulse response matrix is defined as
10
12 1
01
NN
NN
N
hh h
hh h
hh h

 






H




(8)
where , ,....,
n
hn NN

is the impulse response of the
high-pass filter. The output pulse vector
1
[]
T
nnn nN
pp p

pof the UWB pulse generator is
described by the state-space equation
1nn
nn


xAx
pHx
(9)
where the state vector (1)1
N
x
nR
x, the state transition
matrix (1)(1)NxN
R


A. The usual problem is to solve the
state transition matrix for the known signal. We seek the
solution for the inverse problem, where the impulse re-
sponse matrix H is known but the state vector must
obey specific constraints.
By the singular value decomposition (SVD) of the
impulse response matrix we haveT
H=UΣV, where
matrices


...12 N+1 Uuuu and


12 N+1
V=v v ...v
contain the left and right singular vectors (column vec-
tors) of the matrix H and matrix


12 1
diag... N 

Σthe singular values. MatricesU
and Vare unitary, i.e. -1 T
U=Uand -1 Τ V=V. The
eigenvectors are orthogonal, i.e.
1 for
0 for
kl
kl





TT
kl kl
uu vv (10)
We may deduce



12 1
... Nk  



12 N+1
12N+1k k
HV = UΣHvv...v
uuu Hv=u
(11)
for 1,2,..., 1kN

. The state vector n xcan be repre-
sented by a linear combination of the right singular vec-
tor basis as
12 1N


n12 N+1
x=v+v+ ... +v (12)
where (1,2,...,1)
kkN


are scalar coefficients. Now
the UBW pulse vector can be written as
112 211
11
11
NN
NN
kkkkk
kk
w
 







nn 12N+1
pH x=u+u+ ... +u
uu (13)
where kkk w



. Equation (13) shows that the UWB
pulse vector is constructed from a weighted summation
of the left singular vectors.
The trade-off in the present method is that several
singular values in the singular value decomposition of
the impulse response matrix H are very low. As an
example, if we select 32N

and 0.6
c

 , 13 of
the singular values are tiny. The corresponding left sin-
gular vectors are not appropriate for the UWB pulse
waveforms. However, 20 of the 33 singular values are at
the adequate level. The invalid eigenvectors can be re-
jected from the summation by defining their weighting
coefficients 0
k
w

for 1,..., 1kM N

. The weight-
ing sequence carries the information, which can be re-

H. OLKKONEN ET AL. 852
constructed via the orthogonal condition
1; 1,2,...,
M
kl
k
wwl M



TT
nl kl
pu uu (14)
For binary weighting sequence[1,1]
k
w  one
UWB pulse wavefo rm may in volve
M
bits of informa-
tion.
3. Results
A numerical study was carried out on the effect of win-
dow on the Fourier spectrum of the UWB pulse wave-
forms. Compared with the rectan gular window the cosin e
window had a significant improvement in the stop-band.
The

parameter was varied between 0.50,…, 0.54. The
best attenuation in the stop-band was obtained at
0.54

. The number of valid singular vectors was not
dependent on the

parameter. The stop band (0-3.1
GHz) attenuation was typically 170-180 dB surpassing
clearly the FCC criteria [1].
The UWB pulses were synthesized by a program m a ble
RF pulse generator containing a 12 bit DAC and a RF
tranceiver unit. The information coded to the UWB pulse
was a 12 bit word[1,1]
k
w 1,2,..., 16k. The meas-
urements were carried out using a high-speed memory
oscilloscope equipped with an RF antenna probe. In
practical measurements, noise arising in electronic cir-
cuits interferes with the results. Figure 1 describes the
transmitted and received UWB pulse waveforms in the
presence of low noise interference. Figure 2 illustrates
the reconstructed 16-bit word. In Figure 3 the measure-
ment was repeated in heavy noise environment. The peak
-to-peak noise level was about 50 % of the peak-to-peak
value of the transmitted UWB pulse waveform. The re-
constructed 16-bit word is still perfectly recovered (Fig-
ure 4).
4. Discussion
The distinct advantage of the use of the orthogonal UWB
pulse vectors compared with the correlation methods is
involved in their compact support enabling reconstruc-
tion via dot products.
The power spectral density of the UWB pulses clearly
overrides the FCC criteria. In previous works (see e.g.
[5]) the prolate spheroidal functions have been applied to
design the orthogonal eigenvectors, which are used as
UWB pulses. Our method based on the state-space mod-
elling of the UWB pulse generator yields UWB pulse
waveforms, which comprise of the weighted summation
of the left singular vectors of the impulse response ma-
trix. The weighting sequence carries out the transmitted
information. Compared with the Gaussian monocycle
pulse, which carries only one bit of information, the
Figure 1. The transmitted and received UWB pulse wave-
forms in low noise environment.
Figure 2. The reconstructed 16-bit word.
Figure 3. The UWB pulse waveform measured in heavy
noise environment.

H. OLKKONEN ET AL. 853
Figure 4. The coded and reconstructed 16-bit word.
adaptation of the UWB pulse waveform increases the bit
rate by a factor of
M
(14). In this work we used a bi-
nary 16-bit word as coded information. Our experiments
have warranted that the reconstruction succeeds in heavy
noise environment (Figures 3 and 4). On the other hand,
the radiation waste is reduced in the 1/16 ratio.
In many wireless sensor applications the RF circuits
with digital logic would be preferable to analog circuits.
The construction of the UWB pulse generator can be
made in VLSI environment [6]. The coding of the binary
information needs only an adder and the register bank for
left singular vectors. The output of the adder circuit can
be directly interfaced to the DAC and the RF transmitter
unit. The transmitted 16-bit word can comprise of the
device address, information bits and error detection bits.
This simplifies the design of the wireless multi-sensor
measurements compared with one bit per pulse transmis-
sion systems using impulse streams [7-10].
5. Acknowledgements
We are indebted to the reviewers’ comments, which im-
proved the manuscript significantly.
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