1754 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 9, DECEMBER 2002
Performance of Ultra-Wideband Communications
With Suboptimal Receivers in Multipath Channels
John D. Choi, Student Member, IEEE, and Wayne E. Stark, Fellow, IEEE
Abstract—The performance of a single-user ultra-wideband
(UWB) communication system employing binary block-coded
pulse-position modulation (PPM) and suboptimal receivers in
multipath channels is considered. The receivers examined include
a rake receiver with various diversity combining schemes and
an autocorrelation receiver, which is used in conjunction with
transmitted reference (TR) signaling. A general framework is pro-
vided for deriving the performance of these receivers in multipath
channels corrupted by additive white Gaussian noise (AWGN).
By employing previous measurements of indoor UWB channels,
we obtain numerical results for several cases which illustrate the
tradeoff between performance and receiver complexity.
Index Terms—Autocorrelation receiver, diversity combining,
on–off keying (OOK), pulse-position modulation (PPM), rake
receiver, transmitted reference (TR), ultra-wideband (UWB).
I. INTRODUCTION
U
LTRA-WIDEBAND (UWB) communications involves
the transmission of short pulses with a relatively large
fractional bandwidth [1], [2]. More specifically, these pulses
possess a
10-dB bandwidth which exceeds 500 MHz or 20%
of their center frequency [3] and is typically on the order of
one to several gigahertz. The large bandwidth occupancy of
UWB signals primarily accounts for both the advantages and
disadvantages associated with UWB communication systems
[4], [5]. For instance, the large bandwidth of UWB signals in
conjunction with appropriate spreading techniques [5]–[13]
provides robustness to jamming, as well as a low probability
of intercept and detection. These favorable characteristics are
offset by the fact that UWB communication systems must co-
exist with narrowband and wideband systems already operating
in dedicated frequency bands. In order to minimize interference
to these systems, UWB systems must follow strict regulations
[3] which limit the achievable data rates [4], [14], transmission
range, and implementation of power control. The presence of
multiple interfering signals also necessitates additional receiver
complexity, even with the potentially large processing gains of
UWB spread-spectrum (SS) systems.
This duality regarding UWB signaling also manifests itself in
the effects of the channel. Because UWB signals possess such
Manuscript received December 21, 2001; revised August 27, 2002. This
work was supported in part by the Office of Naval Research under Grant
N00014-00-1-0224 and in part by the United States Army Research Office
under Grant DAAH04-96-1-0377. This paper was presented in part at the
International Conference on Communications (ICC), New York, NY, April
28–May 2, 2002 and in part at the Ultra-Wideband Systems and Technologies
Conference (UWBST), Baltimore, MD, May 20–May 23, 2002.
The authors are with the Department of Electrical Engineering and Com-
puter Science, University of Michigan, Ann Arbor, MI 48105 USA (e-mail: jd-
choi@eecs.umich.edu; stark@eecs.umich.edu).
Digital Object Identifier 10.1109/JSAC.2002.805623
a large bandwidth, the channel is extremely frequency-selective
and the received signal contains a significant number of resolv-
able multipath components [15]–[19]. The fine time resolution
of UWB signals reduces the fading caused by several multi-
path components from different propagation paths overlapping
in time and adding destructively [20]. However, each multipath
component (or more appropriately, pulse [23]) associated with a
particular path collectively exhibits distortion after reflections,
diffractions, and scattering and does not resemble the ideal re-
ceived signal corresponding to the line-of-sight (LOS) path [1],
[2], [21]–[23]. This heightened sensitivity of UWB signals to
differentscatterers makesthem particularly well-suitedfor radar
applications [1], [2], [22], while making it more difficult for
practical communications receivers to fully exploit the multi-
path diversity inherent in the received signal [16].
Among the main claimed advantages of UWB communi-
cation systems is the availability of technology to implement
low-cost transceivers which can operate over such large band-
widths [5], [24]. In general, current embodiments of UWB
receivers [4], [25], [26] sacrifice performance for low-com-
plexity operation and a large discrepancy in performance exists
between these implementations and the theoretically optimal
receiver for most indoor and outdoor environments.
The most common receiver implementations cited in UWB
literature include threshold detectors [4], [24], [26], correlation
or rake receivers [1], [4]–[8], [10], [16]–[19], [21], [27]–[31],
and autocorrelation receivers [1], [9], [11], [22], [32]. The rela-
tive performance of these receivers in multipath and jamming
channels and the inherent tradeoff between performance and
complexity have not been fully examined.
The majority of the performance analyses of UWB com-
munication systems assumes the use of correlation receivers
or rake receivers [10], [16]–[19], [21], [27]–[31]. Because of
the large number of resolvable multipath components present
in the received signal, practical UWB rake receivers must
select, process, and combine only a small subset of these
components (hence, employing hybrid selection combining
(H-SC) [33]–[35] or suboptimal variations). The energy capture
of UWB rake receivers can be relatively low for a moderate
number of fingers [16] and is highly dependent upon the choice
of tap delays. This selection takes on increased significance
because UWB receivers generally do not employ a local
oscillator [36] nor do they explicitly perform phase estimation
and compensation in the demodulation process for a given tap
delay.
An alternative approach to exploiting the multipath diversity
is the use of an autocorrelation receiver which correlates the
received signal with a previously received signal [1], [9],
[11], [22], [32]. This receiver can capture the entire received
0733-8716/02$17.00 © 2002 IEEE
CHOI et al.: PERFORMANCE OF UWB COMMUNICATIONS WITH SUBOPTIMAL RECEIVERS IN MULTIPATH CHANNELS 1755
signal energy for slowly varying channels without requiring
channel estimation. The primary drawback of autocorrelation
receivers, however, is the performance degradation associated
with employing noisy received signals as reference signals
in demodulation [37], [38]. Autocorrelation receivers have
been typically used as suboptimal differential detectors for
differential phase-shift keying (DPSK) in narrowband systems
[37], [39]–[43]. The application of autocorrelation receivers to
UWB systems was proposed for radar target detection purposes
in [22]. The signaling and detection scheme described in this
work [22] falls under the classification of a transmitted refer-
ence (TR) system [38]. TR communication systems operate
by transmitting a pair of unmodulated and modulated signals
and employing the former to demodulate the latter [9], [38].
Similar to differential modulation and pilot symbol-assisted
modulation schemes, TR systems in essence transmit reference
signals to generate side information regarding the channel.
A TR-SS communication system employing UWB pulses or
noise for signaling and a hybrid of time-hopping (TH) and
direct-sequence (DS) spreading techniques has been recently
proposed in [9], [11], [32].
This paper examines the performance of rake and auto-
correlation receivers with varying degrees of complexity. We
consider these receivers in the context of a single-user UWB
system employing binary block-coded pulse-position modula-
tion (PPM) in multipath channels corrupted by additive white
Gaussian noise (AWGN). We extend previous work involving
block-coded PPM and rake reception [8], [17], [21], [27], [28],
[30] by providing a more general analytical framework and
considering various suboptimal diversity combining schemes.
These schemes select a subset of the received multipath
components, either optimally or suboptimally, and combine
them with maximal ratio combining (MRC) or square-law
combining (SLC). Furthermore, we examine a TR system
employing block-coded PPM and an autocorrelation receiver
which averages previously received reference pulses as a
means of noise suppression. It is noted that although UWB
systems encounter narrowband and wideband jamming, the
effect of such interference upon the system performance is not
taken into consideration for analytical simplicity. In order to
obtain numerical results, the performance of both the rake and
autocorrelation receivers is evaluated with measured indoor
channel data [15].
The paper is organized as follows. In Section II, the system
model and performance analysis involving rake receivers are
presented. Likewise, Section III details the performance anal-
ysis associated with the TR system. The performance of these
two systems is evaluated for various cases by employing indoor
UWB channel measurements in Section V.
II. P
ERFORMANCE OF RAKE RECEIVERS
In this section, we consider a single-user UWB system em-
ploying binary block-coded PPM in a multipath channel with
AWGN. We first describe the system model and then analyze
the performance of a rake receiver with arbitrary tap delays and
either MRC or SLC. Because of the assumptions made in this
section, the results are independent of any TH which might be
employed to help smoothen the power spectral density of the
transmitted signal [44] or in a SS multiple-access system. It is
also noted that some of the assumptions stated in the system
model will apply to the TR system considered in Section III.
A. System Model
The user employs binary signaling in which the transmitted
signals consist of a low duty cycle sequence of
UWB pulses
with energy . The signal waveforms correspond to code
words of a binary block code with length and Hamming distance
in which each code element is pulse-position modulated.
The signals
are equally likely to be transmitted and
are given by
(1)
where
denotes a unit energy pulse with time
duration
is the average pulse repetition period
is the th binary code element associated with the th
code word being modulated, and
is the delay associated with
PPM. In general, overlapped PPM refers to the case in which
, while orthogonal PPM corresponds to the case in
which
.
In order to simplify the ensuing analysis, the modulated code
words are assumed to have equal weight, thereby implying that
is even. For concreteness, the binary code elements are de-
fined to be
and . In addition,
the cumulative effect of the transmit and receive antennas has
been implicitly incorporated into the definition of
for nota-
tional simplicity.
The multipath channel is modeled as a linear, randomly
time-varying filter which is time-invariant over a symbol du-
ration with impulse response
and maximum excess delay
spread
. A tapped-delay-line representation of
the channel impulse response is assumed with a tap spacing
much less than
and random tap weights. Furthermore,
the pulse repetition period is chosen to be sufficiently large
to preclude intersymbol and intrasymbol
interference.
Assuming without any loss of generality that
is trans-
mitted, the received signal is
(2)
(3)
where
is a zero mean, AWGN random process with two-
sided power spectral density
. The substitution of (1) into
(2) yields the last expression (3). This expression can be simpli-
fied by employing the relation
as follows:
(4)
Thus, the received signal energy in the absence of noise
when conditioned upon the channel is
, where
1756 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 9, DECEMBER 2002
and is the time duration of a received
UWB pulse
.
Because UWB signals occupy such a large bandwidth, the
channel is extremely frequency-selective. The received signal
consequently contains a significant number of resolvable mul-
tipath components and in order to exploit the multipath diver-
sity, a rake receiver is considered. Because of complexity con-
straints, however, therake receiver processes only a subset of the
total number of received multipath components. The receiver
operates by passing
through a tapped-delay-line and per-
forming cross correlations with two reference signals at
tap
delays
, where .
The normalized reference signals
corresponding to
the two possible transmitted signals are given by
(5)
In general, the template pulse comprising the reference sig-
nals need not be equivalent to
because of either limitations
in receivercomplexity or the fact that UWB pulses which are re-
flected, diffracted, or scattered during propagation may not re-
semble the ideal received pulse corresponding to the LOS path
[1], [2], [21], [22], [45]. The use of an orthonormal set of tem-
plate pulses has been proposed in [16], [18], and [19] without
accompanying performance results.
The operation of the rake receiver can be viewed from an al-
ternative implementation in which the received signal is corre-
lated with delayed versions of the reference signals [46]. By
assuming that the tap delays are chosen such that
, the reference signals comprising
and , respectively, form two separate sets of
orthonormal basis functions. The output of the correlator corre-
sponding to the
th finger of the rake receiver is given by
(6)
Prior to evaluating the equation for
, we define the cross-
correlation function between
and as
(7)
where
if or . The cross-correlation
function is simply denoted as
in the remainder of the paper
for notational simplicity.
By substituting (4) and (5) into (6) and employing the pre-
vious definition (7), the output of the correlator becomes
(8)
where
. The assumption that the
modulated code words have equal weight is used in the deriva-
tion of
in (8). It is noted that and
represent two separate sets of independent, zero-mean Gaussian
random variables with variance
. Collectively, these two
sets of random variables are correlated or uncorrelated de-
pending upon
.
Because the diversity combining schemes of interest ma-
nipulate these correlator outputs differently, their operation
and associated performance are described in the ensuing two
subsections.
B. MRC
The optimal linear combining technique isMRCwhichyields
the maximum output signal-to-noise ratio (SNR) [46], [47]. For
the rake receiver under consideration, the combiner appropri-
ately weights the correlator outputs according to their SNR prior
to summing. The performance and optimality of MRC conse-
quently depend upon the receiver’s knowledge of the channel.
Assuming that the receiver can perfectly estimate these optimal
weights, the maximal ratio combiner output corresponding to
each decision hypothesis is
(9)
The substitution of (8) into (9) then leads to the following ex-
pansion:
(10)
where
and
.
For notational simplicity, we collectively denote the channel-
dependent random variables
as in this
section. By conditioning upon
, it can be shown that the de-
cision statistics
are jointly Gaussian random variables
which are either correlated or independent depending on
.In
order to derive the conditional probability of error, we define the
random variable
which is conditionally Gaussian
with the following mean and variance:
(11)
(12)
In (12),
denotes the normalized autocorrelation function
of
and is given by
(13)
From (11) and (12), the conditional probability of error is
simply
transmitted (14)
(15)
CHOI et al.: PERFORMANCE OF UWB COMMUNICATIONS WITH SUBOPTIMAL RECEIVERS IN MULTIPATH CHANNELS 1757
where . As evidenced by the
derivations for the conditional mean and variance of
, the
resulting equation for the conditional error probability for
binary block-coded PPM is cumbersome in general.
It is noted that the term
in (11) and (12) de-
notes the energy capture of the rake receiver. The normalized
energy capture
, where is defined in (4), is
simply the normalized energy of the signal space representation
of the noiseless received signal. This notion of energy capture
is a loose interpretation of the definition provided in [16] which
specifies the tap delays
to be optimal. The impact of
selecting optimal, as well as suboptimal, delays upon the perfor-
mance of rake receivers is discussed in Section IV. In addition,
the normalized energy capture equals one when the noiseless
received signal can be expressed completely in terms of one of
the two sets of orthonormal basis functions and corresponds to
the optimal receiver for the case of MRC.
Although the PPM delay
is typically chosen to be less than
one or a few multiples of
, we consider the case in which
the two possible received signals without noise are orthogonal
. Substituting this value of into (15) and noting that
for , we obtain the following simplification:
(16)
It is noted that this particular case of block-coded orthogonal
PPM is equivalent to one in which the code elements of a dif-
ferent set of equal weight code words of length and Hamming
distance
are modulatedby on–offkeying(OOK) with a pulse
repetition period of
.
C. SLC
A suboptimal, reduced-complexity diversity combining
scheme is SLC which does not require an estimate of the
optimal weights. This scheme is commonly employed in
conjunction with orthogonal modulation and noncoherent
reception in wideband and multichannel systems. For the
previously described rake receiver structure, the square-law
combiner operates by squaring and then summing the correlator
outputs
associated with each possible transmitted signal.
The decision statistics after SLC are thus
(17)
In general, SLC is not well-suited for overlapped PPM and must
be used with orthogonal PPM
. In order to exploit the
multipath diversity of the received signal and obtain reasonable
performance, the delay
is chosen such that .
The substitution of the correlator outputs (8) into (17) yields
the following:
(18)
where the simplification for
results from the fact that for the
being considered, .
Conditioned upon
, the decision statistics are both indepen-
dent noncentral chi-square random variables with
degrees of
freedom for
, whereas becomes a
central chi-square random variable for
. Although the
conditional probability of error can be derived for these two
cases (see Appendix A), the resulting equations do not provide
much insight into the relative performance of SLC with respect
to MRC.
As a result, we consider the case in which the number of
combined multipath components
grows large and the deci-
sion statistics in (18) can be modeled as independent Gaussian
random variables when conditioned upon
. Similar to the case
of MRC, we define the conditionally Gaussian random variable
, which possesses the following mean and vari-
ance:
(19)
(20)
Thus, the approximate conditional error probability of SLC
for the case of large
is obtained by simply substituting the
above equations into the previous derivation (15). As might be
expected, the energy capture parameter is present in
for
SLC through (19) and (20).
Furthermore, we note that the first term of the conditional
variance (20) corresponds to the noise-on-noise term which oc-
curs when expanding the equations in (18). This term degrades
the performance of the rake receiver with SLC whenever the re-
ceived SNR is small relative to
. It is emphasized that prior
to making a comparison between SLC and MRC or any two
schemes, the potential differences in data rates arising from the
selection of
must be taken into account.
The performance gap between SLC and MRC is more clearly
illustrated by again considering the case in which
.For
this designation, the conditional mean and variance of
sim-
plify because
, in (19) and (20). The con-
ditional probability of error under the Gaussian approximation
then becomes
(21)
This equation indicates that binary block-coded OOK with SLC
suffers a 3-dB loss relative to the same signaling scheme with
MRC (16) when the Gaussian approximation is valid and the
SNR is very large. The noise-on-noise term manifests in the
second term of the denominator in (21) and has a deleterious
effect for low to moderate SNR.
III. P
ERFORMANCE OF TR SYSTEM WITH
AUTOCORRELATION RECEIVER
An alternative approach to collecting the received multipath
signal energy is the use of autocorrelation receivers which
1758 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 9, DECEMBER 2002
perform differential detection suboptimally. In this section,
we consider a TR system employing binary block-coded PPM
and an autocorrelation receiver which suppresses the noise
by averaging the reference pulses prior to demodulation. The
analysis below does not take into consideration the effect of
TH for simplicity.
A. System Model
The user employs a variation of the binary signaling scheme
described in Section II. In particular, the TR system transmits
a reference pulse which carries no information prior to each
pulse-position modulated data pulse. The modulated data pulses
correspond to code elements of equal weight, binary code words
of length and Hamming distance
(thereby implying that
is even). The signals are equally likely to be
transmitted and are given by
(22)
where
, and
. Unless otherwise stated, the previous assump-
tions regarding the parameters in (22) still hold. In addition, the
user is assumed to havebeen transmitting continuously since the
previous symbol duration.
We employ the same channel model as before with the no-
table exception being that the channel is assumed to be time-
invariant over two symbol durations, as opposed to a single
symbol duration.
Assuming without any loss of generality that
is trans-
mitted, the received signal is
(23)
(24)
where
and (24) employsthe substitution of (22)
into (23), as well as the same relation specified in (4). We also
define
and as before in (4).
The autocorrelation receiver under consideration first passes
through an ideal bandpass filter with bandwidth and
center frequency
. The bandwidth of the filter is chosen to be
sufficiently wide
such that negligible intersymbol
and intrasymbol interference results [37]. The received signal
after filtering is expressed as
(25)
where
is a zero mean, Gaussian random process. The au-
tocorrelation function of the filtered noise [37] is given by
(26)
where
.
In order to demodulate the
th data pulse, the receiver first
multiplies
during the time frame
with an appropriately delayed average of the
previously received reference pulses. The receiver then
integrates this product over time duration
.
The autocorrelator outputs corresponding to the two possible
transmitted code elements for the
th received data pulse are
thus
(27)
where the term inside the brackets corresponds to the averaged
reference pulses. Next, by making the substitution
and taking into account the previously received
symbol
, the outputs can be reexpressed as
(28)
We next define
and employ the previous
result (28) to obtain the following expansion:
(29)
where
(30)
(31)
(32)
(33)
We note that the last two terms in (29) reflect the performance
degradation associated with correlating the received data pulse
with averaged reference pulses which contain noise.
The decision statistic
is obtained by summing over all of
the differences between the autocorrelation outputs as follows:
(34)
(35)
(36)
