UWB Signal Detection by Cyclic Features · of UWB systems and WiMAX systems, where the UWB devices as secondary users are able to detect the presence of WiMAX systems by their cyclic

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UWB Signal Detection by Cyclic FeaturesYiyin Wang†∗, Xiaoli Ma‡, Cailian Chen†, and Xinping Guan†

†Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China‡School of ECE, Georgia Institute of Technology, Atlanta, GA30332-0250, USA

Abstract—Ultra-wideband (UWB) impulse radio (IR) systemsare well known for low transmission power, low probability ofdetection, and overlaying with narrowband (NB) systems. Thesemerits in fact make UWB signal detection challenging, sinceseveral high-power wireless communication systems coexist withUWB signals. In the literature, cyclic features are exploited forsignal detection. However, the high computational complexity ofconventional cyclic feature based detectors burdens the receivers.In this paper, we propose computationally efficient detectorsusing the specific cyclic features of UWB signals. The closed-form relationships between the cyclic features and the systemparameters are revealed. Then, some constant false alarm rate de-tectors are proposed based on the estimated cyclic autocorrelationfunctions (CAFs). The proposed detectors have low complexitiescompared to the existing ones. Extensive simulation results indi-cate that the proposed detectors achieve a good balance betweenthe detection performance and the computational complexity invarious scenarios, such as multipath environments, colored noise,and NB interferences.

Index Terms—Cyclostationarity, feature detection, ultra-wideband (UWB) communications

EDICS: SSP-CYCS Cyclostationary signal analysis, SSP-DETC Detection, SPC-UWBC Ultra wideband communica-tions

I. I NTRODUCTION

Ultra wideband technology is adopted by the IEEE802.15.4a standard [1] for wireless personal area networks(WPANs) to provide low-power communications and preciseranging capabilities. It is featured by sharing the spectrumwith other communication systems to efficiently use rarespectrum resources [2]–[5]. For example, according to theIEEE 802.15.4a standard, the specified UWB signal mayoccupy the same spectrum as the signal specified by the IEEE802.16 standard [6] (also named as worldwide interoperabilityfor microwave access (WiMAX)). Therefore, UWB systemsusually work in a heterogeneous wireless environment. Thefirst fundamental task of a UWB receiver is to detect thetransmitted UWB signal regardless of interferences in hetero-geneous environments. Conventional energy detectors failtodistinguish different signals from each other. Moreover, UWBchannel environments are rich in multipaths and subject tovarying noise. Hence, the detectors based on known signaland noise statistics, such as matched filters, are impracticalfor implementation.

Part of the work was supported by the National Nature ScienceFoundationof China (No. 61301223), the Nature Science Foundation of Shanghai (No.13ZR1421800), and the Georgia Tech Ultra-wideband Center of Excellence(http://www.uwbtech.gatech.edu/). Some preliminary results of this work werepresented at the IEEE International Conference on Ultra-Wideband, Sydney,Australia, September 2013

Recently, cyclostationarity is of interest for detecting thesignal in cognitive radio (CR) systems [7]–[11], in whichsecondary users detect the presence of primary users and makeuse of the unoccupied spectrum efficiently. Cyclostationaritydescribes the periodicity of the statistical properties ofasignal, and exists in almost all modulated signals [12]–[15].The signal parameters, e.g. modulation types, symbol rates,carrier frequencies, and periods of spreading codes, determinecyclic features of a signal. Since these parameters are differentfor different types of signals, the distinct cyclic featurescan be exploited for signal detection. In addition, the cyclicfeature detectors are robust to noise uncertainty. For CRs,theDandawate-Giannakis detector [16] is employed by secondaryusers to detect various primary signals, such as orthogonalfrequency division multiplexing (OFDM) signals, Gaussianminimum shift keying (GMSK) modulated signals, and codedivision multiple access (CDMA) signals [7], [17], [18]. Amulti-cycle extension of the Dandawate-Giannakis detector aswell as its computationally efficient modifications are proposedto detect the OFDM signals in [19]. Consequentially, a collab-orative detection among secondary users with censoring is alsodeveloped in [19]. Furthermore, a multi-antenna extensionofthe Dandawate-Giannakis detector is designed in [20] to takeadvantage of the diversity gain of multiple antennas.

Since the computational complexities of the Dandawate-Giannakis detector and its inheritors are relatively high,severalheuristic cyclic detectors considering noise uncertaintyaredesigned in [21]–[23] to reduce the complexity. A single-cycle single-lag detector is proposed in [21] to detect OFDMsignals of WiMAX systems. A multi-cycle single-lag detectoris further developed in [22] to perceive the OFDM signals.Taking colored Gaussian noise into account, a multi-cyclemulti-lag cyclic feature detector is derived in [23], and itcanaccommodate multiple antennas as well. Although some ofthese proposed detectors can be adapted for UWB signals,such as the ones in [19], [23], [24], they either maintain highcomplexity or do not take advantage of the unique propertiesof UWB signals. For example, a Dandawate-Giannakis typedetector is employed in [24] to detect a UWB signal underthe coexistence of a GMSK signal. However, it still suffersfrom high computational complexity. Furthermore, a wideband spectrum sensing method based on recovered sparse 2-D cyclic spectrum from compressive samples is proposed in[25]. This sub-Nyquist scheme is attractive, as the Nyquistrate of the UWB signal is notably high. However, a 2-Dcyclic spectrum recovery is not necessary here, as some priorknowledge of the cyclic features of the UWB signal hasbeen assumed. Moreover, a detection and avoidance (DAA)scheme is proposed in [21], [26] to facilitate the coexistence

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of UWB systems and WiMAX systems, where the UWBdevices as secondary users are able to detect the presenceof WiMAX systems by their cyclic features, and avoid thetransmission in the occupied spectrum. Different from [21],[26], our work focuses on the receiver to detect the UWBsignal in heterogeneous environments, not on the transmitterto sense the available spectrum.

As a result, multi-cycle multi-lag detectors based on cyclicfeatures are proposed for UWB receivers to recognize theUWB signals of interest in this paper. At first, the UWBsignal structure is specified by following the IEEE 802.15.4astandard. Sequentially, the cyclic features of the UWB signalare investigated. Although the cyclic features of non-standardUWB signals have also been analyzed in [24], [27], [28],where the symbol rate plays the main role, due to a hybridmodulation and scrambling codes, the cyclic features of thestandard UWB signal do not simply appear at consecu-tive multiples of the symbol rate. The closed-form relation-ships between the cyclic features and the system parametersare further revealed. Furthermore, constant false alarm rate(CFAR) detectors are proposed based on the estimated cyclicautocorrelation functions (CAFs). Thanks to the ultra widebandwidth of the signal and the resolvable multipath channelcomponents, the proposed detectors take advantage of multiplecyclic frequencies (CFs) and multiple time lags (TLs). Theircomputational complexities are significantly less than theonesof the Dandawate-Giannakis type detectors in [19], and arecomparable to the complexity of the detector proposed in[23], which deals with colored Gaussian noise. Note that thedetector [23] fails under the case of interference corruption, asit oversimplifies the covariance estimation. On the other hand,the proposed detector composed of the single-cycle single-lag Dandawate-Giannakis test statistics can still deal with theinterferences, and it achieves a tradeoff between detectionperformance and computational complexity.

The rest of the paper is organized as follows. The pre-liminary knowledge of the cyclostationarity is reviewed inSection II. The model of the IEEE 802.15.4a UWB signal andits cyclic features are described and analyzed, respectively, inSection III. Consequentially, CFAR detectors are developedbased on the estimated CAFs to exploit the specific cyclicfeatures of the UWB signal in Section IV. The comparisonof the computational complexities between the proposed de-tectors and the existing ones is carried out in Section V. Ex-tensive simulations validate the detection performance undervarious scenarios in Section VI. The conclusions are drawn inSection VII.

II. PRELIMINARIES OF CYCLOSTATIONARITY

In this section, we briefly review the preliminary knowledgefor cyclostationary processes and introduce the notations.More comprehensive details can be found in [12]–[15].

A cyclostationary process is characterized by the cyclicallyvaried statistical properties of a signal with respect to (w.r.t.)time. A special case of cyclostationary signals is the wide-sense cyclostationary signal, whose second-order statistics isperiodic in time. Hence, the autocorrelation function of a zero-

mean wide-sense cyclostationary signals(t) is given by

Υss(t, τ)=E[s(t + τ/2)s∗(t− τ/2)]=Υss(t+ nTf , τ), (1)

∀n ∈ Z

where∗ denotes the complex conjugate,Z is the integer set,τ is the time lag (TL), andTf is the fundamental period. Asa result,Υss(t, τ) can be decomposed into Fourier series as

Υss(t, τ) =∑

αk∈A

Rss(αk, τ)ej2παkt, (2)

where A = αk|Rss(αk, τ) 6= 0 is the set of cyclicfrequencies (CFs), andαk is related to the fundamental periodasαk = k/Tf , k ∈ Z. The Fourier coefficientsRss(αk, τ) canbe calculated as

Rss(αk, τ) = limT→∞

1

T

∫ T/2

−T/2

Υss(t, τ)e−j2παktdt, (3)

which is also named the cyclic autocorrelation function (CAF),andT is the observation period. Consequently, the spectrumcorrelation density (SCD) function is defined as the Fouriertransform of the CAF

Sss(αk, f) =

∫ ∞

−∞

Rss(αk, τ)e−j2πfτdτ. (4)

The counterparts of these functions are the conjugate ones.Letus start with the conjugate autocorrelation function givenby

Υss∗(t, τ)=E[s(t+ τ/2)s(t− τ/2)]. (5)

Its Fourier coefficients are the conjugate CAFRss∗(αk, τ).Hence, the Fourier transform ofRss∗(αk, τ) is the conjugateSCDSss∗(αk, f). To combine all these definitions, we denotethem asΥ

ss(∗)(t, τ), R

ss(∗)(αk, τ) andS

ss(∗)(αk, f), where

(∗) represents two options (nonconjugate and conjugate).

III. T HE IEEE 802.15.4A UWB SIGNAL MODEL AND ITS

CYCLIC FEATURES

A. Signal Model

According to the IEEE 802.15.4a standard [1], the equiv-alent baseband model for the UWB PHY transmitting signalcan be written as

x(t) =

+∞∑

k=−∞

ak

Ncpb−1∑

n=0

cn+kNcpb(6)

×p(t− kTdsym − bkTBPM − h(k)Tburst − nTc − ǫ),

which is modulated by a combination of burst position mod-ulation (BPM) and binary phase-shift keying (BPSK). Theparameters and notations in (6) are listed in Table I. The UWBsymbol structure is shown in Fig. 1, where a UWB symbolis composed of a burst of UWB pulses, whose amplitudesare modulated by the data symbolak and the scramblingsequencecn+kNcpb

, and whose positions are modulated bythe data symbolbk and the burst hopping sequenceh(k). Notethat cn+kNcpb

and h(k) are different for each symbol. Theyfacilitate the multiuser interference suppression and spectralsmoothing, and are generated from a common pseudo-random

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p(t) the transmitted UWB pulse of lengthTp

ǫ the unknown deterministic timing offsetNcpb the number of chips per burstNburst the number of burst per symbolNhop the number of hopping burst per symbolTc the chip interval

Tburst the burst interval, whereTburst = NcpbTc,TBPM the position shift for the BPM, whereTBPM = Tdsym/2Tdsym the symbol period, whereTdsym = NburstTburst,ak the kth symbol modulates the burst amplitude, whereak ∈ ±1bk the kth symbol modulates the burst position, wherebk ∈ 0, 1

cn+kNcpbthe scrambling code for thekth symbol, wherecn+kNcpb

∈ ±1, n = 1, . . . , Ncpb

h(k) the burst hopping code for thekth symbol, whereh(k) ∈ 0, 1, . . . , Nhop − 1TABLE I

PARAMETERS FOR THEIEEE 802.15.4A UWB SIGNAL [1] IN (6)

Fig. 1. The UWB symbol structure according to the IEEE 802.15.4a standard [1]

bit stream (PRBS) scrambler. Consequentially, the scramblingsequence is given by [1]

cn = cn−14 ⊕ cn−15, (7)

where⊕ denotes modulo-two addition, and the hopping se-quence is generated as [1]

h(k) = 20cn+kNcpb+ · · ·+ 2m−1cm−1+kNcpb

, (8)

wherem = log2(Nhop). Since the polynomial order of thePRBS scrambler is high andm takes values from1, 3, 5,the scrambling sequencecn+kNcpb

and the hopping sequenceh(k) can be assumed as wide-sense stationary (WSS) andindependent from each other. Furthermore,ak and bk arealso assumed to be WSS and independent from each other.Therefore,cn+kNcpb

and ak take values from±1 withequal probability, andbk selects values from0, 1 withequal probability as well. Moreover, the hopping sequenceh(k) chooses values from0, 1, . . . , Nhop − 1 with equalprobability.

B. Cyclic Features of IEEE 802.15.4a UWB Signals

In this subsection, we investigate the cyclic features ofthe IEEE 802.15.4a UWB signalx(t) in (6). As BPSK andscrambling code are employed,x(t) is a zero-mean randomsignal. Sincex(t) is a real signal,Υxx(t, τ) andΥxx∗(t, τ) areequivalent in this case. Without loss of generality,Υxx(t, τ) isderived by plugging (6) into (1) as (see Appendix A for moredetails)

Υxx(t, τ) = αx1

+∞∑

q=−∞

ej2παxq (t−ǫ)β(αx

q )η(αxq )

×φp(αxq , τ)w

(Tcα

xq , Ncpb

), (9)

where αxq = q/Tdsym, q ∈ Z, β(f) = E[e−j2πbkTBPMf ],

η(f) = E[e−j2πh(k)Tburstf ],

φp(αxq , τ) =

∫P (z + αx

q )P∗(z)ej2πτ(z+αx

q/2)dz

= p(τ)e−j2πταxq /2 ⊗ p∗(−τ)ej2πτα

xq /2,(10)

with ⊗ denoting convolution,P (f) being the Fourier trans-form of p(t), and

w(ρ,H) =H−1∑

n=0

e−j2πρn =sin(πρH)

sin(πρ)e−jπρ(H−1). (11)

In (9), Υxx(t, τ) is decomposed into the Fourier series using1/Tdsym (αx

1 ) as the fundamental CF w.r.t.t. The Fouriercoefficient ofΥxx(t, τ) is the CAFRxx(α

xq , τ). Recall thatbk

andh(k) select values from0, 1 and 0, 1, . . . , Nhop − 1with equal probability, respectively, and they are independentwith each other. The CAFRxx(α

xq , τ) is simplified as

Rxx(2αxq , τ) =

αx1

Nhopw

(2q

NburstNcpb, NhopNcpb

)

×φp(2αxq , τ)e

−j4παxq ǫ, q ∈ Z. (12)

Please refer Appendix B for more details about this derivation.Several remarks are due here.

Remark 1: The function w (2q/(NburstNcpb), NhopNcpb)reaches local maxima, whenq = kNburstNcpb/2 with k ∈ Z.Meanwhile it equals zero, whenq = k′Nburst/(2Nhop)with k′ ∈ Z and k′ 6= kNhopNcpb/2. Therefore, there areNcpbNhop − 1 zeros between two adjacent local maximumvalues ofw (2q/(NburstNcpb), NhopNcpb). As a result, thenonzero pattern ofw (2q/(NburstNcpb), NhopNcpb) w.r.t. q isrelated to the productNburstNcpb/2 andNburst/(2Nhop).

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Remark 2: According to (12), the nonzeropattern of Rxx(2α

xq , τ) w.r.t. q is determined by

w (2q/(NburstNcpb), NhopNcpb), and thus also relatedto the productNburstNcpb/2 andNburst/(2Nhop). Moreover,the ranges of the nonzero support ofRxx(2α

xq , τ) w.r.t. q and

τ are all determined byφp(2αxq , τ), which is related to the

bandwidth of the UWB pulse.Remark 3: Although we do not take the multipath channel

into the derivation, its impact on the CAF can be analyzed. Letus assume the multipath channelh(t) does not change duringthe detection. Due to the multipath channel, the received pulseshape is thatp(t) = p(t) ⊗ h(t). Hence, the CAF of theUWB signal via a multipath channelRxx(2α

xq , τ) is still given

by (12), but replacingφp(αxq , τ) with φp(α

xq , τ), which is as

follows

φp(αxq , τ) =

∫P (z + αx

q )P∗(z)ej2πτ(z+αx

q/2)dz, (13)

where P (f) = H(f)P (f) and H(f) is the Fouriertransform of the multipath channelh(t). As a result,the nonzero pattern ofRxx(2α

xq , τ) is still decided by

w (2q/(NburstNcpb), NhopNcpb). The nonzero values ofRxx(2α

xq , τ) are related toφp(α

xq , τ). Moreover, the range of

the nonzero support ofRxx(2αxq , τ) w.r.t. τ increases, since

p(t) may contain many multipath components. For notationsimplicity, we do not consider channel in the detector design,but we show the channel effect in the simulation.

IV. UWB SIGNAL DETECTION USING ITS CYCLIC

FEATURES

According to the analysis in the previous section, the CAFof the UWB signal is nonzero at several CFs and for a range ofTLs. Hence, multi-cycle multi-lag detectors can be exploitedto take full advantage of their cyclic features. Moreover, boththe CAF and the conjugate CAF can be used here, as both ofthem indicate the cyclic features. In this section, we proposeconstant false alarm rate (CFAR) detectors based on theestimated CAFs to tradeoff between computational complexityand detection performance. Since our proposed detectors arecomposed of the single-cycle single-lag Dandawate-Giannakistest statistics, we would first briefly review the general formof the Dandawate-Giannakis detector in [16], [19] in Sec-tion IV-A. Consequentially, the CFAR detectors are proposedin Section IV-B. In order to facilitate fair comparisons, fourkinds of existing detectors in [19], [23] are summarized inSection IV-C.

The estimated CAFs at the interested TLs and CFs are usedto calculate the test statistics. Hence, they are collectedin arow vectorr

xx(∗)as

rxx(∗)

=[r1xx(∗)

. . . rMxx(∗)

], (14)

whereM is the number of the CFs of interest, the row vectorrixx(∗)

is defined as

rixx(∗)

=[Re

R

xx(∗)(αi, τi,1)

, . . . ,Re

R

xx(∗)(αi, τi,Ni

),

ImR

xx(∗)(αi, τi,1)

, . . . , Im

R

xx(∗)(αi, τi,Ni

)]

,

and

Rxx(∗)

(αi, τi,l)=1

K

K−1∑

n=0

x[n]x(∗)[n+ τi,l]e−j2παin, (15)

∀i ∈ 1, . . . ,M, ∀l ∈ 1, . . . , Ni,

with K being the number of samples of the UWB signal (x(t)),αi being the CF of interest,τi,l being thelth TL of interestfor αi, andNi being the total number of TLs forαi. Thus,2J = 2

∑Mi=1 Ni is the total length ofr

xx(∗). Note thatx[n]’s

are the discrete samples ofx(t).The detection of the UWB signal is a binary-hypothesis test.

The two hypotheses are given as follows:

H0 : rxx(∗)

= e,

H1 : rxx(∗)

= rxx(∗)

+ e,(16)

where rxx(∗)

is the true nonrandom CAF vector, ande isthe estimation error row vector, which is asymptotically zero-mean Gaussian distributed with covariance matrixσ2I. Forthe above binary-hypothesis test, several existing detectors arereviewed for comparison, and low-complexity CFAR detectorsare proposed in the following subsections.

A. The multi-cycle multi-lag Dandawate-Giannakis detector

For the binary-hypotheses test (16), the multi-cycle multi-lag detector [19], which is a natural extension of the originalDandawate-Giannakis detector [16], is given by

TDG(

∗) = K r

xx(∗)Σ

−1

xx(∗)

(rxx(∗)

)T

, (17)

where(·)T denotes the transpose, andΣxx(∗)

is the estimatedasymptotic covariance matrix following the method in [16],[19]. The true covariance matrixΣ

xx(∗)can be divided into

M2 blocks of size2Ni × 2Nℓ, ∀i, ∀ℓ ∈ M, whereM =1, . . . ,M. The blockΣ

xx(∗)(i, ℓ) of size2Ni × 2Nℓ is the

covariance matrix for the CF pair(αi, αℓ). Thus, it is givenby [16], [19]

Σxx(∗)

(i, ℓ)=

Re

Q

(∗)

i,ℓ +P(∗)

i,ℓ

2

Im

Q

(∗)

i,ℓ −P(∗)

i,ℓ

2

Im

Q

(∗)

i,ℓ +P(∗)

i,ℓ

2

Re

P

(∗)

i,ℓ −Q(∗)

i,ℓ

2

,

whereQ(∗)

i,ℓ andP(∗)

i,ℓ are defined as

Q(∗)

i,ℓ (k, l) = S(∗)

τℓ,kτi,l(αi + αℓ, αℓ), (18)

P(∗)

i,ℓ (k, l) = S(∗),∗

τℓ,kτi,l(αi − αℓ,−αℓ), (19)

and the estimates ofS(∗)

τℓ,kτi,l(αi + αℓ, αℓ) andS(∗),∗

τℓ,kτi,l(αi −αℓ,−αℓ) are calculated using the frequency smoothed cyclic

5

periodograms, respectively

S(∗)

τℓ,kτi,l(αi + αℓ, αℓ) (20)

=1

KL

(L−1)/2∑

s=−(L−1)/2

W (s)F(∗)

τi,l

(αi −

s

K

)F

(∗)

τℓ,k

(αℓ +

s

K

),

S(∗),∗

τj,mτi,n(αi − αℓ,−αℓ) (21)

=1

KL

(L−1)/2∑

s=−(L−1)/2

W (s)(F

(∗)

τi,l

(αi +

s

K

))∗

F(∗)

τℓ,k

(αℓ +

s

K

),

whereF (∗)

τ (α) =∑K−1

n=0 x[n]x(∗)[n + τ ]e−j2παn, andW (s)is the normalized spectral smoothing window function withlength L. Under the null hypothesisH0, the distribution ofTDG(

∗) asymptotically converges to the centralχ2

2J distri-bution with 2J degrees of freedom. Therefore, a thresholdγDG can be decided by a constant false alarm rate asPfa =Prob(T

DG(∗) ≥ γDG|H0).

B. Proposed CFAR detectors

The computational complexity of the detector (17) including(20) and (21) is notably high. To reduce the complexity, wepropose a computationally efficient cyclic detector as

T I

prop(∗)=

M∑

i=1

YI

i(∗), (22)

where

YI

i(∗)= max

l=1,...,Ni

TDG(

∗)(αi, τi,l). (23)

The test statisticTDG(

∗)(αi, τi,l) is a single-cycle single-lag

Dandawate-Giannakis detector, and thus a special case of (17).It is given by

TDG(

∗)(αi, τi,l) = K r

i,l

xx(∗)Σ

−1

xx(∗)(i, i, l, l)(ri,l

xx(∗)

)T

, (24)

whereri,lxx(∗)

=[Re

R

xx(∗)(αi, τi,l)

, Im

R

xx(∗)(αi, τi,l)

]

and

Σxx(∗)

(i, i, l, l)

=

Re

Q

(∗)

i,i (l, l) +P(∗)

i,i (l, l)

2

Im

Q

(∗)

i,i (l, l)−P(∗)

i,i (l, l)

2

Im

Q

(∗)

i,i (l, l) +P(∗)

i,i (l, l)

2

Re

P

(∗)

i,i (l, l)−Q(∗)

i,i (l, l)

2

.

Furthermore, the mapping between the CF set and the TLset can also be described in another way as follows: for eachTL τu, u = 1, . . . , N with N being the total number of TLsof interest, the CFs of interest areαu,v, v = 1, . . . , Mu, whereMu is the total number of CFs for the TLτu. Note that2J = 2

∑Nu=1 Mu = 2

∑Mi=1 Ni is the total length ofr

xx(∗).

Hence, a nature variation of the test statisticT I

prop(∗)in (22)

is proposed as

T II

prop(∗)=

N∑

u=1

YII

u(∗), (25)

where

YII

u(∗)= max

v=1,...,Mu

TDG(

∗)(αu,v, τu). (26)

Both T I

prop(∗)(22) andT II

prop(∗)(25) are motivated by the rich

cyclic features of the UWB signal, since multiple CFs andTLs could provide the diversity gain to combat the multipathfading.

As the methods to calculate the thresholds forT I

prop(∗)and

T II

prop(∗)are the same, the calculation of the threshold for

T I

prop(∗)is exemplified. Under the null hypothesisH0, it is

known thatTDG(

∗)(αi, τi,l) asymptotically follows the central

χ22 distribution. The probability of density function (pdf)pi(y)

of YI

i(∗)(y , YI

i(∗)) can be computed as

f Ii (y) = NiF (y)Ni−1f(y), ∀i ∈ M, (27)

whereF (y) andf(y) are the cumulative distribution function(cdf) and the pdf of the centralχ2

2 given by

F (y) = 1− e−y2 , y ≥ 0, (28)

f(y) =1

2e−

y2 , y ≥ 0. (29)

Making use of the binomial expansion, we arrive at

f Ii (y)=

Ni

2

Ni−1∑

k=0

(−1)k(

Ni − 1k

)e−

(k+1)y2 ,

y ≥ 0, ∀i ∈ M. (30)

SinceYI

i(∗)are independent of each other, the pdf of the test

statistic ofT I

prop(∗)should be a multi-dimensional convolution

of all f Ii (y), ∀i ∈ M. Therefore, we can achieve the pdf of

T I

prop(∗)numerically and compute a thresholdγI

prop according

to a constant false alarm rate asPfa = Prob(T I

prop(∗)≥

γIprop|H0). In the case thatNi = Nℓ, ∀i, ∀ℓ ∈ M and M

is small, it is possible to obtain a closed-form pdf ofT I

prop(∗).

For example, whenM = 2 andN1 = N2 = 2, the pdf andthe cdf ofT I

prop(∗)(Let us reloady asy , T I

prop(∗)) are given

by, respectively

f(y) = (4 + y)(e−y + e−y/2), (31)

and

F (y) = 1 + (4− 2y)e−y/2 − (5 + y)e−y. (32)

As a result, the thresholdγIprop can be found in a lookup table

calculated offline in advance.

C. Other existing detectors

There are several variations of the multi-cycle multi-lagDandawate-Giannakis detectorT

DG(∗) in (17). Regardless of

the correlation between different CFs, the estimated covariancematrix can be simplified as a block diagram matrix. Denotethe corresponding test statistic asT

sum DG(∗) , which is given

by

Tsum DG(

∗) =

M∑

i=1

TDG(

∗)(αi), (33)

6

where

TDG(

∗)(αi) = K ri

xx(∗)Σ

−1

xx(∗)(i, i)(ri

xx(∗))T . (34)

Note thatTDG(

∗)(αi) is the test statistic for a single CF with

multiple TLs. When only a single TL is employed for eachCF, the test statisticT

sum DG(∗) is equivalent toT I

prop(∗).

Another variation is to choose the maximum test statisticamongT

DG(∗)(αi), i ∈ M, and compare it with a threshold.

Let us denote this test statistic as

Tmax DG(

∗) = max

i=1,...,MTDG(

∗)(αi). (35)

The test statisticsTsum DG(

∗) andT

max DG(∗) are more com-

putationally efficient thanTDG(

∗) . Furthermore, it has been

shown in [19] that the detection performance ofTsum DG(

∗)

is close toTDG(

∗) .

Moreover, an ad hoc detectorTad hoc is proposed in [23]to take colored Gaussian noise into account for the nullhypothesis. In order to facilitate a fair comparison in thesimulation section, we review the detector proposed in [23]using the notations defined in this paper. The autocorrelationof the colored Gaussian noise is assumed to be nonzero in therange of[−Ln, Ln] (Rxx(m) 6= 0, |m| ≤ Ln). Thus, the teststatistic based on the CAF estimates is derived as

Tad hoc = 2K

M∑

i=1

Ni∑

l=1

|Rxx(αi, τi,l)|2

γxx(αi), (36)

where

γxx(αi) =

Ln∑

s=−Ln

|ˆRxx(s)|

2e−j2παis, (37)

ˆRxx(s) =

1

K − s

K−s−1∑

n=0

x[n]x∗[n+ s], s ≥ 0, (38)

and ˆRxx(−s) =

ˆR

∗

xx(s). On the other hand, the test statisticbased on the conjugate CAF estimates is adapted as

Tad hoc∗ = 2KM∑

i=1

Ni∑

l=1

|Rxx∗(αi, τi,l)|2

γxx∗(αi, τi,l), (39)

where

γxx∗(αi, τi,l) (40)

=

Ln∑

s=−Ln

(ˆR

2

xx(s) +ˆRxx(s+ τi,l)

ˆRxx(s− τi,l)

)ej2παis.

The correlation among the CAF estimates with different CF-TL pairs are neglected in (36) and (39). Moreover, the teststatisticT

ad hoc(∗)asymptotically follows a centralχ2

2J dis-tribution under the null hypothesis.

Last but not the least, the energy detector is widely used inthe signal detection due to its simplicity. The test statistic forenergy detector is given by

TED =1

K

K−1∑

n=0

|x[n]|2, (41)

where

H0 : TED ∼ N

(ς2n,

2ς4nK

), (42)

according to the central limit theorem, andς2n is the varianceof the observation noise.

V. THE COMPUTATIONAL COMPLEXITY ANALYSIS

In this section, we analyze the computational complexity ofthe proposed detectors, and compare them with other detectorsreviewed in Section IV. The computational complexity countsfor the number of operations to calculate the test statistics.The complexities of additions, subtractions and comparisonare trivial compared to multiplications and divisions, andthusthey are neglected for simplicity. We further assume that thecomplexities of the multiplication and the division are thesame, thus the divisions are counted as the multiplicationsas well. The complexities of the elements to calculate the teststatistics is first explored. The computational complexitytocalculateR

xx(∗)(αi, τi,l) is 2K multiplications. The calcula-

tion of Σxx(∗)

(i, ℓ) involvesNiNℓ(6K +4L) multiplications.

Hence, to achieveΣxx(∗)

needs (6K + 4L)(∑M

i=1 N2i +∑M−1

i=1

∑Mℓ=i+1 NiNℓ) multiplications. The inverse of an

A × A matrix using the Gauss-Jordan elimination requiresA3 + 6A2 multiplications. As a result, the inverse ofΣ

xx(∗)

andΣxx(∗)

(i, i) count for8J3+24J2 and8N3i +24N2

i multi-plications, respectively. Moreover, the product of anA×B ma-trix and aB×C matrix needsABC multiplications. Therefore,the test statisticT

DG(∗)(αi, τi,l) itself for a single(αi, τi,l) pair

asks for6 multiplications, and the calculation ofTDG(

∗)(αi)

needs4N2i +2Ni multiplications. To compute the test statistic

TDG(

∗) requires4J2+2J multiplications. Considering the test

statisticsTad hoc(∗)

, the computation of the correlation termˆRxx(s) involves approximatelyK multiplications. There areLn + 1 TLs. Thus, the correlation coefficientsγxx(αi) andγxx(αi, τi,ℓ) need2(Ln + 1) and 3(Ln + 1) multiplications,respectively. The test statisticT

ad hoc(∗)itself asks for2J

multiplications.Based on the above analysis, the results are summarized

in Table II. Recall thatJ =∑M

i=1 Ni =∑N

u=1 Mu andKis the number of samples. ThereforeK ≫ J, L, Ln. Theterms related toK in Table II are the most significant ones.The proposed detectorsT I

prop(∗)andT II

prop(∗)have comparable

complexities as the detectorTad hoc(∗)

. All of them need muchless operations than the detectorT

DG(∗) and its variations

(Tsum DG(

∗) and T

max DG(∗)). The detectorT

DG(∗) requires

the most computational resources.

VI. N UMERICAL EXAMPLES

In this section, the performance of the proposed detectorsare evaluated under several scenarios by Monte-Carlo simula-tions. An8th order Butterworth pulse with a3 dB bandwidthof 500MHz [1] is used as the baseband UWB pulse. The IEEE802.15.4a multipath channel model CM1 - indoor LOS [29]is employed for simulations. The channel changes randomlyin each Monte-Carlo run. The average power of the channels

7

Detectors Σxx(∗)

(i, i, l, l)/Σxx(∗)

(i, ℓ) / Σxx(∗)

Matrix inverse TDG(

∗) (αi, τi,l) / T

DG(∗) (αi)

T I

prop(∗), T II

prop(∗)J(6K + 4L) 32J 6J

TDG(

∗) (6K + 4L)

M∑

i=1

N2i +

M−1∑

i=1

M∑

ℓ=i+1

NiNℓ

8J3 + 24J2 -

Tsum DG(

∗) , T

max DG(∗) (6K + 4L)

M∑

i=1

N2i

M∑

i=1

(

8N3i + 24N2

i

)

M∑

i=1

(

4N2i + 2Ni

)

Detectors Total number of operationsT I

prop(∗), T II

prop(∗)2JK + J(6K + 4L) + 38J

TDG(

∗) 2JK + (6K + 4L)

M∑

i=1

N2i +

M−1∑

i=1

M∑

ℓ=i+1

NiNℓ

+ 8J3 + 28J2 + 2J

Tsum DG(

∗) , T

max DG(∗) 2JK +

M∑

i=1

(

N2i (6K + 4L) + 8N3

i + 28N2i + 2Ni

)

Tad hoc(∗)

2JK + (Ln + 1)K + 2M(Ln + 1) + 2J

TABLE IITHE COMPUTATIONAL COMPLEXITY OF DIFFERENT DETECTORS

0 0.5 1 1.5 20

1

2

3

4

5

6x 10

−3

α (GHz)

The

est

imat

e of

Rxx

(α,τ

)

2α1x

16α1x

14α1x

10α1x

6α1x

Fig. 2. An example of the amplitude of the estimated CAFRxx(α, τ) vs.α without noise and multipath channel, andτ = 2ns.

are normalized. Thus, the received baseband UWB signalis a complex signal due to the multipath channel effect.Furthermore, in order to avoid the cyclic spectrum aliasing, thesampling rate is set to1GHz in the simulations. Without lossof generality, we are interested in detecting the UWB signalwith the highest symbol rate (31.2MHz). The parameters areset asNburst = 8, Nhop = 2, Ncpb = 2 and Tc = 2ns[1]. Therefore, the fundamental cyclic frequency of the UWBsignal is that2αx

1 = 2/Tdsym = 62.4MHz according to thederivations in Section III-B. The spectrum smoothing windowis selected as the Kaiser window of length65 with β being1 to avoid a higherPfa than the expectation based on theremarks in [23]. The length of the observation window (To)is 10µs. Thus, the total number of samples is about10, 000(K = 10, 000). The signal-to-noise ratio (SNR) is defined asSNR = 10log10σ

2x/σ

2n similarly as [19], whereσ2

x and σ2n

are the power of the UWB signal and the complex Gaussiannoise, respectively. The plotted detection performance curvesare averaged over1, 000 Monte-Carlo runs.

A. The cyclic features of the UWB signal

In order to clearly illustrate the cyclic features of the UWBsignal, the sampling rate is2GHz, and the TL is assigned

0 0.5 1 1.5 20

0.5

1

1.5x 10

−3

α (GHz)

The

est

imat

e of

Rxx

(α,τ

)

2α1x

−2α1x

−14α1x

−16α1x

16α1x

14α1x

10α1x

6α1x

Fig. 3. An example of the amplitude of the estimated CAFRxx(α, τ) vs.α under a noiseless multipath channel, andτ = 2ns.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−3

α (GHz)

The

est

imat

e of

Rxx

∗(α,

τ)

2α1x

−2α1x

−6α1x

−10α1x

14α1x

16α1x

−16α1x

−14α1x

Fig. 4. An example of the amplitude of the estimated conjugate CAFRxx∗(α, τ) vs. α under a noiseless multipath channel, andτ = 2ns.

to 2 ns in this subsection. Fig. 2 shows the amplitude of theestimated CAFRxx(α, τ) of the UWB signal without anynoise and multipath effects. AsRxx(α, τ) in the range of[1, 2]GHz is equivalent to the one in the range of[−1, 0]GHz,we denote the estimated CAFRxx(α, τ) in the range of[1, 2]GHz (or [0, 1]GHz) as Rxx(α

−, τ) (or Rxx(α+, τ)).

Without the multipath channel effect, the UWB signal is areal signal. Therefore,|Rxx(α

−, τ)| and |Rxx(α+, τ)| are

8

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pd

To = 10 µs, Ω

1

To = 10 µs, Ω

2

To = 20 µs, Ω

1

To = 20 µs, Ω

2

Tprop I

Tprop II

Tsum_DG

TED

, ∆ = 0 dB

TED

, ∆ = 1 dB

TED

, ∆ = 2 dB

Fig. 5. The comparison ofPd among different detectorsT Iprop (22), T II

prop(25), Tsum DG (33), andTED (41) using different CF and TL sets withdifferent observation window lengths,Pfa = 0.01.

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pd

Ω1

Ω2

Ω3

TDG

Tsum_DG

Tmax_DG

Fig. 6. The comparison ofPd among different detectorsTx (17),Tsum DG

(33), andTm (35) using different CF and TL sets,To = 10 µs, Pfa = 0.01.

symmetric w.r.t.1 GHz. Significant peaks of|Rxx(α, τ)| are atthe expected CFs±2αx

1 ,±6αx1 ,±10αx

1 ,±14αx1 ,±16αx

1 . Theamplitudes of the peaks are in the tendency to decrease as thecycle frequency increases. Since the bandwidth of the UWBsignal is500 MHz, no significant peaks are observed beyond500 MHz.

When the transmitted UWB signal goes through a noiselessmultipath channel, the amplitudes of the estimated CAFs of thereceived UWB signal are depicted in Fig. 3. Some expectedpeaks disappear due to the multipath channel effects. Notethat |R

xx(∗)(α−, τ)| and |R

xx(∗)(α+, τ)| are not symmetric

anymore. The amplitude of the estimated conjugate CAFsare illustrated in Fig. 4 as well. Clear peaks at multiple CFsare observed in these figures. Hence, both the CAF and theconjugate CAF estimates can be used to detect the UWBsignal.

B. Multi-cycle multi-lag detection under multipath channels

In this subsection, we investigate the detection performanceof various detectors under multipath channels with complexGaussian noise and colored Gaussian noise, respectively. Thefalse-alarm ratePfa is fixed to0.01 for all the detectors.

1) Complex Gaussian noise: Two sets of CFs and TLs arechosen. The first setΩ1 = ±2αx

1 , 2 ns, 4 ns, and the secondoneΩ2 = ±2αx

1 ,±6αx1 , 2 ns, 4 ns. Every CF inΩ1 (or Ω2)

shares the same set of TLs (2 ns, 4 ns). Thus, there are intotal four and eight CF-TL pairs forΩ1 andΩ2, respectively.

In Fig. 5, the proposed detectorsT Iprop andT II

prop are eval-uated under multipath channels and with different observationwindow lengths (denoted byTo in Fig. 5). Moreover, theyare compared with the detectorsTsum DG andTED, which arereviewed in (33) and (41), respectively. Witnessed by Fig. 5,employing more CFs for the proposed detectors does notsubstantially improve the detection performance. On the otherhand, due to the increased degree of freedom, the detectionperformance of the proposed detectors using the parametersetΩ2 (the dotted and the dash-dot lines) is worse than theone usingΩ1 (the solid and the dashed lines). In general,the Pd curves ofT I

prop and T IIprop are close to each other.

Longer observation time facilities the improvement of thedetection. In addition, the detection performance ofTsum DG

with To = 10µs is depicted in Fig. 5 as well. It is slightlybetter than the performance of the proposed detectors withthe same parameter settings. Furthermore, the energy detectorTED with perfect knowledge of the noise variance (∆ = 0)achieves the best detection performance, whenTo = 10µs.There is about8 dB performance gain ofTED over theproposed detectors. However, the energy detector is sensitiveto noise uncertainty. When the employed noise variance isuniformly distributed between±∆ 6= 0 of the perfect noisevariance, the detection performance of the energy detectordegrades dramatically. Additionally, the energy detectorbarelyguarantees the expectedPfa due to the noise uncertainty at lowSNR.

In Fig. 6, the multi-cycle multi-lag Dandawate-Giannakisdetector TDG in (17) is compared with its two varia-tions (Tsum DG in (33) andTmax DG in (35)). Besides thesets Ω1 and Ω2, another set applied here is thatΩ3 =±2αx

1 ,±6αx1 , 2 ns. With both Ω1 and Ω3, the detection

curves ofTsum DG closely follow the ones ofTDG, whileTmax DG always performs the worst. In the agreement withFig. 5, TLs help more than CFs w.r.t. the detection perfor-mance. Since thePd of the detectorTDG using the setΩ2 atlow SNR is much higher than the expectedPfa, the detectorsemploying the setΩ1 performs the best overall. Moreover,the detectorTsum DG can replace the detectorTDG in thefavor of the computational complexity. Hence, the detectorTsum DG represents the Dandawate-Giannakis type detectorsfor comparison in the following subsections.

2) Colored Gaussian noise: The colored Gaussian noiseis generated in the same way as in [23], where a complexGaussian noise goes through a three-tap linear filter with coef-ficients0.3, 1, 0.3. The parameterLn is set to5 similarly asin [23]. The detection performance of several detectors (T I

prop,T IIprop, Tsum DG, andTad hoc) vs. SNR is compared in Fig. 7.

The detectorTad hoc reviewed in (36) marginally outperformsthe rest in Fig. 7, as it is dedicated to deal with coloredGaussian noise. The detectors employing the parameter setΩ1 are better than the ones usingΩ2. Major performancedifferences exist at the medium SNR range, and the detectionperformance converge at high SNR.

In order to show more insight about the performance ofthe detectors, the receiver operating characteristics (ROCs) of

9

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pd

Ω1

Ω2

TpropI

TpropII

Tsum_DG

Tad_hoc

Fig. 7. The comparison ofPd among different detectorsT Iprop (22), T II

prop(25), Tsum DG (33), andTad hoc (36) using different CF and TL sets withcolored Gaussian noise,To = 10µs, Pfa = 0.01.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pfa

Pd

SNR = −2 dB,Gaussian noiseSNR = 2 dB,Gaussian noiseSNR = −2 dB,colored noiseSNR = 2 dB,colored noise

TpropI

TpropII

Tsum_DG

Tad_hoc

Fig. 8. The comparison of ROC among different detectorsT Iprop (22),

T IIprop (25),Tsum DG (33), andTad hoc (36) using the setΩ1 with complex

Gaussian noise and colored Gaussian noise, respectively,To = 10 µs

different detectors are evaluated in Fig. 8 withSNR = −2 dBand SNR = 2 dB, respectively. The ROC performance ofall detectors under complex Gaussian noise outperforms thatunder colored Gaussian noise as indicated in Fig. 8. The higherSNR is, the better ROC performance is. In general, the ROCcurves of all the detectors are close to each other. The ROCsof Tsum DG and Tad hoc are slightly better than that of theproposed detectors with higher complexity.

C. Multi-cycle multi-lag detection with narrow band interfer-ences under multipath channels

According to the IEEE 802.15.4a standard [1], one possiblesignal occupying the same frequency bands as the UWB signalis from the IEEE 802.16 standard [6] systems (e.g. WiMAXsystems). Therefore, the OFDM signals based on the IEEE802.16 standard are used as a narrow band interference. In thissubsection, we evaluate the probability of detecting the UWBsignal, which coexists with the OFDM signal under multipathchannels.

According to the IEEE 802.16 standard [6], the baseband

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

Expected Pfa

Act

ual P

fa

E[Pfa

]

Tprop

∗I

Tprop

∗II

Tsum_DG

∗

Tad_hoc

∗

INR = 20 dB

INR = 0 dB

INR = −20 dB

Fig. 9. The comparison of the simulatedPfa vs. the expectedPfa

among different detectorsT Iprop∗ (22), T II

prop∗ (25), Tsum DG∗ (33), andTad hoc∗ (39) using the parameter setΩ1, whenTo = 10 µs, and INR =−20 dB, 0 dB, 20 dB.

cyclic prefix OFDM signal can be written as

s(t) =

∞∑

l=−∞

Nused/2∑

n=−Nused/2,n6=0

dn,lg(t− lTs − ζ)

×ej2πn∆f(t−lTs−ζ), (43)

where the parameters in (43) and their values assigned in thesimulations are listed in Table III. It is well known that thefundamental cyclic frequencyαs

1 of the OFDM signal is thesymbol rate1/Tsym, and significant CAF values manifest atthe TLs±Td [17], [21]. Thus, the OFDM signal has differentcyclic features from the UWB signal. Furthermore, the OFDMsignal does not have the conjugate cyclic features when it issampled at the rate1/(NcTd). Note that the received signalis sampled at1 GHz in order to avoid spectrum aliasing forthe UWB signal. Due to this oversampling, the OFDM signalcontributes to the conjugate CAF as well. Nevertheless, theestimated conjugate CAFs are still be applied here to calculatethe test statistic.

Moreover, the bandwidth of the OFDM signal is20MHz.Its carrier frequencyf ′

c is randomly generated in the rangeof [−240, 240]MHz in each Monte-Carlo run. The OFDMsignal goes through a Rayleigh fading channel of20 taps withan exponentially decaying power delay profile. The averagepower of the channel is normalized. The signal-to-interferenceratio (SIR) is defined as SIR= 10log10(σ

2x/σ

2s), whereσ2

s isthe power of the OFDM signal. The parameterLn for Tad hoc∗

is set to30 to deal with the OFDM interference as a colorednoise.

1) Complex Gaussian noise: As the OFDM interference isjust in the band of interest, there may be circumstances thatonly the OFDM interference and the complex Gaussian noiseexist. Under such circumstance, the actualPfa may be differentfrom the expectation. Let us define the interference-to-noiseratio (INR) as INR= 10log10(σ

2s/σ

2n). Consequentially, the

difference between the actualPfa’s (the simulated ones) andthe expectedPfa’s of various detectors using the conjugateCAF estimates is explored in Fig. 9. The parameter setΩ1

is employed. When INR =−20 dB, the noise dominates.The actualPfa values of the detectors (the dotted lines)

10

g(t) the pulse function of lengthTs (e.g. the rectangular function)ζ the unknown deterministic timing offsetNc the number of subcarriers,Nc = 256

Nused the number of subcarriers used for data,Nused = 200∆f the carrier separation,∆f = 20MHz/Nc = 78.125KHzTd the length of the data block, whereTd = 1/∆fρ the ratio of the cyclic prefix block to the data block,ρ = 0.25

Tcp the length of the cyclic prefix block,Tcp = ρTd

Tsym the length of the OFDM symbol, whereTsym = Td + Tcp = (1 + ρ)Td

dn,l the lth data symbol modulates thenth carrier, QPSK modulation,dn,l ∈ ±1/√2± j/

√2

TABLE IIIPARAMETERS FOR THEIEEE 802.16 WIMAX OFDM SIGNAL [6].

−10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pd

Ω1

Ω2

Tprop

∗I

Tprop

∗II

Tsum_DG

∗

Tad_hoc

∗

TED

Fig. 10. The comparison ofPd among the detectorsT Iprop∗ (22), T II

prop∗

(25),Tsum DG∗ (33),Tad hoc∗ (39) andTED (41) using different CF and TLsets with the OFDM interference,To = 10µs, Pfa = 0.01, SIR = −5dB.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pfa

Pd

SIR = −5 dB, SNR = 0 dBSIR = 0 dB, SNR = 0 dBSIR = −5 dB, SNR = 5 dBSIR = 0 dB, SNR = 5 dB

Tprop

∗I

Tprop

∗II

Tsum_DG∗

Fig. 11. The comparison of ROC among the detectorsT Iprop∗ (22), T II

prop∗

(25), Tsum DG∗ (33), Tad hoc∗ (39) andTED (41) using the parameter setΩ1 with the OFDM interference,To = 10µs.

generally follow the expectedPfa (the dash-dotted line) values.However, they deviate to some extent from the expectation,when the expectedPfa increases. When INR increases to0 dB, the actualPfa of the detectorTad hoc∗ is larger than theexpectation obviously in the range of small expectedPfa’s. Itconverges to the expectation after the expectedPfa = 0.2.Moreover, the actualPfa of the detectorT I

prop∗ is slightlysmaller the expectation, which is harmless. ThePfa’s of otherdetectors are still around the predesigned values. When INR=20 dB, the interference dominates. The actualPfa’s of T II

prop∗

and Tsum DG∗ closely follow the benchmark. On the otherhand, the actualPfa’s of T I

prop∗ and Tad hoc∗ are below thebenchmark, thus the thresholds are overestimated. ThePd’s

−10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Pd

Ω1

Ω2

Tprop

∗I

Tprop

∗II

Tsum_DG

∗

Tad_hoc

∗

TED

Fig. 12. The comparison ofPd among the detectorsT Iprop∗ (22), T II

prop∗

(25), Tsum DG∗ (33), Tad hoc∗ (39) andTED (41) using different CF andTL sets with OFDM interference under colored Gaussian noise, To = 10µs,Pfa = 0.01, SIR = −5 dB.

would decrease as well. Especially, thePfa curve ofTad hoc∗

is almost flat and far below the benchmark. The degradationof its detection performance would be obvious.

Now let us evaluate the detection performance under theOFDM interference and complex Gaussian noise. The de-tection performance of various detectors are indicated inFig. 10, where SIR= −5 dB. The OFDM interferencedramatically changes the performance of the detectors. ThedetectorTad hoc∗ is the worst. It fails to detect the signal ofinterest due to its oversimplified covariance estimation. Thisresult is consistent with the prediction based on Fig 9. Re-gardless of the correlation between different CFs, the detectorTsum DG∗ exploits the correlation ofRxx∗(αi) using the sameCF but different TLs, and it performs best. The proposeddetectors (T I

porp∗ andT IIprop∗ ) neglect the correlation between

Rxx∗(αi, τi,l) of different CF-TL pairs, and thus suffer fromthe performance degradation. It trades the detection perfor-mance with the low complexity. Moreover, the energy detectorTED fails to distinguish the interference and the signal ofinterest, and thus itsPd is always1 by mistakenly regardingthe interference as the signal.

In Fig. 11, the ROC is investigated under different com-binations of SIR and SNR using the parameter setΩ1. Thehigher SIR and SNR are, the higher probability of detection is.The detection performancePd benefits more from the higherSNR than the higher SIR. The detectorTsum DG∗ is obviouslysuperior to the proposed detectors in good agreement withFig. 10.

11

2) Colored Gaussian noise: In the case of the OFDMinterference and the colored Gaussian noise, the performanceof detectors further degrades as shown in Fig. 12, wherethe colored Gaussian noise is generated in the same way asin Section VI-B2. Different detectors behave in the similartendency as the ones in Fig. 10. Note that the detectorTad hoc∗

designed for the colored noise cannot detect due to the OFDMinterference. Since the ROC performance of the proposeddetectors and the detectorTsum DG∗ under colored Gaussiannoise maintains the same properties as the one under complexGaussian noise, it is not shown in this paper to save the space.

VII. C ONCLUSIONS

In this paper, we propose multi-cycle multi-lag cyclic fea-ture detectors for UWB receivers in heterogeneous environ-ments. The unique cyclic features of UWB signals based onthe IEEE 802.15.4a standard are analyzed. Due to the ultrawide bandwidth, the UWB signal manifests itself at severalcyclic frequencies. Furthermore, the multipath channel effectsincrease the range of its cyclic features w.r.t. the time lag. Theconstant false alarm rate detectors based on cyclic features areproposed accordingly. Their computational complexities aresignificantly less than the ones of the conventional Dandawate-Giannakis detector and its variations. Extensive simulationresults indicate that the proposed detectors introduce tradeoffsbetween the detection performance and the computationalcomplexity in various scenarios, such as multipath channels,colored Gaussian noise and OFDM interferences.

APPENDIX ADERIVATION OF Rxx(t, τ)

Making use of the Fourier transform pairp(t) =∫∞

−∞P (f)ej2πftdf , whereP (f) is the Fourier transform of

p(t), the signalx(t) (6) can be written as

x(t)=

+∞∑

k=−∞

ak

Ncpb−1∑

n=0

cn+kNcpb

∫ P (f)ej2πf(t−ǫ−nTc)

×e−j2πfkTdsyme−j2πfbkTBPM e−j2πfh(k)Tburst

df. (44)

Sinceck andak take values from±1 with equal probability,we obtain thatE[aka

∗k−l] = δ(l) and E[ckc

∗k−l] = δ(l),

where δ(l) is the delta function. Furthermore, let us defineβ(f) = E[e−j2πfbkTBPM ] and η(f) = E[e−j2πfh(k)Tburst ].Using these definitions, and plugging (44) into (1), we arriveat

Υxx(t, τ) (45)

=

+∞∑

k=−∞

Ncpb−1∑

n=0

∫ ∫ P (y)P ∗(z)β(y − z)η(y − z)

×ej2π((y−z)(t−ǫ)+τ(y+z)/2−Tdsym(y−z)k−Tc(y−z)n)dydz.

According to the Poisson sum formula+∞∑

k=−∞

e−j2πkfT =

1

T

+∞∑

q=−∞

δ(f −q

T), and denotingαx

q = q/Tdsym, q ∈ Z, the

equation (45) can be further rewritten w.r.t.k summations as(9).

APPENDIX BDERIVATION OF Rxx(α

xq , τ)

Recalling thatTburst = NcpbTc andTdsym = NburstTburst

in Table I, the CAFRxx(αxq , τ) as the Fourier coefficient of

Υxx(t, τ) is given by

Rxx(αxq , τ) = αx

1e−j2παx

q ǫβ(αxq )η(α

xq )φp(α

xq , τ)

×w (q/(NburstNcpb), Ncpb) . (46)

Note that bk and h(k) select values from0, 1 and0, 1, . . . , Nhop − 1 with equal probability, respectively, andthey are independent with each other. Making use ofTBPM =Tdsym/2 in Table I, we arrive at

β(αxq ) =

1 + e−j2πTBPMαxq

2

=1 + (−1)q

2=

1 q = 0,±2,±4, . . .0 q = ±1,±3, . . .

,(47)

η(αxq ) =

1

Nhop

Nhop−1∑

n=0

e−j2πTburstαxqn

=1

Nhopw (q/Nburst, Nhop) . (48)

As β(αxq ) is nonzero only whenq is an even number, the

fundamental CF is doubled as2αx1 . For the sake of brevity,

we abuseq ∈ Z. Plugging (47) and (48) into (46), the CAFRxx(2α

xq , τ) of x(t) can be written as (12).

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