4386 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009

Performance of BICM–OFDM Systems in UWB InterferenceAmir Nasri, Member, IEEE, Robert Schober, Senior Member, IEEE, and Lutz Lampe, Senior Member, IEEE

Abstract—In this paper, we develop an analytical frame-work for performance analysis of generic bit–interleaved codedmodulation orthogonal frequency–division multiplexing (BICM–OFDM) systems impaired by ultra–wideband (UWB) interfer-ence. For practical relevance we consider multi–band OFDM(MB–OFDM), direct–sequence UWB (DS–UWB), and impulse–radio UWB (IR–UWB) interference formats following recentIEEE/ECMA standards or standard proposals. Besides the exactanalysis we calculate the bit error rate (BER) for the case whenthe UWB interference is modeled as additional Gaussian noise.Our results show that in general the BER of the BICM–OFDMsystem strongly depends on the UWB format and the OFDMsub–carrier spacing. While the Gaussian approximation is veryaccurate for DS–UWB, it may severely over- or underestimatethe true BER for MB–OFDM and IR–UWB interference. Ouranalysis is applicable to e.g. IEEE 802.11 wireless local area net-works (WLANs), IEEE 802.16 wireless access systems (WiMAX),and 4th generation mobile communication systems. Furthermore,since the ECMA MB–OFDM standard is also based on theBICM–OFDM concept, our analysis can also be used to evaluatethe impact of other UWB signals on ECMA MB–OFDM UWBsystems.

Index Terms—Bit-interleaved coded modulation (BICM),orthogonal frequency division multiplexing (OFDM), ultra-wideband (UWB), no–Gaussian interference, multi–band OFDM(MB-OFDM), direct-sequence UWB (DS-UWB), impulse-radioUWB (IR-UWB).

I. INTRODUCTION

IN recent years, ultra–wideband (UWB) has emerged asan important technology for short–range high data rate

wireless communication and soon UWB devices will be ubiq-uitous. Since UWB systems will operate as spectral underlaysystems, the effect of UWB on existing narrowband andwideband systems occupying the same frequency resourceshas been studied extensively in the past few years, cf. e.g. [1]–[5], and references therein.

In this paper, we provide an analytical framework thatallows us to accurately calculate the bit error rate (BER)of a generic bit–interleaved coded modulation orthogonalfrequency–division multiplexing (BICM–OFDM) system im-paired by UWB interference and additive white Gaussian noise(AWGN). BICM–OFDM has become the most important airinterface in wireless communications and will likely also beadopted by most future wireless communication standards.For practical relevance, we consider interference from multi–band (MB) OFDM, direct–sequence (DS) UWB, and impulse–

Manuscript received October 12, 2007; revised February 25, 2008 and April10, 2008; accepted April 15, 2008. The associate editor coordinating thereview of this letter and approving it for publication was R. Mallik.

The authors are with the Department of Electrical and Computer Engi-neering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada(e-mail: {amirn, rschober, lampe}@ece.ubc.ca).

This work was supported in part by an NSERC Strategic Project Grant(STPGP 350451).

This work has been accepted for presentation in part at the IEEE Interna-tional Conference on Communications (ICC), Bejing, 2008.

Digital Object Identifier 10.1109/TWC.2009.071125

radio (IR) UWB following the ECMA standard [6], the IEEE802.15.3a standard proposal [7], and the IEEE 802.15.4astandard [8], respectively. We also derive a simple Gaussianapproximation (GA) for the exact BER which is easy to com-pute. We note that related previous work was either simulationbased, cf. e.g. [2]–[4], or considered only uncoded single–carrier victim systems, cf. e.g. [1], [5], [9] and referencestherein.

The remainder of this paper is organized as follows. InSection II, the system model is presented, and in Section III themoment generating functions (MGFs) of the considered UWBformats are provided. The performance of BICM–OFDM inUWB interference is analyzed in Section IV. In Section V,this performance analysis is used to study the impact of theconsidered UWB formats on BICM–OFDM, and conclusionsare drawn in Section VI.

Notation: In this paper, [⋅]∗, [⋅]𝑇 , ℰ{⋅}, ℜ{⋅}, ℑ{⋅}, and ⊗denote complex conjugation, transposition, statistical expecta-tion, the real part of a complex number, the imaginary part ofa complex number, and convolution, respectively. In addition,an 𝒩 (𝜇, 𝜎2) distributed random variable (RV) is a GaussianRV with mean 𝜇 and variance 𝜎2.

II. SYSTEM MODEL

In this section, we present the mathematical models for allcomponents of the system. We note that for convenience allsignals and systems are represented by their complex basebandequivalents.

A. BICM–OFDM System

We assume that the victim system employs the popularBICM–OFDM concept, e.g. [10], [11]. Therefore, coding isperformed along the frequency axis over the 𝑁𝑠 sub–carriersof a single OFDM symbol using the concatenation of aconvolutional encoder of rate 𝑅𝑐, an interleaver, and a mem-oryless mapper. In particular, the elements of the codeword𝒄 ≜ [𝑐1, 𝑐2, . . . , 𝑐𝑚𝑁𝑠 ] are interleaved, and the interleaved bitsare broken up into sub–sequences of 𝑚 bits each, which aresubsequently mapped to symbols 𝑥[𝑘] from a constellation𝒳 of size ∣𝒳 ∣ ≜ 𝑀 = 2𝑚 to form the transmit sequence𝒙 ≜ [𝑥[−𝑁𝑠/2], 𝑥[−𝑁𝑠/2+1], . . . , 𝑥[𝑁𝑠/2−1]] (𝑁𝑠 is even).Subsequently, the transmit symbols 𝑥[𝑘] are modulated ontothe 𝑁𝑠 OFDM sub–carriers resulting in the baseband transmitsignal

𝑠(𝑡) =

𝑁𝑠2 −1∑

𝑘=−𝑁𝑠2

𝑥[𝑘]𝜙𝑘(𝑡), (1)

where 𝜙𝑘(𝑡) ≜ 𝑒𝑗2𝜋Δ𝑓𝑠𝑘𝑡 𝑤𝑠(𝑡) is the 𝑘th OFDM sub–carrierwaveform with sub–carrier spacing Δ𝑓𝑠. Here, 𝑤𝑠(𝑡) is arectangular pulse with 𝑤𝑠(𝑡) =

√Δ𝑓𝑠, −𝑇cp ≤ 𝑡 ≤ 1/Δ𝑓𝑠,

and 𝑤𝑠(𝑡) = 0 otherwise, where 𝑇cp denotes the length

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009 4387

of the cyclic prefix. The duration of one OFDM symbol is𝑇𝑠 = 𝑇cp + 1/Δ𝑓𝑠.𝑠(𝑡) is transmitted over a multipath channel with an impulse

response ℎ(𝑡) which is zero outside the interval [0, 𝑇cp]. Thereceived signal 𝑟(𝑡) is impaired by AWGN 𝑛(𝑡) and 𝐼 UWBinterferers 𝑖𝜈(𝑡), 1 ≤ 𝜈 ≤ 𝐼 . Consequently, the received signalcan be modeled as

𝑟(𝑡) = 𝑠(𝑡)⊗ℎ(𝑡)+𝑛(𝑡)+𝐼∑

𝜈=1

𝑔𝜈(𝑡)⊗ 𝑒𝑗2𝜋𝑓𝜈 (𝑡−𝜏𝜈)𝑖𝜈(𝑡− 𝜏𝜈),(2)

where 𝑔𝜈(𝑡), 𝜏𝜈 , and 𝑓𝜈 are the (causal) channel impulseresponse (CIR) of length 𝑇𝑔 , the delay, and the frequencyoffset of the 𝜈th UWB signal, respectively. Both 𝜏𝜈 and 𝑓𝜈are defined relative to the victim signal. The typical OFDMreceiver processing involving low–pass filtering, sampling, anddiscrete–time Fourier transform (DFT) can be equivalentlyrepresented by filtering of the received signal with a bankof matched filters 𝜓𝑘(𝑡) = 𝜙∗𝑘(−𝑡), −1/Δ𝑓𝑠 ≤ 𝑡 ≤ 0,−𝑁𝑠/2 ≤ 𝑘 ≤ 𝑁𝑠/2 − 1, and 𝜓𝑘(𝑡) = 0 otherwise,and subsequent sampling. Assuming coherent reception, thesampled output of the 𝑘th matched filter is obtained as

𝑟[𝑘] ≜ 𝑒−𝑗Θ𝐻 [𝑘] 𝑟(𝑡)⊗𝜓𝑘(𝑡)∣𝑡=0 = 𝛼[𝑘]𝑥[𝑘]+𝑛[𝑘]+𝑖[𝑘], (3)

for −𝑁𝑠

2 ≤ 𝑘 ≤ 𝑁𝑠

2 − 1, where 𝑖[𝑘] is the (effective) UWBinterference, 𝐻 [𝑘] ≜

∫ 𝑇cp

0 ℎ(𝑡)𝑒−𝑗2𝜋Δ𝑓𝑠𝑘𝑡 d𝑡 = 𝛼[𝑘]𝑒𝑗Θ𝐻 [𝑘]

is the gain of the 𝑘th sub–carrier with magnitude 𝛼[𝑘] andphase Θ𝐻 [𝑘], and 𝑛[𝑘] is AWGN. The UWB interference isgiven by

𝑖[𝑘] = 𝑒−𝑗Θ𝐻 [𝑘]𝜓𝑘(𝑡)⊗𝐼∑

𝜈=1

𝑔𝜈(𝑡)⊗ 𝑒𝑗2𝜋𝑓𝜈 (𝑡−𝜏𝜈)𝑖𝜈(𝑡− 𝜏𝜈)∣𝑡=0

≈𝐼∑

𝜈=1

𝑔𝜈 [𝑘]𝑖𝜈 [𝑘] (4)

with 𝑔𝜈[𝑘] = 𝑒−𝑗Θ𝐻 [𝑘] 𝑒−𝑗2𝜋𝑓𝜈𝜏𝜈 𝐺𝜈(𝑘Δ𝑓𝑠) and 𝑖𝜈 [𝑘] =𝜓𝑘(𝑡) ⊗ 𝑒𝑗2𝜋𝑓𝜈𝑡 𝑖𝜈(𝑡 − 𝜏𝜈)∣𝑡=0, where we have assumed thatthe frequency response of the UWB channel 𝐺𝜈(𝑓) ≜∫ 𝑇𝑔

0𝑔𝜈(𝑡) 𝑒

−𝑗2𝜋𝑓𝑡 d𝑡 is approximately constant over the rangeof frequencies where 𝜓𝑘(𝑡) has significant energy. Throughoutthis paper we assume that the 𝑖𝜈 [𝑘], 𝑔𝜈 [𝑘], and 𝜏𝜈 of differentUWB interferers 𝜈 are independent RVs. Furthermore, ascustomary in the literature [11], in this paper, we assumethat the 𝐻 [𝑘] and the 𝛼[𝑘] are independent and identicallydistributed (i.i.d.) 𝒩 (0, 1) and Rayleigh RVs, respectively,where the independence is due to the bit–interleaving whichis assumed to be ideal. The effect of non–i.i.d. 𝐻 [𝑘] and 𝛼[𝑘]will be studied in Section V via simulations. The delays 𝜏𝜈are modeled as uniformly distributed in [0, 𝑇𝑠). For the UWBinterference channel we consider two different models:

C1) In the first case, we model 𝑔𝜈 [𝑘] as i.i.d. 𝒩 (0, 𝜎2𝑔,𝜈)

RVs (Rayleigh fading), where 𝜎2𝑔,𝜈 ≜ ℰ{∣𝑔𝜈 [𝑘]∣2}. We notethat different UWB interferers may have different variances𝜎2𝑔,𝜈 due to e.g. different distances from the victim receiver.The assumption of i.i.d. Rayleigh fading along the frequencyaxis is a good model for typical UWB channels, cf. [12].

C2) In the second case, we assume that the UWB channel isfrequency flat and constant. Hence, the phase of 𝐻 [𝑘] rendersthe 𝑔𝜈 [𝑘] i.i.d. RVs that can be modeled as 𝑔𝜈 [𝑘] = 𝑔𝜈 𝑒𝑗Θ𝜈 [𝑘],

where 𝑔𝜈 ≜ ∣𝑔𝜈 [𝑘]∣ is constant and Θ𝜈 [𝑘] is an i.i.d. RV whichis uniformly distributed in [−𝜋, 𝜋). Assuming a frequency flatinterference channel is popular in the literature, e.g. [13], as itallows an unobscured view of the degradation caused by theinterference signal.

B. UWB Signal Model

In this paper, we consider the following UWB signalmodels: MB–OFDM [6], DS–UWB [7], and IR–UWB [8]. Forsimplicity of notation, we drop the subscript 𝜈 in the UWBsignal 𝑖𝜈(𝑡) in the following if no confusion arises.

1) MB–OFDM Signal Model: The adopted MB–OFDMmodel closely follows the ECMA standard [6]. In particular,the MB–OFDM signal is modeled as

𝑖(𝑡) =1√𝑁𝑖

∞∑𝜅=−∞

𝑁𝑖/2∑𝑛=−𝑁𝑖/2

𝑛 ∕=0

𝑎𝑛[𝜅] 𝑒𝑗2𝜋𝑛Δ𝑓𝑖(𝑡−𝜅𝑇𝑖)

×𝑤𝑖(𝑡− 𝜅𝑇𝑖) e𝑗2𝜋𝑓MB [𝜅]𝑡, (5)

where 𝑁𝑖, Δ𝑓𝑖, 𝑇𝑖, and 𝑎𝑛[𝜅] ∈ {±1± 𝑗} denote the numberof sub–carriers, the sub–carrier spacing, the symbol duration,and the data symbols of the MB–OFDM signal, respectively.The MB–OFDM pulse shape 𝑤𝑖(𝑡) is given by 𝑤𝑖(𝑡) = 1,𝑇𝑃 ≤ 𝑡 ≤ 𝑇𝑖 − 𝑇𝐺, and 𝑤𝑖(𝑡) = 0 otherwise, where 𝑇𝑃 and𝑇𝐺 ≜ 𝑇𝑖− (𝑇𝑃 +1/Δ𝑓𝑖) denote the zero–prefix duration andthe guard interval, respectively. Moreover, in (5), 𝑓MB[𝜅] ∈{(𝑛𝑏 − 1)𝑁𝑖Δ𝑓𝑖∣𝑛𝑏 ∈ {1, 2, . . . , 𝑁𝐵}} is a periodic, time–dependent frequency offset used to switch between the 𝑁𝐵

MB–OFDM frequency bands.

2) DS–UWB Signal Model: For DS–UWB we closelyfollow the IEEE 802.15.3a standard proposal [7], where bothbinary phase–shift keying (BPSK) and 4–ary bi–orthogonalkeying (4–BOK) are considered for modulation. For 4–BOKsignaling the UWB interference signal is given by

𝑖(𝑡) =

∞∑𝜅=−∞

𝑎[𝜅] 𝑝𝑏[𝜅](𝑡− 𝜅𝑇𝑖), (6)

where 𝑎[𝜅] ∈ {±1} and 𝑏[𝜅] ∈ {0, 1} represent the 4–ary data symbol and 𝑇𝑖 is the symbol duration. Both 𝑎[𝜅]and 𝑏[𝜅] are i.i.d. sequences. The pulse shape is defined as𝑝𝑏(𝑡) ≜

∑𝐿−1𝑙=0 𝑐𝑙,𝑏 𝑝𝑐(𝑡 − 𝑙𝑇𝑐) with orthogonal spreading

sequences 𝑐𝑙,0 and 𝑐𝑙,1 of length 𝐿. 𝑝𝑐(𝑡) and 𝑇𝑐 denote thechip waveform and the chip duration, respectively. For DS–UWB with BPSK signaling (6) is still valid with 𝑏[𝜅] = 0,∀𝜅, and the simplified notation 𝑝0(𝑡) ≜ 𝑝(𝑡), and 𝑐𝑙,0 ≜ 𝑐𝑙 isadopted.

3) IR–UWB Signal Model: For IR–UWB we adopt thesignal model proposed by the IEEE 802.15.4a standardiza-tion committee [8] which employs a combination of BPSKand binary pulse position modulation (BPPM). An IR–UWBsymbol has duration 𝑇𝑖 and consist of 𝑁𝑏 burst of duration𝑇𝑏 = 𝑇𝑖/𝑁𝑏. Similarly, a burst consist of 𝐿 chips of duration𝑇𝑐 = 𝑇𝑏/𝐿. The IR–UWB signal can be modeled as

𝑖(𝑡) =

∞∑𝜅=−∞

𝐿−1∑𝑙=0

𝑎[𝜅] 𝑝[𝜅, 𝑙]

×𝑝𝑐(𝑡− 𝜅𝑇𝑖 − ℎ[𝜅]𝑇𝑏 − 𝑙𝑇𝑐 − 𝑏[𝜅]𝑇PPM), (7)

4388 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009

TABLE ICONDITIONAL MGFS Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) FOR CONSIDERED UWB FORMATS. FOR MB–OFDM 𝑛𝑙 , 𝑛𝑢 , 𝜅𝑙 , AND 𝜅𝑢 ARE CHOSEN SUCH THAT 𝛽𝑛[𝜅, 𝑘] ≈ 0FOR 𝑛 ∕∈ [𝑛𝑙, 𝑛𝑢] OR/AND 𝜅 ∕∈ [𝜅𝑙, 𝜅𝑢]. FOR DS–UWB AND IR–UWB THE LIMITS 𝜅𝑙 , 𝜅𝑢 ARE CHOSEN TO ENSURE 𝛽𝑙[𝜅, 𝑘] ≈ 0 AND 𝛽𝑙[𝜅, 𝑘, 𝜗[𝜅]] ≈ 0FOR 𝜅 ∕∈ [𝜅𝑙, 𝜅𝑢], RESPECTIVELY. THE INTEGRALS REQUIRED FOR CALCULATION OF 𝛽𝑙[𝜅, 𝑘] AND 𝛽𝑙[𝜅, 𝑘, 𝜗[𝜅]] FOR DS–UWB AND IR–UWB CAN BE

EASILY EVALUATED NUMERICALLY, RESPECTIVELY.

MB–OFDM

Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) ≃ ∏𝜅𝑢𝜅=𝜅𝑙

∏𝑛𝑢𝑛=𝑛𝑙

cosh (𝑠ℜ{𝑔[𝑘]𝛽𝑛[𝜅, 𝑘]}) cosh (𝑠ℑ{𝑔[𝑘]𝛽𝑛[𝜅, 𝑘]})𝛽𝑛[𝜅, 𝑘] =

√Δ𝑓𝑠

𝑗𝜉𝑛[𝑘]𝑒𝑗𝜉𝑛[𝑘](𝜏+𝜅𝑁𝐵𝑇𝑖)

(𝑒𝑗𝜉𝑛[𝑘]𝑙𝑢 − 𝑒𝑗𝜉𝑛[𝑘]𝑙𝑙

)

𝜉𝑛[𝑘] = 2𝜋(𝑓 −Δ𝑓𝑠𝑘 + 𝑛Δ𝑓𝑖)

𝑙𝑢 = min{1/Δ𝑓𝑠 − 𝜏 − 𝜅𝑁𝐵𝑇𝑖, 𝑇𝑖 − 𝑇𝐺}, 𝑙𝑙 = max{−𝜏 − 𝜅𝑁𝐵𝑇𝑖, 𝑇𝑃 }𝜅𝑙 = ⌈(𝑇𝐺 − 𝑇𝑖 − 𝜏)/(𝑁𝐵𝑇𝑖)⌉, 𝜅𝑢 = ⌊(1/Δ𝑓𝑠 − 𝑇𝑃 − 𝜏)/(𝑁𝐵𝑇𝑖)⌋

DS–UWB

Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏)≃∏𝜅𝑢

𝜅=𝜅𝑙cosh(𝑠ℜ{𝑔[𝑘]∑𝐿−1

𝑙=0 𝑐𝑙𝛽𝑙[𝜅, 𝑘]}) (BPSK)

Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) ≃ ∏𝜅𝑢𝜅=𝜅𝑙

12

∑1𝑏=0 cosh(𝑠ℜ{𝑔[𝑘]∑𝐿−1

𝑙=0 𝑐𝑙,𝑏𝛽𝑙[𝜅, 𝑘]}) (4–BOK)

𝛽𝑙[𝜅, 𝑘] =√Δ𝑓𝑠𝑒𝑗2𝜋(𝑓−Δ𝑓𝑠𝑘)(𝜏+𝜅𝑇𝑖+𝑙𝑇𝑐)

∫ 𝑙𝑢𝑙𝑙

𝑒𝑗2𝜋(𝑓−Δ𝑓𝑠𝑘)𝑡𝑝𝑐(𝑡) d𝑡

𝑙𝑢 = 1/Δ𝑓𝑠 − 𝜏 − 𝜅𝑇𝑖 − 𝑙𝑇𝑐, 𝑙𝑙 = −𝜏 − 𝜅𝑇𝑖 − 𝑙𝑇𝑐

IR–UWB

Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) ≃ ∏𝜅𝑢𝜅=−𝜅𝑙

12𝑁ℎ

∑𝑁ℎ−1ℎ=0

∑1𝑏=0

∏𝐿−1𝑙=0 cosh(𝑠ℜ{𝑔[𝑘]𝛽𝑙[𝜅, 𝑘, ℎ𝑇𝑏 + 𝑏𝑇PPM]})

𝛽𝑙[𝜅, 𝑘, 𝜗[𝜅]] =√Δ𝑓𝑠 𝑒𝑗2𝜋(𝑓−Δ𝑓𝑠𝑘)(𝜏+𝜅𝑇𝑖+𝑙𝑇𝑐+𝜗[𝜅])

∫ 𝑙𝑢𝑙𝑙

𝑒𝑗2𝜋(𝑓−Δ𝑓𝑠𝑘)𝑡𝑝𝑐(𝑡) d𝑡

𝑙𝑢 = 1/Δ𝑓𝑠 − 𝜏 − 𝜅𝑇𝑖 − 𝑙𝑇𝑐 − 𝜗[𝜅], 𝑙𝑙 = −𝜏 − 𝜅𝑇𝑖 − 𝑙𝑇𝑐 − 𝜗[𝜅]

where 𝑎[𝜅] ∈ {±1} and 𝑏[𝜅] ∈ {0, 1} denote the i.i.d. BPSKand BPPM data symbols, respectively. Furthermore, 𝑝𝑐(𝑡) isthe chip waveform and 𝑇PPM is the delay for BPPM. 𝑝[𝜅, 𝑙] ∈{±1} and ℎ[𝜅] ∈ {0, 1, . . . , 𝑁ℎ−1} are the i.i.d. scramplingsequence and the i.i.d. hopping sequence, respectively, where𝑁ℎ denotes the number of hopping positions.

C. MGF of MB–OFDM Interference

The MGF of MB–OFDM interference has already beenstudied for single–carrier victim systems in [5]. It is shown in[5] that, because of the hopping between different frequencybands, the repetition in time of the MB–OFDM data symbols𝑎𝑛[𝜅] for data rates of less than 320 Mb/s does not affectthe MGF. The conjugate symmetry of the MB–OFDM datasymbols for data rates of 55 and 80 Mb/s does have an impacton the MGF but the impact on the (uncoded) BER was foundto be minor [5]. Therefore, in this paper, we concentrate onMB–OFDM data rates of more than 80 Mb/s but note that allour results can be easily extended to lower data rates, cf. [5].Adjusting the derivations in [5] to the problem at hand yieldsthe conditional MGF given in Table I.

III. MGF OF UWB INTERFERENCE SIGNAL

For the error rate analysis presented in Section IVthe MGF Φ𝑖𝜈 [𝑘](𝑠∣𝑔𝜈 [𝑘], 𝜏𝜈) ≜ ℰ{𝑒−𝑠ℜ{𝑔𝜈 [𝑘]𝑖𝜈 [𝑘]}∣𝑔𝜈 [𝑘], 𝜏𝜈} of ℜ{𝑔𝜈[𝑘]𝑖𝜈 [𝑘]} conditioned on 𝑔𝜈 [𝑘] and𝜏𝜈 plays a major role. Therefore, in this section, we calculateΦ𝑖𝜈 (𝑠∣𝑔𝜈 [𝑘], 𝜏𝜈) for the considered UWB formats. For thispurpose, it is useful to note that 𝑖𝜈 [𝑘] in (4) can be simplifiedto

𝑖𝜈 [𝑘] =√Δ𝑓𝑠

1/Δ𝑓𝑠∫0

𝑒𝑗2𝜋(𝑓𝜈−Δ𝑓𝑠𝑘)𝑡𝑖𝜈(𝑡− 𝜏𝜈) d𝑡. (8)

For simplicity of notation we will drop in the remainder ofthis section the subscript 𝜈, 1 ≤ 𝜈 ≤ 𝐼 , in 𝑓𝜈 , 𝜏𝜈 , 𝑔𝜈 [𝑘], and𝑖𝜈(𝑡), respectively. For convenience, the results of this sectionare summarized in Table I.

A. MGF of DS–UWB Interference

Exploiting (6) and (8) we first rewrite 𝑖[𝑘] for BPSK DS–UWB as

𝑖[𝑘] =

∞∑𝜅=−∞

𝑎[𝜅]

𝐿−1∑𝑙=0

𝑐𝑙𝛽𝑙[𝜅, 𝑘], (9)

where 𝛽𝑙[𝜅, 𝑘] is defined in Table I. Based on (9) and the factthat the 𝑎[𝜅] ∈ {±1} are i.i.d., Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) can be obtainedas given in Table I. Further details can be found in [9].

Similarly, for 4–BOK the interference signal can be ex-pressed as

𝑖[𝑘] =

∞∑𝜅=−∞

𝑎[𝜅]

𝐿−1∑𝑙=0

𝑐𝑙,𝑏[𝜅]𝛽𝑙[𝜅, 𝑘], (10)

where 𝛽𝑙[𝜅, 𝑘] is identical to that of the BPSK case. Averagingthe MGF first over 𝑎[𝜅] we obtain

Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏, 𝑏[𝜅]) =∞∏

𝜅=−∞cosh

(𝑠ℜ{𝑔[𝑘]

𝐿−1∑𝑙=0

𝑐𝑙,𝑏[𝜅]𝛽𝑙[𝜅, 𝑘]

}). (11)

The final result for Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) given in Table I is obtainedby averaging Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏, 𝑏[𝜅]) in (11) with respect to 𝑏[𝜅]and noting that 𝑏[𝜅] ∈ {0, 1} is an i.i.d. RV. Eq. (11) isin the form of an infinite product. However, convergence ofthis product is not problematic since 𝛽𝑙[𝜅, 𝑘] is essentiallythe convolution of the UWB chip pulse shape 𝑝𝑐(𝑡) and the

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009 4389

OFDM sub–carrier matched filter 𝜓𝑘(𝑡). Thus, in practice,𝛽𝑙[𝜅, 𝑘] ≈ 0 holds for 𝜅 ∕∈ {𝜅𝑢, 𝜅𝑙}, where 𝜅𝑢 and 𝜅𝑙 aresuitably chosen limits. Therefore, the infinite product in (11)can be truncated to a finite one as shown in Table I.

B. MGF of IR–UWB Interference

For calculation of the MGF of IR–UWB interference it isconvenient to rewrite (7) as

𝑖(𝑡) =

∞∑𝜅=−∞

𝐿−1∑𝑙=0

�̃�[𝜅, 𝑙] 𝑝𝑐(𝑡− 𝜅𝑇𝑖 − 𝑙𝑇𝑐 − 𝜗[𝜅]), (12)

where �̃�[𝜅, 𝑙] ≜ 𝑎[𝜅]𝑝[𝜅, 𝑙] and 𝜗[𝜅] ≜ ℎ[𝜅]𝑇𝑏+𝑏[𝜅]𝑇PPM. Wenote that �̃�[𝜅, 𝑙] ∈ {±1} is i.i.d. and 𝜗[𝜅] is i.i.d. and has pdf

𝑝𝜗(𝜗) =1

2𝑁ℎ

𝑁ℎ−1∑ℎ=0

1∑𝑏=0

𝛿(𝜗− ℎ𝑇𝑏 − 𝑏 𝑇𝑃𝑃𝑀 ). (13)

From (8) we obtain for the effective discrete–time interfer-ence

𝑖[𝑘] =

∞∑𝜅=−∞

𝐿−1∑𝑙=0

�̃�[𝜅, 𝑙]𝛽𝑙[𝜅, 𝑘, 𝜗[𝜅]], (14)

with 𝛽𝑙[𝜅, 𝑘, 𝜗[𝜅]] as defined in Table I. Using now 𝑖[𝑘] from(14) and averaging over �̃�[𝜅, 𝑙], which is i.i.d. with respect toboth 𝜅 and 𝑙, we obtain

Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏, 𝜗[𝜅])=∞∏

𝜅=−∞

𝐿−1∏

𝑙=0

cosh (𝑠ℜ{𝑔[𝑘]𝛽𝑙[𝜅, 𝑘, 𝜗[𝜅]]}) . (15)

The final result for Φ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏) given in Table I isobtained by exploiting that 𝜗[𝜅] is an i.i.d. RV when averagingΦ𝑖[𝑘](𝑠∣𝑔[𝑘], 𝜏, 𝜗[𝜅]) with respect to 𝜗[𝜅]. Note that usingsimilar arguments as for DS–UWB, the infinite product in(15) can again truncated to a finite one, cf. Table I.

IV. BER OF BICM–OFDM SYSTEMS

In this section, we derive a tight approximation for theexact BER of a BICM–OFDM victim receiver as well as acorresponding simple and easy–to–evaluate GA.

A. Conditional MGF of Metric Difference

As customary, we assume that the BICM–OFDM systememploys Viterbi decoding with branch metric [10]

𝜆𝑖[𝑘] ≜ min𝑥[𝑘]∈𝒳 𝑖

𝑏

{∣𝑟[𝑘]− 𝛼[𝑘]𝑥[𝑘]∣2} (16)

for the 𝑖th bit of the 𝑘th sub–carrier. Here, 𝒳 𝑖𝑏 denotes the

subset of all symbols in constellation 𝒳 whose label has value𝑏 ∈ {0, 1} in position 𝑖. For BER calculation the MGF of themetric difference Δ(𝑥[𝑘], 𝑧[𝑘]) ≜ ∣𝑟[𝑘]− 𝛼[𝑘]𝑥[𝑘]∣2 − ∣𝑟[𝑘]−𝛼[𝑘]𝑧[𝑘]∣2 is of interest where 𝑥[𝑘] and 𝑧[𝑘] ∕= 𝑥[𝑘] denotethe transmitted symbol and another symbol in 𝒳 , respectively.Δ(𝑥[𝑘], 𝑧[𝑘]) can be simplified to

Δ(𝑥[𝑘], 𝑧[𝑘]) = 2𝛼[𝑘]𝑑𝑥𝑧[𝑘]ℜ{𝑒−𝑗Θ𝑑[𝑘](𝑖[𝑘] + 𝑛[𝑘])}−𝛼2[𝑘]𝑑2𝑥𝑧[𝑘] (17)

with 𝑑𝑥𝑧[𝑘]𝑒𝑗Θ𝑑[𝑘] ≜ 𝑥[𝑘] − 𝑧[𝑘], where 𝑑𝑥𝑧[𝑘] and

Θ𝑑[𝑘] denote the magnitude and phase, respectively.

Introducing the definitions 𝒈[𝑘] ≜ [𝑔1[𝑘] . . . 𝑔𝐼 [𝑘]] and𝝉 ≜ [𝜏1 . . . 𝜏𝐼 ], we can express the conditional MGFΦΔ(𝑥[𝑘],𝑧[𝑘])(𝑠∣𝒈[𝑘], 𝝉 , 𝛼[𝑘]) ≜ ℰ {𝑒−𝑠Δ(𝑥[𝑘],𝑧[𝑘])

}of

Δ(𝑥[𝑘], 𝑧[𝑘]) as

ΦΔ(𝑥[𝑘],𝑧[𝑘])(𝑠∣𝒈[𝑘], 𝝉 , 𝛼[𝑘]) = 𝑒𝛼2[𝑘]𝑑2𝑥𝑧[𝑘](1+𝑠𝜎

2𝑛)𝑠

×𝐼∏

𝜈=1

Φ𝑖𝜈 [𝑘](2𝛼[𝑘]𝑑𝑥𝑧[𝑘]𝑠∣𝑔𝜈 [𝑘], 𝜏𝜈),(18)

where we have exploited the rotational invariance of 𝑖[𝑘] and𝑛[𝑘], cf. Section II, the MGF Φ𝑛(𝑠) = ℰ{𝑒−𝑠𝑛[𝑘]} = 𝑒𝑠

2𝜎2𝑛 of

𝑛[𝑘], and the mutual independence of the UWB interferers.

B. Tight Approximation for Exact BER

The union bound for the BER of a convolutional code ofrate 𝑅𝑐 = 𝑘𝑐/𝑛𝑐 (𝑘𝑐 and 𝑛𝑐 are integers) is given by

𝑃𝑏 ≤ 1

𝑘𝑐

∞∑𝑑= 𝑑min

𝑤𝑐(𝑑)𝑃 (𝒄 → �̂�), (19)

where 𝒄 and �̂� are two distinct code sequences with Hammingdistance 𝑑 that differ only in 𝑙 ≥ 1 consecutive trellis states.Furthermore, 𝑤𝑐(𝑑) and 𝑑min denote the total input weightof error events at Hamming distance 𝑑 and the minimumHamming distance of the code, respectively. 𝑃 (𝒄 → �̂�) isthe pairwise error probability (PEP), i.e., the probability thatthe decoder erroneously chooses the code sequence �̂� whenthe code sequence 𝒄 is transmitted (assuming that 𝒄 and �̂� arethe only possible code sequences). It is convenient to expressthe PEP in terms of an MGF as [10]

𝑃 (𝒄 → �̂�) =1

2𝜋𝑗

𝑐+𝑗∞∫𝑐−𝑗∞

ℰ𝝉{Ψ(𝑠∣𝝉 )} d𝑠

𝑠, (20)

where 𝑐 is a small positive constant that lies in the region ofconvergence of the integral. The MGF Ψ(𝑠∣𝝉 ) is conditionedon 𝝉 since the delay 𝝉 is constant for one OFDM symbol, andthus the bit de–interleaving does not result in an averagingwith respect to 𝝉 . Following the same steps as in [10] forthe AWGN channel and exploiting (18), we obtain Ψ(𝑠∣𝝉 ) =(Ψ̃(𝑠∣𝝉 ))𝑑 with

Ψ̃(𝑠∣𝝉 ) = 1

𝑚2𝑚𝑁𝑠

𝑁𝑠2 −1∑

𝑘=−𝑁𝑠2

𝑚∑𝑖=1

1∑𝑏=0

∑𝑥[𝑘]∈𝒳 𝑖

𝑏

ℰ𝛼{𝑒𝛼

2𝑑2𝑥𝑧[𝑘](1+𝑠𝜎2𝑛)𝑠

𝐼∏𝜈=1

ℰ𝑔𝜈{Φ𝑖𝜈 [𝑘](2𝛼𝑑𝑥𝑧[𝑘]𝑠∣𝑔𝜈 , 𝜏𝜈)}},

(21)

where 𝑧[𝑘] represents the nearest neighbor of 𝑥[𝑘] in 𝒳 𝑖�̄�

with�̄� being the bit compliment of 𝑏, i.e., we invoke the BICMexpurgated bound from [10]. We note (21) is valid for channelmodels C1 and C2 discussed in Section II, respectively. Theonly difference between both cases as far as (21) is concernedis the different statistics of 𝑔𝜈 , 1 ≤ 𝜈 ≤ 𝐼 .

For sufficiently high SNR a tight approximation for theexact BER of a BICM–OFDM victim receiver impaired byUWB interference and AWGN can be obtained by truncating

4390 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009

the union bound in (19) and using (20) in combinationwith (18) and (21), cf. e.g. Fig. 1.1 Thereby, the integral in(20) can be efficiently evaluated using a Gauss–Chebyshevquadrature rule, cf. [14] for details. The conditional MGFsΦ𝑖𝜈 [𝑘](2𝛼[𝑘]𝑑𝑥𝑧[𝑘]𝑠∣𝑔𝜈 [𝑘], 𝜏𝜈) in (18) can be found in TableI.

The averaging over 𝝉 in (21) involves an 𝐼–dimensionalintegration which is potentially problematic for large 𝐼 . How-ever, numerical evidence shows that Ψ(𝑠∣𝝉 ) is almost inde-pendent of 𝝉 and assuming that 𝜏𝜈 , 1 ≤ 𝜈 ≤ 𝐼 , is uniformlydistributed in the BICM–OFDM symbol interval, it suffices toaverage 𝜏𝜈 over few (e.g. 5) values in [0, 𝑇𝑠) to get accurateBER results. The averaging over 𝛼 in (21) is not problematicas it only involves a one–dimensional integral. The same istrue for the averaging over 𝑔𝜈 in (21) since only 𝐼 independentone–dimensional integrals have to be evaluated which can bedone efficiently numerically.

C. Gaussian Approximation

If the UWB interference signals 𝑖𝜈 [𝑘] are modeled as𝒩 (0, 𝜎2𝑖,𝜈 [𝑘]) distributed with 𝜎2𝑖,𝜈 [𝑘] ≜ ℰ{∣𝑖𝜈[𝑘]∣2}, thevariance of 𝑛[𝑘] + 𝑖[𝑘] conditioned on 𝑔𝜈[𝑘] is given by𝜎2[𝑘] = 𝜎2𝑛 +

∑𝐼𝜈=1 ∣𝑔𝜈 [𝑘]∣2 𝜎2𝑖,𝜈 [𝑘]. In this case, the PEP

in (20) can be simplified to

𝑃 (𝒄 → �̂�) =1

2𝜋𝑗

𝑐+𝑗∞∫𝑐−𝑗∞

(1

𝑚2𝑚𝑁𝑠

𝑁𝑠2 −1∑

𝑘=−𝑁𝑠2

𝑚∑𝑖=1

1∑𝑏=0

∑𝑥[𝑘]∈𝒳 𝑖

𝑏

Ψ̃(𝑠∣𝑑𝑥𝑧[𝑘]))𝑑

d𝑠

𝑠(22)

where

Ψ̃(𝑠∣𝑑𝑥𝑧 [𝑘]) ≜ ℰ𝛼{𝑒𝛼

2𝑑2𝑥𝑧[𝑘][1+𝑠𝜎2𝑛]𝑠

×𝐼∏

𝜈=1

ℰ𝑔𝜈{𝑒𝛼2𝑑2𝑥𝑧[𝑘]∣𝑔𝜈 ∣2𝜎2

𝑖,𝜈 [𝑘]𝑠2}}. (23)

For the two interference models C1 and C2 outlined in SectionII, Ψ̃(𝑠∣𝑑𝑥𝑧[𝑘]) can be further simplified.

C1) If the 𝑔𝜈 [𝑘] are i.i.d. 𝒩 (0, 𝜎2𝑔,𝜈) RVs, we obtain

Ψ̃(𝑠∣𝑑𝑥𝑧 [𝑘]) = ℰ𝛼{

𝑒𝛼2𝑑2𝑥𝑧[𝑘][1+𝑠𝜎

2𝑛]𝑠∏𝐼

𝜈=1(1 + 𝑠2𝛼2𝑑2𝑥𝑧[𝑘]𝜎

2𝑔,𝜈𝜎

2𝑖,𝜈 [𝑘])

}.

(24)C2) If ∣𝑔𝜈 [𝑘]∣ is constant, we can simplify Ψ̃(𝑠∣𝑑𝑥𝑧[𝑘]) to

Ψ̃(𝑠∣𝑑𝑥𝑧[𝑘]) = 1

1 + 𝑠𝑑2𝑥𝑧[𝑘][1 + 𝑠(𝜎2𝑛 +

∑𝐼𝜈=1 𝑔

2𝜈𝜎

2𝑖,𝜈 [𝑘])]

.

(25)An approximation for the BER of a BICM–OFDM receiver

can be obtained by combining (19) and (22)–(25). The integralin (22) can again be efficiently evaluated using a Gauss–Chebyshev quadrature rule.

1Note that because of the truncation of the sum in (19) to a finite numberof terms as well as the numerical evaluation of (20), we cannot guaranteethat the resulting expression for the BER is a true upper bound.

3 4 5 6 7 8 9 10 11 1210

−7

10−6

10−5

10−4

10−3

10−2

10−1

TheorySimulationGA

BER

IR-UWB (Δfs = 4.125 MHz)

MB-OFDM (Δfs = 1 MHz)

BPSK DS-UWB (Δfs = 4.125 MHz)

MB-OFDM (Δfs = 4.125 MHz)

IR-UWB (Δfs = 2 MHz)

SIR [dB]

Fig. 1. BER vs. SIR for IR–UWB (𝑁𝑏 = 32, 𝐿 = 16), BPSK DS–UWB(𝐿 = 24), and MB–OFDM. Case C2 for channel, 𝐼 = 1, SNR → ∞,4–PSK, and 𝑅𝑐 = 1/2.

V. RESULTS AND DISCUSSION

Throughout this section we consider a BICM–OFDM sys-tem with 𝑁𝑠 = 64 sub–carriers. We found however thatthe performance is almost independent of 𝑁𝑠 for 𝑁𝑠 ≥ 16,which will be true for most practical systems. For BICMwe adopted the rate 1/2 standard convolutional code withgenerator polynomials [133, 171] (octal representation) andhigher code rates were obtained via puncturing. In the follow-ing, signal–to–noise ratio (SNR) and signal–to–interferenceratio (SIR) refer to the SNR and SIR per information bit,respectively. Unless stated otherwise, 𝑓𝜈 = 2Δ𝑓𝑖 = 8.25MHz2, 1 ≤ 𝜈 ≤ 𝐼 , 4–PSK modulation with 𝑅𝑐 = 1/2 is used,and only 𝐼 = 1 interferer is present. The various parametersfor the UWB signals were taken from [6]–[8]. Specific valuesadopted include: MB–OFDM: 𝑁𝑖 = 128, Δ𝑓𝑖 = 4.125 MHz,𝑁𝐵 = 3; DS–UWB: 𝑇𝑐 = 0.762 ns, 𝐵𝑖 = 1.3 GHz; IR–UWB: 𝑇𝑐 = 2.02 ns, 𝐵𝑖 = 499.2 MHz. For both DS–UWBand IR–UWB 𝑝𝑐(𝑡) was a square root Nyquist pulse.

In Fig. 1, we show the BER vs. SIR for MB–OFDM, BPSKDS–UWB (𝐿 = 24), and IR–UWB (𝑁𝑏 = 32, 𝐿 = 16)interference for different BICM–OFDM sub–carrier spacingsΔ𝑓𝑠 and channel model C2. An interference limited scenariois considered (i.e., SNR → ∞). Fig. 1 includes simulationresults, theoretical results obtained by evaluating the analyticalexpressions in Section IV, and the GA. For the theoreticalresults the union bound in (19) was truncated after the first 8terms. For the simulations ideal bit–interleaving is assumed,i.e., the frequency domain gains 𝛼[𝑘] of the victim channelwere simulated as i.i.d. Rayleigh RVs. As can be observedfrom Fig. 1, for relevant BERs (e.g. BER ≤ 10−4), where theunion bound becomes tight, the theoretical results are in per-fect agreement with the simulations for all considered UWBformats and sub–carrier spacings. We note that although theanalytical BER expressions involve integrals, which have to

2We note that, when plotted as a function of the SIR, the performance ofBICM–OFDM is practically independent of the frequency offset as long asthe BICM–OFDM signal lies fully in the UWB frequency band.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009 4391

105

106

107

10−6

10−5

10−4

IR−UWB (Nb = 8)

DS−UWB (L = 24)MB−OFDMGA

4−BOK

BPSK

BER

L = 2

L = 64L = 512

Δfs [Hz]

Fig. 2. BER vs. Δ𝑓𝑠 for IR–UWB (𝑁𝑏 = 8), BPSK DS–UWB (𝐿 =24), 4–BOK DS–UWB (𝐿 = 24), and MB–OFDM. Case C2 for channel,𝐼 = 1, SNR = 15 dB, SIR = 10 dB, 4–PSK, and 𝑅𝑐 = 1/2. Numericalresults.

be evaluated numerically, it took only minutes to compute thetheoretical BER curves in Fig. 1. In contrast, the simulationsfor Fig. 1 took several days to finish3. We observe from Fig. 1that the GA is only a good approximation for DS–UWB butthat all BER curves have the same asymptotic slope as theGA.

In Fig. 2, we show the BER vs. Δ𝑓𝑠 and investigate theimpact of the UWB format on the validity of the GA forSNR = 15 dB, SIR = 10 dB, and channel model C2.The BER was evaluated numerically based on the analyticalexpressions in Section IV. Fig. 2 shows that for IR–UWBwith 𝑁𝑏 = 8 the validity of the GA strongly depends onΔ𝑓𝑠 and on the data rate of the IR–UWB system. For datarates of 0.1 Mb/s (𝐿 = 512) and 0.85 Mb/s (𝐿 = 64) IR–UWB is highly impulsive and the GA is not accurate. Forvery high IR–UWB data rates (𝐿 = 2) the GA is fairlyaccurate in the considered Δ𝑓𝑠 range since the IR–UWB datasignal is less impulsive in this case. Similarly, the GA is veryaccurate for BPSK and 4–BOK DS–UWB. We note that weonly show results for the lower operating band specified in[7] and 𝐿 = 24 (corresponding to the lowest DS–UWB datarates). However, the results for smaller 𝐿 (corresponding tohigher data rates) and the higher operating band are verysimilar and are also in excellent agreement with the GA. ForMB–OFDM interference the GA may over- or underestimatethe performance depending on Δ𝑓𝑠 and is accurate only forΔ𝑓𝑠 ≤ 100 kHz.

To get a basic understanding of the influence of puncturingon the shape of the BER curves, we chose in Fig. 3 the SIRfor different code rates such that the GA yields the same BERfor all considered code rates. We show again numerical BERresults and consider channel model C2 for SNR → ∞. Fig. 3shows that the impact of the non–Gaussianity of the UWB

3Both the simulations and numerical evaluations were performed on thesame computer (with two Intel Xeon 3.6 GHz processors). The simulationprogram was written in C, whereas MATLAB was used for the numericalevaluations.

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BER

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Fig. 3. BER vs. Δ𝑓𝑠 for IR–UWB, 4–BOK DS–UWB, and MB–OFDM.Code rates of 𝑅𝑐 = 1/2 (SIR = 8.6 dB), 𝑅𝑐 = 2/3 (SIR = 11.2dB), 𝑅𝑐 = 3/4 (SIR = 13.7 dB), and 𝑅𝑐 = 5/6 (SIR = 16.8 dB) areconsidered. Case C2 for channel, IEEE 802.16 puncturing patterns, 𝐼 = 1,SNR → ∞, and 4–PSK. Numercial results.

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IR−UWB (analysis)DS−UWB (analysis)MB−OFDM (analysis)GA (analysis)MB−OFDM (C1, i.i.d. victim channel, simulation)MB−OFDM (C1, B = 10, simulation)IR−UWB (C2, B = 20, simulation)IR−UWB (C2, B=7, simulation)MB−OFDM (C2, i.i.d. victim channel, simulation)MB−OFDM (C2, B = {5, 9, 20, 40}, simulation)

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Fig. 4. BER vs. Δ𝑓𝑠 for IR–UWB (𝑁𝑏 = 32, 𝐿 = 16), 4–BOK DS–UWB (𝐿 = 24), and MB–OFDM. Results for C1 and C2 are compared.𝐼 = 1, SNR = 15 dB, SIR = 10 dB, 4–PSK, and 𝑅𝑐 = 1/2.

interference is less pronounced for higher code rates. We notethat higher code rates correspond to a code with a smallerminimum distance 𝑑min and thus, a smaller diversity order[10]. Therefore, our results are in agreement with the findingsin [5] where it was shown that (for an uncoded single–carriersystem) the impact of the non–Gaussianity of the interferenceon the BER is larger for channels with higher diversity order.

In Fig. 4, we study the impact of the victim and theinterference channel on the performance for SNR = 15 dBand SIR = 10 dB. In particular, we compare numericalresults obtained with the analysis presented in Section IVfor C1 (severely frequency–selective UWB channel) and C2(frequency–flat UWB channel). Remarkably, the general shapeof the BER curves is not affected by the interference channel.However, a frequency–selective interference channel leadsto a performance loss for all UWB formats and the non–

4392 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 9, SEPTEMBER 2009

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Fig. 5. BER vs. Δ𝑓𝑠 for IR–UWB, 4–BOK DS–UWB, and MB–OFDM.𝐼 = 1, 2, 5, 10, 100 i.i.d. interferers are considered. Case C2 for channel,SNR → ∞, SIR = 8 dB, 4–PSK, and 𝑅𝑐 = 1/2. The markers (“∇”)indicate simulation results for MB–OFDM and 𝐼 = 2.

Gaussian behavior of the BER curves is also slightly morepronounced in this case. Besides the numerical results whichassume that the frequency domain gains 𝛼[𝑘] of the victimchannel are i.i.d. Rayleigh RVs, we also show in Fig. 4simulation results (markers) for both i.i.d. and non–i.i.d. victimchannels. For the non–i.i.d. case we assume that the victimchannel impulse response ℎ(𝑡) has 10 i.i.d. taps and codingand interleaving is performed over 𝐵 OFDM symbols. Thechannel gains in different OFDM symbols are assumed to bestatistically independent due to e.g. frequency hopping insidethe bandwidth of the UWB signal. Thus, for large 𝐵 thefrequency domain fading gains 𝛼[𝑘] are approximately i.i.d. asfar as the dominant error events of the code are concerned.In contrast, for small 𝐵 some correlation will be left and thei.i.d. assumption may lead to overly optimistic performancepredictions. However, the results in Fig. 4 show that thesimulation results agree well with the numerical results evenfor relatively small 𝐵.

In Fig. 5, we investigate the effect of multiple i.i.d. in-terferers on the BER for channel model C2. To facilitatethis comparison SIR = 8 dB (SNR → ∞) was assumedindependent of the number of interferers 𝐼 . As can be observedfrom Fig. 5, for MB–OFDM and IR–UWB the total interfer-ence becomes more Gaussian as the number of interferersgrow. This behavior can be explained by the central limittheorem. However, while for MB–OFDM the GA is accuratefor Δ𝑓𝑠 < 2 MHz for 𝐼 ≥ 5 interferers, for the considered IR–UWB format even 100 interferers are not sufficient to make theGA tight in the considered range of sub–carrier spacings. ForDS–UWB increasing the number of interferers has almost noeffect since, in this case, the interference signal is practicallyGaussian already for 𝐼 = 1. To confirm the numerical BERresults, we have also included simulation points for MB–OFDM with 𝐼 = 2. Here, the frequency domain channel gains𝛼[𝑘] were simulated as i.i.d. Rayleigh RVs.

VI. CONCLUSIONS

In this paper, we have provided an analytical performanceevaluation framework for generic BICM–OFDM systems im-paired by UWB interference. The considered UWB formatsclosely follow IEEE/ECMA standards or standard proposals.Besides the accurate analysis we have also proposed a simpleGA where the UWB interference is treated as additionalGaussian noise. Our results show that while for DS–UWBthe GA is applicable for all practically relevant BICM–OFDM sub–carrier spacings (e.g. Δ𝑓𝑠 < 10 MHz), forMB–OFDM and IR–UWB the GA becomes tight only forΔ𝑓𝑠 < 100 kHz. For larger sub–carrier spacings the GA mayseverely underestimate the BER for IR–UWB interferenceemploying the mandatory data rate of 0.85 Mb/s specifiedin the IEEE 802.15.4a standard. In contrast, for MB–OFDMthe GA overestimates the BER for Δ𝑓𝑠 < 2 MHz butunderestimates it for Δ𝑓𝑠 > 2 MHz. For example, for IEEE802.11a WLANs (Δ𝑓𝑠 = 312.5 kHz) the GA overestimatesthe BER for MB–OFDM interference but underestimates it forIR–UWB interference. In contrast, if an MB–OFDM system(Δ𝑓𝑠 = 4.125 MHz) is interfered by other UWB systems,the GA underestimates the BER for both IR–UWB and MB–OFDM interference.

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