Modified pulse repetition coding boosting energy detector performance in low data rate systems

Abstract

We consider ultra-wideband impulse radio (UWB-IR) low data rate (LDR) applications where a more complex cluster head (CH) communicates with many basic sensors nodes (SN). At receiver side, noncoherent energy detectors (ED) operating at low sampling clock, i.e., below 300 kHz, are focused. Drawback is that EDs suffer from significant performance losses with respect to coherent receivers. Pulse repetition coding (PRC) is a known solution to increase receiver performance at the expense of more transmit power. But in LDR systems known PRC is very inefficient due to the low receiver sampling clock. Boosting transmit power is not possible due to Federal Communications Commission's (FCC) power constraints. Hence, we present a modified PRC scheme solving this problem. Modified repetition coded binary pulse position modulation (MPRC-BPPM) fully exploits FCC power constraints and for EDs of fixed integration duration is optimal with respect to bit error rate (BER). Furthermore, MPRC-BPPM combined with ED outperforms SRAKE receivers at the expense of more transmit power and makes ED's performance robust against strong channel delay spread variations.

Full text

Page 1

Modified Pulse Repetition Coding Boosting Energy Detector
Performance in Low Data Rate Systems
Florian Troesch, Frank Althaus, and Armin Wittneben
Swiss Federal Institute of Technology (ETH) Zurich
Communication Technology Laboratory, CH-8092 Zurich, Switzerland
Email: troesch@nari.ee.ethz.ch
Abstract—We consider Ultra-Wideband Impulse Radio (UWB-IR) Low
Data Rate (LDR) applications where a more complex Cluster Head (CH)
communicates with many basic Sensors Nodes (SN). At receiver side,
noncoherent Energy Detectors (ED) operating at low sampling clock, i.e.,
below 300kHz, are focused. Drawback is that EDs suffer from significant
performance losses with respect to coherent receivers. Pulse Repetition
Coding (PRC) is a known solution to increase receiver performance at
the expense of more transmit power. But in LDR systems known PRC is
very inefficient due to the low receiver sampling clock. Boosting transmit
power is not possible due to Federal Communications Commission’s
(FCC) power constraints. Hence, we present a modified PRC scheme
solving this problem. Modified Repetition Coded Binary Pulse Position
Modulation (MPRC-BPPM) fully exploits FCC power constraints and
for EDs of fixed integration duration is optimal with respect to Bit Error
Rate (BER). Furthermore, MPRC-BPPM combined with ED outperforms
SRAKE receivers at the expense of more transmit power and makes ED’s
performance robust against strong channel delay spread variations.
I. INTRODUCTION
Recently, Ultra-Wideband Impulse Radio (UWB-IR) technology
has gained strong interest as a very promising technology for future
indoor wireless communication. Key applications for which UWB-IR
technology is considered an interesting candidate are Low Data Rate
(LDR) communication systems requiring rates below 1Mbps [1].
UWB-IR transmitters produce very short time domain pulses
of up to 7.5GHz bandwidth without the need for an additional
Radio Frequency (RF) mixing stage due to their essentially baseband
nature. This leads to significant complexity reduction at transmitter
and receiver side with respect to conventional radio systems. This
advantage makes UWB-IR a well suited candidate for low cost
LDR applications. On the other hand, channel investigations [2]
show that UWB-IR indoor channel energy is spread over a large
number of multipath components. This highly increases complexity of
coherent receivers as energy has to be re-combined by a large number
of RAKE fingers. Furthermore, UWB-IR systems are intended to
operate over a large bandwidth, overlaying bands of many other
services. They are thus rigorously power constrained by regulations,
as e.g., by the Federal Communications Commission (FCC), to
minimize interference to victim receivers. These regulations impose
hard performance limits to UWB-IR communication systems as
energy per pulse is restricted very stringently.
In this work, we focus on UWB-IR LDR applications where a more
complex Cluster Head (CH) communicates with many basic Sensor
Nodes (SN). An example could be a wireless control system where
only very small amount of data is transmitted from and to the SNs.
At sensor side, only simple hardware structures are affordable. While
the design of simple UWB-IR transmitters seems a minor problem,
this is not the case for simple receivers. Only non-coherent receivers
seem reasonable, which suffer from significant performance losses
with respect to coherent receivers as channel energy is spread over a
large number of multipath components.
Hence, we consider non-coherent Energy Detectors (ED) operating
at very low sampling clock, i.e., below 300 kHz, as a reasonable
choice and investigate signaling schemes to efficiently increase
performance of ED. The low sampling clock is applied to relax
requirements on receiver sampling accuracy and to reduce power
consumption.
Pulse Repetition Coding (PRC) is a known solution in asymmetric
sensor networks to increase receiver performance of SNs at the
expense of more transmit power at CH side. With PRC a bit is loaded
on several consecutive pulses, as e.g., it is often applied in Time-
Hopping (TH) Pulse Position Modulation (PPM). In LDR systems,
classic PRC has two major drawbacks. First, throughput is further
decreased and secondly, it does not exploit FCC power constraints
efficiently.
In this paper, we present a Modified PRC (MPRC) coding scheme
for LDR systems with receiver sampling rates of below 300kHz. This
MPRC scheme maximizes transmit power, if FCC power constraints
have to be respected. For an ED of fixed integration duration, men-
tioned precoding scheme is optimal, i.e., it minimizes Bit Error Rate
(BER) by fully exploiting FCC power constraints and transmitting
maximized power most efficiently. Furthermore, it is well known
that performance of EDs strongly depends on the appropriate choice
of the integration duration. MPRC, which requires Channel State
Information (CSI) neither at transmitter nor at receiver side, mainly
decouples receiver performance from integration duration. This has
major advantages. First, performance of the ED becomes extremely
robust against strong delay spread variations. Secondly, constraints
on the integration duration, e.g., fixed large size due to circuit
design aspects, can be compensated. Finally, jitter robustness can be
increased by choosing a large integration duration. Presented MPRC,
without any CSI, achieves performance of a complex Selective RAKE
(SRAKE), at the expense of more transmit power. Presented results
are based on BER performance analysis incorporating simulations
using UWB channels from different measurement campaigns. Al-
though, FCC power constraints are considered, only, results are easily
adaptable to other regulations.
Applied Modified Pulse Repetition Coded Binary Pulse Position
Modulation (MPRC-BPPM) scheme equals an orthogonal BPPM
scheme of equivalent pulses, where each equivalent pulse consists of a
sequence of equidistant copies of a basic pulse waveform, as shown in
Fig. 1. The extension to dithered temporal pulse separation is straight
Fig. 1. Principle difference between BPPM (Left) and MPRC-BPPM (Right)
forward, but was omitted for convenience. The different copies are
multiplied by an arbitrary phase in order to flatten the spectrum of the
transmit signal and to minimize interference to other users, i.e., MAC.

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The phases are totally ignored by the ED at the moment, but might
be useful for synchronization purpose in future work. All equidistant
copies of the basic waveform have the same energy Ep, which equals
maximally allowed pulse energy, if a single pulse was transmitted.
The pulse separation is chosen as small as possible without violating
peak power constraint. Transmitter and receiver require no CSI. The
transmitter is allowed to use as much power as admitted by the
FCC, as we consider the FCC power constraints as binding enough.
Due to its simplicity and great advantages, this scheme seems a very
promising candidate for realization in real world LDR systems.
In the following section, the signal model is introduced and BER is
analyzed. In Section III, impact of FCC power constraints on MPRC-
BPPM is discussed, followed by analytic and simulation results in
Section IV. In Section V, we conclude with a short summary.
II. SIGNAL MODEL AND BER ANALYSIS
A. MPRC-BPPM Transmitter
The MPRC-BPPM signal sent by the transmitter is described by:
sNP
tx(t) =
?Ep
∞
?
k=−∞
NP−1
?
n=0
βiw(t − kTf− αkδ − nτp),
(1)
where t is the transmitter’s clock time and w(t) the real transmitted
bandpass pulse of width Tw. The pulse is energy normalized, i.e.,
?∞
Depending on αk ∈ {0,1}, a sequence of NP repeated pulses is
either transmitted at beginning of a frame or delayed by δ. The frame
repetition rate Rf = 1/Tf equals the nominal BPPM pulse rate. Pulse
separation τp ? Tf is chosen such that peak power according to FCC
power constraints is not increased with respect to single pulse trans-
mission. To avoid Intersymbol Interference (ISI) between consecutive
symbols and to maintain BPPM orthogonality in presence of large
channel delay spread τc, conditions δ ≥ (NP − 1)τp+ τc+ Tw and
Tf ≥ δ + (NP − 1)τp + τc + Tw are respected. The coefficients
βi ∈ {−1,1} with i = kNP + n are chosen randomly or according
to a Direct Sequence (DS). They are applied to smooth the spectrum
of the transmit signal. Thereby, transmit power can be increased,
while interference to other users is kept small. In this work, the βs
are ignored at receiver side. Although application of TH is straight
forward, it is omitted for convenience. Especially, as very short
channel occupation times of MPRC compared to classic PRC allow
for other MAC schemes, as e.g., TDMA, if combined with spectral
smoothing DS.
−∞w2(t)dt = 1, and Ep is the single pulse energy. During each
frame repetition time Tf, one BPPM symbol αk is transmitted.
B. Energy Detector Receiver and BER Analysis
A schematic description of an ED as used in this paper can be seen
in Fig. 2. The input filter f(t) is assumed to be an ideal bandpass
Fig. 2. Signal model of energy detector receiver
filter of bandwidth Bpb≥ B, with B the bandwidth of the transmit
pulse. Hence, the receive signal is not influenced by the filter. The
received signal after the bandpass filter f(t) equals:
sNP
rx(t)=
?Ep
?Ep
∞
?
∞
?
k=−∞
NP−1
?
n=0
βihw(t − kTf− αkδ − nτp)
(2)
=
k=−∞
hE,k(t − kTf − αkδ),
(3)
with hw(t) the convolution of the energy normalized transmit wave-
form and the real channel, i.e., hw(t) = w(t) ∗ h(t). We call
hE,k(t) the equivalent channel. For analysis of the uncoded BER, it
is sufficient to consider only a single received frame. Therefore, we
focus in the following on the k-th frame and omit index k. Assuming
that αk= 1 was sent, the outputs of the two integrator units at time
ts are:
r(1)(ts) =
ts
?ts+TI
where ˜ n(t) = f(t) ∗ n(t) is filtered zero-mean Additive White
Gaussian Noise (AWGN) n(t) with two-sided power spectral density
N0/2. We rewrite expression (4) as:
?ts+TI
??EphE(t) + ˜ n(t)
?2
dt
(4)
r(0)(ts) =
ts
˜ n2(t − δ)dt,
(5)
r(1)(ts)=ν(1)(ts) + ω(1)(ts) + ζ(1)(ts).
?ts+TI
(6)
Thereby, ν(1)(ts) = Ep
ergy collected by the ED and ζ(1)(ts) =?ts+TI
ω(1)(ts) = 2?Ep
r(0)(ts)
with ζ(0)(ts) =?ts+TI
Assuming Maximum Likelihood (ML) detection, based on the
statistics of r = r(1)−r(0), the BER conditioned on a certain channel
realization h(t), an integration duration TI and a sampling instance
ts is given by:
ts
h2
E(t)dt equals the signal en-
˜ n2(t)dt is the
ts
pure (quadratic) noise term. The mixed signal-noise term is
?ts+TI
=
ts
hE(t)˜ n(t)dt. Due to the lack of a signal
component, expression (5) simply equals:
ζ(0)(ts)
(7)
ts
˜ n2(t − δ)dt. For convenience, we omit the
time ts in the following.
Pe|h,TI,ts
=P(ν(1)< ζ(0)− ζ(1)− ω(1)).
(8)
In the following, we approximate z = ζ(0)−ζ(1)−ω(1)as a Gaussian
random variable. Applying quadrature sampling expansion at Nyquist
rate Bpb [3] and central-limit theorem, we achieve:
ζ(α)∼ N?BpbTIN0,BpbTIN2
for the pure quadratic noise terms and
?
for the mixed signal-noise term. It can be shown that the correlation
between ω(1), ζ(1)and ζ(0)is approximately zero [4]. Hence, z can
be approximated as Gaussian random variable:
?
The ML performance given a certain channel realization h(t), an
integration duration TI and a sampling instance ts is now [5]:

0
?, α ∈ {0,1}
?ts+TI
(9)
ω(1)
∼N
0,2N0Ep
ts
h2
E(t)dt
?
(10)
z
∼N
0,2BpbTIN2
0+ 2N0ν(1)?
.
(11)
Pe|h,ts,TI
=
1
2erfc

?
(ν(1))2
0+ 4N0ν(1)
4BpbTIN2

.
(12)
It is noteworthy that (12) strongly depends on the bandwidth of the
receiver’s input filter due to the term 4BpbTIN2
0.

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III. FCC POWER CONSTRAINTS
A. Maximal Average and Peak Power of Antipodal Signal
A device operating under FCC’s provisions of UWB indoor devices
[6], has to occupy a total 10 dB bandwidth of at least 500 MHz
between 3.1 and 10.6 GHz. Additionally, the emitted signal has
to respect average and peak power constraint. Average power Pav
measurements are based on spectrum analyzers with Resolution
Bandwidth (RBW) set to Bav = 1MHz, RMS detector and average
time window Taw = 1ms. For all center frequencies f0 of the
resolution filter within 3.1 to 10.6 GHz, maximal average power
Pmax
to [4] and [6], is best measured with a RBW of Bp = 50 MHz.
For all center frequencies f0 within 3.1 to 10.6GHz, maximal peak
power Pmax
p
must not exceed PFCC
p
Accordingto [4],maximal
of anantipodal signal ofequidistant
s(t) =?Ep?∞
Pmax
av has to be below PFCC
av = −41.25 dBm. Peak power, according
= 0 dBm.
average and
pulses
be
peak
defined
approximated
power
by
n=−∞βnw(t − n/Rf), can
very tightly as:
av(Rf,f0) = 2EpW2(f0)BavRf
Rf ≥
1
Taw,
(13)
Pmax
p
(Rf,f0) =
?
2EpW2(f0)B2
0.452
2EpW2(f0)R2
p
Rf <
Rf >
Bp
0.45
Bp
0.45,
f
(14)
with W(f) the Fourier transform of the pulse waveform w(t). The
two regimes in (14) origin from the fact that for low frame repetition
frequencies resolution filtered pulses do not overlap and add up
linearly in power, while for higher frequencies, they overlap and
add up linear in amplitude. Next, we set the maximal powers Pmax
and Pmax
p
equal to the maximally allowed powers PFCC
and solve for EpW2(f0). In so doing, maximally allowed single
pulse spectral energy at frequency f0 with respect to average and
peak power constraint is achieved, as shown in Fig. 3. Two different
av
av
and PFCC
p
10
4
10
6
10
8
10
−26
10
−24
10
−22
10
−20
10
−18
10
−16
Pulse Repetition Frequency (Rf) [Hz]
Maximally allowed single pulse spectral energy
regimes can be distinguished. Peak power regime for Rf < Bav and
average power regime for Rf ≥ Bav. From Fig. 3 and Eq. (14),
it follows that maximal peak power does not increase for Rf <
Bp/0.45. This is an important property we will use several times in
the following section.
EpW2(f0) [J/Hz]
max. Ep W2(f0) from Pav
max. Ep W2(f0) from Pp
Active Power Constraint
NP
Fig. 3.
B. Impact of FCC Average and Peak Power Constraint on LDR
MPRC-BPPM
For the rest of this paper, LDR MPRC-BPPM schemes with frame
repetition rate Rf ≤ 300kHz, BPPM modulation shift δ ≤ 1/(2Rf),
and minimal temporal pulse distance τp ≥ 10ns are focused, i.e.,
UWB-IR operating in peak power regime. Under above system
specifications, the uncoded LDR BPPM signal and the antipodal one
from previous section show approximately the same maximal average
and peak power, if same Rf is applied [4]. For Rf ≤ 300kHz, the
resolution filtered pulses are non-overlapping for both BPPM and the
antipodal signal. Hence, average power is dominated by the number
of pulses that fall into averaging duration Taw = 1ms. As this
number is the same for both, average power is approximately equal.
Maximal peak power stays unchanged as minimal pulse separation
is larger than Tmin = 10ns > 0.45/Bp. From same argumentation,
it follows that MPRC-BPPM with τp ≥ Tmin = 10ns shows the
same peak power as corresponding antipodal signal. For maximal
MPRC-BPPM average power, an accurate upper bound can be found
assuming that each doubling of MPRC pulses increases BPPM
average power by 6dB. This assumption is equivalent to assuming
average power resolution filtered pulses as totally overlapping and
therefore, adding up perfectly in amplitude. As average power filtered
pulses extend over about 2µs and due to MPRC are separated by only
a few nanoseconds, this is a reasonable assumption. Hence, maximal
average and peak power for LDR MPRC-BPPM of NP pulses equals:
Pmax
av(Rf,f0)=2EpW2(f0)BavN2
2EpW2(f0)B2
0.452
PRf
(15)
Pmax
p
(Rf,f0)=
p
(16)
and the maximally allowed single pulse spectral energy:
EMPRC
p,avW2(f0)=
PFCC
av
PRfBav
0.452PFCC
2B2
2N2
(17)
EMPRC
p,p W2(f0)=
p
p
.
(18)
According to Fig. 3, FCC power constraints are fully exploited,
if maximally allowed single pulse spectral energies in (17) and
(18) are equal, i.e., if average power is increased to its maximally
allowed value, while keeping peak power constant. By equating the
two expressions and solving for NP, maximal number of precoding
pulses is found which can be applied without violating FCC power
constraints [7]:
??
0.452Rf
It is remarkable that (19) scales with 1/?Rf, which is due to the
Examples of (19) are: 2 pulses at 200kHz, 3 at 100kHz and 6
at 20kHz. The number of MPRC pulses that can be applied is
quite restricted, all the same significant performance improvement
is possible.
IV. RESULTS
A. Normalization
For the BER curves presented, we apply an unusual Signal-to-
Noise Ratio (SNR) normalization. We normalize the SNR to the
total received energy, if a single pulse of 500 MHz bandwidth is
transmitted in the band from 3.1 to 3.6GHz. Hence, SNR is defined
as:
?∞
with w500(t) the normalized pulse shape and Ep,500the energy of the
transmit pulse of 10 dB bandwidth 500 MHz. Note that a 500 MHz
pulse has approximate energy Ep,500 ≈ BE0, while a pulse of
7.5 GHz bandwidth has approximately 15 times more. Hence, we
do not normalize SNR to the total received energy, as it would
be necessary to show BER curves as a function of receive SNR.
Nmax
P
=
1
B2
Bav
p
PFCC
av
PFCC
p
?
,
(19)
fact that average power in (13) for Rf ≤ 1MHz scales with Rf.
ξ =Eh,500
N0
=Ep,500
N0
−∞
??∞
−∞
w500(τ)h(t − τ)dτ
?2
dt
(20)

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Justification is that in UWB-IR radiated power is not the dominant
factor in system power consumption but is rigorously limited by
power constraints1. With this normalization, additional receive power
due to increased number of MPRC pulses as well as increased
bandwidth appears as BER improvements. The conditional BER is
now:
1
2erfc
Pe|h,ts,TI
=


ξ
ν(1)
Eh,500
?
4BpbTI+ 4ξ
ν(1)
Eh,500

.
(21)
B. Optimal Temporal Pulse Separation for MPRC
In this Section, it is shown for fixed integration duration TI
that a pulse separation of τp = Tmin is optimal. From (21), it is
evident that for fixed TI, BER depends only on instantaneous SNR
or more precisely, on x =
ξ
Eh,500
the exact pairwise error probability is straight forward [8], aver-
age SNR investigations are considered as meaningful enough, i.e.,
E {P (e|h(t))} is approximated by P?e|E?h2(t)??
SNR shows little variation over a small area. Due to the high
multipath resolution inherent in UWB Channel Impulse Responses
(CIR), this is a reasonable assumption. For x > 0, the expression
within the brackets of (21) is a monotonic growing function of
x. Hence, average receive energy E
optimization of MPRC pulse separation τp, if FCC power constraints
are considered. While maximally allowed number of MPRC pulses is
limited by the average power constraint, optimal pulse separation is
determined by the peak power constraint. Taking into account FCC
peak power constraint and assuming, that maximally allowed power
is radiated by the transmitter, captured signal energy per frame can
be described as:
?
argmaxt,f0


=KcEpE
ts
The second line of (22) equals signal energy collected by the ED, if
all NP pulses are transmitted with maximally allowed single pulse
energy. Doing so is allowed for τp ≥ Tmin, only. Kc ≤ 1 describes the
correction factor by which transmit pulse energy has to be reduced,
if τp is chosen smaller than Tmin. According to the FCC [4], [6],
bandwidth of the spectrum analyzer’s sweeping filter gf0(t) is set to
50 MHz. Center frequency f0 is swept from 3.1 to 10.6 GHz.
In detail, Kc describes the ratio between maximal single pulse and
maximal MPRC peak power. It equals 1 for τp ≥ Tmin and is smaller
than 1 for τp < Tmin. By assuming the spectral energy of the transmit
pulse w(t), i.e., |W(f)|2, to be constant over its supported band, Kc
can be approximated by:
?
ν(1)
?
. Although, evaluation of
with E {·} the
expectation operator. This approximation is close, if instantaneous
?
ν(1)?
shall be maximized by
E
ν(1)?
=
argmaxt,f0
?????NP−1
?|w(t) ∗ gf0(t)|2?
βnw(t − nτp) ∗ gf0(t)
?NP−1
n=0
??ts+TI
n=0
???
2?
·EpE

?ts+TI
ts
?
h2
E(t)dt
βnhw(t − nτp)dt
?
?2

(22)
(23)
.
Kc ≈˜ Kc =
1
argmaxt
?????NP−1
n=0
˜ g(t − nτp)
???
2?,
(24)
with ˜ g(t) a sweeping filter of center frequency f0 = 0 and peak
amplitude 1. From FCC power constraint discussion, it is evident that
1Plotting BER over receive SNR cancels out gains due to higher transmit
power.
the coefficients βn have little impact on Kc [4], and are therefore
omitted. As a typical example for ˜ g(t), we take a Gaussian filter,
which is often used in spectrum analyzers. As a typical CIR, we
consider a Gaussian Random Process (GRP) with exponentially
decaying Average Power Delay Profile (APDP):
hw(t) = ˆ e−γ
2tv(t),
(25)
where v(t) is a zero-mean white GRP of two-sided power spectral
density 1, filtered by an ideal bandpass filter of bandwidth B, i.e.,
σ2
?
As B ?
[4]:
??ts+TI
for single pulse transmission. For MPRC, we achieve:
?ts+TI
n=0
with EMPRC the energy that an ED would collect if no peak power
had to be respected, i.e., if Etx = NPEp.
To achieve more intuition into the behavior of (28) and its impact,
we consider two different scenarios and upper bound EMPRC by the
energy E(u)
i.e., γ = 0. We define Nfit as the maximal number of pulses that fit
into an integration duration TI, i.e., Nfit = ?TI/τp?.
Scenario 1: We consider both integration time TI and number of
MPRC pulses NP ≤ Nfit as fixed. Then E(u)
a function of ∆τp :
v= 2B, γ is a decay coefficient and
ˆ e−x=
e−x
0
if x ≥ 0
else
.
(26)
1
γ, the energy collected by the ED can be approximated as
EpE
ts
h2
w(t)dt
?
≈ 2BEp
?ts+TI
ts
ˆ e−γtdt
(27)
E
?
ν(1)?
≈ 2BEp˜ Kc
ts
NP−1
?
ˆ e−γ(t−nτp)dt =˜ KcEMPRC (28)
MPRC, that would be collected if hw(t) had uniform APDP,
MPRCincreases linearly as
∆E(u)
MPRC(∆τp) =
NP−1
?
n=0
n∆τp =NP(NP − 1)
2
∆τp,
(29)
where new pulse separation equals τnew
E(u)
with decreasing τp. On the other hand, (24) stays constant for
τp ≥ Tmin and decreases quadratically with decreasing τp < Tmin.
Hence, τp = Tmin is the best choice.
Scenario 2: We assume that the transmitter always sends Nfit
pulses. Then, we can formulate an upper found for E(u)
p
= τold
p
− ∆τp. Due to
MPRC≥ EMPRC, it is evident that EMPRC increases at most linearly
MPRC:
E(u)
MPRC(τp) ≤
Nfit
?
n=0
nτp =Nfit(Nfit+ 1)
2
τp ≈T2
I
2τp+TI
2.
(30)
It is evident that E(u)
inspection of (24) and (30) for reasonable values, it becomes evident
that˜ Kcstill decreases faster than E(u)
Hence, in this scenario, τp = Tmin is the best choice for MPRC, too.
MPRC(τp) grows faster now. All the same, by
MPRC(τp) increases, for τp < Tmin.
In Fig. 4, 1/˜ Kc is plotted for Nfit MPRC pulses, where Nfit depends
on τp. The correction factor ˜ Kc, which does not depend on γ,
decreases drastically for τp < Tmin. Compared to FCC or NTIA
[9] evaluations, the minimal pulse separation seems too restrictive.
In Fig. 4, it is Tmin ≈ 15ns, while evaluation from FCC power
constraint shows Tmin = 10ns. This is due to the assumptions made
in (24). All the same, the impact of power constraints is shown very
clearly. In Fig. 5, EMPRC gains with respect to EMPRC at τp = 50ns
are shown for different exponentially decaying channels defined by

Page 5

012345
x 10−8
0
5
10
15
Temporal Pulse Separation τp [s]
Inverse correction factor 1/˜ Kc
1/Kc [dB]
Fig. 4.
012345
x 10
−8
−2
0
2
4
6
8
10
12
Temporal Pulse Separation τp [s]
EMPRCgain due to decreasing τp
EMPRC Gain [dB]
γ = 0.04 e9s
γ = 0.16 e9s
γ = 0.63 e9s
γ = 2.5 e9s
Fig. 5.
(25). The integration duration of the ED was fixed to TI = 60ns.
The plots confirm our result, that for decreasing τp < Tmin, 1/Kc
grows much faster than EMPRC. In Fig. 6, the BERs are plotted
as a function of τp for SNR = 15 dB. The optimum τp = Tmin
can be nicely identified. These BER curves have been obtained
without the need of approximations and approve the reasonability
of our approximations used for above discussion. The error floors
for γ = 2.5 · 109s and γ = 6.3 · 108s origin from the fact that
the corresponding CIR have very small delay spread and are non-
overlapping for τp ≥ 20ns. Then significant BER changes occur,
only, if the number of pulses within the integration window decreases.
Hence, the high degradations at τp = 20ns and τp = 30ns occur
because energy of an overall CIR output slips out of the integration
window. BER curves for small γs are significantly worse than for
larger ones as a higher percentage of total channel output energy
falls out of the integration window for smaller γs.
Summarizing results, we argue in the following that MPRC-BPPM
with τp = Tmin is optimal for EDs of fixed integration duration TI.
First, recall that for fixed TI only the amount of energy concentrated
012345
x 10
−8
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Temporal Pulse Separation τp [s]
BER of ED for different τp and γ at SNR = 15 dB
BER
γ = 0.04 e9s
γ = 0.16 e9s
γ = 0.63 e9s
γ = 2.5 e9s
Fig. 6.
in TI influences the BER and that the specific shape of the receive
signal is irrelevant. From Scenario 2, we know that for Nmax
τp = Tminis optimal. Whether Nmax
P
on the frame repetition rate. For fixed Nmax
that τp = Tmin is optimal, as well. Furthermore, as the BER of EDs
depends only on the amount of energy in fixed TI, it is evident that
pulsing with NP > Nmax
P
decreases performance with respect to
NP = Nmax
P
as energy per pulse has to be reduced to comply with
average power constraint. Hence, we have following result:
For EDs of fixed integration duration TI, MPRC-BPPM with pulse
separation τp = Tmin and number of MPRC pulses:
?
is optimal.
P
≥ Nfit,
≥ Nfitis satisfied or not depends
P
< Nfit, we have shown
NP
=
Nfit
Nmax
P
for Nmax
for Nmax
P
≥ Nfit
< Nfit
P
(31)
C. Simulation Results
BER performance results for LDR MPRC-BPPM systems are
presented based on different measured UWB CIRs with bandwidth
B from 500MHz up to 7.5GHz. They are obtained by evaluating
the captured energy ν(1)= Ep
ts
and averaging at least over 100 CIRs. Transmit pulse is a Gaussian
bandpass pulse.
Most of used UWB channels are taken from a UWB measurement
campaign performed at ETHZ [7], [10] in a SPIN (Sensor, Positioning
and Identification Network) or warehouse like scenario, i.e., in a rich
scattering environment similar to [11]. The equipment is restricted
to a frequency range of 3 to 6GHz. A total of 4500 CIRs in 22
different LOS and NLOS areas has been measured.
In order to demonstrate MPRC-BPPM over channels extending
over the entire UWB bandwidth, we simulate also using channels
taken from a measurement campaign at IMST [12]. These measure-
ments were performed with a network analyzer of frequency range
1 to 11GHz in an office building and were among others basis for
well-known IEEE 802.15a UWB channel model.
The LDR MPRC-BPPM scheme considered has frame repetition
rate Rf ≤ 300kHz, BPPM modulation shift δ ≤ 1/(2Rf) and
τp ≥ Tmin = 10ns.
In Fig. 7 and Fig. 8, MPRC-BPPM in conjunction with an ED
is compared to single pulse transmission combined with a coherent
Selective RAKE (SRAKE) of 20 fingers. This number of fingers was
chosen as a reasonable upper limit for realistic RAKE receivers.
Simulations are performed with a transmit pulse of 2.9 GHz band-
width using NLOS channels from ETHZ. In Fig. 7, the SRAKE is
compared to an ED applying integration duration which is optimally
adjusted to the channel. As expected, the ED suffers from significant
performance losses with respect to the SRAKE, i.e., about 5 dB in
SNR at BER = 10−3. But the ED combined with MPRC-BPPM
of only NP = 3 pulses outperforms the SRAKE in conjunction
with single pulse transmission. Thus, MPRC-BPPM without any CSI
performs better than the very complex SRAKE, at the expense of
more transmit power. As perfect window adjustment is still involved,
we compare an ED of fixed large integration duration TI = 200ns,
in Fig. 8. Confirming intuition, the ED of fixed TI performs even
worse than the optimal one. It is remarkable though that combined
with MPRC-BPPM, it strongly improves its performance and for
NP = 4 outperforms the SRAKE at high SNR. In Fig. 9, SNR
at BER = 10−3is shown as a function of number of MPRC pulses
NP and pulse bandwidth B. It is important to note that the transmit
energy within one frame, scales with both B and NP according to
(21), i.e., Etx ≈ NPBE0. The channels used for this simulation, are
?ts+TI
h2
E(t)dt, plugging it into (21)