Noise suppression in UWB transmitted reference systems

Abstract

Transmitted reference (TR) systems have recently been proposed for ultra wideband (UWB) communications. They considerably simplify synchronization and channel estimation, which are known to be difficult problems in UWB communications. We extend existing receivers for TR-UWB systems by replacing the correlation operation by a linear combination of specific parts of the correlation and weighting the parts that have a small noise contribution more than parts that have a large noise contribution. This turns out to improve the performance considerably.

Full text

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Fifth IEEE Workshop on Signal Processing Advances in Wireless Communications, Lisboa, Portugal, July 11-14, 2004
Noise Suppression in UWB Transmitted Reference Systems1
Geert Leus and Alle-Jan van der Veen
Delft University of Technology
Faculty of Electrical Engineering, Mathematics and Computer Science
Mekelweg 4, 2628 CD Delft, The Netherlands
Email: {leus,allejan}@cas.et.tudelft.nl
Abstract — Transmitted reference (TR) systems have
recently been proposed for ultra wideband (UWB)
communications. They considerably simplify synchro-
nization and channel estimation, which are known to be
difficult problems in UWB communications. In this pa-
per, we extend existing receivers for TR-UWB systems
by replacing the correlation operation by a linear com-
bination of specific parts of the correlation and weight-
ing the parts that have a small noise contribution more
than parts that have a large noise contribution. This
turns out to improve the performance considerably.
I. Introduction
The transmitted reference (TR) approach has been envi-
sioned a long time ago as an effective means to avoid syn-
chronization and channel estimation (see for instance [2]). In
[3], it has been re-introduced in the field of ultra wideband
(UWB) communications, since for this application, synchroniza-
tion and channel estimation constitute major challenges. Many
researchers picked up on the idea, and by now a plethora of
TR-UWB systems have been proposed.
We can basically distinguish two possible types of TR-UWB
systems. One type uses a single pulse per frame, where some
frames represent a reference and others represent a data sym-
bol. The corresponding receivers generate a template using the
reference frames only [5] or using both the reference and data
frames [1], and employ this template to correlate it with the
frame containing the data symbol that we want to detect. The
second type uses multiple pulses per frame, where all pulses to-
gether make up the data symbol. The corresponding receivers
correlate the frame with a number of delayed versions thereof,
and process all outputs to obtain an estimate of the data sym-
bol [3, 4, 6]. Note that the first type of TR-UWB systems has a
lower spectral efficiency than the second type of TR-UWB sys-
tems. In addition, the first type has to implement longer delays
than the second type, but has a better performance than the
second type.
Generally, we can assume that the delay spread of the re-
ceived pulse is much smaller than the frame duration. As a
result, a large part of the frame only consists of noise. However,
none of the above receivers exploit this property. In this pa-
per, we try to exploit it by replacing the correlation operation
by a linear combination of specific parts of the correlation and
giving the parts that have a small noise contribution a higher
weight than parts that have a large noise contribution. How-
ever, since these noise contributions are unknown, the weighting
coefficients are derived in a blind fashion based on the received
signal.
1This research was supported in part by NWO-STW under the
VICI program (DTC.5893).
To simplify the presentation, we will only focus on the
Hoctor-Tomlinson TR-UWB system [3, 4], which is a special
case of the second type of TR-UWB systems. However, the
proposed ideas can be extended to other TR-UWB systems as
well.
Notation: We use upper (lower) bold face letters to de-
note matrices (column vectors). SuperscriptsT,H, and†rep-
resent transpose, Hermitian, and pseudo-inverse, respectively.
Continuous-time (discrete-time) variables are denoted as x(·)
(x[·]). Finally, sign(·) denotes the sign operator and E(·) rep-
resents the expectation. All other notation should be self-
explanatory.
II. Data Model
As mentioned in the introduction, we consider the Hoctor-
Tomlinson TR-UWB system [3, 4] in this work. Each data sym-
bol is represented by Nf frames of duration Tf. Each frame
consists of a reference pulse p(t) of duration Tp and a modu-
lated/delayed version thereof. More specifically, the reference
pulse is modulated with b[?n/Nf?]c[n] and delayed by d[n]Td,
where b[?n/Nf?] ∈ {+1,−1} is the data symbol related to the
nth frame, c[n] ∈ {+1,−1} and d[n] ∈ {1,2,...,D} are the
user codes with spreading gain Nf, and Td is the minimal delay
between a reference pulse and a data pulse. Hence, the trans-
mitted signal is given by
s(t) =
∞
?
n=−∞
p(t − nTf) + b[?n/Nf?]c[n]p(t − nTf − d[n]Td).
Assuming g(t) is the propagation channel with delay spread Tg,
the received signal is given by
y(t) =
∞
?
n=−∞
h(t−nTf)+b[?n/Nf?]c[n]h(t−nTf−d[n]Td)+ν(t),
where h(t) = p(t) ? g(t) is the composite channel with delay
spread Th = Tp+ Tg and ν(t) is the additive noise, which in-
cludes the received signals from the other users. In the following,
we assume that there is no interframe interference between y(t)
and y(t + DTd), i.e., Tf ≥ 2DTd+ Th. For the sake of sim-
plicity, we also assume that there is no interpulse interference,
i.e., Td ≥ Th, as done in [3]. However, in case there is inter-
pulse interference, we can adapt the proposed ideas following
the approach introduced in [4].
III. Autocorrelation Receiver
Under the assumptions made in the previous section, we can
use the well-known autocorrelation receiver [3] to detect the
data. This receiver correlates each frame with a shifted version
thereof:
??+(n+1)Tf
z[n] =
?+nTf
y(t)y(t + d[n]Td)dt,(1)

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:
????????
:
????? ???
:
????????
:
????? ???
t
?????????????????? ????????
?
???????
?
? ??!?#"$?%???
&
??')(
Figure 1: Splitting of the integrand of the correlation into
multiple amplitude sections (option 1).
where ? is the timing offset. If this timing offset is chosen such
that ? ∈ [−Tf + DTd+ Th,−DTd), it is easy to show that
z[n] = αb[?n/Nf?]c[n] + ν[n],
where α is given by
?+∞
and ν[n] can be written as ν[n] = ν(1)[n]b[?n/Nf?]c[n]+ν(2)[n],
with ν(1)[n] and ν(2)[n] given by
α =
−∞
h2(t)dt, (2)
ν(1)[n] =
??+(n+1)Tf
??+(n+1)Tf
??+(n+1)Tf
??+(n+1)Tf
??+(n+1)Tf
?+nTf
h(t − nTf − d[n]Td)ν(t + d[n]Td)dt
+
?+nTf
h(t − nTf)ν(t)dt,(3)
ν(2)[n] =
?+nTf
h(t − nTf)ν(t + d[n]Td)dt
+
?+nTf
h(t − nTf + d[n]Td)ν(t)dt
+
?+nTf
ν(t)ν(t + d[n]Td)dt.(4)
When ν(t) is considered to be a zero-mean i.i.d. random vari-
able, ν(1)[n] and ν(2)[n] are mutually independent zero-mean
i.i.d. random variables, and thus ν[n] is a zero-mean i.i.d. ran-
dom variable.Hence, applying a matched filter to z[n] will
yield the best performance. Denoting c[k] = [c[kNf],...,c[(k +
1)Nf− 1]]Tand z[k] = [z[kNf],...,z[(k + 1)Nf− 1]]T, an esti-
mate for b[k] is therefore computed as:
ˆb[k] = sign{cT[k]z[k]}.
IV. Proposed Receiver
Since the delay spread of the composite channel Th is much
smaller than the frame duration Tf, a large part of the frame
only consists of noise. However, the autocorrelation receiver
:
*?+?, -?.
:
*?/?, -?.
:
*?0?,-?.
:
*?1?,-?.
t
2?34?5?2?3?4?6?7
,-?.8?9
5
:
6
-?8?;
:
6 3
-
6#<$5
8?;
8?;>=??
Figure 2: Splitting of the integration interval of the corre-
lation into multiple time sections (option 2).
does not exploit this. In this paper, we try to use this knowledge
by replacing the correlation operation by a linear combination
of specific parts of the correlation and weighting the parts that
have a small noise contribution more than parts that have a
large noise contribution. More specifically, we compute z[n] as
z[n] =
Q−1
?
q=0
aqzq[n], (5)
where zq[n] represents a specific part of the correlation in (1).
Two options are investigated.
Option 1: We can divide the correlation of (1) into different
parts by splitting the integrand into multiple amplitude sections
(see Figure 1). In other words, we define zq[n] as
??+(n+1)Tf
where


if q = 0,1,...,Q − 2, and
?
if q = Q − 1, with A an indication of the maximum amplitude
of the composite channel h(t).
Option 2: We can also divide the correlation of (1) into dif-
ferent parts by splitting the integration interval into multiple
time sections. In other words, we define zq[n] as
??+nTf+(q+1)Tf/Q
Note that in order for this to work, we have to make sure that
for each n the peak in the integrand of (1) always appears on
the same position within the window [? + nTf,? + (n + 1)Tf),
irrespective of d[n].By choosing the integrand as y(t)y(t +
d[n]Td), as we do in this work, this clearly is the case.
zq[n] =
?+nTf
fq(y(t)y(t + d[n]Td))dt,
fq(x) =

0,
x − sign(x)qA2/Q, if qA2/Q ≤ |x| < (q + 1)A2/Q
sign(x)A2/Q, if |x| ≥ (q + 1)A2/Q
if |x| < qA2/Q
,
fq(x) =
0,
x − sign(x)qA2/Q, if |x| ≥ qA2/Q
if |x| < qA2/Q
,
zq[n] =
?+nTf+qTf/Q
y(t)y(t + d[n]Td)dt.

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A combination of the two options is of course also a possi-
bility. Note that if aq = 1, for q = 0,1,...,Q − 1, options 1
and 2 fall back to the autocorrelation receiver. However, by
weighting terms that have a small noise contribution more than
parts that have a large noise contribution, we can introduce
some noise suppression effect, and therefore outperform the au-
tocorrelation receiver as shown in Section VI.
As for the autocorrelation receiver, we can show that if the
timing offset is chosen such that ? ∈ [−Tf + DTd+ Th,−DTd),
we obtain
zq[n] = αqb[?n/Nf?]c[n] + νq[n],
where αq is given as in (2), reducing the integrand to the qth
amplitude section (option 1) or reducing the integration interval
to the qth time section (option 2), and νq[n] can again be written
as νq[n] = ν(1)
given as in (3) and (4), reducing the integrand of each term to
the qth amplitude section (option 1) or reducing the integration
interval of each term to the qth time section (option 2). From
(5) and (6), we obtain
(6)
q [n]b[?n/Nf?]c[n]+ν(2)
q [n], with ν(1)
q [n] and ν(2)
q [n]
z[n] =
?Q−1
= αb[?n/Nf?]c[n] + ν[n].
?
q=0
aqαq
?
b[?n/Nf?]c[n] +
Q−1
?
q=0
aqνq[n]
For the two options introduced earlier, it is again possible to
derive that if ν(t) is considered to be a zero-mean i.i.d. random
variable, ν[n] also is a zero-mean i.i.d. random variable, and thus
applying a matched filter to z[n] will again result in the best
performance. Denoting c[k] = [c[kNf],...,c[(k + 1)Nf − 1]]T
and z[k] = [z[kNf],...,z[(k + 1)Nf − 1]]T, an estimate for b[k]
is therefore again computed as:
ˆb[k] = sign{cT[k]z[k]}.
The question remains how to determine the weighting co-
efficients aq. Since the noise contribution to each term zq[n]
is unknown, we will estimate the weighting coefficients blindly
based on the received signal, as outlined in the next section.
V. Receiver Optimization
Assume a burst of K data symbols is transmitted: b =
[b[0],...,b[K − 1]]T. Defining z = [z[0],...,z[KNf − 1]]Tand
ν = [ν[0],...,ν[KNf − 1]]T, we can write
z = αCb + ν,(7)
where
C =



c[0]
...
c[K − 1]

.

Defining zq = [zq[0],...,zq[KNf − 1]]T, we can write
z = Za,
where Z = [z0,...,zQ−1] and a = [a0,...,aQ−1]T. Hence, (7)
can be expressed as
Za = αCb + ν.
We now optimize a by minimizing the least squares error be-
tween Za and αCb with unknowns a and αb:
min
a,αb?Za − αCb?2. (8)
−30 −25−20
SNR
−15−10
10
−3
10
−2
10
−1
10
0
BER
autocorrelation
option 1
option 2
Q=4
Q=8
Q=12
Figure 3: Performance comparison for the frequency-flat
non-fading channel case.
It is clear that we need an extra constraint to avoid the trivial
solution a = 0Q×1 and αb = 0K×1. If only a constraint on a is
applied, we can first solve (8) for αb, which leads us to
?
αb = C†Za = 1/NfCHZa.
Substituting this solution in (8), we obtain
min
a?(IKNf− 1/NfCCH)Za?2.
Now the question is what kind of constraint we should put on
a. It is important to note that we should choose a constraint
that prevents a from lying in the right null-space of Z. Since
Z has rank 1 in the noiseless case, a unit-norm or monic con-
straint is not a good option. Therefore, we choose a unit-energy
constraint: ?Za?2= 1, which clearly prevents a from lying in
the right null-space of Z. Hence, we will solve
min
a?(IKNf− 1/NfCCH)Za?2, s.t. ?Za?2= 1. (9)
Defining the ‘economy size’ singular value decomposition (SVD)
of Z as Z = UΣVH, where U and V are orthogonal matrices
of size KNf × Q and Q × Q, respectively, and Σ is a diagonal
matrix of size Q × Q, and denoting ˜ a = ΣVHa, we can rewrite
(9) as
min
˜ a?(IKNf− 1/NfCCH)U˜ a?2, s.t. ?˜ a?2= 1.
The solution for ˜ a is then given by the right singular vector cor-
responding to the lowest singular value of (IKNf−1/NfCCH)U.
In the next section, we will illustrate that this procedure assigns
higher weights to parts with a small noise contribution than to
parts with a large noise contribution.
VI. Simulation Results
In this section, we compare the proposed receiver with the
autocorrelation receiver. We consider a Hoctor-Tomlinson TR-
UWB system where each data symbol is represented by Nf = 10
frames of duration Tf = 50 ns. As pulse, we adopt the first

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01234567
−0.2
0
0.2
q
weight
01234567
−0.2
0
0.2
q
weight
SNR=−10 dB
SNR=−15 dB
SNR=−20 dB
max/min
ε=−Tf/4
ε=−3Tf/4
Figure 4: Weighting coefficients related to option 1 for two
different timing offsets.
derivative of a Gaussian: p(t) = e−s2s, where s = (t−Tp/2)5/Tp
in order for the pulse to have a duration of about Tp. We select
this duration to be Tp = 0.5 ns. We consider D = 4 possible
delays between the reference pulse and the data pulse, with a
minimal delay of Td = 2 ns.
consider a single user system and model the additive noise ν(t)
as additive white Gaussian noise (AWGN). The signal-to-noise
ratio (SNR) is defined as SNR = Ef/(Tfσ2), where Ef is the
average signal energy in a frame: Ef = 2E(?+∞
Test case 1: We first consider the frequency-flat non-fading
channel case: g(t) = 1. Figure 3 shows the performance of
the autocorrelation receiver and the two options of the pro-
posed receiver. We consider Q = 4, Q = 8, and Q = 12.
The BER results are obtained from 100 Monte Carlo runs
of K = 100 data symbols, where in each run we consider a
new data, code, and noise realization, and a new timing offset
? ∈ [−Tf+DTd+Th,−DTd). First of all, we observe that both
options outperform the autocorrelation receiver, especially at
high SNR, and that option 1 is outperformed by option 2 at
low to medium SNR. However, the slope of the BER curve for
option 1 seems to be higher than for option 2, which means that
at some high SNR value, option 1 could outperform option 2.
In addition, we observe that option 1 does not really improve
with increasing Q, whereas option 2 does, but the improvement
gradually diminishes. The latter is caused by the fact that as
Q increases, the number of unknowns in the blind receiver op-
timization algorithm also increases. To illustrate that the pro-
posed receiver optimization algorithm indeed puts more empha-
sis on low noise parts than on high noise parts of the correlation,
Figures 4 and 5 show the behavior of the weighting coefficients
aq for options 1 and 2, respectively. To generate these plots, we
consider Q = 8. The results are obtained from 100 Monte Carlo
runs of K = 100 data symbols, where in each run we consider a
new data, code, and noise realization, but fix the timing offset
to a specific value within the range [−Tf + DTd+ Th,−DTd).
We plot the mean coefficient profile as well as the maximum and
minimum coefficient profiles for two timing offsets and different
For the sake of simplicity, we
−∞h2(t)dt), and
σ2is the variance of the AWGN ν(t).
01234567
−0.05
0
0.05
q
weight
01234567
−0.05
0
0.05
q
weight
SNR=−10 dB
SNR=−15 dB
SNR=−20 dB
max/min
ε=−Tf/4
ε=−3Tf/4
Figure 5: Weighting coefficients related to option 2 for two
different timing offsets.
SNRs. As expected, the coefficient profile related to option 1 is
independent of the timing offset ?, whereas the coefficient pro-
file related to option 2 puts a higher weight on the time section
where the peak of the intergand of (1) is present. On the other
hand, we observe that the coefficient profile of option 2 simply
scales with the SNR, whereas the coefficient profile of option
1 changes shape with the SNR, putting a high weight on the
amplitude section that is important at that SNR. Note that at
very low SNR, the gap between the mean coefficient profile and
the maximum or minimum coefficient profile related to option
1 is large, which basically means that in this case all kinds of
coefficient profiles are possible.
Test case 2:Let us now consider a frequency-selective
Rayleigh fading channel case:
g(t) =
L
?
l=0
glδ(t − τl),
where gl is Gaussian distributed with mean 0 and variance
e−5l/L, and τl = lTp/10 (i.e., Tg = LTp/10 and Th = Tp +
LTp/10). Figures 6 and 7 shows the performance of the autocor-
relation receiver with the two options of the proposed receiver
for L = 10 (i.e., Tg = 0.5 ns and Th = 1 ns) and L = 20 (i.e.,
Tg = 1 ns and Th = 1.5 ns), respectively. Note that for both
values of L, we have no interpulse interference, i.e., Td≥ Th. We
again consider Q = 4 and Q = 8. As a benchmark, we also show
the performance of the optimal receiver, which corresponds to
applying a matched filter to y(t). The BER results are obtained
from 100 Monte Carlo runs of K = 100 data symbols, where in
each run we consider a new data, code, and noise realization,
a new timing offset ? ∈ [−Tf + DTd+ Th,−DTd), and a new
channel realization. Again, we observe that both options per-
form better than the autocorrelation receiver, and that option
1 is outperformed by option 2. However, this time the perfor-
mance improvement for option 1 is small, and the slopes of the
BER curves for the two options in the considered SNR range are
more or less the same. We also observe, as before, that option
1 does not really improve with increasing Q, whereas option 2
does, but only up to a certain value of Q. Next, we notice that

Page 5

−30−25−20
SNR
−15−10
10
−3
10
−2
10
−1
10
0
BER
autocorrelation
option 1
option 2
Q=4
Q=8
Q=12
optimal
Figure 6:
selective Rayleigh fading channel case (L = 10).
Performance comparison for the frequency-
−30−25−20
SNR
−15−10
10
−3
10
−2
10
−1
10
0
BER
autocorrelation
option 1
option 2
Q=4
Q=8
Q=12
optimal
Figure 7:
selective Rayleigh fading channel case (L = 20).
Performance comparison for the frequency-
there is still a big gap between the performance of the proposed
receiver and the optimal receiver. However, this is the price we
have to pay for relaxing the synchronization and channel esti-
mation requirements related to the optimal receiver. Finally,
observe that the performance of all receivers slightly improves
with increasing delay spread, due to the increasing multipath
diversity. Of course, this only holds as long as Td≥ Th. As we
already indicated, when interpulse interference is present, we
can adapt the proposed receiver following the ideas introduced
in [4]. This will be a topic for future research.
VII. Conclusions
In this paper, we have extended the autocorrelation receiver
for the Hoctor-Tomlinson TR-UWB system by replacing the
correlation operation by a linear combination of parts of the
correlation and weighting the parts that have a small noise con-
tribution more than parts that have a large noise contribution.
Two options were studied. The first option splits the integrand
of the correlation into multiple amplitude sections, whereas the
second option splits the integration interval of the correlation
into multiple time sections. The weighting coefficients are de-
rived in a blind fashion based on the received signal.
options clearly improve the performance compared to the auto-
correlation receiver.
Both
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