# 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.

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**ABSTRACT:**In this paper a peak power reduction scheme by pulse repetition for an ultra-wide band (UWB) impulse radio (IR) system is experimentally examined for interference immunity to RF sensitive systems such as medical devices. The effectiveness of the scheme is shown in terms of an improvement of bit error rate (BER) performance. - SourceAvailable from: dtic.mil[Show abstract] [Hide abstract]

**ABSTRACT:**Ultra-WideBand (UWB) radio is a revolutionary, power-limited technology for its unprecedented system bandwidth and the potential of low-cost and low-power. The most pressing challenge is, however, how to reduce the transceiver complexity of coherent reception caused by the high sampling rate and stringent timing requirements. The proposed system paradigm uses time-reversal with noncoherent detection as an alternative to coherent reception. It exploits the hostile, rich-multipath channel as part of the receiver chain. This new method also integrates time-reversal with Multiple Input Multiple Output (MIMO), the most promising approach to use spectrum and transmission power. As a result, time-reversal trades the huge bandwidth of UWB radio and the high power efficiency of MIMO for the noncoherent detection of extremely low cost. This proposed new system paradigm is to take advantages of the impulse nature of UWB signals, a new dimension of a communication channel, through time-reversed MIMO. The new dimension of impulsive time-reversal adds more degrees of freedom in exploiting the spatiotemporal dimensions. In this project we investigate open problems in the framework of UWB. Our focus is to exploit rich multipath by using time reversal, to simplify the system structure. Experiments and analysis are performed. This final report serves as two purposes: (1) it proposes a big master plan of research for the next two years; (2) it summarizes the results obtained in the last year by the support of the project. The significance of the finished work is clearer in the framework of a three-year plan.08/2005; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper a peak power reduction scheme by pulse repetition for ultra-wide band (UWB) impulse radio (IR) scheme is experimentally examined for obtaining interference immunity to RF sensitive systems such as medical devices. The effectiveness of the scheme is shown in terms of an improvement of bit error rate (BER) performance.01/2009;

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.

Page 2

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.

Page 3

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)

Page 4

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)