theoretical performance of the fft filter bank-based

Theoretical Performance of the FFT Filter Bank-Based Summation Detector

Sichun Wang and Robert Inkol

Defence R&D Canada √ Ottawa

TECHNICAL REPORT DRDC Ottawa TR 2005-153

December 2005

Theoretical Performance of the FFT FilterBank-Based Summation Detector

Sichun WangDefence R&D Canada – Ottawa

Robert InkolDefence R&D Canada – Ottawa

Defence R&D Canada – Ottawa

Technical Report

DRDC Ottawa TR 2005-153

December 2005

Author

Sichun Wang, Robert Inkol

Approved by

Bill KatsubeHead/Communications and Navigation Electronic Warfare Section

Approved for release by

Bert Bridgewater/Document Review Panel

c© Her Majesty the Queen as represented by the Minister of National Defence, 2005

c© Sa Majeste la Reine, representee par le ministre de la Defense nationale, 2005

Abstract

This technical report derives the probabilities of detection and false alarm for theFFT filter bank-based summation detector when the received signal is a complexpure tone embedded in additive white Gaussian noise. These results are useful forthe performance analysis of the FFT filter bank-based summation detector and canbe used to set up the detector for operation at a desired constant false alarm rateand predict the detection performance.

Resume

Le present rapport technique presente une derivation des probabilites de detection etde fausses alarmes pour le detecteur de sommation fonde sur les bancs de filtres a TFRlorsque le signal recu est un son pur complexe integre a du bruit blanc gaussien additif.Les resultats obtenus sont utiles pour l’analyse de la performance d’un detecteur desommation fonde sur les bancs de filtres a TFR et peuvent etre utilises pour reglerle fonctionnement du detecteur au taux de fausses alarmes constant voulu et pourpredire la performance de detection.

DRDC Ottawa TR 2005-153 i

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ii DRDC Ottawa TR 2005-153

Executive summary

Theoretical Performance of the FFT Filter Bank-BasedSummation Detector

Sichun Wang, Robert Inkol; DRDC Ottawa TR 2005-153; Defence R&D Canada –Ottawa; December 2005.

Channelized receivers are useful for performing the fast detection and center frequencyestimation of narrowband signals. The application of FFT filter bank techniques hasattracted considerable interest as a result of the computational efficiency of the FFTalgorithm and the availability of function specific FFT hardware [1]-[8]. One of thepractical limitations of FFT filter bank detectors is that the restriction of the FFTlength to powers of two for common implementations of the algorithm limits thechoice of the channel spacing, particularly when the sampling rate of the analog-to-digital converters, used to convert an analog signal from a sensor to a digital signalfor subsequent processing, is fixed for technical reasons.

The FFT filter bank-based summation detector is a modification of the basic FFTfilter bank detector. Whereas the basic FFT filter bank detector performs detectionby comparing the power computed for each FFT bin to a detection threshold, the FFTsummation detector groups the FFT bins to correspond to the desired channelizationand forms an estimate of the power contained in each channel by summing the powercomputed for the individual bins. If the FFT length is sufficiently large, an arbitrarychannelization scheme can be approximated if the number of FFT bins assigned toeach channel is allowed to vary. A further idea is that the bandwidth of each detectorcan be adjusted by summing the power for a subset of the bins assigned to eachchannel. In many practical applications, the enhanced flexibility in defining channelbandwidths and center frequencies is very useful.

Although the performance of the FFT summation detector has been discussed inprevious reports, a complete derivation of the performance analysis has not beenpreviously published. This report presents a thorough treatment of the theoreticalderivation of the formulas for probabilities of detection and false alarm. These resultsare very useful for analyzing the performance bounds of practical sensor systems andare extensible to similar problems.

DRDC Ottawa TR 2005-153 iii


iv DRDC Ottawa TR 2005-153

Sommaire

Theoretical Performance of the FFT Filter Bank-BasedSummation Detector

Sichun Wang, Robert Inkol; DRDC Ottawa TR 2005-153; R & D pour la defenseCanada – Ottawa; decembre 2005.

Les recepteurs multicanaux sont utiles pour detecter rapidement et estimer la frequencecentrale des signaux en bande etroite. L’application de techniques fondees sur lesbancs de filtres a TFR (transformee de Fourier rapide) a suscite un interet considerableen raison de l’efficacite de calcul de l’algorithme TFR et de la disponibilite du materielTFR a fonction specifique [1]-[7]. Une des limites concretes des detecteurs a bancs defiltres TFR est qu’en raison de la longueur restreinte de la TFR a des puissances dedeux pour les applications communes de l’algorithme, le choix en matiere d’espace-ment entre canaux est limite, particulierement lorsque le taux d’echantillonnage desconvertisseurs analogiques/numeriques, qui servent a convertir un signal analogiqueprovenant d’un capteur en un signal numerique pour fins de traitement ulterieur, estfixe pour des raisons d’ordre technique.

Le detecteur de sommation fonde sur les bancs de filtres a TFR est un detecteurde bancs de filtres a TFR de base auquel des modifications ont ete apportees. Ledetecteur de bancs de filtres a TFR de base remplit sa fonction en comparant lapuissance calculee pour chaque intervalle TFR a un seuil de detection, tandis que ledetecteur de sommation TFR regroupe les intervalles TFR pour qu’ils correspondentau decoupage en canaux voulu et donne une estimation de la puissance de chaquecanal en faisant la somme de la puissance calculee pour chacun des intervalles indivi-duels. Si la longueur de la TFR est suffisante, on peut estimer un plan de decoupage encanaux arbitraire lorsque le nombre d’intervalles TFR assignes a chaque canal peut va-rier. Une autre possibilite consisterait a regler la largeur de bande de chaque detecteuren totalisant la puissance d’un sous-ensemble d’intervalles assignes a chaque canal.Dans beaucoup d’applications pratiques, la souplesse accrue en matiere de definitionde largeur de bande des canaux et de frequences centrales est tres utile.

Bien que la performance du detecteur de sommation TFR ait fait l’objet d’une dis-cussion dans des rapports anterieurs, la derivation complete de l’analyse de perfor-mance n’avait pas ete publiee. Le present rapport traite en profondeur de la derivationtheorique des formules utilisees pour calculer les probabilites de detection et les faussesalarmes. Les resultats obtenus sont tres utiles pour analyser les limites de perfor-mance des systemes de detection pratiques et peuvent etre appliques a la resolutionde problemes similaires.

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vi DRDC Ottawa TR 2005-153

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 The FFT Filter Bank-Based Summation Detector . . . . . . . . . . . . . . 2

3 Eigenvalues of the Covariance Matrix H = E(ZnZ

Hn

). . . . . . . . . . . 4

4 Probability of Detection (0 < γ ≤ 1/2, N ≥ 1, L ≥ 1) . . . . . . . . . . . . 12

5 Probability of Detection (γ = 0, N > 1, L > 1) . . . . . . . . . . . . . . . . 25

6 Probability of Detection (γ = 0, K = N = 1, L ≥ 1) . . . . . . . . . . . . . 32

7 Probability of False Alarm (0 < γ ≤ 1/2, N ≥ 1, L ≥ 1) . . . . . . . . . . . 34

8 Probability of False Alarm (γ = 0, N > 1, L > 1) . . . . . . . . . . . . . . 34

9 Probability of False Alarm (γ = 0, K = N = 1, L ≥ 1) . . . . . . . . . . . . 35

10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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viii DRDC Ottawa TR 2005-153

1 Introduction

Channelized receivers are useful for performing the fast detection and center frequencyestimation of narrowband signals. The application of FFT filter bank techniques hasattracted considerable interest as a result of the computational efficiency of the FFTalgorithm and the availability of function specific FFT hardware [1]-[8]. One of thepractical limitations of FFT filter bank detectors is that the restriction of the FFTlength to powers of two for common implementations of the algorithm limits thechoice of the channel spacing, particularly when for technical reasons the samplingrate of the analog-to-digital converters, used to convert an analog signal from a sensorto a digital signal for subsequent processing, is fixed.

The FFT filter bank-based summation detector is a modification of the basic FFTfilter bank detector. Whereas the basic FFT filter bank detector performs detectionby comparing the power computed for each FFT bin to a detection threshold, the FFTsummation detector groups the FFT bins to correspond to the desired channelizationand forms an estimate of the power contained in each channel by summing the powercomputed for the individual bins. If the FFT length is sufficiently large, an arbitrarychannelization scheme can be approximated if the number of FFT bins assigned toeach channel is allowed to vary. A further idea is that the bandwidth of each detectorcan be adjusted by summing the power for a subset of the FFT bins assigned to eachchannel. In many practical applications, the enhanced flexibility in defining channelbandwidths and center frequencies is very useful.

In [11], [12], the theoretical performance of the FFT majority and summation detec-tors designed to provide a constant false-alarm rate (CFAR) was compared. It wasfound that the FFT majority detector performed within 1 dB of the performanceof the FFT summation detector for CFAR operation. This conclusion was derivedfrom closed-form algebraic formulas for the probability of false alarm Pfa and theprobability of detection Pd for the FFT summation detector, with the received signalbeing a pure tone embedded in additive white Gaussian noise.

Due to space limitations, the full derivation for Pfa and Pd was omitted in [11],[12]. In this report, we present the full derivation for Pfa and Pd. The formulas forPd and Pfa presented in this report provide the basis for comparing the theoreticalperformance of the FFT summation detector and other closely related detectors. Itis noted that the formulas for Pd and some of the formulas for Pfa presented herehave not been derived elsewhere and therefore our results extend and complementthe literature on FFT filter bank-based CFAR detectors.

This report is organized as follows. In section 2, notation and definitions used inthis report are introduced. In section 3, a Hermitian covariance matrix is computed,the distinct eigenvalues of which play a central role in computing Pfa and Pd for the

DRDC Ottawa TR 2005-153 1

FFT summation detector. In sections 4, 5 and 6, formulas for Pd are obtained foroverlapping and non-overlapping data blocks. In sections 7, 8 and 9, formulas for Pfa

are derived as corollaries from the corresponding formulas for Pd obtained in sections4, 5 and 6. Finally, in section 10, results obtained in this report are summarized. Forreaders not interested in the mathematical details, they can go directly to section 10to locate the relevant formulas they are interested in.

2 The FFT Filter Bank-Based SummationDetector

Various aspects of the FFT filter bank-based summation detector have already beeninvestigated (see [3], [9]-[15], and the references therein). To be consistent with thereport [12], we shall use similar notation and terminology. In this section, we brieflydescribe how the power level for a channel in the FFT filter bank-based summationdetector is computed. The reader is referred to the references at the end of this reportfor additional details ([3], [9]-[15]).

Assume there are M channels in the sampled bandwidth, with the channels beinguniformly distributed in frequency. Assume an integer number of K FFT bins areassigned per channel. Out of the K FFT bins per channel, N center bins are usedto estimate the power. Hence a FFT of length M × K is needed to compute thepower for the M channels. In this report, without loss of generality, it is assumedthat the integers K and N are both a power of 2 and K − N is an even integer.Let w = [w0, · · · , wMK−1]

t be a linear phase FIR filter of length MK, where thesuperscript t denotes matrix or vector transposition. Dividing the input vector r =[r(L−1)(1−γ)MK+MK−1, · · · , r1, r0]t into L overlapping sample vectors R0, · · ·, RL−1 asfollows :

Rm = [rm(1−γ)MK+MK−1, rm(1−γ)MK+MK−2, · · · , rm(1−γ)MK ]t (1)

where each vector Rm has γMK samples in common with the preceding vector Rm−1

and 0 ≤ γ < 1 is the overlapping ratio. The vectors Rm are windowed by thewindowing sequence w, resulting in the windowed sample vectors Xm:

Xm = [w0rm(1−γ)MK+MK−1, w1rm(1−γ)MK+MK−2, · · · , wMK−1rm(1−γ)MK ]t (2)

The vectors Xm are then transformed by the inverse discrete Fourier transform matrixF of dimensions MK×MK to yield a corresponding sequence of FFT sample vectorsYm:

Ym = FXm = [y0,m, y1,m, · · · , yMK−1,m]t (3)

2 DRDC Ottawa TR 2005-153

where

F =

⎡⎢⎢⎢⎢⎣

1 1 · · · 1· · · · · · · · · · · ·1 exp 2πjl

MK· · · exp 2πj(MK−1)l

MK· · · · · · · · · · · ·1 exp 2πj(MK−1)

MK· · · exp 2πj(MK−1)(MK−1)

MK

⎤⎥⎥⎥⎥⎦ (4)

and j =√−1. We remark here that even though it is the inverse FFT matrix F that

is used in the FFT filter bank, we simply call the vectors Ym FFT sample vectorsand a sample in Ym a FFT bin. Corresponding to each sample vector Ym, a decisionvector zm = [z0,m, · · · , zM−1,m]t of dimension M is formed by summing the powerfrom the N center FFT bins in each channel:

zn,m =

N−1∑l=0

|ynK+ K−N2

+l,m|2, n = 0, 1, 2, · · · ,M − 1 . (5)

That is, the power from the N FFT bins with indices nK+ K−N2

+0, nK+ K−N2

+1,· · ·, nK+ K−N

2+(N −1) is summed to form the power for the n-th channel which is

assigned the K FFT bins with indices nK + 0, nK + 1, · · ·, nK +K − 1. The finaldecision vector z = [z0, z1, · · · , zM−1]

t is obtained by summing the individual decisionvectors zm :

z =L−1∑m=0

zm (6)

We have

zn =

L−1∑m=0

zn,m =

L−1∑m=0

N−1∑l=0

|ynK+ K−N2

+l,m|2 , n = 0, 1, 2, · · · ,M − 1 . (7)

Given a probability of false alarm Pfa, a threshold T is first computed such thatPr {zn > T} = Pfa, where the power level zn for the n-th channel is computed fromthe input data record r = [r(L−1)(1−γ)MK+MK−1, · · · , r1, r0]t that is assumed to containadditive white Gaussian noise (AWGN) only. The FFT filter bank-based summationdetector declares the presence of a signal in the n-th channel if zn > T and declaresthe absence of a signal in the n-th channel if zn ≤ T . In the rest of this report, forbrevity, this detector will simply be called the L-block FFT summation detector orsimply the FFT summation detector.

The signal power zn computed for the n-th channel (c.f. (7)) is a quadratic form inthe input random variables. In fact,

zn,m =

N−1∑l=0

|ynK+ K−N2

+l,m|2 =

N−1∑l=0

(ynK+ K−N2

+l,m)(ynK+ K−N2

+l,m)∗

= ZHn,mZn,m (8)


where

Zn,m = [ynK+ K−N2

,m, ynK+ K−N2

+1,m, · · · , ynK+ K−N2

+N−1,m]t

= FnWRm (9)

In (9), Rm is the (m + 1)-th sample vector defined by (1), W is the MK ×MKdiagonal matrix with its diagonal elements defined by the windowing sequence w:

W =

⎡⎢⎢⎣w0 0 · · · · · · 00 w1 · · · · · · 0· · · · · · · · · · · · · · ·0 · · · · · · · · · wMK−1

⎤⎥⎥⎦ (10)

and Fn is the N ×MK matrix consisting of the N rows of the MK ×MK inversediscrete Fourier transform matrix F with row indices nK + K−N

2, nK + K−N

2+1, · · ·,

nK + K−N2

+N − 1:

Fn =

⎡⎢⎢⎢⎢⎢⎣

1 exp 2πjIn

MK· · · exp 2πj(MK−1)In

MK· · · · · · · · · · · ·1 exp 2πj(In+l)

MK· · · exp 2πj(MK−1)(In+l)

MK

· · · · · · · · · · · ·1 exp 2πj(In+N−1)

MK· · · exp 2πj(MK−1)(In+N−1)

MK

⎤⎥⎥⎥⎥⎥⎦ (11)

where In = nK + K−N2

. In matrix form, the signal power zn for the n-th channel isa quadratic form defined by:

zn =

L−1∑m=0

zn,m =

L−1∑m=0

ZHn,mZn,m =

⎡⎢⎢⎢⎢⎢⎢⎣

Zn,0

Zn,1

· · ·Zn,m

· · ·Zn,L−1

⎤⎥⎥⎥⎥⎥⎥⎦

H ⎡⎢⎢⎢⎢⎢⎢⎣

Zn,0

Zn,1

· · ·Zn,m

· · ·Zn,L−1

⎤⎥⎥⎥⎥⎥⎥⎦

=

L−1∑m=0

RHmWFH

n FnWRm (12)

3 Eigenvalues of the Covariance MatrixH = E

(ZnZ

Hn

)Assume the input sample vectors Rm (c.f.(1)) contain zero-mean additive white Gaus-sian noise only; specifically, assume the input samples r0, r1, · · · , rk, · · · , are indepen-dent, identically distributed, zero-mean Gaussian random variables with E(|rk|2) =


σ2. For 0 ≤ n ≤M − 1, define the Gaussian random vector Zn by

Zn =

⎡⎢⎢⎢⎢⎢⎢⎣

Zn,0

Zn,1

· · ·Zn,m

· · ·Zn,L−1

⎤⎥⎥⎥⎥⎥⎥⎦

(13)

where Zn,m is defined by (9). As will be shown in the following sections, the eigen-values of the covariance matrix H = E

(ZnZ

Hn

)play an important role in the com-

putation of the probability of false alarm Pfa and the probability of detection Pd forthe FFT summation detector. Here in this section, we compute the matrix H for thepractically important case γ ≤ 1/2 and then investigate its eigenvalue distribution.We shall see that the eigenvalues of H roughly cluster into N groups and each grouphas L eigenvalues which are approximately evenly spaced. This property of eigenval-ues of H is very useful in the computation of threshold T for the FFT summationdetector.

The covariance matrix H is given by

H = E(ZnZ

Hn

)(14)

= E

⎛⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎣

Zn,0

Zn,1

· · ·Zn,m

· · ·Zn,L−1

⎤⎥⎥⎥⎥⎥⎥⎦[ZH

n,0, ZHn,1, · · · , ZH

n,m, · · · , ZHn,L−1

]⎞⎟⎟⎟⎟⎟⎟⎠

=

⎡⎢⎢⎢⎢⎣E(Zn,0Z

Hn,0

)E(Zn,0Z

Hn,1

) · · · · · · E(Zn,0Z

Hn,L−1

)· · · · · · · · · · · · · · ·E(Zn,mZH

n,0

)E(Zn,mZH

n,1

) · · · · · · E(Zn,mZH

n,L−1

)· · · · · · · · · · · · · · ·E(Zn,L−1Z

Hn,0

)E(Zn,L−1Z

Hn,1

) · · · · · · E(Zn,L−1Z

Hn,L−1

)

⎤⎥⎥⎥⎥⎦

For any l, m, 0 ≤ l ≤ m ≤ L− 1,

E(Zn,lZ

Hn,m

)= E((FnWRl) (FnWRm)H

)= E

(FnWRlR

HmWHFH

n

)= FnWE

(RlR

Hm

)WFH

n (15)

where

E(RlR

Hm

)= (16)


σ2

⎡⎢⎢⎢⎢⎣

δ(m−l)(1−γ)MK · · · δ(m−l)(1−γ)MK−MK+1

· · · · · · · · ·δ(m−l)(1−γ)MK+p−1 · · · δ(m−l)(1−γ)MK−MK+p

· · · · · · · · ·δ(m−l)(1−γ)MK+MK−1 · · · δ(m−l)(1−γ)MK

⎤⎥⎥⎥⎥⎦

In (16), δk = 0, if k �= 0 and δk = 1 if k = 0. Since γ ≤ 12, it follows that if |m− l| ≥ 2,

then|(m− l)(1 − γ)MK| ≥ 2(1 − γ)MK ≥MK

Therefore, (16) and (15) imply that if |m−l| ≥ 2, then E(RlR

Hm

)= 0, E

(Zn,lZ

Hn,m

)=

0 and the covariance matrix H = E(ZnZ

Hn

)in (14) becomes the following tridiagonal

L× L block matrix

H = E(ZnZ

Hn

)=

⎡⎢⎢⎢⎢⎢⎢⎣

A B 0 · · · 0 0BH A B 0 · · · 00 BH A B · · · 0· · · · · · · · · · · · · · · · · ·0 · · · 0 BH A B0 · · · 0 · · · BH A

⎤⎥⎥⎥⎥⎥⎥⎦

(17)

where A = E(Zn,0 ZH

n,0

)and B = E

(Zn,0 ZH

n,1

)are N ×N matrices.

We now consider the two separate cases γ = 0 and 0 < γ ≤ 12.

1. γ = 0. In this case, B = 0 and the covariance matrix of Zn is the diagonalL× L block matrix

H = E(ZnZ

Hn

)=

⎡⎢⎢⎢⎢⎢⎢⎣

A 0 0 · · · 0 00 A 0 0 · · · 00 0 A 0 · · · 0· · · · · · · · · · · · · · · · · ·0 · · · 0 0 A 00 · · · 0 · · · 0 A

⎤⎥⎥⎥⎥⎥⎥⎦

(18)

where

A = E(Zn,0Z

Hn,0

)= FnWE

(R0R

H0

)WFH

n = σ2FnW2FH

n (19)

With In = nK + K−N2

, we have

FnW2 =⎡

⎢⎢⎢⎢⎢⎣

w20 w2

1 exp 2πj(In+0)MK

· · · w2MK−1 exp 2πj(MK−1)(In+0)

MK· · · · · · · · · · · ·w2

0 w21 exp 2πj(In+l)

MK· · · w2

MK−1 exp 2πj(MK−1)(In+l)MK· · · · · · · · · · · ·

w20 w2

1 exp 2πj(In+N−1)MK

· · · w2MK−1 exp 2πj(MK−1)(In+N−1)

MK

⎤⎥⎥⎥⎥⎥⎦ (20)


and

A = σ2FnW2FH

n = σ2 × (21)⎡⎢⎢⎢⎢⎢⎣

w20 w2

1 exp 2πjIn

MK· · · w2

MK−1 exp 2πj(MK−1)In

MK· · · · · · · · · · · ·w2

0 w21 exp 2πj(In+l)

MK· · · w2

MK−1 exp 2πj(MK−1)(In+l)MK· · · · · · · · · · · ·

w20 w2

1 exp 2πj(In+N−1)MK

· · · w2MK−1 exp 2πj(MK−1)(In+N−1)

MK

⎤⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎣

1 · · · 1· · · · · · · · ·

exp −2πjlIn

MK· · · exp −2πjl(In+N−1)

MK· · · · · · · · ·exp −2πj(MK−1)In

MK· · · exp −2πj(MK−1)(In+N−1)

MK

⎤⎥⎥⎥⎥⎦

= σ2

⎡⎢⎢⎢⎢⎣τ1,1 τ1,2 · · · τ1,q · · · τ1,N

· · · · · · · · · · · · · · · · · ·τp,1 τp,2 · · · τp,q · · · τp,N

· · · · · · · · · · · · · · · · · ·τN,1 τN,2 · · · τN,q · · · τN,N

⎤⎥⎥⎥⎥⎦

where, for 1 ≤ p ≤ q ≤ N ,

τp,q =MK−1∑

l=0

w2l exp

2πjl(p− q)

MK(22)

2. 0 < γ ≤ 12. In this case A is computed by (21) and it remains to compute B.

In fact,

E(R0R

H1

)= σ2

[0γMK×(1−γ)MK IγMK×γMK

0(1−γ)MK×(1−γ)MK 0(1−γ)MK×γMK

](23)

where 0γMK×(1−γ)MK , 0(1−γ)MK×(1−γ)MK and 0(1−γ)MK×γMK are zero matricesof dimensions γMK× (1−γ)MK, (1−γ)MK× (1−γ)MK and (1−γ)MK×γMK respectively and IγMK×γMK is the identity matrix of dimensions γMK×γMK. Let

W1 =

⎡⎢⎢⎣w0 0 · · · 00 w1 0 · · ·· · · · · · · · · · · ·0 0 · · · wγMK−1

⎤⎥⎥⎦ ,W2 =

⎡⎢⎢⎣w0 0 · · · 00 w1 0 · · ·· · · · · · · · · · · ·0 0 · · · w(1−γ)MK−1

⎤⎥⎥⎦

(24)


W3 =

⎡⎢⎢⎣wγMK 0 · · · 0 00 wγMK+1 0 · · · 0· · · · · · · · · · · · · · ·0 0 · · · 0 wMK−1

⎤⎥⎥⎦ (25)

W4 =

⎡⎢⎢⎣w(1−γ)MK 0 · · · 0 00 w(1−γ)MK+1 0 · · · 0· · · · · · · · · · · · · · ·0 0 · · · 0 wMK−1

⎤⎥⎥⎦ (26)

We have

B = E(Zn,0Z

Hn,1

)= FnWE

(R0R

H1

)WFH

n (27)

= σ2Fn

[W1 00 W3

] [0γMK×(1−γ)MK IγMK×γMK


]×[

W2 00 W4

]FH

n

= σ2Fn

[0γMK×(1−γ)MK W1


] [W3 00 W4

]FH

n

= σ2Fn

[0γMK×(1−γ)MK W1W4


]FH

n

Let

F0n =

⎡⎢⎢⎢⎢⎢⎣

1 exp 2πjIn

MK· · · exp 2πj(γMK−1)In

MK· · · · · · · · · · · ·1 exp 2πj(In+p−1)

MK· · · exp 2πj(γMK−1)(In+p−1)

MK· · · · · · · · · · · ·1 exp 2πj(In+N−1)

MK· · · exp 2πj(γMK−1)(In+N−1)

MK

⎤⎥⎥⎥⎥⎥⎦ (28)

and

F1n =

⎡⎢⎢⎢⎢⎢⎣

exp 2πj(1−γ)MKIn

MK· · · exp 2πj(MK−1)In

MK· · · · · · · · ·exp 2πj(1−γ)MK(In+p−1)

MK· · · exp 2πj(MK−1)(In+p−1)

MK· · · · · · · · ·exp 2πj(1−γ)MK(In+N−1)

MK· · · exp 2πj(MK−1)(In+N−1)

MK

⎤⎥⎥⎥⎥⎥⎦ (29)

(27) can be simplified to yield:

B = E(Zn,0Z

Hn,1

)= σ2Fn

[0 W1W4

0 0

]FH

n (30)

= σ2 F0n W1W4

(F1

n

)H= σ2C


where

C = F0n W1W4

(F1

n

)H=

⎡⎢⎢⎢⎢⎣γ1,1 γ1,2 · · · γ1,q · · · γ1,N

· · · · · · · · · · · · · · · · · ·γp,1 γp,2 · · · γp,q · · · γp,N

· · · · · · · · · · · · · · · · · ·γN,1 γN,2 · · · γN,q · · · γN,N

⎤⎥⎥⎥⎥⎦ (31)

with

γp,q =

γMK−1∑l=0

exp2πjl(In + p− 1)

MK× (32)

wlwl+(1−γ)MK exp−2πj(l + (1 − γ)MK)(In + q − 1)

MK

= e−2πj(1−γ)(In+q−1)

γMK−1∑l=0

wlwl+(1−γ)MK exp2πjl(p− q)

MK

To gain insights into the distribution of eigenvalues of the covariance matrixE(ZnZ

Hn

),

we first compute the characteristic polynomial PL(λ) = det(E(ZnZ

Hn

)− λI), where

I denotes the identity matrix of dimensions LN ×LN . The characteristic polynomialPL(λ) can be expressed in terms of the two N ×N matrices A and B. We considerthe two separate cases γ = 0 and 0 < γ ≤ 1

2.

1. γ = 0. In this case the covariance matrix H = E(ZnZ

Hn

)is given by (18) and

the characteristic polynomial PL(λ) is computed by

PL(λ) = det(E(ZnZ

Hn

)− λI)

= (det(A − λI))L (33)

Obviously, each eigenvalue of H has a multiplicity of L.

2. 0 < γ ≤ 12. In this case we assume that B is an invertible matrix of dimensions

N × N (c.f. (30)). As an illustration, the characteristic polynomial PL(λ) iscomputed here for the two cases L = 2 and L = 3. If L = 2, it follows from(17) that the covariance matrix E

(ZnZ

Hn

)is equal to[

A BBH A

]

and hence

P2(λ) = det

([A − λI B

BH A − λI

])(34)

Let

X = − (A − λI)(BH)−1

(35)


It follows that

P2(λ) = (36)

det

([A− λI B

BH A − λI

])= det

([I X0 I

] [A − λI B

BH A − λI

])

= det

([0 B− (A − λI)

(BH)−1

(A− λI)BH A− λI

])

= (−1)N det

([BH A− λI

0 B− (A − λI)(BH)−1

(A− λI)

])

= (−1)N det(BH)det(B− (A − λI)

(BH)−1

(A− λI))

= (−1)N det(BH)×

det[BBH

(BH)−1 − (A − λI)

((BH)−1

ABH − λI) (

BH)−1]

= (−1)N det(BH)×

det[BBH − (A− λI)

((BH)−1

ABH − λI)]

det[(

BH)−1]

= (−1)N det[BBH − (A− λI)

((BH)−1

ABH − λI)]

= det[(A − λI)

((BH)−1

ABH − λI)− BBH

]If the entries of B are relatively small , it should be reasonable to assume thatBBH ≈ 0 and therefore

P2(λ) = det((A − λI)

((BH)−1

ABH − λI)− BBH

)≈ det

((A − λI)

((BH)−1

ABH − λI))

= det (A − λI)2

Hence we can expect the eigenvalues of H to be close to the roots ofdet (A− λI)2 = 0. In other words, the eigenvalues of H should cluster intoN groups with each group consisting of L = 2 closely spaced eigenvalues.

Similarly, if L = 3 and let X be defined by (35), then

P3(λ) = det

⎛⎝⎡⎣ A − λI B 0

BH A − λI B0 BH A − λI

⎤⎦⎞⎠ (37)

= det

⎛⎝⎡⎣ I X 0

0 I 00 0 I

⎤⎦⎡⎣ A − λI B 0

BH A − λI B0 BH A − λI

⎤⎦⎞⎠

= det

⎛⎝⎡⎣ 0 B + X (A− λI) XB

BH A − λI B0 BH A− λI

⎤⎦⎞⎠


= (−1)N det

⎛⎝⎡⎣ BH A − λI B

0 B + X (A− λI) XB0 BH A− λI

⎤⎦⎞⎠

= (−1)N det(BH)det

([B + X (A − λI) XB

BH A − λI

])

Let

Y = − (B + X (A − λI))(BH)−1

(38)

It follows that

det

([B + X (A − λI) XB

BH A − λI

])(39)

= det

([I Y0 I

] [B + X (A− λI) XB

BH A − λI

])

= det

([0 XB + Y (A − λI)

BH A− λI

])

= (−1)N det

([BH A− λI0 XB + Y (A − λI)

])= (−1)N det

(BH)det (XB + Y (A − λI))

and

XB + Y (A − λI) (40)

=(− (A − λI)

(BH)−1)

B +(− (B + X (A− λI))

(BH)−1)

(A− λI)

= − (A − λI)(BH)−1

B − B(BH)−1

(A − λI)

−X (A − λI)(BH)−1

(A − λI)

= − (A − λI)(BH)−1

B − B(BH)−1

(A − λI)

+ (A − λI)(BH)−1

(A− λI)(BH)−1

(A − λI)

= − (A − λI)(BH)−1

B − BBH((

BH)−2

A(BH)2 − λI

) (BH)−2

+ (A − λI)((

BH)−1

ABH − λI)((

BH)−2

A(BH)2 − λI

)(BH)−2

= − (A − λI)(BH)−1

BBHBH(BH)−2

−[BBH − (A − λI)

((BH)−1

ABH − λI)]

×[(BH)−2

A(BH)2 − λI

] (BH)−2

Combining (37), (39) and (40) and making simplifications, we obtain

P3(λ) = (41)


det[− (A− λI)

(BH)−1

BBHBH + Q2

[(BH)−2

A(BH)2 − λI

]]= det

[Q2

[(BH)−2

A(BH)2 − λI

]− (A− λI)

(BH)−1

BBHBH]

where

Q2 = (A− λI)((

BH)−1

ABH − λI)−BBH (42)

withP2(λ) = det (Q2)

Since the entries of B are relatively small, we have BBH ≈ 0 and P3(λ) ≈det(A− λI)3. Hence we can expect the eigenvalues of H to be small perturba-tions of the roots of det (A − λI)3 = 0. In other words, the eigenvalues of Hshould cluster into N groups with each group consisting of L = 3 closely spacedeigenvalues.

In general, the characteristic polynomial PL(λ) for L ≥ 4 can also be computedin terms of A and B in the same manner and if the entries of B are relativelysmall, we have BBH ≈ 0, PL(λ) ≈ det(A − λI)L and we can expect theeigenvalues of H to be small perturbations of the roots of det (A − λI)L = 0.In other words, eigenvalues of H should roughly cluster into N groups witheach group consisting of L closely spaced eigenvalues.

To illustrate these observations, for N = 5, L = 4, n = 1, γ = 12, we group

the 20 eigenvalues of H = E(ZnZ

Hn

)for the normalized Blackman window as

follows:

Group 1 2.8652 2.7812 2.6807 2.6015Group 2 1.7164 1.6139 1.4916 1.3960Group 3 0.8358 0.6843 0.5041 0.3600Group 4 0.2884 0.1802 0.1727 0.1258Group 5 0.0638 0.0531 0.0517 0.0149

Clearly, the eigenvalues of H can be roughly divided into N = 5 groups ofL = 4 eigenvalues each and each group with L = 4 eigenvalues approximatelyform an arithmetic progression.

4 Probability of Detection(0 < γ ≤ 1/2, N ≥ 1, L ≥ 1)

In this section, we compute the probability of detection Pd for the FFT summationdetector under the assumption that the overlapping ratio 0 < γ ≤ 1/2 and the


received signal is a complex pure tone embedded in additive white gaussian noise.Specifically, the received signal samples are given by

rk = A exp(2πjfk) + uk = sk + uk , k = 0, 1, 2, · · · ,

where A and f are respectively the amplitude and normalized frequency of the puretone, uk is the additive white gaussian noise sample and sk = A exp(j2πfk). Thereceived signal sample vector Rm can now be written as the sum of the signal com-ponent Sm and the noise component Nm

Rm = Sm + Nm , 0 ≤ m ≤ L− 1 ,

where Sm and Nm are defined by:{Sm =

[sm(1−γ)MK+MK−1, sm(1−γ)MK+MK−2, · · · , sm(1−γ)MK

]tNm =

[um(1−γ)MK+MK−1, um(1−γ)MK+MK−2, · · · , um(1−γ)MK

]t (43)

The additive white Gaussian noise samples uk are independent and identically dis-tributed with E(uku

∗l ) = σ2δk,l, where δk,l = 1 if k = l and δk,l = 0 if k �= l. It can be

verified that for 0 ≤ m, l ≤ L− 1, E (NmNtl) = 0. In fact, let 0 ≤ n ≤ M − 1 and

define

Vm = Zn,m − E (Zn,m) = FnWRm − FnWSm = FnWNm (44)

We have

E

⎛⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎣

V0

V1

· · ·Vm

· · ·VL−1

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

V0

V1

· · ·Vm

· · ·VL−1

⎤⎥⎥⎥⎥⎥⎥⎦

t⎞⎟⎟⎟⎟⎟⎟⎠

(45)

=

⎡⎢⎢⎢⎢⎣E (V0V

t0) E (V0V

t1) · · · · · · E

(V0V

tL−1

)· · · · · · · · · · · · · · ·E (VmVt

0) E (VmVt1) · · · · · · E

(VmVt

L−1

)· · · · · · · · · · · · · · ·E (VL−1V

t0) E (VL−1V

t1) · · · · · · E

(VL−1V

tL−1

)

⎤⎥⎥⎥⎥⎦

= 0

since for 0 ≤ m, l ≤ L− 1,

E(VmVt

l

)= E

(FnWNm (FnWNl)

t) = E(FnWNmNt

lWFtn

)(46)

= FnWE(NmNt

l

)WFt

n = 0


(46) implies that the components of the Gaussian random vector

Zn =

⎡⎢⎢⎢⎢⎢⎢⎣

Zn,0

Zn,1

· · ·Zn,m

· · ·Zn,L−1

⎤⎥⎥⎥⎥⎥⎥⎦

satisfy the two constraints (1) and (2) of [16]. Note that here the Gaussian randomvector Zn is no longer zero mean owing to the presence of the complex sinusoidalsignal. It follows from [16] that the characteristic function φ(t) of the decision statisticzn, which is the quadratic form defined by (12), can be computed using the formula(4a) in [16]:

φ(t) = (det (I − jtH))−1 exp(−ZH

n H−1(I − (I− jtH)−1) Zn

)(47)

= (det (I − jtH))−1 exp(ZH

n H−1jtH (I − jtH)−1 Zn

)= (det (I − jtH))−1 exp

(jtZH

n (I − jtH)−1 Zn

)where H is the positive definite Hermitian matrix defined by (17) and Zn is the meanvalue or signal component of Zn given by

Zn = E (Zn) (48)

=

⎡⎢⎢⎢⎢⎢⎢⎣

E (Zn,0)E (Zn,1)· · ·E (Zn,m)· · ·E (Zn,L−1)

⎤⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

FnWS0

FnWS1

· · ·FnWSm

· · ·FnWSL−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

Let U be a complex unitary matrix of dimensions LN × LN such that

UHU−1 = UHUH =

⎡⎢⎢⎢⎢⎢⎢⎣

λ1 0 · · · 0 · · · 00 λ2 0 · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 · · · λm · · · 0· · · · · · · · · · · · · · · · · ·0 0 · · · 0 · · · λLN

⎤⎥⎥⎥⎥⎥⎥⎦

(49)

where λ1, λ2, · · · , λLN are the LN distinct eigenvalues of H. Then

(det (I − jtH))−1 =

(LN∏m=1

(1 − jtλm)

)−1

(50)


and

ZHn (I− jtH)−1 Zn = (51)(UZn

)H ×⎡⎢⎢⎢⎢⎢⎢⎣

(1 − jtλ1)−1 0 · · · 0

0 (1 − jtλ2)−1 0 · · ·

· · · · · · · · · · · ·0 · · · (1 − jtλm)−1 · · ·· · · · · · · · · · · ·0 · · · 0 (1 − jtλLN)−1

⎤⎥⎥⎥⎥⎥⎥⎦

UZn

=

LN∑m=1

|αm|21 − jtλm

where

UZn = (α1, α2, · · · , αLN)t (52)

Combining (50), (51), (52), we obtain the following expression for the characteristicfunction of the random variable zn:

φ(t) = (det (I − jtH))−1 exp(jtZH

n (I − jtH)−1 Zn

)(53)

=

(LN∏m=1

(1 − jtλm)

)−1

expLN∑m=1

jt|αm|21 − jtλm

It can be verified that (LN∏m=1

(1 − jtλm)

)−1

=LN∑m=1

Am

1 − jtλm

(54)

where Am =λLN−1

m∏l �=m(λm − λl)

and

LN∑m=1

jt|αm|21 − jtλm

=

LN∑m=1

(jtλm − 1 + 1) |αm|2λm

1 − jtλm(55)

= −LN∑m=1

|αm|2λm

+

LN∑m=1

|αm|2λm

1 − jtλm

Hence the characteristic function φ(t) can be rewritten as

φ(t) =

(LN∑m=1

Am

1 − jtλm

)exp

(−

LN∑m=1

|αm|2λm

+

LN∑m=1

|αm|2λm

1 − jtλm

)(56)


=

(exp

(−

LN∑m=1

|αm|2λm

))(LN∑m=1

Am

1 − jtλm

)exp

LN∑m=1

|αm|2λm

1 − jtλm

= γ0

LN∑m=1

Am

1 − jtλm

expLN∑l=1

βl

1 − jtλl

where βl = |αl|2λl

, 1 ≤ l ≤ LN , and γ0 = exp(−∑LN

m=1|αm|2λm

)= exp(−∑LN

m=1 βm).

Now we are ready to compute the probability of detection Pd for a given thresholdT > 0. The derivations in the sequel of this report will not depend on the channelindex n explicitly. To avoid any potential confusion, we now call the readers’ attentionto the fact that we shall from now on use the letter n freely to denote one of the indicesin the rest of this document.

Let ε > 0 be a small positive real number ( 0 < ε < 1max1≤l≤LN λl

) and let L = {z =

t − jε : −∞ < t < ∞}. L is a straight line that is in parallel with and below thereal axis and the distance between these two lines is ε. Let p(x) be the probabilitydensity function of the random variable zn. We have

p(x) =1

2π

∫ ∞

−∞φ(t)e−jxtdt

and it can be verified, using elaborate contour integral arguments from the theoryof functions of one complex variable and also Fubini’s Theorem from measure theory(details omitted), that

Pd =

∫ ∞

T

p(x) dx =1

2π

∫ ∞

T

∫ ∞

−∞φ(t)e−jxtdt dx (57)

=1

2π

∫ ∞

T

∫Lφ(z)e−jxzdz dx =

1

2π

∫L

∫ ∞

T

φ(z)e−jxzdx dz

=1

2πj

∫L

φ(z)e−jzT

zdz

=1

2πj

∫L

γ0

∑LNm=1

(Am

1−jzλmexp∑LN

l=1βl

1−jzλl

)e−jzT

zdz

=γ0

2πj

LN∑m=1

Am

∫L

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz

Let Cl, 1 ≤ l ≤ LN , be mutually disjoint small circles centered respectively at 1jλl

and oriented clockwise. Using the Residue Theorem from the theory of functions ofone complex variable, one obtains, for any m, 1 ≤ m ≤ LN ,∫

L

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz (58)


=LN∑n=1

∫Cn

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz

=∑n �=m

∫Cn

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz +

∫Cm

(1

1 − jzλm

expLN∑l=1

βl

1 − jzλl

)e−jzT

zdz

Let

ψn(z) = exp∑l �=n

βl

1 − jzλl

(59)

If n �= m, we have∫Cn

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz (60)

=

∫Cn

1

1 − jzλm

e−jzT

zψn(z) exp

βn

1 − jzλndz

=

∫Cn

1

1 − jzλm

e−jzT

zψn(z)

∞∑p=0

1

p!

(βn

1 − jzλn

)p

dz

=∞∑

p=0

1

p!

(βn

−jλn

)p ∫Cn

1

1 − jzλm

e−jzT

zψn(z)

1(z − 1

jλn

)pdz

=

∞∑p=1

1

p!

(βn

−jλn

)p ∫Cn

1

1 − jzλm

e−jzT

zψn(z)

1(z − 1

jλn

)pdz

To simplify (60), we first develop the power series expansions of 11−jzλm

e−jzT

zand ψn(z)

around the point 1jλn

. We have

1

1 − jzλm=

1

1 − j(z − 1jλn

+ 1jλn

)λm

=λn

λn−λm

1 − j(z − 1jλn

) λmλn

λn−λm

(61)

=λn

λn − λm

∞∑r=0

(j

(z − 1

jλn

)λmλn

λn − λm

)r

=λn

λn − λm

∞∑r=0

(jλmλn

λn − λm

)r (z − 1

jλn

)r

1

z=

1

z − 1jλn

+ 1jλn

= jλn1

1 + (z − 1jλn

)jλn

(62)


= jλn

∞∑s=0

(−jλn)s

(z − 1

jλn

)s

e−jzT = exp

(−jT

(z − 1

jλn

)− T

λn

)(63)

= exp

(− T

λn

)exp

(−jT

(z − 1

jλn

))

= exp

(− T

λn

) ∞∑t=0

(−jT )t

t!

(z − 1

jλn

)t

and for any l �= n,

exp

(βl

1 − jzλl

)= exp

(βlλn

λn − λl

1

1 − j(z − 1jλn

) λlλn

λn−λl

)(64)

= 1 +∞∑

q=1

1

q!

(βlλn

λn − λl

)q(

1

1 − j(z − 1jλn

) λlλn

λn−λl

)q

= 1 +∞∑

q=1

1

q!

(βlλn

λn − λl

)q ∞∑u=0

(u+ q − 1q − 1

)(jλlλn

λn − λl

)u(z − 1

jλn

)u

= 1 +∞∑

u=0

[ ∞∑q=1

1

q!

(βlλn

λn − λl

)q (u+ q − 1q − 1

)](jλlλn

λn − λl

)u(z − 1

jλn

)u

=∞∑

u=0

[ ∞∑q=0

1

q!

(u+ q − 1q − 1

)(βlλn

λn − λl

)q](

jλlλn

λn − λl

)u(z − 1

jλn

)u

=

∞∑u=0

Bu

(βlλn

λn − λl

)(jλlλn

λn − λl

)u(z − 1

jλn

)u

where

(qp

)is the binomial coefficient satisfying the following identities:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(qp

)= q(q−1)···(q−p+1)

p!= q!

p!(q−p)!, q > p > 0 ,(

pp

)= 1 , p is any integer ,(

p0

)= 1 , p > 0 ,(

qp

)= 0 , p < 0 , q > p ,

(65)


and

Bu(x) =∞∑

q=0

1

q!

(u+ q − 1q − 1

)xq (66)

Combining (59), (61) -(64), we obtain

1

1 − jzλm

e−jzT

zψn(z) (67)

=

(λn

λn − λm

∞∑r=0

(jλmλn

λn − λm

)r (z − 1

jλn

)r)

×(jλn

∞∑s=0

(−jλn)s

(z − 1

jλn

)s)

×(

exp

(− T

λn

) ∞∑t=0

(−jT )t

t!

(z − 1

jλn

)t)

×

∏l �=n

( ∞∑ul=0

Bul

(βlλn

λn − λl

)(jλlλn

λn − λl

)ul(z − 1

jλn

)ul

)

=

( −jλ2n

λm − λn

exp

(− T

λn

)) ∞∑v=0

dv(m,n)

(z − 1

jλn

)v

where

dv(m,n) (68)

=∑

r+s+t+P

l�=n ul=v

(jλmλn

λn − λm

)r

(−jλn)s (−jT )t

t!×

∏Bul

(βlλn

λn − λl

)(jλlλn

λn − λl

)ul

= (−j)v∑

r+s+t+P

l�=n ul=v

(λmλn

λm − λn

)r

λsn

T t

t!

∏Bul

(βlλn

λn − λl

)(λlλn

λl − λn

)ul

= (−j)vλvnfv(m,n)

with

fv(m,n) (69)

=∑

r+s+t+P

l�=n ul=v

(λm

λm − λn

)r

(Tλn

)t

t!

∏Bul

(βlλn

λn − λl

)(λl

λl − λn

)ul


=

v∑s=0

∑r+t+

Pl�=n ul=v−s

(λm

λm − λn

)r

(Tλn

)t

t!

∏Bul

(βlλn

λn − λl

)(λl

λl − λn

)ul

=

v∑s=0

v−s∑t=0

(Tλn

)t

t!

∑r+

Pl�=n ul=v−s−t

(λm

λm − λn

)r∏Bul

(βlλn

λn − λl

)(λl

λl − λn

)ul

Substituting (67) into (60), we obtain, for n �= m,∫Cn

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz (70)

=

∞∑p=1

1

p!

(βn

−jλn

)p(λn

λm − λnexp

(− T

λn

))×

∞∑v=0

(−jλn)1+v fv(m,n)

∫Cn

(z − 1

jλn

)v1(

z − 1jλn

)pdz

= −2πj

(λn

λm − λn

exp

(− T

λn

)) ∞∑p=1

1

p!βp

nfp−1(m,n)

Next let us compute ∫Cm

(1

1 − jzλm

expLN∑l=1

βl

1 − jzλl

)e−jzT

zdz (71)

We have

1

z=

1

z − 1jλm

+ 1jλm

= jλm1

1 + (z − 1jλm

)jλm

(72)

= jλm

∞∑s=0

(−jλm)s

(z − 1

jλm

)s

and

e−jzT = exp

(−jT

(z − 1

jλm

)− T

λm

)(73)

= exp

(− T

λm

) ∞∑t=0

(−jT )t

t!

(z − 1

jλm

)t

Following (64), we have, for any l �= m,

exp

(βl

1 − jzλl

)=

∞∑u=0

Bu

(βlλm

λm − λl

)(jλlλm

λm − λl

)u(z − 1

jλm

)u

(74)


It follows from (72)-(74) that(1

1 − jzλmexp∑l �=m

βl

1 − jzλl

)e−jzT

z(75)

=

(1

−jλm

)1

z − 1jλm

×(jλm

∞∑s=0

(−jλm)s

(z − 1

jλm

)s)

×(

exp

(− T

λm

) ∞∑t=0

(−jT )t

t!

(z − 1

jλm

)t)

×

∏l �=m

∞∑ul=0

Bul

(βlλm

λm − λl

)(jλlλm

λm − λl

)ul(z − 1

jλm

)ul

=− exp

(− T

λm

)z − 1

jλm

∞∑v=0

d∗v(m)

(z − 1

jλm

)v

where

d∗v(m) =∑

s+t+P

l�=m ul=v

(−jλm)s (−jT )t

t!

∏Bul

(βlλm

λm − λl

)(jλlλm

λm − λl

)ul

= (−j)v∑

s+t+P

l�=m ul=v

λsm

T t

t!

∏Bul

(βlλm

λm − λl

)(λlλm

λl − λm

)ul

= (−j)vλvmgv(m) (76)

with

gv(m) =∑

s+t+P

l�=m ul=v

(T

λm

)t

t!

∏Bul

(βlλm

λm − λl

)(λl

λl − λm

)ul

(77)

=v∑

s=0

v−s∑t=0

(T

λm

)t

t!

∑P

l�=m ul=v−s−t

∏Bul

(βlλm

λm − λl

)(λl

λl − λm

)ul

Utilizing (75)-(77), we finally obtain

∫Cm

[1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

]e−jzT

zdz (78)

= − exp

(− T

λm

)×


∫Cm

[1

z − 1jλm

∞∑v=0

d∗v(m)

(z − 1

jλn

)v] ∞∑

p=0

(βm

−jλm

)p(z − 1

jλm

)−p

dz

= − exp

(− T

λm

)×∫Cm

[1

z − 1jλm

∞∑v=0

λvmgv(m)(−j)v

(z − 1

jλn

)v]×

[ ∞∑p=0

1

p!

(βm

−jλm

)p(z − 1

jλm

)−p]dz

= 2πj exp

(− T

λm

) ∞∑p=0

1

p!βp

mgp(m)

Combining (57), (58), (70) and (78), we obtain the probability of detection Pd asfollows:

Pd =γ0

2πj

LN∑m=1

Am

∫L

(1

1 − jzλmexp

LN∑l=1

βl

1 − jzλl

)e−jzT

zdz (79)

=γ0

2πj

LN∑m=1

Am

∑n �=m

−2πj

(λn

λm − λn

exp

(− T

λn

)) ∞∑p=1

1

p!βp

nfp−1(m,n)

+γ0

2πj

LN∑m=1

Am

[2πj exp

(− T

λm

) ∞∑p=0

1

p!βp

mgp(m)

]

= γ0

LN∑m=1

Am

∑n �=m

(λnβn

λn − λm

exp

(− T

λn

)) ∞∑p=0

1

(p+ 1)!βp

nfp(m,n)

+γ0

LN∑m=1

Am exp

(− T

λm

) ∞∑p=0

1

p!βp

mgp(m)

The formula (79) can be further simplified. In fact, it can be shown (details omitted)that, if u ≥ 1, Bu(x) can be written as

Bu(x) =x

u!

du

dxu

(xu−1 ex

)= Pu(x)e

x (80)

where Pu(x) is a polynomial of degree u with non-negative coefficients and⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

P0(x) = 1P1(x) = xP2(x) = x(1 + 1

2x)

Pu(x) =x

u

u−1∑k=0

(u

k + 1

)xk

k!, u ≥ 1 .

(81)


As an illustration, we have⎧⎨⎩

B0(x) = ex

B1(x) = xex

B2(x) = (x+ 12x2)ex

(82)

From (77), (69) and (82), it follows that⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

g0(m) = exp∑

l �=mβlλm

λm−λl

= exp∑

l �=mβl(λm−λl)+βlλl

λm−λl=[exp∑

l �=m βl

]exp∑

l �=mβlλl

λm−λl

= e−βm

γ0exp∑

l �=mβlλl

λm−λl

f0(m,n) = exp(∑

l �=nβlλn

λn−λl

)= g0(n)

= e−βn

γ0exp∑

l �=nβlλl

λn−λl, n �= m

(83)

Also we have, for n �= m and v ≥ 1,

fv(m,n) (84)

=∑

r+s+t+P

l�=n ul=v

(λm

λm − λn

)r

(Tλn

)t

t!

∏Bul

(βlλn

λn − λl

)(λl

λl − λn

)ul

= eP

l�=nβlλn

λn−λl ×∑

r+s+t+P

l�=n ul=v

(λm

λm − λn

)r

(Tλn

)t

t!

∏Pul

(βlλn

λn − λl

)(λl

λl − λn

)ul

= g0(n)∑

r+s+t+P

l�=n ul=v

(λm

λm − λn

)r

(Tλn

)t

t!

∏Pul

(βlλn

λn − λl

)(λl

λl − λn

)ul

and for 1 ≤ m ≤ LN and v ≥ 1,

gv(m) (85)

=∑

s+t+P

l�=m ul=v

(T

λm

)t

t!

∏Bul

(βlλm

λm − λl

)(λl

λl − λm

)ul

= g0(m)∑

s+t+P

l�=m ul=v

(T

λm

)t

t!

∏Pul

(βlλm

λm − λl

)(λl

λl − λm

)ul


Substituting (83), (84) and (85) into (79), we obtain the following formula for Pd:

Pd =

γ0

LN∑m=1

Am

∑n �=m

(λnβn

λn − λmexp

(− T

λn

)) ∞∑p=0

1

(p+ 1)!βp

ng0(n) ×

p∑s=0

∑r+t+

Pl�=n ul=p−s

(λm

λm − λn

)r

(Tλn

)t

t!×

∏Pul

(βlλn

λn − λl

)(λl

λl − λn

)ul

+γ0

LN∑m=1

Am exp

(− T

λm

) ∞∑p=0

1

p!βp

mg0(m) ×

p∑s=0

∑t+

Pl�=m ul=p−s

(T

λm

)t

t!

∏Pul

(βlλm

λm − λl

)(λl

λl − λm

)ul

=

LN∑m=1

Am

∑n �=m

(λnβn

λn − λmγ0 g0(n) exp

(− T

λn

)) ∞∑p=0

1

(p+ 1)!βp

n ×

p∑s=0

∑r+t+

Pl�=n ul=p−s

(λm

λm − λn

)r

(Tλn

)t

t!×

∏Pul

(βlλn

λn − λl

)(λl

λl − λn

)ul

+LN∑m=1

Am γ0 g0(m) exp

(− T

λm

) ∞∑p=0

1

p!βp

m ×

p∑s=0

∑t+

Pl�=m ul=p−s

(T

λm

)t

t!

∏Pul

(βlλm

λm − λl

)(λl

λl − λm

)ul

=LN∑m=1

Am

∑n �=m

[λnβn

λn − λm

e−βn

[exp∑l �=n

βlλl

λn − λl

]exp

(− T

λn

)]

×∞∑

p=0

1

(p+ 1)!βp

n

p∑s=0

∑r+t+

Pl�=n ul=p−s

(λm

λm − λn

)r

(Tλn

)t

t!×

∏Pul

(βlλn

λn − λl

)(λl

λl − λn

)ul


+LN∑m=1

Am e−βm

(exp∑l �=m

βlλl

λm − λl

)exp

(− T

λm

) ∞∑p=0

1

p!βp

m ×

p∑s=0

∑t+

Pl�=m ul=p−s

(T

λm

)t

t!

∏Pul

(βlλm

λm − λl

)(λl

λl − λm

)ul

=LN∑m=1

Am

∑n �=m

[λnβn

λn − λm

(exp∑l �=n

βlλl

λn − λl

) ] ∞∑p=0

1

(p+ 1)!βp

ne−βn

×p∑

s=0

∑r+t+

Pl�=n ul=p−s

(λm

λm − λn

)r

(Tλn

)t

t!e−

Tλn ×

∏Pul

(βlλn

λn − λl

)(λl

λl − λn

)ul

+

LN∑m=1

Am

(exp∑l �=m

βlλl

λm − λl

) ∞∑p=0

1

p!βp

m e−βm ×

p∑s=0

∑t+

Pl�=m ul=p−s

(T

λm

)t

t!e−

Tλm

×∏

Pul

(βlλm

λm − λl

)(λl

λl − λm

)ul

(86)

5 Probability of Detection (γ = 0, N > 1, L > 1)

In this section, we compute the probability of detection Pd for the FFT summationdetector under the assumption that the received signal is a complex pure tone em-bedded in additive white Gaussian noise, the overlapping ratio γ is zero and N > 1,L > 1. Since γ = 0, the data blocks are not overlapped and it follows from (18), (33),(47) and (53) that the characteristic function φ(t) is given by

φ(t) = γL0

(1∏N

m=1(1 − jtλm)

)L

ePN

l=1Lβl

1−jtλl (87)

Here λm, 1 ≤ m ≤ N , are the N distinct eigenvalues of the Hermitian matrix

A defined by (21) and (22), βm = α2m

λm, with αm defined by (52), where L = 1,

H = A, γ0 = e−PN

m=1 βm . The rational function(

1QN

m=1(1−jtλm)

)L

admits the following


fractional decomposition:(1

(1 − jtλ1)(1 − jtλ2) · · · (1 − jtλN )

)L

=

N∑m=1

L∑k=1

Am,k1

(1 − jtλm)k(88)

where it can be shown that⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Am,L =

⎛⎜⎜⎝ λN−1

m∏l �=m

(λm − λl)

⎞⎟⎟⎠

L

, 1 ≤ m ≤ N ,

Am,k = Am,L

∑P

l�=m kl=L−k

∏( L+ kl − 1kl

)(λl

λl − λm

)kl

,

1 ≤ k ≤ L− 1 , 1 ≤ m ≤ N .

(89)

As in the case of overlapping data blocks, we have

Pd =

∫ ∞

T

p(x) dx =1

2π

∫ ∞

T

∫ ∞

−∞φ(t)e−jxtdt dx

=1

2π

∫ ∞

T


1

2π

∫L

∫ ∞

T

φ(z)e−jxzdx dz

=1

2πj

∫L

φ(z)e−jzT

zdz

=1

2πj

∫L

(γL

0

N∑m=1

L∑k=1

Am,k

(1 − jzλm)k

)exp

N∑l=1

Lβl

1 − jzλl

e−jzT

zdz

=γL

0

2πj

N∑m=1

L∑k=1

Am,k

∫L

(1

1 − jzλm

)k(

exp

N∑l=1

Lβl

1 − jzλl

)e−jzT

zdz (90)

Let Cl, 1 ≤ l ≤ N , be mutually disjoint small circles centered respectively at 1jλl

andoriented clockwise. Using the Residue Theorem from the theory of functions of onecomplex variable, we obtain, for any m, k, 1 ≤ m ≤ N , 1 ≤ k ≤ L,∫

L

(1

1 − jzλm

)k(

exp

N∑l=1

Lβl

1 − jzλl

)e−jzT

zdz (91)

=N∑

n=1

∫Cn

(1

1 − jzλm

)k(

expN∑

l=1

Lβl

1 − jzλl

)e−jzT

zdz

=∑n �=m

∫Cn

(1

1 − jzλm

)k(

expN∑

l=1

Lβl

1 − jzλl

)e−jzT

zdz +

∫Cm

(1

1 − jzλm

)k(

exp

N∑l=1

Lβl

1 − jzλl

)e−jzT

zdz


Let

Ψn(z) = exp∑l �=n

Lβl

1 − jzλl

(92)

If n �= m, we have

∫Cn

(1

1 − jzλm

)k(

expN∑

l=1

Lβl

1 − jzλl

)e−jzT

zdz (93)

=

∫Cn

(1

1 − jzλm

)ke−jzT

zΨn(z) exp

Lβn

1 − jzλn

dz

=

∫Cn

(1

1 − jzλm

)ke−jzT

zΨn(z)

∞∑p=0

1

p!

(Lβn

1 − jzλn

)p

dz

=∞∑

p=0

1

p!

(Lβn

−jλn

)p ∫Cn

(1

1 − jzλm

)k (e−jzT

z

)Ψn(z)

1(z − 1

jλn

)pdz

=∞∑

p=1

1

p!

(Lβn

−jλn

)p ∫Cn

(1

1 − jzλm

)k (e−jzT

z

)Ψn(z)

1(z − 1

jλn

)pdz

To evaluate (93), we first develop the power series expansions of(

11−jzλm

)ke−jzT

zand

ψn(z) around the point 1jλn

. We have

(1

1 − jzλm

)k

=

(1

1 − j(z − 1jλn

+ 1jλn

)λm

)k

=

(λn

λn−λm

1 − j(z − 1jλn

) λmλn

λn−λm

)k

=

(λn

λn − λm

)k ∞∑r=0

(r + k − 1k − 1

)(j

(z − 1

jλn

)λmλn

λn − λm

)r

=

(λn

λn − λm

)k ∞∑r=0

(r + k − 1k − 1

)(jλmλn

λn − λm

)r (z − 1

jλn

)r

(94)

where

(r + k − 1k − 1

)is a binomial coefficient defined by (65),

1

z= jλn

∞∑s=0

(−jλn)s

(z − 1

jλn

)s

(95)

and

e−jzT = exp

(− T

λn

) ∞∑t=0

(−jT )t

t!

(z − 1

jλn

)t

(96)


and from (64) it follows that, for any l �= n,

exp

(Lβl

1 − jzλl

)=

∞∑u=0

Bu

(Lβlλn

λn − λl

)(jλlλn

λn − λl

)u(z − 1

jλn

)u

(97)

where

Bu(x) =

∞∑q=0

1

q!

(u+ q − 1q − 1

)xq = Pu(x)e

x

with Pu(x) defined by (80) and (81). Combining the preceding equations, we obtain

(1

1 − jzλm

)k (e−jzT

z

)ψn(z) (98)

=

[(λn

λn − λm

)k ∞∑r=0

(r + k − 1k − 1

)(jλmλn

λn − λm

)r (z − 1

jλn

)r]

×[jλn

∞∑s=0

(−jλn)s

(z − 1

jλn

)s]×

[exp

(− T

λn

) ∞∑t=0

(−jT )t

t!

(z − 1

jλn

)t]×

∏l �=n

∞∑ul=0

Bul

(Lβlλn

λn − λl

)(jλlλn

λn − λl

)ul(z − 1

jλn

)ul

= jλn

(λn

λn − λm

)k

exp

(− T

λn

) ∞∑v=0

Dv(k,m, n)

(z − 1

jλn

)v

where

Dv(k,m, n) (99)

=∑

r+s+t+P

l�=n ul=v

(r + k − 1k − 1

)(jλmλn

λn − λm

)r

(−jλn)s (−jT )t

t!

×∏

Bul

(Lβlλn

λn − λl

)(jλlλn

λn − λl

)ul

= (−j)vλvnFv(k,m, n)

and

Fv(k,m, n) (100)


=

v∑s=0

∑r+t+

Pl�=n ul=v−s

(r + k − 1k − 1

)(λm

λm − λn

)r

(Tλn

)t

t!×

∏Bul

(Lβlλn

λn − λl

)(λl

λl − λn

)ul

=v∑

s=0

v−s∑t=0

(Tλn

)t

t!

∑r+


(r + k − 1k − 1

)(λm

λm − λn

)r

×

∏Bul

(Lβlλn

λn − λl

)(λl

λl − λn

)ul

= eP

l�=nLβlλnλn−λl

v∑s=0

v−s∑t=0

(Tλn

)t

t!

∑r+


(r + k − 1k − 1

)(λm

λm − λn

)r

×∏

Pul

(Lβlλn

λn − λl

)(λl

λl − λn

)ul

= eP

l�=n Lβl eP

l�=nLβlλlλn−λl

v∑s=0

v−s∑t=0

(Tλn

)t

t!

∑r+


(r + k − 1k − 1

)(λm

λm − λn

)r

×∏

Pul

(Lβlλn

λn − λl

)(λl

λl − λn

)ul

Substituting (98) into (93), we finally obtain, for n �= m,

∫Cn

(1

1 − jzλm

)k(

expN∑

l=1

Lβl

1 − jzλl

)e−jzT

zdz (101)

= −∞∑

p=1

1

p!

(Lβn

−jλn

)p(λn

λn − λm

)k (exp

(− T

λn

))×

∞∑v=0

(−jλn)1+v Fv(k,m, n)

∫Cn

(z − 1

jλn

)v1(

z − 1jλn

)pdz

= 2πj

(λn

λn − λm

)k (exp− T

λn

) ∞∑p=1

1

p!(Lβn)pFp−1(k,m, n)

It remains to compute

∫Cm

(1

1 − jzλm

)k(

exp

N∑l=1

Lβl

1 − jzλl

)e−jzT

zdz (102)


We have

1

z= jλm

∞∑s=0

(−jλm)s

(z − 1

jλm

)s

(103)

and

e−jzT = exp

(− T

λm

) ∞∑t=0

(−jT )t

t!

(z − 1

jλm

)t

(104)

From (64) it follows that, for any l �= m,

exp

(Lβl

1 − jzλl

)=

∞∑u=0

Bu

(Lβlλm

λm − λl

)(jλlλm

λm − λl

)u(z − 1

jλm

)u

(105)

The preceding identities imply that

(1

1 − jzλm

)k(

exp∑l �=m

Lβl

1 − jzλl

)e−jzT

z(106)

=

(1

−jλm

)k(

1

z − 1jλm

)k

×(jλm

∞∑s=0

(−jλm)s

(z − 1

jλm

)s)

×(

exp

(− T

λm

) ∞∑t=0

(−jT )t

t!

(z − 1

jλm

)t)

×

∏l �=m

∞∑ul=0

Bul

(Lβlλm

λm − λl

)(jλlλm

λm − λl

)ul(z − 1

jλm

)ul

=− exp

(− T

λm

)(−jλm)k−1

∞∑v=0

D∗v(m)

(z − 1

jλm

)v−k

where

D∗v(m) (107)

=∑

s+t+P

l�=m ul=v

(−jλm)s (−jT )t

t!

∏Bul

(Lβlλm

λm − λl

)(jλlλm

λm − λl

)ul

= (−j)v∑

s+t+P

l�=m ul=v

λsm

T t

t!

∏Bul

(Lβlλm

λm − λl

)(λlλm

λl − λm

)ul

= (−j)vλvmGv(m)


and

Gv(m) =∑

s+t+P

l�=m ul=v

(T

λm

)t

t!

∏Bul

(Lβlλm

λm − λl

)(λl

λl − λm

)ul

(108)

=v∑

s=0

v−s∑t=0

(T

λm

)t

t!

∑P

l�=m ul=v−s−t

∏Bul

(Lβlλm

λm − λl

)(λl

λl − λm

)ul

= eP

l�=m Lβl eP

l�=mLβlλl

λm−λl ×v∑

s=0

v−s∑t=0

(T

λm

)t

t!

∑P

l�=m ul=v−s−t

∏Pul

(Lβlλm

λm − λl

)(λl

λl − λm

)ul

Combining (106)-(108) yields∫Cm

(1

1 − jzλm

)k(

expN∑

l=1

Lβl

1 − jzλl

)e−jzT

zdz (109)

=− exp

(− T

λm

)(−jλm)k−1

×∫Cm

( ∞∑v=0

(−jλm)vGv(m)

(z − 1

jλm

)v−k)

×( ∞∑

p=0

(Lβm

−jλm

)p(z − 1

jλm

)−p)dz

= − exp

(− T

λm

)×∫Cm

( ∞∑v=0

(−jλm)v−k+1Gv(m)

(z − 1

jλm

)v−k)

×( ∞∑

p=0

1

p!

(Lβm

−jλm

)p(z − 1

jλm

)−p)dz

= 2πj exp

(− T

λm

) ∞∑p=0

1

p!(Lβm)p Gp+k−1(m)

Substituting (101) and (109) into (91) and (90), we obtain:

Pd = e−LPN

m=1 βm × (110)N∑

m=1

L∑k=1

Am,k exp

(− T

λm

) ∞∑p=0

1

p!(Lβm)p Gp+k−1(m)

+e−LPN

m=1 βm ×N∑

m=1

L∑k=1

Am,k

∑n �=m

(λn

λn − λm

)k (exp− T

λn

) ∞∑p=1

1

p!(Lβn)pFp−1(k,m, n)


=N∑

m=1

eP

l�=mLβlλl

λm−λl

L∑k=1

Am,k

∞∑p=0

1

p!(Lβm)pe−Lβm ×

p+k−1∑s=0

p+k−1−s∑t=0

(T

λm

)t

t!e−

Tλm

∑P

l�=m ul=p+k−1−s−t

∏Pul

(Lβlλm

λm − λl

)(λl

λl − λm

)ul

+N∑

m=1

L∑k=1

Am,k

∑n �=m

(λn

λn − λm

)k

eP

l�=nLβlλlλn−λl

∞∑p=1

1

p!(Lβn)pe−Lβn ×

p−1∑s=0

p−1−s∑t=0

(Tλn

)t

t!e−

Tλn

∑r+

Pl�=n ul=p−1−s−t

(r + k − 1k − 1

)(λm

λm − λn

)r

×

∏Pul

(Lβlλn

λn − λl

)(λl

λl − λn

)ul

6 Probability of Detection(γ = 0,K = N = 1, L ≥ 1)

In this section, we compute the probability of detection Pd for the FFT summationdetector under the assumption that the received signal is a complex pure tone em-bedded in additive white Gaussian noise and γ = 0, K = N = 1, L ≥ 1. Since γ = 0,the data blocks are non-overlapped and it follows from (18), (33), (47) and (53) thatthe characteristic function φ(t) is given by

φ(t) = γL0

(1

1 − jtλ1

)L

eLβ1

1−jtλ1 (111)

where λ1 = σ2∑M−1

l=0 w2l is the single eigenvalue of the 1× 1 Hermitian matrix A de-

fined by (21) and (22), γ0 = e−β1 , β1 = |α1|2λ1

and α1 = A∑M−1

l=0 wl exp (−2πjl(f − fn))with fn = n

M. Here the integer n, 0 ≤ n ≤M − 1, denotes the channel index (or bin

index) since only one FFT bin is assigned to each channel (K = N = 1). As in theprevious two sections, it can be shown that

Pd =

∫ ∞

T

p(x) dx =1

2π

∫ ∞

T

∫ ∞

−∞φ(t)e−jxtdt dx (112)

=1

2π

∫ ∞

T


1

2π

∫L

∫ ∞

T

φ(z)e−jxzdx dz

=1

2πj

∫LγL

0

(1

1 − jzλ1

)L

eLβ1

1−jzλ1e−jzT

zdz


We have

1

z= jλ1

∞∑s=0

(−jλ1)s

(z − 1

jλ1

)s

(113)

and

e−jzT = exp

(− T

λ1

) ∞∑t=0

(−jT )t

t!

(z − 1

jλ1

)t

(114)

It follows that(1

1 − jzλ1

)L

eLβ1

1−jzλ1e−jzT

z(115)

=

(1

−jλ1

)L (z − 1

jλ1

)−L[

+∞∑p=0

1

p!

(Lβ1

1 − jzλ1

)p]×

jλ1

∞∑s=0

(−jλ1)s

(z − 1

jλ1

)s

× exp

(− T

λ1

) ∞∑t=0

(−jT )t

t!

(z − 1

jλ1

)t

= − exp

(− T

λ1

)(−jλ1)

−L+1

[+∞∑p=0

1

p!

(Lβ1

−jλ1

)p(z − 1

jλ1

)−p−L]×

+∞∑v=0

[ ∑s+t=v

(−jλ1)s (−jT )t

t!

](z − 1

jλ1

)v

and hence

Pd =γL

0

2πj

∫L

(1

1 − jzλ1

)L

eLβ1

1−jzλ1e−jzT

zdz (116)

= γL0 exp

(− T

λ1

) +∞∑p=0

(Lβ1)p

p!

∑s+t=p+L−1

(Tλ1

)t

t!

=

+∞∑p=0

(Lβ1)p

p!e−Lβ1

p+L−1∑t=0

(Tλ1

)t

t!e− T

λ1

where

β1 =|α1|2λ1

=

∣∣∣A∑M−1l=0 wl exp (−2πjl(f − fn))

∣∣∣2σ2∑M−1

l=0 w2l

(117)

We remark here that A is the amplitude of the complex pure tone defined in section4 in the paragraph before (43) and fn = n

Mis the normalized center frequency of the

n-th channel, 0 ≤ n ≤ M − 1.


7 Probability of False Alarm(0 < γ ≤ 1/2, N ≥ 1, L ≥ 1)

For a given threshold T > 0, the corresponding probability of false alarm Pfa for theFFT summation detector with 0 < γ ≤ 1/2 can be immediately obtained from theformula (86). When the amplitude A of the complex pure tone s(t) = A exp(2πjfFst)vanishes (Fs is the sampling frequency), the received signal samples rk contain whiteGaussian noise only. In this case, the formula (86) actually computes the probabilityof false alarm Pfa for a given threshold T and the α sequence defined by (52) and

the β sequence defined by βl = |αl|2λl

, 1 ≤ l ≤ LN , both vanish. It follows from the

formula (86) that for a given threshold T > 0, the corresponding probability of falsealarm Pfa for the FFT summation detector with 0 < γ ≤ 1/2 is given by

Pfa =LN∑l=1

Ale− T

λl (118)

where λl, 1 ≤ l ≤ LN , are the LN distinctive eigenvalues of the covariance matrix

H = E(ZnZ

Hn

)given by (17) and Al =

λLN−1lQ

m�=l(λl−λm), 1 ≤ l ≤ LN . Note that the

formula (118) was derived in [13] via a different method.

8 Probability of False Alarm(γ = 0, N > 1, L > 1)

Using basically the same argument as in the previous section, we can derive theformula for the probability of false alarm Pfa for the FFT summation detector withγ = 0, N > 1, L > 1. In fact, it follows from (110) that, for a given thresholdT > 0, the probability of false alarm Pfa for the FFT summation detector withγ = 0, N > 1, L > 1 is given by

Pfa =

N∑m=1

L∑k=1

Am,k

k−1∑t=0

(T

λm

)t

t!e−

Tλm (119)


where the constants Am,k, defined by (89), are reproduced here for easy reference⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Am,L =

⎛⎜⎜⎝ λN−1

m∏l �=m

(λm − λl)

⎞⎟⎟⎠

L

, 1 ≤ m ≤ N ,

Am,k = Am,L

∑P

l�=m kl=L−k

∏( L+ kl − 1kl

)(λl

λl − λm

)kl

,

1 ≤ k ≤ L− 1 , 1 ≤ m ≤ N .

(120)

Here the numbers λl, 1 ≤ l ≤ N , are the N distinct eigenvalues of the Hermitianmatrix A computed by (21).

9 Probability of False Alarm(γ = 0,K = N = 1, L ≥ 1)

Finally, from (116) it follows that for a given threshold T > 0, the probability of falsealarm Pfa for the FFT summation detector for γ = 0, K = N = 1, L ≥ 1 is given by

Pfa =

L−1∑t=0

(Tλ1

)t

t!e− T

λ1 (121)

where

λ1 = σ2

M−1∑l=0

w2l

We remark here that the formula (121) was also derived in [3] and [13], using differenttechniques.

10 Summary

We have presented a full derivation of formulas for the probability of detection Pd andthe probability of false alarm Pfa for the FFT filter bank-based summation detectorwhen the received signal is a complex pure tone embedded in additive white Gaussiannoise. Specifically, the formulas for the probability of detection Pd are given in theidentities (86), (110), (116) and the formulas for the probability of false alarm Pfa

are given in the identities (118), (119), (121). The formulas (118) and (121) werederived in [13] by different approaches but they are presented here as corollaries of


formulas for Pd for the sake of completeness. The results presented in this reportprovide the basis for evaluating the performance of the FFT summation detector fordifferent scenarios.

References

[1] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, 1993.

[2] F. J. Harris, “On the Use of Windows for Harmonic Analysis with the DiscreteFourier Transform,” Proceedings of the IEEE, vol. 66, no.1, January 1978.

[3] R. S. Walker, “The Detection Performance of FFT Processors for NarrowbandSignals,” D.R.E.A. Technical Memorandum 82/A, February 1982.

[4] G. A. Zimmerman and S. Gulkis, “Polyphase-Discrete Fourier TransformSpectrum Analysis for the Search for the Extraterrestrial Intelligence SkySurvey,” TDA progress report 42-107, Jet Propulsion Laboratory, Pasadena,CA., pp. 141-154, July-September, 1991.

[5] H. C. So, Y. T. Chan, Q. Ma, P. C. Ching, “Comparisons of VariousPeriodograms for Sinusoidal Detection and Frequency Estimation,” IEEETransactions on Aerospace and Electronic Systems, vol. 35, no.3, pp. 945-952,July 1999.

[6] B.M. Bass, “Low-Power High-Performance 1024-Point FFT Processor,” IEEEJournal of Solid-State Circuits, vol. 34, no. 3, March 1999.

[7] S. He and M. Torkelson,“A New Approach to Pipeline FFT Processor,” Proc.10th International Parallel Processing Symposium, IPPS April 1996.

[8] M. Frigo, S.G. Johnson, “The Design and Implementation of FFTW3,”Proceedings of the IEEE, vol. 93, no. 2, pp. 216–231, 2005.

[9] B. H. Maranda, “On the False Alarm Probability for an Overlapped FFTProcessor,” IEEE Transactions on Aerospace and Electronic Systems, vol. 32,no.4, pp. 1452-1456, October, 1996.

[10] R. Inkol and S. Wang, “A Comparative Study of FFT-Summation andPolyphase-FFT CFAR Detectors,” Canadian Conference on Electrical andComputer Engineering, May 2004.

[11] S. Wang and R. Inkol, “FFT Filter Bank-Based Majority and SummationCFAR Detectors: A Comparative Study,” Canadian Conference on Electricaland Computer Engineering, May 2004.


[12] S. Wang and R. Inkol, “Operating Characteristics of the Wideband FFT FilterBank J-out-of-L CFAR Detectors,” DRDC Ottawa TR 2003-235, DefenceR & D Canada – Ottawa, December 2003.

[13] F. Patenaude and D. Boudreau, “ CFAR Detection Based on FFT andPolyphase FFT Filter Banks: Known SNR,” Technical Memorandum,VPCS22-97, Communications Research Centre, Industry Canada, December,1997.

[14] F. Patenaude, S. Wang and B. Kozminchuk, “The Wideband FFT andPolyphase FFT Filter Bank-Based CFAR Detectors with Direction Finding:Comparisons and Recommendations,” Technical Memorandum, VPSAT 16/00,Communications Research Centre, Industry Canada, October 2000.

[15] W.J.L. Read, “Detection of Frequency Hopping Signals in Digital WidebandData,” DRDC Ottawa TR 2002-162, Defence R & D Canada – Ottawa,December 2002.

[16] G.L. Turin, “The Characteristic Function of Hermitian Quadratic Forms inComplex Normal Variables,” Biometrika, vol. 47, pp. 199-201, June 1960.


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Theoretical Performance of the FFT Filter Bank-Based Summation Detector

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Wang, Sichun ; Inkol, Robert

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December 2005

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This technical report derives the probabilities of detection and false alarm for the FFT filter bank-based summation detector when the received signal is a complex pure tone embedded in additivewhite Gaussian noise. These results are useful for the performance analysis of the FFT filter bank-based summation detector and can be used to set up the detector for operation at a desired constantfalse alarm rate and predict the detection performance.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a document and could behelpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment modeldesignation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from apublished thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to selectindexing terms which are Unclassified, the classification of each should be indicated as with the title).

Fast Fourier Transform, probability of detection, probability of false alarm, summation detector

theoretical performance of the fft filter bank-based

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