IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO.

4, JULY 2007 2363

Correspondence
NBI Mitigation for UWB Systems Using Multiple Several research efforts have investigated the impact of NBI on
Antenna Selection Diversity UWB. An analysis of the effect of tone jammers on UWB systems
is presented in [3], while the effect of partial-band interference on
Jihad Ibrahim, Student Member, IEEE, and UWB is analyzed in [4]. In [5], the effect of NBI on direct-sequence-
R. Michael Buehrer, Senior Member, IEEE based impulse radio (DS-UWB) is derived, where NBI is modeled
as the sum of sinusoidal signals with variable power and frequency.
The performance of a generalized Rake receiver in the presence of
Abstract—In this paper, a narrowband interference (NBI) mitigation multiple access interference and NBI is analyzed in [6]. The impact
scheme for ultrawide bandwidth (UWB) signals using multiple receive
antennas is examined. The low spatial fading of UWB signals compared to of NBI on DS-UWB as well as single and multicarrier multiband
NBI signals is exploited to provide “interference selection diversity (SD).” UWB is discussed in [7] and [8]. The effect of the Global System for
Whereas classical SD is designed to maximize the desired received signal Mobile Communications and the Universal Mobile Telecommuni-
power, the aim of interference SD is to minimize the effective NBI power. cations System/Wideband Code-Division Multiple-Access bands on
The resulting distribution of the signal-to-interference ratio at the receiver
is derived for both Rayleigh and Ricean NBI fading scenarios. Expressions
UWB is studied in [9], where it is shown by simulation that perfor-
for the probability of error of the SD scheme are also derived for a perfect mance degradation is most severe when the NBI bandwidth overlaps
receiver (where the correlator template is matched to the received UWB with the UWB nominal center frequency.
pulse shape) as well as for a Rake receiver with a limited number of Some NBI mitigation techniques for UWB systems have been pro-
fingers. It is shown that doubling the number of antennas results in a 3-dB posed in the literature. Various pulse-shaping methods that introduce
performance improvement for the Rayleigh fading case. Less substantial
gains are observed under Ricean fading. The method provides a simple nulls in the UWB spectrum where NBI occurs are investigated in
but effective technique to mitigate NBI effects in UWB systems. [10]–[13]. However, these methods assume knowledge of the NBI
Index Terms—Multipath channel, multiple antennas, narrowband spectrum. NBI mitigation based on the optimization of the pulse-
interference (NBI), ultrawide bandwidth (UWB). position modulation delay parameter is suggested in [14] along with
two other methods based on passing the UWB signal through a notched
filter and on a minimum mean-square error (MMSE) Rake receiver,
I. I NTRODUCTION
respectively. NBI mitigation based on MMSE Rake is also studied in
Ultrawide bandwidth (UWB) systems must share their large band- [15] and [16]. In [17], NBI is modeled by a sine wave of unknown
width with coexisting narrowband applications. Although UWB amplitude, frequency, and phase, and these three parameters are first
signals may enjoy inherently high spreading gain, stringent Federal estimated, and then used to reconstruct and cancel the NBI wave. A
Communications Commission power restrictions [1] make them sus- spectrally encoded system for NBI mitigation is introduced in [18],
ceptible to strong narrowband interference (NBI), which can severely where spectral nulls are introduced in the UWB spectrum using surface
degrade performance. The extent of performance degradation de- acoustic wave (SAW) devices. The use of SAW filters for transform-
pends on the number, power, and spatial distribution of the inter- domain processing is also suggested in [19]. Another technique is
ferers relative to the UWB signal. NBI may be tens of decibels suggested in [19], where NBI is digitally estimated, and a radio
stronger than the UWB signal [2] and can completely overwhelm frequency estimate is produced to perform the NBI cancellation in the
the receiver front end. Severe NBI could thus render the acquisition analog domain (also studied in [20]).
process impossible, and front-end interference mitigation techniques In this paper, we develop and extend the novel NBI mitigation
are desirable. Some NBI mitigation techniques have been proposed method introduced in [21]. In [21], NBI mitigation for UWB systems
for UWB. However, most of these techniques are based on traditional based on multiple receive antennas is proposed. The method takes
spread-spectrum methods, which might not be suitable for UWB advantage of the great immunity to multipath fading that UWB signals
applications. The design of NBI mitigation algorithms tailored for exhibit when compared to narrowband signals (see [23]–[27]). In
UWB systems is thus still largely an open research issue. In this fact, in indoor environments, whereas the total captured UWB energy
paper, we propose a technique that exploits the low spatial fading varies only slightly over a relatively small area, the NBI energy
of UWB signals compared to NBI signals to provide “interference level tends to vary wildly. This leads to a SD scheme where the
selection diversity (SD).” The method may be applied at the receiver signal corresponding to the receive antenna with the least measured
front end and does not require prior knowledge of the NBI’s spectral power is selected (Fig. 1). The rationale behind this choice is the
characteristics. following: The UWB signal power is approximately constant across
antennas, while the interference power observed is independent from
antenna to antenna when the multipath angle spread is high. Thus,
Manuscript received December 10, 2005; revised May 19, 2006 and any increase in received power level is due to a higher NBI power.
September 8, 2006. This work was supported in part by the Defense Advanced Therefore, the effective received signal-to-interference ratio (SIR) is
Research Projects Agency NETworking in EXtreme Environments project, by maximized by selecting the antenna with the lowest measured power.
the National Science Foundation under Grant CCF-0515019, and by the Office
of Naval Research under Grant N00014051179. The review of this paper was This is a potentially attractive method, since it does not assume
coordinated by Prof. R. Qiu. signal synchronization or knowledge of the NBI’s spectral content
The authors are with the Mobile and Portable Radio Research Group, or statistical properties prior to interference mitigation. Moreover, it
Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA does not require high sampling rates at the receiver’s front end. This
(e-mail: jibrahim@vt.edu; buehrer@vt.edu).
Color versions of one or more of the figures in this paper are available online
paper is further developed in [22], where additional antenna combining
at http://ieeexplore.ieee.org. methods [equal gain combining (EGC) and maximum ratio combining
Digital Object Identifier 10.1109/TVT.2007.897660 (MRC)] are investigated. It is shown that those methods yield no or

0018-9545/$25.00 © 2007 IEEE

2364 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

Fig. 1. Proposed SD NBI mitigation receiver.

negligible gain compared to SD at the expense of a more complex to guarantee a target error rate drops with an increasing number of
design.1 antennas. Simulation and numerical results are included in Section VI.
The previous work in [21] and [22] assumes that the NBI under- In addition to providing validation of the theoretical findings of the
goes classical flat Rayleigh fading. Also, it is assumed that perfect previous sections, the system’s performance is compared to the Rake
knowledge of the UWB channel is available at the receiver, and perfect MMSE receiver, which is the most popular NBI mitigation technique
UWB energy capture is obtained by using the received pulse shape available in the literature. Finally, Section VII concludes this paper.
as a correlator template. In this paper, we first provide a rigorous
mathematical model of the proposed system (which is not included II. S PATIAL E NERGY V ARIATION OF UWB AND NBI S IGNALS
in [21]) and then extend the analysis to the more general NBI Ricean
fading case. Moreover, we also examine the case where a realistic A number of indoor nonline-of-sight (NLOS) UWB measurements
UWB receiver is used rather than the ideal correlator matched to were taken as part of a campaign conducted at Virginia Tech.2 The
the received UWB pulse shape. More specifically, we study a Rake details of the measurement campaign can be found in [25].
receiver that combines the energy available in a subset of the multipath Measurements were taken at multiple locations in an indoor office
components. We derive the probability of error expression for the environment. Consider a specific location, or position, and define its
proposed SD method under Rayleigh and Ricean fading. We also position set as all the impulse responses recorded at that position.
provide insight into the effective SIR gain brought by this method Measurements were grouped into separate position sets each consist-
under both fading scenarios. ing of 49 channel impulse responses. Each position set holds measured
The rest of this paper is organized as follows: The spatial energy channel impulse responses at 49 uniformly distributed points in a
variation characteristics of narrowband (NB) and UWB signals are 1-m2 local area. For a specific position set, the total energy captured
discussed in Section II. It is shown that while UWB energy remains al- by a receiver is measured and recorded. The 49 energy-capture values
most constant over a local area, NB energy varies substantially. Spatial are then divided by the mean energy-capture value over the position
diversity gains are thus possible, where NBI mitigation is performed set. Energy averaging is performed because we are interested in the
by selecting the received signal corresponding to the lowest measured energy variation around the mean value and not in the mean value
power. Such a diversity system, based on multiple antennas, is mathe- itself. The procedure is repeated over all position sets, and data from all
matically formulated in Section III. NBI is assumed to undergo either position sets are analyzed. The histogram of the total energy captured
flat Rayleigh or Ricean fading. SD is discussed in Section IV, and the by a receiver over a 1-m2 spatial grid at multiple locations for a pulse
probability of error of the SD system employing M receive antennas width of 500 ps is shown in Fig. 2. The variance of the captured energy
is derived for both NBI Rayleigh and Ricean fading, first for the was found to be only 0.0035. Thus, the total energy capture may be
general Rake receiver employing F fingers, and then for the “ideal” assumed to be constant over this local area emphasizing the low spatial
receiver (a perfect Rake or All-Rake receiver), where perfect channel fading of UWB signals. While we base our technique on results from
knowledge is assumed. It is shown that doubling the number of anten- [25], it should be noted that the lack of spatial fading for UWB is a
nas potentially results in a 3-dB performance boost for the Rayleigh well-documented phenomenon (for example, see [26] and [27]).
case. Less substantial gains are seen for the Ricean case. Section V In another experiment, separate transmissions of NBI and UWB
derives the effective SIR gain brought by this method under both signals over NLOS channels were simulated based on actual chan-
fading scenarios. Specifically, we show that the average SIR required nel measurements [23]. Two UWB monocycles of different duration
(250 ps and 2 ns) were employed. The received energy content of the
1 Note that synchronization is not required for the antenna selection process NBI and the UWB signal over a 70-cm linear segment was recorded
in the SD system. However, synchronization prior to NBI mitigation is required and appears in Fig. 3. Note that the energy over multiple distance
for EGC and MRC, since these two techniques are based on the combination values is normalized such that it has unit mean over the segment. The
of decision statistics from different antennas. Additionally, SD requires only a
single front end (a single correlator circuit). Thus, in this paper, we concentrate
on the SD method. For an illustration of the method applied to EGC and MRC, 2 Measurements taken within the context of the DARPA NETworking in
the reader is referred to [22]. EXtreme (NETEX) environments program.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007 2365

Fig. 2. Histogram of the normalized received UWB signal energy over a 1-m2
area. Perfect receiver. Fig. 4. Normalized received UWB signal energy over a 1-m2 area. Ten-finger
Rake receiver.

across antennas, while the interference power varies independently

from antenna to antenna, according to a certain statistical distribution.
The independence of NBI fading assumes a large multipath angular
spread (which is typical for an indoor environment) and that the
distance between adjacent antennas is greater than λ/2, where λ is
the wavelength of the NBI signal. Thus, an increase in the SIR of the
system is equivalent to less interference power and consequently lower
overall received power.
Note that, in the proposed model, the antenna with the lowest power
is selected, and then signal detection is performed, based on a decision
statistic obtained at the output of a correlator receiver. In the case
of perfect channel knowledge (i.e., the desired received pulse shape
is known, or equivalently a perfect Rake with infinite resources is
used), the entire UWB signal energy is captured, and the desired UWB
component in the decision statistic is constant across all antennas
(since it is a direct function of the total energy capture). However,
if only a subset of the multipath components delays and amplitudes
is available, or the receiver complexity is limited, a more realistic
Rake receiver structure (with a limited number of fingers) is used,
Fig. 3. Received signal energies at different antenna separations.
and only a fraction of the available energy is captured. In this case,
more UWB energy variation may be expected across antennas in the
received NBI energy varies wildly over the linear region as compared decision statistic. The energy capture for a Rake receiver employing
to the received energy of the UWB signal. Further, the 250-ps UWB F fingers was thus also analyzed using the same procedure in Fig. 3.
pulse shows less variation in energy across the grid than the 2-ns pulse Note that it is assumed that the position of the F strongest multipath
due to the presence of more resolvable paths (i.e., larger bandwidth). components is known. It is shown that the Rake energy level varies
Thus, in terms of total received energy, UWB signals exhibit great more significantly than the total energy-capture case. For example, the
immunity to fading when compared to narrowband signals. We would energy variance for a ten-finger Rake when normalized to unit mean
like to exploit this unique characteristic of UWB signals in our NBI is equal to 0.0273, compared to a variance of 0.0035 for the perfect
mitigation system. energy-capture case. However, the energy variance is still relatively
In a multiple receive antenna system, the total measured UWB small compared to the NBI. Moreover, it was found that the energy
power can thus be assumed nearly constant from antenna to antenna, capture d across multiple antennas may be approximated by a Laplace
whereas the NBI power is not. This leads to our proposed method. In random variable (Fig. 4) with distribution given by
conventional SD systems, diversity is achieved by feeding the signal √
2|x−d̄F |
with the highest average received power to the demodulator. This 1 −
fd (x) = √ e σd
F (1)
is effective due to the fact that the noise power is constant from σdF 2
antenna to antenna, while the desired signal varies; thus, the signal
with the greatest received power is most likely to have the largest where d¯F is the average energy capture, σd2F is the energy-
desired signal power. In the proposed interference diversity system, capture variance, and |.| is the absolute value operator. A
the weakest signal is chosen to be fed to the demodulator. This can be Kolmogorov–Smirnov statistical goodness of fit test was performed
explained as follows: Assuming adequate antenna separation in a mul- to test the Laplace distribution hypothesis, and the hypothesis passed
tipath environment, the UWB signal power is approximately constant the test for five, 10, 15, and 20 fingers (Table I). Thus, we will use a
2366 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

TABLE I We restrict our analysis to the time interval [0, Tf ] for ease of
KOLMOGOROV–SMIRNOV TEST FOR LAPLACIAN FIT OF RAKE RECEIVER notation. The received signal corresponding to bit b0 at the jth receive

POINTS. SIGNIFICANCE LEVEL = 0.05.
ENERGY CAPTURE. 580 SAMPLE
antenna can be written as
TEST THRESHOLD = −1/(2 × 580) log(0.05/2) = 0.0564


rj (t) = b0 Ep w(t) ∗ hj,d (t) + sNBI (t) ∗ hj,NBI (t) + n(t) (3)

where hj,d (t) is the impulse response of the channel between the
UWB transmitter and the jth receive antenna, hj,NBI (t) is the impulse
response of the channel between the NBI transmitter and the jth
receive antenna,4 and n(t) is an additive white Gaussian noise process
Laplacian model for the captured Rake received energy with a variance with double-sided power spectral density N0 /2.
dependent on the number of Rake fingers.3 We assume that the transmitted NBI signal is a continuous wave
We now briefly describe the proposed model in Fig. 1 in light of tone5 with power PI , center frequency fc , and phase Θ:
the above discussion. A power measurement device first measures the

total power of the signals at each of the M receive antennas. Let Pj sNBI (t) = 2PI cos(2πfc t + Θ). (4)
be the measured power at the jth antenna. In the next section, we
Moreover, we assume that the NBI undergoes a frequency-
will show that Pj may be expressed as the sum of the UWB total
nonselective slowly fading channel. Then, rj (t) can be written as
received power (which may be assumed constant across antennas,
based on the discussion in this section), the noise power, and the NBI


rj (t) = b0 Ep wj,rx (t) + αj 2PI cos(2πfc t + φj ) + n(t) (5)
received power. The signal corresponding to the minimum measured
power is selected. Acquisition, modulation, and signal detection are where wj,rx (t) = w(t) ∗ hj,d (t) is the received UWB pulse shape at
then performed on the selected signal. If a perfect Rake receiver the jth antenna, φj is a random phase offset uniformly distributed over
is employed, complete energy capture is achieved, and the desired [−π, π], which models the resulting NBI phase after passing through
UWB component d in the correlator decision statistic may be assumed the channel, and αj is the multiplicative fading term. We analyze NBI
constant across antennas. If a Rake receiver with F fingers is used, the under both Rayleigh and the more general Ricean fading cases. In the
desired component of the decision statistic is modeled by a Laplacian case of Rayleigh fading, αj is modeled by a Rayleigh random variable
random variable with mean d¯F and variance σd2F , where d¯F and σdF with probability density function (pdf) [29]
depend on F . Thus, antenna selection is based on the entire received
signal power, whereas data detection depends on some fraction of the α − α22
fαj (α) = e 2σ , α≥0 (6)
received power based on the receiver structure used. The next section σ2
formulates a mathematical model for this system. where σ depends on the average NBI received power.
It is important to note that the proposed method assumes slow In the case of Ricean fading, the pdf of αj is given by
NBI fading. That is, the NBI fading coefficients on the receiver an-
α − s2 +α 2
s
 
tennas remain constant during the power measurement–data detection
fαj (α) = e 2σ 2 I0 α 2 , α≥0 (7)
cycle. This is a valid assumption for indoor environments, where σ2 σ
the coherence time is estimated at around 30 ms for low-mobility
where I0 is a modified Bessel function of the first kind, and s is the
environments [28].
noncentrality parameter. The ratio of the line-of-sight (LOS) to the
NLOS power is defined as K = s2 /2σ 2 .
III. G ENERAL S YSTEM M ODEL
We assume a UWB system with one transmit antenna and A. Power Measurement
M (M ≥ 1) receive antennas. For mathematical simplicity, we assume Recall that the SD algorithm is based on selecting the signal corre-
perfect synchronization at the receiver. The UWB system employs sponding to the antenna with lowest measured power. The measured
binary pulse amplitude modulation (PAM) modulation and is corrupted power from the jth antenna in the interval [0, Tf ] can be written as
by an NBI signal sNBI (t) with center frequency fc .
The transmitted UWB signal can be written as Tf
1
Pj = rj2 (t)dt


∞
Tf
s(t) = Ep bk w(t − kTf ) (2) 0
k=−∞
Tf Tf
Ep 2 2 2
where bk = ±1 is the kth data bit, w(t) is the unit-energy UWB Pj = wj,rx (t)dt + α PI cos2 (2πfc t + φj )dt
T Tf Tf j
transmit pulse of duration Tw ( 0 w w2 (t)dt = 1), and Ep is the trans- 0 0
mit pulse energy. The transmit and receive antenna transfer functions
Tf
are assumed to be included in w(t). Tf is the symbol duration time 2 Ep b0
+ wj,rx (t)n(t)dt
(Tf  Tw ). We assume that Tf is longer than the channel maximum Tf
delay spread, so that any intersymbol interference effect may be 0
ignored. Note that the same analysis may be applied to any UWB
modulation technique. 4 We assume the classical tap delay line channel model in the analysis.
However, note that real channel measurements are used.
5 Since the UWB bandwidth is very large compared to the typical NB
3 Note that the Laplacian model f (x) is clearly an approximation, since bandwidth, any NB system may be modeled as a tone signal. Note that the
d
the Laplacian distribution extends to negative x. However, for small σdF , the analysis may be easily extended to a scenario with multiple NB sources by
ensuing error is negligible. Alternatively, we may use a truncated Laplacian modeling the effective NB signal as a sum of tones with potentially different
distribution (only defined for positive x). powers and different center frequencies.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007 2367

Tf B. Antenna Decision Statistic

2 2PI Ep
+ b0 αj wj,rx (t) cos(2πfc t + φj )dt We assume that a correlator is applied at the receiver after the signal
Tf
0
corresponding to the lowest measured power is selected. The template
of the correlator is given by
√  Tf
2αj 2PI
+ n(t) cos(2πfc t + φj )dt 
F
Tf y0 (t) = γf w(t − τf ). (12)
0
f =1
Tf
1
+ n2 (t)dt. (8) In the case of perfect knowledge of hj,d (t), γf and τf are set to the
Tf amplitude and delay of the f th multipath component of hj,d (t). Then,
0
y0 (t) is matched to the received pulse shape wj,rx (t), and F is equal to
Note that, if the power is averaged over multiple bits, E[b0 ] = 0 the number of multipath components L. The same template structure
(where E[.] is the expectation operator).6 Moreover, we assume that may be used if a more realistic Rake receiver is employed, where γf
signals are passed through a band-limiting filter prior to power mea- and τf are the weight and delay of the f th Rake finger, respectively,
surement, where the filter bandwidth W is large enough not to cause and F ≤ L.
distortion to the UWB signal and to the NBI signal.7 Then, since noise The decision statistic corresponding to the jth antenna after corre-
is zero mean, Pj may be approximated by lation can be written as


Tf Tf rj = Ep b0 dj + ij + nj (13)
Ep 2 2
Pj ≈ 2
wj,rx (t)dt + α PI cos2 (2πfc t + φj )dt + N0 W.
Tf Tf j where
0 0
(9) Tf
Therefore dj = y0 (t)wj,rx (t)dt (14)
Tf 0
Ep
Pj ≈ 2
wj,rx (t)dt + N0 W

Tf
Tf
0 ij = αj 2PI y0 (t) cos(2πfc t + φj )dt (15)
Tf 0
αj2 PI
+ (1 + cos(4πfc t + 2φj )) dt. (10) 
Tf
Tf
0 nj = y0 (t)n(t)dt. (16)
0
Assuming 4πfc Tf is large (a valid assumption considering UWB’s
operating range), then we can write [22] Based on the discussion from Section II, we study the following two
receiver cases.

Tf
Ep 1) In the case of perfect knowledge of hj,d (t), perfect energy
Pj ≈ 2
wj,rx (t)dt + N0 W + αj2 PI . (11)
Tf capture is achieved. The UWB power is then constant over all
0
antennas, and consequently, dj = d¯∀j. This case is equivalent to
Thus, the measured power Pj is the sum of the total received a perfect Rake receiver, which can resolve all the available chan-
UWB power (which is constant from antenna to antenna as seen in nel multipath components. In other words, y0 (t) = wj,rx (t).
Section II), the noise power, and the NBI power αj2 PI , where αj2 , 2) If a Rake receiver with F fingers is employed, only a fraction
1 ≤ j ≤ M , are independent random variables whose distributions of the total available energy is captured, and dj is no longer
depend on the type of NBI fading (Rayleigh or Ricean). This means assumed to be constant across the M antennas but rather follows
that the signal with the lowest NBI power corresponds to the signal a Laplace distribution with mean d¯F and variance σd2F , where
with lowest measured power, as expected. Note that the analysis d¯F and σdF depend on F .
assumes slow NBI fading, i.e., the fading coefficients αj are assumed Moreover, the interference component ij can be written as
constant during the process.
It is important to note that the power measurement procedure is

ij = 2NI [aαj cos φj − bαj sin φj ] (17)
independent of the type of correlator used. That is, the signal with
the lowest measured power is selected for both the perfect correlator where
(matched to the UWB received pulse shape) and the Rake receiver,
since signal selection is done prior to demodulation and detection, and  Tf
is based on total received power. 1
a= y0 (t) cos(2πfc t)dt (18)
Tw
0
6 It can easily be shown that the variance of the third and fourth terms on
the right-hand side of (8) is proportional to 1/N , where N is the number of  Tf
bits over which power is averaged. The number of bits required for adequate 1
estimation depends on the strength of the NBI, but 10–20 bits should be b= y0 (t) sin(2πfc t)dt (19)
Tw
sufficient to effectively limit the term’s effect. 0
7 The design of such a filter is beyond the scope of this paper. We note,
however, that the filter is essential for limiting noise power. NI = PI Tw (20)
2368 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

where NI may be interpreted as the interferer’s one-sided power Assume b0 = −1. Then, a bit error occurs if rm > 0, or
density, as defined in [31].8


− Ep dm + im + nm > 0. (28)
IV. SD P ROBABILITY OF E RROR
Equivalently
A. Rayleigh Fading
dm 1
In this section, we derive the probability of error expression for the <
. (29)
proposed system when a Rake receiver with F fingers is employed im + nm Ep
in the presence of NBI Rayleigh fading. The result may be easily
extended to the perfect energy-capture case (All-Rake), as will be seen. Let Z = dm /(im + nm ). It is trivial to show that nm is a zero-mean
Assume that the signal corresponding to the mth antenna is selected. Gaussian random variable with variance σN 2
= N0 /2. Then, the cdf of
The decision statistic may be written as Z is given by [30]


rm = Ep b0 dm + im + nm (21) ∞ zy √
2|x−d̄F | y2
− σd
− 2
2σ
FZ (z) = cz e F T dx dy (30)
with

y=0 x=0
im = 2NI [aαm cos φm − bαm sin φm ] (22)
√
where cz = 1/2σdF σT π, and σT2
= σI2 + σN 2
. From (29), the prob-
where αm and φm are the NBI Rayleigh fading term and phase at the
ability of error is equal to FZ (1/ Ep ). After some basic manipu-
mth antenna, respectively.
lation (the details are included in Appendix A), we get the following
Based on the power measurement derivation in the previous section,
closed-form expression for the probability of error:
the antenna selection mechanism is equivalent to selecting the mini-
mum of M independent identically distributed (i.i.d.) random variables √ σ2
√
2d̄
T − σ F
α12 , α22 , and αM
2
Ep d¯2F
2d̄F
. Then, based on classical-order statistics [32], the 1 − 1 Ep σd2 d
distribution of αm is given by Pe = Q − e σd
F + e F
F

σT2 4 2
 M −1 
fαm (α) = M 1 − Fαj (α) fαj (α) (23) √
√
− 2σT Ep d¯F 2σT
· Q
−Q −

where Fαj (α) is the cumulative distributive function (cdf) of αj . For Ep σdF σT Ep σdF
Rayleigh fading, we get
σ2
√
2d̄
√
Ep d¯F
T + σ F
2
M − Mα 1 Ep σd2 2σT
+

d
fαm (α) = 2
αe 2σ2 . (24) − e F
F
Q . (31)
2σ 2 σT Ep σdF
Thus, αm is also a Rayleigh random variable, where σ 2 is replaced by
σ 2 /M . Note that, in case of full energy capture (which is equivalent to a
Now, let z = αm cos φm , and w = αm sin φm . Note that the power perfect Rake receiver with complete knowledge of the desired received
measurement process is not affected by the phase, and φm remains UWB pulse shape), the desired component of the decision statistic
uniformly distributed over [−π, π]. Using the basic Jacobian method may be assumed constant, as demonstrated in Section II. Equivalently,
[30], the joint distribution of z and w is found to be σd = 0, and the probability of error becomes

M − M (z2 +w 2)

fz,w (z, w) = 2
e 2σ 2 . (25) Ep d¯2 Ep M d¯2
2πσ Pe = Q =Q
σT2 2NI (a2 + b2 )σ 2 + N0
2
z and w are thus uncorrelated jointly Gaussian random variables.
Consequently, z and w are independent zero-mean Gaussian random


= Q( M × SIRSD ) (32)
variables with equal variance σ 2 /M . Thus, im is also a zero-mean
Gaussian random variable with variance
where SIRSD = Ep d¯2 /(2NI (a2 + b2 )σ 2 + N0 /2) is the SIR at the
 σ 2
output of the correlator. Thus, for the perfect Rake receiver, we expect
σI2 = 2NI E (az − bw)2 = 2NI (a2 + b2 ) . (26)
M a 3-dB performance gain in the probability of error every time we
double the number of antennas M . Moreover, simulation results will
Note that the power selection process does not affect the distribution
show that the gain in the case of the traditional Rake receiver is also
of dm (since power is measured before modulation and detection,
practically equal to 3 dB for a moderate number of fingers because of
and is based on total received power). In other words, dm is selected
the negligible UWB energy variation across antennas (σd2 ≈ 0).
randomly from a group of M i.i.d. Laplace random variables. Then,
the pdf of dm is given by
√ |x−d̄F |
B. Ricean Fading
1 − 2 σ
fdm (x) = √ e dF
. (27) The probability of error for the NBI Ricean fading case may be
σdF 2 obtained following the same analysis performed for Rayleigh fading.
However, the calculations are more cumbersome, and the obtained
8 Here, the interference’s contribution to the decision statistic is expressed as expressions do not always provide intuitive understanding of the gains
a function of NI rather than PI to make the notation consistent with classical yielded by the proposed system. We include the basic strategy for the
jammer analysis in the literature. derivation of Pe and note the fundamental results.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007 2369

Keeping with the notation of the previous section and based on (23), V. D IVERSITY SIR I MPROVEMENT
the distribution of the selected interference fading parameter after SD
for Ricean fading is given by We have studied system gain from a probability of error point of
view in the previous section. We now formulate the improvement
M M −1
 s α 2 2  s in effective SIR provided by the SD method. Specifically, we are
− s +α
fαm (α) = Q , e 2
2σ I0 α (33) interested in the average SIR required to guarantee a target probability
σ2 1 σ σ σ2
of error. It will be shown that the required SIR drops with an increasing
where Q1 (.) is the generalized Marcum’s Q function. number of antennas. Analysis is provided for both Rayleigh and
The Jacobian method used for Rayleigh fading yields complex Ricean fading.
expressions for the Ricean case, because the random variables are no
longer Gaussian. We thus follow a different approach for obtaining Pe .
Note that im may be written as A. Rayleigh Fading
Let us first consider NBI Rayleigh fading. In this case, the inter-
im = Rαm sin(φm + Γ) (34) ference energy at the jth antenna Ij = |αj |2 (where PI is subsumed

into the random variable for notational simplicity) over a local area is
where Γ = tan−1 (−a/b), and R = 2NI (a2 + b2 ). The distribu- central Chi-square distributed with two degrees of freedom and may
tion of βm = R sin(φm + Γ) is given by [30] be expressed as
1 1 −x
fβm (β) =
, |β| < R. (35) fIj (x) = e I (40)
π R2 − β 2 I

Similar to the previous section, the probability of error is given by where I is the average interference energy. Let S̄ be the average UWB
signal energy over the local area. Then, based on the arguments in
√αβ Section II, the UWB energy may be approximated by a constant S̄
∞  R  ∞  Ep (since power measurement is performed prior to demodulation and
Pe = fd (x)fnm (n) detection), and the distribution of the SIR at the jth antenna χj =
α=0 β=−R n=−αβ x=0
S/Ij may be written as
    
· fαm (α)fβm (β)dx dn dα dβ (36) S  d S  χ −χ
fχj (χ) = fIj   = 2e χ (41)
χ dχ χ χ
which may be written as
where χ = S/I is the average SIR.
√αβ
 The cdf of the SIR can then be easily shown to be

∞  R 
∞  Ep
√
2|x−d̄F |
M 1 − σd χ
Pe = e F −χ
πσ 2 πN0 Fχj (χ) = e . (42)
α=0 β=−R n=−αβ x=0

2
1
 s α 2 2
Consider a diversity array with M receive antennas, i.e., M indepen-
n
−N −1 − s +α
·e 0
αQM
1 , e 2 2σ dent fading channel realizations (from the NBI perspective). Let the
(R2 − β 2 ) σ σ instantaneous SIR in each branch be χi . The pdf of χi is given by (41).
 s The probability that a single branch has an instantaneous SIR less than
· I0 α dx dn dα dβ. (37) or equal to some threshold χ is e−χ/χ . Since the diversity branches are
σ2
independent, the probability that all M independent diversity branches
In the case of full energy capture (σd = 0) and ignoring thermal noise receive signals that are simultaneously less than or equal to some
(the case where performance is strictly limited by high-power NBI), specific SIR threshold χ is
the probability of error becomes Mχ
− χ
√ PM (χ) = (Pr[χi ≤ χ])M = e . (43)
Ep d̄
∞ −
 α
The probability that at least one branch exceeds χ is then
Pe = fαm (α)fβm (β)dαdβ. (38) Mχ
− χ
√ −R
PS (χ) = 1 − e . (44)
Ep d̄
R
Assume that we require at least one branch to exceed χ with a
After some manipulation, we get probability p (where 1 − p can be interpreted as an outage probability).
Then, assuming the average SIR is equal to χ, the required number of
∞

M −1
− Ep d¯ π antennas is
Pe = sin +   
πσ 2 Rx 2 χ 1
√ M= log (45)
Ep d̄
R
χ 1−p
 s x 2 2  s
−1 − s +x where log(x) is the natural logarithm of x, and . is the ceil operator.
· xQM
1 , e 2 2σ I0 x dx. (39)
σ σ σ2 Equivalently, with a fixed M , the required χ would be
 
Numerical evaluation of (39) shows that Pe drops with increasing M . χ 1
However, the diversity gains are less substantial as compared to the χ≥ log . (46)
M 1−p
Rayleigh fading case. Moreover, the gain deteriorates for increasing
power ratio K = s2 /2σ 2 . Note that the required χ̄ drops directly with increasing M .
2370 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

B. Ricean Fading
Assume that the NBI goes through Ricean fading. The energy I =
|αj |2 is now distributed following a noncentral Chi-square distribution
with two degrees of freedom:

1 − s2 +x
√ s 
fI (x) = e 2σ 2 I0 x 2 . (47)
2σ 2 σ

Again, let χ = S/I be the SIR. Then, using the same method as for
Rayleigh fading, we get
2 S̄

S̄ 1 − s +2χ S̄ s
fχ (χ) = e 2σ I0 . (48)
2σ 2 χ2 χ σ2

The cdf may then be written as

χ 2 S̄

S̄ 1 − s +2u S̄ s
Fχ (χ) = e 2σ I0 du. (49)
u2 2σ 2 u σ2
0
Fig. 5. SD under NBI Rayleigh fading. Perfect energy capture. One, two, four,
Let v = S̄/u. A basic change of variable gives and eight antennas.

∞ √ s  The proof is included in Appendix B. The above observation is

1 − s2 +v
Fχ (χ) = 2
e 2σ2 I0 v 2 dv. (50) intuitive. Indeed, as K grows, Ricean fading is dominated by the
2σ σ
S̄ deterministic nonvariable LOS component. For K → ∞, the NBI
χ
power is constant across antennas (χ = χ̄), and all diversity gains
Using the well-known formula for the cdf of a Ricean distribution, vanish.
we get Simulation results will show that, although performance im-
  proves with increasing M , the gains are less substantial than the

S̄ Rayleigh case.
s
Fχ (χ) = Q1  , .
χ
(51)
σ σ
VI. S IMULATION R ESULTS

Following the same analysis for Ricean fading and based on (51), the A simulation of the proposed system using UWB 2-PAM modu-
probability that at least one branch exceeds χ is lation was carried out based on actual NLOS channel measurements
recorded at the Mobile and Portable Radio Research Group. For
   M
S̄ a thorough analysis of the channel characterization, the reader is
s
PS (χ) = 1 − Q1  ,  .
χ referred to [25]. Measurements are grouped in separate position sets of
(52)
σ σ 49 channel impulse responses. Each position set holds measured
channel impulse responses at 49 uniformly distributed points in a
Assume that we require at least one branch to exceed χ with a 1-m2 local area. In the simulation, M channel impulse responses
probability p. Then, after some manipulation, we get from the same position set are selected randomly and assigned to
the M receive antennas. The simulation is averaged over a large
 
  number of channel realizations by repeating the procedure over mul-
 log 1
 tiple position sets. NBI is modeled by a 1-GHz tone (which is near
M =
 
1−p
   . (53) the middle of the UWB band) and is assumed to undergo indepen-
 log Q−1 S̄

 
χ
1
s
, dent Rayleigh or Ricean fading over M receive antennas. Power
σ σ
measurements are averaged over multiple bits such that (11) holds.
A 500-ps UWB Gaussian transmit pulse is used. The symbol duration
Note that I¯ = s2 + 2σ 2 . Also, recall that K = s2 /2σ 2 . Then, after Tf is equal to 80 ns, which is longer than the channel maximum
straightforward manipulation, (53) may be written as delay spread.
    Fig. 5 displays the performance of SD versus signal-to-interference-
1
log 1−p plus-noise ratio (SINR) under Rayleigh fading when perfect channel
M =
   √   
. (54)
knowledge and full energy capture are available (i.e., a perfect Rake
 log Q−1
1
χ̄
2K, 2(K + 1) χ 
receiver that captures all the available multipath components). Here,
SINR is defined as the SINR at the output of the correlator. As
If K = 0, it is easy to show that (54) degenerates to (45), which expected, performance matches the theoretical expression of (32).
corresponds to the Rayleigh fading case. Moreover, for K → ∞, we Doubling the number of antennas yields a 3-dB gain.
distinguish the following two cases.
√
The performance of the proposed SD system applied to a 20-finger
1) If χ < χ̄, then limK→∞ Q1 ( 2K, 2(K + 1)χ̄/χ) → 0, and Rake receiver employing MRC combining is tested for NBI Rayleigh
M = 1 satisfies (54). √
fading in Fig. 6. The performance is plotted for simulation results,
2) If χ > χ̄, then limK→∞ Q1 ( 2K, 2(K + 1)χ̄/χ) → 1, and theoretical results assuming a Laplacian distribution for the desired
M > ∞ (in other words, no solution exists). signal component (31), and theoretical results assuming constant
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007 2371

Fig. 6. SD under NBI Rayleigh fading. Twenty-finger Rake. One, two, four, Fig. 7. SD performance under NBI Ricean fading. Perfect energy capture.
and eight antennas. K = 5.

energy capture (32). Notice that, at low SINR values, diversity gains
are practically equal to 3 dB when doubling M , and that (31) and (32)
yield almost the same performance. The variation in energy capture
across antennas is negligible compared to the variation of interfer-
ence energy, and gains are similar to the full energy-capture case.
However, for high SIR values, the energy-capture variation impacts
performance, and the simulated probability of error is closer to (31).
At high SINR, the performance is more heavily influenced by the
fluctuation in UWB energy. Note that since SIR is measured at the
output of the correlator, the average Rake energy capture d¯F is
incorporated into SIR. In other words, we are interested in the variance
of the energy over a local area and not in its mean value.9 Also, it is
important to note the distinction between the SD process applied at
the power selection level and the MRC process that occurs at the Rake
finger combination level; the two processes are independent and must
not be confused. Finally, note that similar diversity gains are observed
for a Rake receiver with a smaller number of fingers (ten fingers or
five fingers).
Performance for Ricean fading (K = 5) under SD and complete
energy capture is displayed in Fig. 7. It is assumed that performance is Fig. 8. SD performance under NBI Ricean fading for varying K. M = 4.
limited by high-power NBI, and Gaussian noise is neglected. Notice Perfect energy capture.
that the simulation results match the theoretical expression in (39).
Moreover, for a specific number of antennas M , Ricean fading yields a
note that we are more concerned with performance at low SIR,
higher probability of error compared to Rayleigh fading. Less diversity
since practical UWB systems will normally operate in the low SIR
gains are observed when increasing the number of antennas (1–2 dB
region.
when doubling M ). This is an intuitive result, since the NBI power
The performance of a 20-finger Rake receiver under Ricean fading
does not vary much across antennas compared to the Rayleigh case
(K = 5) for high-power NBI is shown in Fig. 9. Notice that the
because of the dominant deterministic LOS power component, and
diversity gains are limited compared to the Rayleigh fading case
thus, fewer diversity gains are achievable.
(around 1 dB when M is doubled) because of the dominating LOS
The effect of increasing the power in the Ricean LOS component
interference component.
under full energy capture is illustrated in Fig. 8. Performance degrades
The proposed system applied to a 20-finger Rake receiver is com-
with increasing K at low SIR because of the diminishing diversity
pared to a 20-tap MMSE receiver in Fig. 10. The MMSE receiver is the
gains. At high SIR, the dominant LOS component becomes negligible
most popular NBI mitigation technique in the literature. The MMSE
compared to the UWB power, and the performance of the receiver
receiver has the structure of a Rake receiver, but the individual finger
is better with Ricean fading compared to Rayleigh fading (K = 0).
weights (or filter taps) are computed to minimize the mean square error
As K → ∞, the probability of error curve tends to a step function,
instead of maximizing the energy capture [16]. In this simulation, it
where Pe = 0.5 at SIR < 0 dB, and Pe = 0 at SIR > 0 dB. However,
was assumed that the MMSE receiver has complete knowledge of the
NBI’s center frequency and the NBI average received power, which
9 The Rake average energy capture is an important issue for a practical are not needed for our proposed system. The SIR is fixed at −10 dB,
system, but here we are only interested in the diversity gains achievable by and performance is studied for varying signal-to-noise ratio. Notice
our system. that the MMSE receiver outperforms the proposed receiver by about
2372 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

Fig. 9. SD performance under NBI Ricean fading. Twenty-finger Rake. Fig. 11. PS (χ) (probability that at least one antenna SIR exceeds χ) versus
K = 5. χ for different number of antennas. Rayleigh fading. Average SIR = 5 dB.

Fig. 10. Proposed system versus MMSE receiver. Twenty-finger Rake. Fig. 12. PS (χ) versus χ for different number of antennas. Ricean fading.
Rayleigh fading. K = 5. Average SIR = 5 dB.

2 dB when M = 8. With M = 16, the proposed receiver performs

practically the same as the MMSE receiver. Note that the MMSE
receiver is substantially more complex than the proposed receiver (it
requires the inversion of a 20 × 20 covariance matrix) and simply
provides a performance benchmark in the context of this work.
The probability PS (χ) of at least one antenna SIR exceeding a
threshold χ [see (44)] is plotted versus χ in Fig. 11. Rayleigh fading is
assumed with an average antenna SIR of 5 dB. For a fixed PS (χ), the
threshold χ increases by 3 dB when the number of antennas is doubled.
PS (χ) is plotted for Ricean fading (with the same average antenna
SIR) in Fig. 12. The power ratio K is equal to 5. First, note that for
a fixed χ, Ricean fading yields a lower (worse) PS (χ) compared to
Rayleigh fading. Second, the gains seen from increasing M are less
substantial (about 1 dB when doubling the number of antennas). Due
to the deterministic Ricean LOS power component, the SIR varies less
across different antennas, and fewer diversity gains are possible. This
is illustrated in Fig. 13, where performance degrades for larger K. As
K grows, the interference power across antennas becomes constant,
and all diversity gains vanish. Fig. 13. PS (χ) versus χ for different K. Average SIR = 5 dB. M = 4.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007 2373

VII. C ONCLUSION Moreover

This paper presented a novel NBI mitigation scheme for UWB   √

d¯F
2d̄F
based on multiple receive antennas, where UWB’s low spatial fading 1 − 1 1
FZ (z) = Q − e σd
F +

characteristics were exploited to minimize the effect of NBI. NBI SD zσT 4 2 2πσ 2
T
was studied under Rayleigh and Ricean fading. The probabilities of  
d̄F
error expressions were derived for both perfect (full) energy capture z √ ∞ √
and partial energy capture. Diversity was shown to yield about 3-dB  − 2σ2
y2
2(zy− d¯F )
y2
− 2 2(d¯F − zy) 
·
 e
T + dy − e 2σT + dy
.
performance gain when the number of antennas is doubled for the σdF σdF
Rayleigh fading case. Gains are less substantial for the Ricean fading 0 d̄F
z
case, because Ricean NBI fading provides less variation to exploit
through diversity but can still provide some benefit.
We thus get the following expression:

A PPENDIX A   √ z2 σ2
T
√
2d̄
− σ F
d¯F
2d̄F
1 − 1 σ2 d
The cdf of Z is given by FZ (z) = Q − e σd
F + e dF F

zσT 4 2
∞ zy √
y2
!  √   √ "
1 1 −
2|x−d̄F | − − 2zσT d¯F 2zσT
FZ (z) = √
e σd
F e 2σ 2
T dxdy. · Q −Q −
σdF 2 2πσT2 σdF zσT σdF
y=0 x=0
z2 σ2
T
√
2d̄
+ σ F
 √ 
1 σ2 d d¯F 2zσT
Then, we may write − e dF F
Q + .
2 zσT σdF
 d̄
z y2 zy √

1 1  − 2(x−d̄)
Recall that the probability of error is equal to FZ (1/ Ep ). Then
FZ (z) = √
 e 2σ 2
T e σd
dxdy
2σd 2πσT2
y=0 x=0 √
√ σ2 2d̄
T − σ F
Ep d¯2F
2d̄F
 Pe = Q
1 −
− e σd
F
1 Ep σd2
+ e F
d F
∞ y2 d¯ √ ∞ y2 zy √ σT2 4 2
−
2σ 2
2(x−d̄) −
2σ 2 −
2(x−d̄)

+ e T e σd
dxdy + e T e σd
dxdy 
√
√
x=0 x=d¯ − 2σT Ep d¯F 2σT
y= d̄
z
y= d̄
z . Q
−Q −

Ep σd2F σT Ep σd2F
which is equivalent to √


σ2 2d̄ √
Ep d¯F
T + σ F
 1 Ep σd2 2σT
+

d
d̄F
− e F
F
Q .
 z ! √ √ " 2 σT Ep σd2F
1 1  y2
− 2 2(zy−d̄F ) 2d̄
− σ F
FZ (z) =
 e 2σ
e σd
−e dF
dy
2 2πσ 2 
T F

T
y=0
A PPENDIX B
 ∞
y2 ! √
2d̄
" Recall that the required number of antennas for the Ricean fading
− − σ F
2σ 2
+ e T 1−e dF
dy case was given in (54):
d̄
y= zF    
1
 log 1−p
M =
   √   
.
∞ ! √ "  log Q−1 χ̄

y2
− 2 2(d̄F −zy)
 1 2K, 2(K + 1) χ
+ e 2σ
T 1−e σd
F dy 
.
d̄
y= zF Let us define D as
  
Reordering elements, we get √ χ̄
D = Q1 2K, 2(K + 1) .
χ

d̄F
z √
∞
1 1  −
y2
+
2(zy−d̄F )
−
y2

FZ (z) =
 e 2σ 2 σd
F dy + 2 e 2σ 2
dy Let Z = χ̄/χ. Then, D may be written as
2 2πσ 2 
T T

T
0 d̄F √ √
z D = Q1 ( 2K, 2ZK + 2Z).

√ ∞ ∞ √ √ √
2d̄
− σ F
y2
− 2
y2
− 2 +
2(d̄F −zy)
 Suppose Z > 1. Then, 2K < 2ZK + 2Z. Then, based on [33]
dy 
σd
−e d F e2σ
T dy − e 2σ
T F
.
√ √
0 d̄F ( 2ZK+2Z− 2K)2
z D ≤ e− 2 .
2374 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 4, JULY 2007

It is trivial to show that [12] W. Tao, W. Yong, and C. Kangsheng, “Analyzing the interference power
of narrowband jamming signal on UWB system,” in Proc. PIMRC, 2003,
√ √
( 2ZK + 2Z − 2K)2 √ vol. 1, pp. 612–615.
lim = lim K( Z − 1)2 = ∞. [13] Z. Luo, H. Gao, Y. Liu, and J. Gao, “A new UWB pulse design
K→∞ 2 K→∞ method for narrowband interference suppression,” in Proc. GLOBECOM,
Nov. 29, 2004, vol. 6, pp. 3488–3492.
Then, since D ≥ 0, limK→∞ D = 0, and consequently [14] X. Chu and R. D. Murch, “The effect of NBI on UWB time-hopping
systems,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1431–1436,
1
  Sep. 2004.
log 1−p [15] I. Bergel, E. Fishler, and H. Messer, “Narrowband interference sup-
lim  √   = 0. pression in time-hopping impulse-radio systems,” in Proc. UWBST,
log Q−1
K→∞ χ̄
1 2K, 2(K + 1) χ May 2002, pp. 303–307.
[16] N. Boubaker and K. B. Letaief, “A low complexity MMSE-RAKE re-
ceiver in a realistic UWB channel and in the presence of NBI,” in Proc.
Thus, if Z > 1, M = 1 satisfies (54) as K → ∞. WCNC, Mar. 2003, vol. 1, pp. 233–237.
√ Z <√
Now, suppose 1. Then, there exists a value K0 such that, [17] E. Baccarelli, M. Biagi, and L. Taglione, “A novel approach to in-band in-
for K > K0 , 2K > 2ZK + 2Z. Then, again based on [33], for terference mitigation in ultra wideband radio systems,” in Proc. UWBST,
K > K0 May 2002, pp. 297–301.
[18] C. R. C. M. da Silva and L. B. Milstein, “Spectral-encoded UWB
! √ √ √ √ " communication systems: Real-time implementation and interference sup-
1 −( 2K− 2ZK+2Z)2 ( 2K+ 2ZK+2Z)2 pression,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1391–1401,
−
1− e 2 −e 2 ≤ D. Aug. 2005.
2
[19] B. Donlan, “Ultra-wideband narrowband interference cancellation and
channel modeling for communications,” M.S. thesis, Dept. Elect.
It is trivial to show that Comput. Eng., Virginia Tech, Blacksburg, VA, 2005.
[20] D. R. McKinstry and R. M. Buehrer, “LMS analog and digital narrowband
! √ √ √ √ " rejection system for UWB communications,” in Proc. UWBST, Nov. 2003,
1 −( 2K− 2ZK+2Z)2 ( 2K+ 2ZK+2Z)2
lim 1 − e 2 − e− 2 = 1. pp. 91–95.
K→∞ 2 [21] V. Bharadwaj and R. M. Buehrer, “An interference suppression scheme
for UWB signals using multiple receive antennas,” IEEE Commun. Lett.,
Then, since D ≤ 1, limK→∞ D = 1, and consequently vol. 9, no. 6, pp. 529–531, Jun. 2005.
[22] J. Ibrahim and R. M. Buehrer, “A novel NBI suppression scheme for
1
  UWB communications using multiple receive antennas,” in Proc. Radio
log 1−p Wireless Symp., Jan. 2006, pp. 507–510.
lim  √   = ∞. [23] V. Bharadwaj, “Ultra-wideband for communications: Spatial characteris-
log Q−1
K→∞ χ̄
1 2K, 2(K + 1) χ tics and interference suppression,” M.S. thesis, Dept. Elect. Comput. Eng.,
Virginia Tech, Blacksburg, VA, 2005.
[24] A. H. Muqaibel, “Characterization of ultra wideband communication
Thus, if Z < 1, no value of M satisfies (54) as K → ∞. channels,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Virginia Tech,
Blacksburg, VA, 2003.
R EFERENCES [25] R. M. Buehrer, A. Safaai-Jazi, W. A. Davis, and D. Sweeney, “Ultra-
wideband propagation measurements and modeling,” Elect. Comput.
[1] “Revision of Part 15 of the Commission’s Rules Regarding Ultra- Eng., Virginia Polytechnic Inst. State Univ., Blacksburg, VA, 2004. Final
Wideband Transmission Systems,” First note and Order, Federal Com- Rep. DAPRA NETEX Program. [Online]. Available: http://www.mprg.
munications Commission. ET-Docket 98-153, Adopted Feb. 14, 2002, org/people/buehrer/ultra/darpa-netex.shtml
released Apr. 22, 2002. [26] M. Z. Win and R. A. Scholtz, “On the robustness of ultra-wide bandwidth
[2] J. H. Reed, Ed., An Introduction to Ultra Wideband Communication signals in dense multipath environments,” IEEE Commun. Lett., vol. 2,
Systems. Englewood Cliffs, NJ: Prentice-Hall, 2005. no. 2, pp. 51–53, Feb. 1998.
[3] J. D. Choi and W. E. Stark, “Performance analysis of ultra-wideband [27] M. Z. Win and R. A. Scholtz, “Characterization of ultra-wide bandwidth
spread-spectrum communications in narrowband interference,” in Proc. wireless indoor channels: A communication—Theoretic view,” IEEE J.
MILCOM, Oct. 2002, vol. 2, pp. 1075–1080. Sel. Areas Commun., vol. 20, no. 9, pp. 1613–1627, Dec. 2002.
[4] L. Zhao, A. M. Haimovich, and H. Grebel, “Performance of ultra- [28] A. F. Molisch, Time variance for UWB wireless channels, Nov. 2002.
wideband communications in the presence of interference,” IEEE J. Sel. IEEE P802.15-02/461r0.
Areas Commun., vol. 20, no. 9, pp. 1684–1691, Dec. 2002. [29] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 2001.
[5] L. Piazzo and F. Ameli, “Performance analysis for impulse radio and [30] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic
direct-sequence impulse radio in narrowband interference,” IEEE Trans. Processes. New York: McGraw-Hill, 2002.
Commun., vol. 53, no. 9, pp. 1571–1580, Sep. 2005. [31] M. K. Simon, J. Omura, R. Scholtz, and K. Levitt, Spread Spectrum
[6] J. R. Foerster, “The performance of a direct-sequence spread ultrawide- Communications. Rockville, MD: Computer Science, 1985.
band system in the presence of multipath, narrowband interference, and [32] C. Rose and M. D. Smith, Order Statistics. New York: Springer-Verlag,
multiuser interference,” in Proc. UWST, 2002, pp. 87–91. 2002.
[7] L. Yang and G. B. Giannakis, “Unification of ultra-wideband multiple [33] M. K. Simon and M. S. Alouini, “Exponential-type bounds on the gen-
access schemes and comparison in the presence of interference,” in Proc. eralized Marcum Q-function with application to error probability analysis
Asilomar Conf., 2003, vol. 2, pp. 1239–1243. over fading channels,” IEEE Trans. Commun., vol. 48, no. 3, pp. 359–366,
[8] L. Yang and G. B. Giannakis, “A general model and SINR analysis of Mar. 2000.
low duty-cycle UWB access through multipath with narrowband interfer-
ence and rake reception,” IEEE Trans. Wireless Commun., vol. 4, no. 4,
pp. 1818–1833, Jul. 2005.
[9] M. Hamalainen, V. Hovinen, R. Tesi, J. Iinatti, and M. Latva-aho,
“On the UWB system coexistence with GSM900, UMTS/WCDMA, and
GPS,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1712–1721,
Dec. 2002.
[10] M. S. Iacobucci, M. G. Di Benedetto, and L. De Nardis, “Radio frequency
interference issues in impulse radio multiple access communication
systems,” in Proc. UWBST, 2002, pp. 293–296.
[11] A. Taha and K. M. Chugg, “A theoretical study on the effects of interfer-
ence UWB multiple access impulse radio,” in Proc. Asilomar Conf., 2002,
vol. 1, pp. 728–732.