detection of narrow-band signals through the fft and polyphase fft filter banks: noncoherent versus...

1424 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 59, NO. 5, MAY 2010

Detection of Narrow-Band Signals Through the FFTand Polyphase FFT Filter Banks: Noncoherent

Versus Coherent IntegrationSichun Wang, Senior Member, IEEE, Robert Inkol, Senior Member, IEEE,

Sreeraman Rajan, Senior Member, IEEE, and François Patenaude

Abstract—Formulas are derived for computing performancegain that is achieved by coherent integration over noncoherentintegration in detection schemes based on polyphase fast Fouriertransform (FFT) and FFT filter banks. Numerical computationof the processing gain is then discussed. The crucial role that isplayed by window energy normalization in detection performancecomparisons, which has never been recognized in the literature, isemphasized and analyzed. The numerical results provided for typ-ical implementation parameters are useful for making appropriatetradeoffs in the design of solutions for practical signal detectionproblems.

Index Terms—Coherent integration, constant false-alarm rate(CFAR), detection and estimation, fast Fourier transform (FFT)filter bank, noncoherent integration, polyphase FFT (PFFT) filterbank, spectral analysis.

I. INTRODUCTION

THE fast Fourier transform (FFT) filter bank, a computa-tionally efficient approach for implementing the equiv-

alent of a bank of bandpass filters [1], is well suited forthe detection of narrow-band signals that are embedded inwideband noise. It has been widely used in traditional appli-cations such as civilian spectrum monitoring, military radiosurveillance, search for extraterrestrial intelligence, and instru-mentation and measurement [2]–[8]. The FFT filter bank isa fundamental component in orthogonal-frequency-division-multiplexing-based wireless communication systems and hasrecently been proposed to serve the additional function of spec-tral sensing in advanced cognitive radios [9]–[12]. Fig. 1 showsa basic implementation where input data blocks are directlyprocessed by an FFT, and detection decisions are obtainedby comparing the resultant power spectrum estimates againsta detection threshold that is selected to provide a reasonabletradeoff between sensitivity and the probability of false alarm.An important refinement involves the windowing of input datablocks with a time-domain window function [13]. At the cost

Manuscript received June 30, 2009; revised November 16, 2009. Currentversion published April 7, 2010. The Associate Editor coordinating the reviewprocess for this paper was Prof. Alessandro Ferrero.

S. Wang, R. Inkol, and S. Rajan are with the Defence Research and Develop-ment Canada, Ottawa, ON K1A 0Z4, Canada (e-mail: [email protected]; [email protected]; [email protected]).

F. Patenaude is with the Communications Research Centre Canada, Ottawa,ON K2H 8S2, Canada (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2009.2038294

Fig. 1. Basic FFT detector.

of some processing loss, this suppresses the spectral sidelobesthat would otherwise exist in the spectrum of a strong signaland that could be incorrectly detected as valid signals.

The usable sensitivity for detecting a pure sinusoid is a func-tion of the noise power that is contained in the detection band-width. Since the detection bandwidth is inversely proportionalto the length of the FFT, a 3-dB gain in detection performanceis obtained for each doubling of the FFT length [14], [15].However, once the detection bandwidth becomes small relativeto the signal bandwidth, further increases in the FFT lengthare likely to be of little benefit since the proportion of thesignal power being processed within the detection bandwidthdecreases. This limitation is of considerable importance giventhat most signals of practical interest exhibit a combination ofphase noise, time-varying Doppler frequency shifts, and modu-lation. Furthermore, implementation constraints, such as mem-ory and processor throughput, may also limit the achievablecoherent gain.

Two alternative approaches for improving detection perfor-mance are possible. The first approach, which is called theL-block FFT summation detector and shown in Fig. 2, involvesthe following steps:

• divide the available input data record into L subblocks;• apply a time-domain window function to each subblock;• perform an inverse FFT (IFFT) on each subblock (or per-

form an FFT if the ordering of elements in the subblocksis reversed);

• add the L power spectral estimates;• perform the detection thresholding operation using the

summed power spectral estimates.

0018-9456/$26.00 © 2010 IEEE

WANG et al.: DETECTION OF NARROW-BAND SIGNALS THROUGH THE FFT AND POLYPHASE FFT FILTER BANKS 1425

Fig. 2. L-block FFT summation detector (N = K).

This L-block FFT summation detector, described in[16]–[18] for the single FFT bin case, can be used to processlong data records without incurring the previously mentionedcomplications.

Several points are worth noting. First, there is the ability toadjust the detection bandwidth by grouping the FFT bins andsumming the spectral power from the individual bins of eachgroup, as shown in Fig. 2. This allows the detection bandwidthto be set independently of the integration period and opensup possibilities for approximating channels having arbitrarybandwidths and center frequencies through the selection ofthe FFT bins that are used to form each channel power es-timate. Second, the processing losses incurred by the use oftime-domain window functions can be substantially reducedby overlapping the input data subblocks [19]. Finally, similarideas have been applied in analog signal detection implementa-tions. For example, low-cost radar and radar electronic warfarereceivers often use a simple receiver consisting of an RFbandpass filter, a square law or other nonlinear device, and alow-pass postdetection filter. Decreasing the bandwidth of thelow-pass filter effectively provides noncoherent integration gainand improves the usable sensitivity provided that the signalduration is sufficiently large. An examination of publishedresults for the dependence of the sensitivity on the RF andpostdetection filter bandwidths [20] shows that the effect ofnoncoherent integration on the analog implementation is quitesimilar to that of the FFT summation detector.

The second approach, shown in Fig. 3, is based on thepolyphase FFT (PFFT) filter bank [21]–[23]. The PFFT is anextension of the FFT where multiple input data blocks arepreprocessed by a polyphase network [5], [22]. The magni-tude response associated with the individual PFFT bins canbe adjusted by implementing a sophisticated window functionbased on a finite impulse response (FIR) prototype filter inthe polyphase network. In some respects, the PFFT can beregarded as a computationally efficient approach for increasingthe length of the FFT and decimating the FFT output datavector to provide the desired frequency resolution. An excellentintroduction to the PFFT filter bank can be found in [22], whereit is referred to as the window presum FFT. The PFFT filterbank has applications in radio astronomy [5], [24] and is alsobeing considered for spectrum sensing applications in cognitiveradios [11], [12].

Fig. 3. PFFT detector.

In [25] and [26], the dependence of the detection perfor-mance on L is characterized for the L-block FFT summationdetector. For large L (≥ 512), the FFT summation detectorexhibits roughly a 1.5-dB gain in detection performance foreach doubling of L. If L is relatively small, a case of practicalinterest in some applications, the performance gain exceeds2 dB for each doubling of L. The spectral summation performedby the FFT summation detector, which is a form of noncoherentintegration, differs from the coherent integration that is implicitin the correlation process in the PFFT filter bank. Therefore, theFFT summation and PFFT detectors will have quite differentperformance characteristics when used for the detection ofnarrow-band signals. In this paper, the techniques used in [25]and [26] are adapted to analyze the relative performance ofthe FFT summation and PFFT detectors for a single complexsinusoid embedded in white Gaussian noise. Special cases ofthe results presented in this paper, when the frequency of thecomplex sinusoid coincides with the center frequency of an FFTbin, have been derived in the conference paper [27]. Graphsthat are constructed using the results of this paper are useful forexploring tradeoffs in the design of practical implementationswithout necessitating extensive computer simulations.

This paper is organized as follows. In Section II, the FFTsummation detector and the PFFT detector are described inmathematical terms. In Section III, the performance gain ofthe PFFT detector over the FFT summation detector is derivedfor various scenarios of interest given the assumption thatthe same amount of signal data is processed. In Section IV,numerical computation of the processing gain is discussed. InSection V, simulation results are presented that confirm thetheoretical predictions produced by the derived formulas. InSection VI, the important role of window energy normalizationin the comparison of windows and prototype filters is discussed.Finally, conclusions and remarks are given in Section VII.

II. SIGNAL DETECTION BASED ON NONCOHERENT

INTEGRATION AND COHERENT INTEGRATION

Here, we present a mathematical definition of the FFT sum-mation detector based on noncoherent integration. It is then


shown that the PFFT detector, which is implicitly based oncoherent integration using a single large FFT, is a special caseof the FFT summation detector.

Assume that the input signal processed by the FFT or PFFTfilter bank is a uniformly sampled band-limited baseband com-posite signal. The signal sample sequence, which is denoted byrk, can be written as

rk = sk + uk, k = 0, 1, 2, . . . (1)

where sk denotes the superposition (sum) at the kth samplinginstant of signals (modulated or unmodulated) with differentcenter frequencies in the sampled frequency band, and uk

denotes the additive complex-valued zero-mean white Gaussiannoise with variance E(|uk|2) = σ2 > 0. The real and imag-inary parts of uk are assumed to be uncorrelated Gaussianrandom variables with zero mean and identical variance σ2/2.

Consider M channels that are uniformly distributed acrossthe frequency range contained within the Nyquist bandwidth,where K FFT bins are assigned to each channel, and N centerFFT bins (N ≤ K) in each channel are used to estimate thechannel spectral power. Consequently, an FFT or IFFT of lengthMK is used to compute the power spectral estimates for theM channels. Without loss of generality, assume that K − N isan even integer. Let W = [w0, . . . , wMK−1]t be a symmetricwindow of length MK, where the superscript t denotes matrixtransposition. Consider L consecutive overlapped sample vec-tors Sl constructed as follows (0 ≤ l ≤ L − 1):

Sl =[rl(1−γ)MK+MK−1, . . . , rl(1−γ)MK

]t(2)

where γ is the input data overlap ratio with γMK being aninteger. In practice, γ is often selected to be either 0 (nooverlap) or 1/2 (50% overlap). Note that the length of the wholeinput data record, which consists of the L overlapped datablocks, i.e., Sl, 0 ≤ l ≤ L − 1, is (L − 1)(1 − γ)MK + MKand that (L − 1)(1 − γ)MK + MK = LMK if γ = 0. Win-dowing each of the vectors Sl by W results in the windowedsample vectors Xl, i.e.,

Xl =[w0rl(1−γ)MK+MK−1, . . . , wMK−1rl(1−γ)MK

]t.

The vectors Xl are then transformed by the IFFT matrix FMK

of dimensions MK × MK to yield the FFT filter bank outputsample vectors Yl, i.e.,

Yl = FMKXl = [yl,0, yl,1, . . . , yl,MK−1]t

where the inverse FFT matrix FM of dimensions M × M forany positive integer M ≥ 1 is defined by

FM =

⎡⎢⎢⎢⎣

1 1 · · · 1· · · · · · · · · · · ·1 e

2πjlM · · · e

2πjl(M−1)M

· · · · · · · · · · · ·1 e

2πj(M−1)M · · · e

2πj(M−1)(M−1)M

⎤⎥⎥⎥⎦ . (3)

Note that if the ordering of the signal samples in the input datavectors Sl, 0 ≤ l ≤ L − 1, is reversed (time reversal), then the

FFT rather than the IFFT should be used in the FFT filter bank.These two implementations are completely equivalent. Fromeach vector Yl, an associated vector Zl = [zl,0, . . . , zl,M−1]t

of length M is formed by summing the power from the Ncenter FFT bins in each channel. Specifically, for the mthchannel with indexes Im, Im + 1, . . . , Im + (N − 1), whereIm = mK + (K − N)/2, we have

zl,m =N−1∑n=0

|yl,n+Im|2, 0 ≤ l ≤ L − 1; 0 ≤ m ≤ M − 1.

(4)

For the FFT summation detector, the detection statistic ofthe mth channel is defined by

∑L−1l=0 zl,m. Given a detection

threshold T > 0, a signal is declared to exist in the mth chan-nel if

∑L−1l=0 zl,m ≥ T . For constant false-alarm rate (CFAR)

operation, T is adjusted to yield a desired probability of falsealarm Pfa, which is defined by

Pfa = Pr

{L−1∑l=0

zl,m ≥ T

∣∣∣∣∣ rk = uk, k = 0, 1, . . .

}. (5)

This paper makes the assumption that the noise varianceE(|uk|2) = σ2 is known. To achieve a specified Pfa, a practicalsystem implementation would require provisions for adaptivelyestimating σ2 over time from the channel power output data.

The L-block FFT summation detector based on L overlappedFFTs of size MK reduces to a coherent integration schemewhen the total available input data record, which is of length(L − 1)(1 − γ)MK + MK, is transformed into the frequencydomain by a single FFT. This is closely related to the approachthat is taken in the PFFT detector shown in Fig. 3. Conceptually,the PFFT filter bank is constructed by applying a long FFT toan input data vector windowed by a prototype linear-phase FIRfilter and then forming the channel output vector by the deci-mation of the long FFT output vector by a factor correspondingto the length of the polyphase filters.

Assume that the PFFT filter bank has M channels and eachof the M polyphase filters has P taps. Let the prototype filter ofthe PFFT filter bank, which is denoted by H here, be defined by

H = [h0, h1, . . . , hPM−1]t. (6)

The M polyphase filters, which are denoted by H0,H1, . . . ,HM−1 here, are defined by

Hm = [hm, hM+m, . . . , h(P−1)M+m]t, 0 ≤ m ≤ M − 1.(7)

Let R be the input data record into the PFFT filter bankdefined by

R = [rPM−1, rPM−2, . . . , r1, r0]t. (8)

Partition the vector R into M subvectors, i.e., R0,R1, . . . ,RM−1, each of length P , as follows:

Rm =[r(P−1)M+m, r(P−2)M+m, . . . , rM+m, rm

]t,

0 ≤ m ≤ M − 1. (9)


R0 is the data vector entering the polyphase FIR filter HM−1,R1 is the data vector entering the polyphase FIR filter HM−2,and so on, and RM−1 is the data vector entering the polyphaseFIR filter H0. The output sample xm from the mth polyphaseFIR filter Hm is, therefore, defined by

xm = hmr(P−1)M+M−1−m + hM+mr(P−2)M+M−1−m + · · ·+h(P−1)M+mrM−1−m, 0 ≤ m ≤ M − 1. (10)

The output vector of the PFFT filter bank for the input vectorR, which is denoted by Y here, is then defined by

Y = [y0, y1, . . . , yM−1]t = FMX (11)

where FM is the IFFT matrix of dimensions M × M definedby (3), and

X = [x0, x1, . . . , xM−1]t. (12)

It is instructive to compute the channel output Cm(f) of themth channel in the PFFT filter bank for the complex sinusoidalinput signal defined by rk = e2πjkf , where f ∈ [−1/2, 1/2)and k ≥ 0. We have

Cm(f) =M−1∑k=0

xke2πjmk/M

=M−1∑k=0

[P−1∑l=0

hlM+krM(P−1−l)+M−1−k

]e2πjmk/M

=M−1∑k=0

P−1∑l=0

hlM+ke2πj[M(P−1−l)+M−1−k]fe2πjmk/M

= e2πj[PM−1]fM−1∑k=0

P−1∑l=0

hlM+k

× e−2πj[lM+k]fe2πjmk/M

= e2πj[PM−1]fM−1∑k=0

P−1∑l=0

hlM+k

× e−2πj[lM+k]fe2πj(m(lM+k))/M

= e2πj[PM−1]fPM−1∑

q=0

hqe−2πjqfe2πjmq/M

= e2πj[PM−1]fPM−1∑

q=0

hqe−2πjq(f−m/M). (13)

Hence, the magnitude response of the mth channel in the PFFTfilter bank is given by

|Cm(f)| =

∣∣∣∣∣PM−1∑

q=0

hqe−2πjq(f−m/M)

∣∣∣∣∣=

∣∣∣∣∣PM−1∑

q=0

hqe−2πjq(f−Pm/(PM) )

∣∣∣∣∣ (14)

which is exactly the same as the magnitude response of the(Pm)th channel in the FFT filter bank with its window definedby the prototype filter H given by (6).

The detection statistic for the mth channel in the PFFT filterbank is defined by |ym|2, where ym is defined by (11). Givena detection threshold T > 0, a signal is detected in the mthchannel for the input data record R defined by (8) if |ym|2 ≥ T .For CFAR operation, T is adjusted to yield a desired probabilityof false alarm Pfa defined by

Pfa = Pr{|ym|2 ≥ T

∣∣rk = uk, k = 0, 1, . . .}

. (15)

III. PERFORMANCE GAIN OF COHERENT INTEGRATION

OVER NONCOHERENT INTEGRATION

Here, we derive formulas for the performance gain of thePFFT detector over the FFT summation detector when the sameamount of input signal data is processed. To set up a simpleframework for making comparisons, we shall assume that oneFFT bin is assigned to each channel in the FFT summationdetector (i.e., K = N = 1) and that the input signal is a singlecomplex sinusoid that is embedded in additive complex-valuedzero-mean white Gaussian noise.

There are several motivations for this assumption. First, ifthe signal of interest is a single complex sinusoid, the theo-retical issues are well defined and mathematically tractable.Second, the results directly apply to the problem of detectingsinusoidal signals embedded in white Gaussian noise [5]. Manypractical applications in test and measurement [6] and powersystems [7], [8], involve detection of these signals. Moreover,in applications such as spectrum monitoring [2], [3], electronicwarfare systems [4], and cognitive radios [10], most signalsof interest are constructed by the modulation of sinusoidalcarriers. Typically, uniformly spaced center frequencies areassigned to portions of the RF spectrum, and specifications forthe modulation waveforms are set to limit the signal bandwidthand avoid excessive interference between signals on adjacentcenter frequencies. Ideally, one would set the FFT length andthe sampling rate (i.e., the observation time corresponding toa single detection decision) to result in FFT bins lining upwith the allowed carrier frequencies and providing a frequencyresolution corresponding to the signal bandwidth. As most ofthe signal power falls inside a single FFT bin, such signalscan essentially be treated as simple sinusoids. Finally, formodulated signals that cannot be approximated as describedabove, insights into the relative detection performance of thePFFT and FFT summation detectors can be obtained by varyingthe frequency of the complex sinusoid within the bandwidth ofa typical FFT channel.

Now, assume that the input data stream processed by the FFTand PFFT filter banks is defined by

rk = Aej2πkf + uk = sk + uk, k = 0, 1, 2, . . . (16)

where sk = Aej2πkf is a complex sinusoid, with A and f ,0 ≤ f < 1, being, respectively, its amplitude and frequency(normalized with respect to the sampling frequency), and uk

is the independent identically distributed complex-valued zero-mean Gaussian noise component with E(usu

∗t) = σ2δst. Here,


δst is the Kronecker delta function, and E(|uk|2) = σ2 is thenoise variance. The noise samples uk are further assumed to becircular in the sense that the real and imaginary parts of uk areuncorrelated real-valued Gaussian random variables with zeromean and equal variance σ2/2.

Some of the notation and results from previous work [25],[26] are used here.

Definition 1: For L ≥ 1 and 0 < Pfa < 1, the unique solu-tion of the following in z:

e−z[1 + z + z2/2! + · · · + zL−1/(L − 1)!

]= Pfa (17)

is denoted by TL(Pfa).Definition 2: Let L ≥ 1. The function FL(x, y) is defined by

FL(x, y) = e−x−y∞∑

q=0

xq

q!

q+L−1∑k=0

yk

k!. (18)

It is known [25], [26] that ∂FL/∂x > 0 and ∂FL/∂y < 0 ifx > 0 and y > 0. For any fixed y > 0, FL(x, y), as a functionof x ∈ [0,∞), is strictly increasing with its range defined by theinterval J(y) = [e−y

∑L−1k=0 yk/k!, 1).

Definition 3: For any fixed y > 0, the inverse function ofFL(x, y) that is treated as a function of x is denoted byGL(z, y). The following identity:

FL (GL(z, y), y) = z (19)

holds for y > 0 and z ∈ J(y).Lemma 1 [28, eq. (116)]: Consider the nth channel in the

L-block FFT summation detector with N = K = 1 and γ = 0.Let T > 0 be a given detection threshold. For the input signaldefined by (16), the probability of detection is given by

Pd = FL(Lβ, T/λ) (20)

where FL(x, y) is defined by (18), and the parameters λ, β, andfn are computed by⎧⎪⎪⎨

⎪⎪⎩λ = σ2

∑M−1l=0 w2

l

β =∣∣∣A∑M−1

l=0 wl exp (−2πjl(f − fn))∣∣∣2 /λ

fn = n/M.

(21)

When the input signal rk contains only noise (i.e., rk = uk,k = 0, 1, 2, . . .), the probability of false alarm is given by

Pfa = e−T/λL−1∑k=0

(T/λ)k

/k!. (22)

Solving (22) for T/λ, we obtain

T/λ = TL(Pfa). (23)

Substituting (23) into (20) and solving for β yield

β =

∣∣∣∣∣AM−1∑l=0

wl exp (−2πjl(f − fn))

∣∣∣∣∣2 /

λ

= GL (Pd, TL(Pfa))/ L (24)

which can be rewritten as

|A|2/σ2 =GL (Pd, TL(Pfa))

L∣∣∣∑M−1

l=0 w′l exp (−2πjl(f − fn))

∣∣∣2 (25)

where w′l = wl/

√∑M−1m=0 w2

m, l = 0, 1, . . . ,M − 1. Equation(25) computes the SNR required by the L-block summationdetector to achieve the given probabilities of detection Pd andfalse alarm Pfa for the input signal defined by (16). Note thatthe computed SNR depends only on the window sequence w′

l,which satisfies the constraint

∑M−1l=0 (w′

l)2 = 1.

Definition 4: The window function W = [w0, . . . , wM−1]t

is energy-normalized if its coefficients satisfy the constraint

M−1∑l=0

w2l = 1.

Definition 5: The window function W = [w0, . . . , wM−1]t

is dc-normalized if its coefficients satisfy the constraint

M−1∑l=0

wl = 1.

Definition 6: Consider the nth channel in the L-block FFTsummation detector with N = K = 1 and γ = 0. Assume thatthe window W = {wm}M−1

m=0 in the FFT filter bank is energy-normalized, i.e.,

∑M−1l=0 w2

l = 1. For the input signal defined by(16), the SNR required by the L-block summation detector toachieve the given probability of detection Pd at a given level ofprobability of false alarm Pfa in the nth channel is denoted bySNR(W, Pd, Pfa, L,M, n, f), i.e.,

SNR(W, Pd, Pfa, L,M, n, f)

=GL (Pd, TL(Pfa))

L∣∣∣∑M−1

l=0 wl exp (−2πjl(f − n/M))∣∣∣2 . (26)

Formula (26) is of fundamental importance in the perfor-mance analysis of the L-block summation detector. In fact, thisformula has recently been used to analyze the performance ofthe L-block summation detector and a closely related detector(the FFT majority detector) as a function of L for a givenwindow function [25], [26]. In this paper, we use (26) tocharacterize the difference in performance between the FFTsummation and PFFT detectors. The formulas presented in thefollowing theorems can be used for comparing the detectionperformance of the FFT summation and PFFT detectors. Theyare presented in the FFT summation detector framework sincethe PFFT detector is a special case of the FFT summationdetector. Note that the windows employed in the FFT summa-tion detector are usually derived from a continuous prototypewindow function [13], whereas the prototype filters in the PFFTdetector can be either a classical window function or an FIRlow-pass filter. The latter can be designed by applying a windowfunction to the impulse response of an ideal infinite impulseresponse filter or iterative optimization algorithms, such as theParks–McClellan filter design algorithm.


Definition 7: Let w(x) be a real-valued continuous prototypewindow function defined on the unit interval [0, 1]. For any pos-itive integer M > 1, a length-M energy-normalized window,which is denoted by WM here, is defined by

WM = [wM (0), . . . , wM (M − 1)]t

wM (m) =w(

mM−1

)√

M−1∑k=0

w2(

kM−1

) , 0 ≤ m ≤ M − 1. (27)

Most classical windows are defined in this manner [13] usinga symmetric prototype window function w(x) satisfying thesymmetry constraint w(x) = w(1 − x), 0 ≤ x ≤ 1.

We first consider the case where the window in the FFT sum-mation detector and the prototype filter in the PFFT detectorare both derived from real-valued continuous prototype windowfunctions defined on the unit interval [0, 1].

Theorem 1: Consider the nth channel in the L-block sum-mation detector with K = N = 1 and γ = 0. Assume that Mis large, and the input signal is the complex sinusoid definedby (16). Let w(x) and v(x) be two real-valued continuousprototype window functions defined on [0, 1]. Let WM bedefined by (27), and let VLM = [vLM (0), . . . , vLM (LM −1)]t be the energy-normalized window function of length LMderived from v(x) as defined by (27), i.e.,

vLM (m) =v(

mLM−1

)√

LM−1∑k=0

v2(

kLM−1

) , 0 ≤ m ≤ LM − 1. (28)

Then, for a given probability of false alarm Pfa ∈ (0, 1) and agiven probability of detection Pd ∈ (0, 1), the performance gainof the one-block FFT summation detector employing a singleFFT of size LM and the window function VLM defined by(28) over the L-block FFT summation detector employing Lnonoverlapped FFTs of size M and the window function WM

defined by (27) is given by

gL = DL + 10 log 10 [κL(v, w, fo)] (29)

where

DL = 10 log 10GL (Pd, TL(Pfa))G1 (Pd, T1(Pfa))

(30)

κL(v, w, fo) =

∫ 1

0 w2(x) dx∫ 1

0 v2(x) dx

∣∣∣∫ 1

0 v(x)e−2πjLfox dx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πjfox dx∣∣∣2 (31)

and fo = M(f − n/M) is called the normalized frequencyoffset of the signal (16) from the channel center frequencyn/M . If fo = 0, κL(v, w, fo) simplifies to

κ(v, w) =

∫ 1

0 w2(x) dx∫ 1

0 v2(x) dx

∣∣∣∫ 1

0 v(x) dx∣∣∣2∣∣∣∫ 1

0 w(x) dx∣∣∣2

=

∣∣∣∫ 1

0 v(x) dx∣∣∣2∫ 1

0 v2(x) dx

⎡⎢⎣∣∣∣∫ 1

0 w(x) dx∣∣∣2∫ 1

0 w2(x) dx

⎤⎥⎦−1

. (32)

The right-hand side of (32) is equal to 1 if v(x) = w(x), x ∈[0, 1].

The expression κ(v, w) in (32) is the ratio of the normalizedprocessing gain of the window VLM to that of the windowWM . Following [13, eq. (15)], the processing gain of thelength-M window WM is computed by

PG(WM ) =

∣∣∣∑M−1m=0 wM (m)

∣∣∣2∑M−1m=0 w2

M (m)

=

∣∣∣∑M−1m=0 w

(m

M−1

)1

M−1

∣∣∣21

M−1

∑M−1m=0 w2

(m

M−1

)1

M−1

≈M

∣∣∣∫ 1

0 w(x) dx∣∣∣2∫ 1

0 w2(x) dx.

The normalized processing gain of the window WM , whichis denoted by NPG(w) here, is defined as the proportionalityconstant of the processing gain as a linear function of M , i.e.,

NPG(w) =

∣∣∣∫ 1

0 w(x) dx∣∣∣2∫ 1

0 w2(x)dx.

It follows that κ(v, w) = NPG(v)/NPG(w). Note that thereciprocal of the normalized processing gain NPG(w) isnothing but the equivalent noise bandwidth defined by[13, eq. (11)] after normalization by the FFT bin width 1/M ,i.e., the equivalent noise bandwidth of the window WM, whichis denoted by ENBW(w) here, is computed by

ENBW(w) =

∑M−1

m=0w2

M (m)∣∣∑M−1

m=0wM (m)

∣∣21M

= M

∫ 1

0 w2(x) dx

M∣∣∣∫ 1

0 w(x) dx∣∣∣2 =

1NPG(w)

.

Thus, we can also write κ(v, w) = ENBW(w)/ENBW(v).If L is set to 1 in (29), we can immediately see that DL = 0

and gL reduces to the simple form

g(v, w, fo) = 10 log 10

∫ 1

0 w2(x)dx∫ 1

0 v2(x)dx

∣∣∣∫ 1

0 v(x)e−2πjfoxdx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πjfoxdx∣∣∣2 .

(33)

In other words, formula (33) computes the processing gain ofwindow v over window w for the one-block FFT summationdetector for the input signal defined by (16). It can be verifiedthat the processing gain of window v over window w for theL-block summation detector with γ = 0 is independent of Land also equal to g(v, w, fo).

On the other hand, when L becomes relatively large, ac-cording to the Riemman–Lebesgue Lemma from Fourier analy-sis, κL(v, w, fo) in (31) approaches zero for fo = 0, and10 log 10[κL(v, w, fo)] may assume negative values with alarge absolute value. As will be shown later in this paper, the


processing gain computed by (29) becomes negative for fo

close to ±0.5. This implies that, compared with the L-blocksummation detector, the ability of the PFFT detector to detectsinusoidal signals with frequencies that are close to the channelboundaries fo = ±0.5 diminishes very quickly. This is notsurprising since the frequency response of the prototype filterbecomes highly concentrated near the channel center frequencyas L increases. As the processing gain gL is a function of thenormalized frequency offset fo, when designing the prototypefilter in the PFFT filter bank, the power spectral density char-acteristics of targeted signals should be taken into account,if possible. It is conjectured that the results of this paper areultimately useful for characterizing the performance differenceof the FFT summation and PFFT detectors for modulatedsignals by treating the power spectral density of a modulatedsignal as a probability density function of fo.

A proof of Theorem 1 is sketched next. Applying formula(26) to the nth channel, the performance gain of the one-block summation detector employing the window (28) overthe L-block summation detector employing the window (27)is computed by (34), shown at the bottom of the page, where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

xm = mLM−1 , 0 ≤ m ≤ LM − 1

Δxm = xm+1 − xm = 1LM−1

ym = mM−1 , 0 ≤ m ≤ M − 1

Δym = ym+1 − ym = 1M−1

B(x) = (LM − 1)(f − n/M)xD(y) = (M − 1)(f − n/M)yfo = M(f − n/M).

This completes the proof of (29).Note that in the identity (29), in theory, the approximation

sign ≈, instead of the identity sign =, should be used; however,as the approximation error is completely negligible, the identitysign = is used. This practice will also be followed in subsequenttheorems without further elaboration.

We next extend Theorem 1 to the general case where theprototype filter in the PFFT filter bank is an FIR low-pass filter,and the window in the FFT filter bank is a classical window.

Theorem 2: Consider the nth channel in the L-block sum-mation detector with K = N = 1 and γ = 0. Assume that Mis large, and the input signal is the complex sinusoid definedby (16). Let WM and WLM be energy-normalized windows,of length M and LM , respectively, derived from the prototypewindow function w(x) as defined by (27). Let H be the energy-normalized linear-phase FIR filter of length-(LM) defined by

H = [h0, h1, . . . , hLM−1]t, whereLM−1∑m=0

h2m = 1. (35)

Then, for a given probability of false alarm Pfa ∈ (0, 1) anda given probability of detection Pd ∈ (0, 1), the performancegain of the one-block FFT summation detector employing asingle FFT of size LM and the window function H definedby (35) over the L-block FFT summation detector employing Lnonoverlapped FFTs of size M and the window function WM

defined by (27) is computed by

gL = DL + 10 log 10 [κL(H, w,M, fo)] (36)

where DL is defined by (30)

κL(H, w,M, fo)=

∣∣∣∫ 1

0 w(x)e−2πjLfoxdx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πjfoxdx∣∣∣2

×

∣∣∣∣LM−1∑m=0

hme−2πjm foM

∣∣∣∣2

∣∣∣∣LM−1∑m=0

wLM (m)e−2πjm foM

∣∣∣∣2 (37)

gL = 10 log 10SNR(WM , Pd, Pfa, L,M, n, f)

SNR(VLM , Pd, Pfa, 1, LM, nL, f)

= 10 log 10GL (Pd, TL(Pfa))G1 (Pd, T1(Pfa))

+ 10 log 10

∣∣∣∑LM−1m=0 vLM (m) exp (−2πjm(f − n/M))

∣∣∣2L∣∣∣∑M−1

m=0 wM (m) exp (−2πjm(f − n/M))∣∣∣2

= DL + 10 log 10

∣∣∣∑LM−1m=0 Δxmv(xm) exp (−2πjB(xm))

∣∣∣2∣∣∣∑M−1m=0 Δymw(ym) exp (−2πjD(ym)

∣∣∣2 + 10 log 10∑M−1

m=0 Δymw2(ym)∑LM−1m=0 Δxmv2(xm)

+ 10 log 10LM − 1

L(M − 1)

≈DL + 10 log 10

∫ 1

0 w2(x)dx∫ 1

0 v2(x)dx

∣∣∣∫ 1

0 v(x)e−2πjB(x)dx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πjD(x)dx∣∣∣2 ≈ DL + 10 log 10

∫ 1

0 w2(x)dx∫ 1

0 v2(x)dx

∣∣∣∫ 1

0 v(x)e−2πj(LM−1)

LM Lfoxdx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πj(M−1)

M foxdx∣∣∣2

≈DL + 10 log 10

∫ 1

0 w2(x)dx∫ 1

0 v2(x)dx

∣∣∣∫ 1

0 v(x)e−2πjLfoxdx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πjfoxdx∣∣∣2 = DL + 10 log 10(κL (v, w, fo)) (34)


and fo = M(f − n/M) is the normalized frequency offset ofthe complex sinusoid defined by (16). If fo = 0, κL(H, w,M, fo) simplifies to

κ(H,WLM ) =

∣∣∣∣LM−1∑m=0

hm

∣∣∣∣2

∣∣∣∣LM−1∑m=0

wLM (m)∣∣∣∣2 . (38)

If H is the length-LM energy-normalized window VLM de-rived from the continuous prototype window function v(x),κ(H,WLM ) = κ(v, w).

Formula (36) follows again from an application of (26).In fact

gL = 10 log 10GL (Pd, TL(Pfa))G1 (Pd, T1(Pfa))

+ 10 log 10

∣∣∣∣LM−1∑m=0

hme−2πjm(f−n/M)

∣∣∣∣2

L

∣∣∣∣M−1∑m=0

wM (m)e−2πjm(f−n/M)

∣∣∣∣2

=DL + 10 log 10

∣∣∣∣LM−1∑m=0

wLM (m)e−2πjm(f−n/M)

∣∣∣∣2

L

∣∣∣∣M−1∑m=0

wM (m)e−2πjm(f−n/M)

∣∣∣∣2

×

∣∣∣∣LM−1∑m=0


∣∣∣∣2

∣∣∣∣LM−1∑m=0


∣∣∣∣2

≈DL + 10 log 10

∣∣∣∫ 1

0 w(x)e−2πj(LM−1)(f−n/M)xdx∣∣∣2∣∣∣∫ 1

0 w(x)e−2πj(M−1)(f−n/M)xdx∣∣∣2

×

∣∣∣∣LM−1∑m=0


∣∣∣∣2

∣∣∣∣LM−1∑m=0


∣∣∣∣2

≈DL + 10 log 10

∣∣∣∫ 1



×

∣∣∣∣LM−1∑m=0


∣∣∣∣2

∣∣∣∣LM−1∑m=0


∣∣∣∣2

=DL + 10 log 10

∣∣∣∫ 1



×

∣∣∣∣LM−1∑m=0

hme−2πjm foM

∣∣∣∣2

∣∣∣∣LM−1∑m=0


∣∣∣∣2 . (39)

It can be shown that κL(H, w,M, fo) defined by (37) re-duces to κL(v, w, fo) defined by (31) if the prototype filterH defined by (35) is derived from the continuous prototypewindow function v(x), i.e., H is the same as the energy-normalized window VLM defined by (28).

To understand the relative performance of the PFFT detectorand the L-block FFT summation detector, it suffices to inves-tigate the variation of the expression κL(H, w,M, fo) as afunction of fo ∈ [−0.5, 0.5]. The expression κL(H, w,M, fo)can be written as the product of two terms respectivelydefined by ⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

t1 =∣∣∫ 1

0w(x)e−2πjLfoxdx

∣∣2∣∣∫ 1

0w(x)e−2πjfoxdx

∣∣2

t2 =

∣∣∣∣LM−1∑m=0

hme−2πjm

foM

∣∣∣∣2

∣∣∣∣LM−1∑m=0

wLM (m)e−2πjm

foM

∣∣∣∣2 .

(40)

The first term t1 depends only on L, fo, and the prototypewindow function w(x). The second term t2 is the square of theratio of the magnitude response of H to that of WLM computedat fo/M .

When fo = 0, t1 = 1 and t2 is equal to the ratio of theprocessing gains of H and WLM , i.e.,

t2 = κ(H,WLM ) =

∣∣∣∣LM−1∑m=0

hm

∣∣∣∣2

∣∣∣∣LM−1∑m=0

wLM (m)∣∣∣∣2 .

When designing the energy-normalized prototype filter H, areasonable strategy is to maximize its processing gain. Themaximum of the processing gain of H can be easily computedby applying the method of Lagrange’s multipliers. In fact,letting

φ =LM−1∑m=0

hm + μ

(LM−1∑m=0

h2m − 1

)

and setting ∂φ/∂hm to 0, m = 0, 1, . . . , LM − 1, yield

1 + 2μhm = 0, 0 ≤ m ≤ LM − 1.

This system of equations implies that

h0 = h1 = h2 = · · · = hLM−1.

Hence, among all energy-normalized prototype filters Hof length LM , the energy-normalized rectangular windowachieves the maximum processing gain LM . Thus, the max-imum value of t2 at fo = 0 for a fixed prototype windowfunction w is given by

maxfo=0

t2 =LM∣∣∣∣LM−1∑

m=0wLM (m)

∣∣∣∣2 ≥ 1.


Fig. 4. t1 as a function of fo (Blackman window).

Fig. 5. t2 as a function of fo (Blackman and rectangular windows).

This inequality implies that, in theory, for the sinusoid (16) withfo relatively close to 0, a PFFT detector can always be designedto outperform the L-block summation detector defined by anonrectangular prototype window function w, as the term DL

in (36) is always nonnegative (cf. Fig. 8).When fo is relatively far away from 0 in the interval [−0.5,

0.5], both terms t1 and t2 are less than 1. To obtain an idea of thevariation of t1 and t2 as a function of fo, for L = 1, 2, 3, 4 andM = 1024, t1 is plotted in Fig. 4 for the prototype Blackmanwindow defined by

w(x) = 0.42 − 0.5 cos(2πx) + 0.08 cos(4πx)

and t2 is plotted in Fig. 5 for the prototype Blackman win-dow w(x) and the prototype filter H defined by the energy-normalized rectangular window. Careful inspection of Figs. 4and 5 shows that 10 log 10(t1) is close to 0 when fo ≈ 0,whereas 10 log 10(t2) is positive for fo ≈ 0. Moreover, whenfo = 0, 10 log 10(t2) is equal to the difference (in decibels)in the processing gains between the rectangular window andthe Blackman window. When fo moves away from 0, both10 log 10(t1) and 10 log 10(t2) become negative, and, although

not shown in these figures, 10 log 10(t2) can approach −50 dBfor fo close to ±0.5. These plots imply that the L-block sum-mation detector may be able to detect certain weak sinusoidalsignals close to the channel boundaries fo = ±0.5, whereasthe PFFT detector most likely will fail to detect them when asimple window function is used for the prototype filter. This isconsistent with the observation that the magnitude response ofthe prototype filter is more concentrated near the channel centerthan that of the window that is used in the FFT filter bank.

The preceding two theorems have been formulated fornonoverlapped input data blocks (γ = 0). As discussed inSection I, overlapped data blocks can be employed to mitigateprocessing losses that are incurred by data windowing in theFFT summation detector. A natural question of practical inter-est is the following: Is it possible to compute or approximate theperformance gain of the PFFT detector over the FFT summationdetector if the input data blocks processed by the summationdetector are overlapped with γ = 0.5? The answer is affirma-tive. For an input data record of total length LM , if the overlapratio γ = 0.5 is applied, a total of 2L − 1 overlapped input datablocks each of size M can be constructed. This implies thatan additional group of L − 1 data blocks have been formed.Since the correlation between successive channel power spec-tral estimates computed from these 2L − 1 overlapped inputdata blocks is negligible for many commonly used windows[13], [29], [30], we are effectively applying a (2L − 1)-blocksummation detector without data overlap for signal detection,although the actual input data record length is only LM .Thus, the processing gain from the use of overlapped datablocks should at least partially offset the processing loss thatwould otherwise be expected from noncoherent averaging. Theprevious two theorems can both be reformulated appropriately.However, since Theorem 2 includes Theorem 1 as a specialcase, only Theorem 2 is reformulated here as Theorem 3.

Theorem 3: Consider the nth channel in the (2L − 1)-blocksummation detector with K = N = 1 and γ = 0.5. Assumethat M is large, and the input signal is the complex sinusoiddefined by (16). Let WM = {wM (m)}M−1

m=0 and W(2L−1)M ={w(2L−1)M (m)}(2L−1)M−1

m=0 be energy-normalized windows, oflength M and (2L − 1)M , respectively, derived from the pro-totype window function w(x) via (27). More precisely, wM (m)is defined by (27) and

w(2L−1)M (m) =w(

m(2L−1)M−1

)√

(2L−1)M−1∑k=0

w2(

k(2L−1)M−1

) ,

0 ≤ m ≤ (2L − 1)M − 1. (41)

Let H be the energy-normalized prototype filter defined by(35). Then, for a given probability of false alarm Pfa ∈ (0, 1)and a given probability of detection Pd ∈ (0, 1), the perfor-mance gain of the one-block FFT summation detector em-ploying a single FFT of size LM and the window functionH defined by (35) over the (2L − 1)-block FFT summationdetector employing (2L − 1) overlapped FFTs of size M with


overlap ratio γ = 0.5 and the window function WM defined by(27) is denoted by go

L and computed by

goL = D2L−1 + 10 log 10 [τL(H, w,M, fo)] (42)

where D2L−1 is defined by (30) with L replaced by 2L − 1

τL(H, w,M, fo) =

∣∣∣∫ 1

0 w(x)e−2πj(2L−1)foxdx∣∣∣2∣∣∣∫ 1


×

∣∣∣∣LM−1∑m=0

hme−2πjm foM

∣∣∣∣2

∣∣∣∣∣(2L−1)M−1∑

m=0w(2L−1)M (m)e−2πjm fo

M

∣∣∣∣∣2 (43)

and fo = M(f − n/M) is the normalized frequency offset ofthe signal (16). If fo = 0, τL(H, w,M, fo) simplifies to

τL

(H,W(2L−1)M

)=

∣∣∣∣LM−1∑m=0

hm

∣∣∣∣2

∣∣∣∣∣(2L−1)M−1∑

m=0w(2L−1)M (m)

∣∣∣∣∣2

≈ L

2L − 1

∣∣∣∣LM−1∑m=0

hm

∣∣∣∣2

∣∣∣∣LM−1∑m=0

wLM (m)∣∣∣∣2 . (44)

The proof of (42) is essentially the same as that of (36) and isomitted here. The approximation in (44) immediately followsfrom the following two approximations:∣∣∣∣∣∣

(2L−1)M−1∑m=0

w(2L−1)M (m)

∣∣∣∣∣∣2

≈ (2L − 1)M

∣∣∣∫ 1

0 w(x)dx∣∣∣2∫ 1

0 w2(x)dx

∣∣∣∣∣LM−1∑m=0

wLM (m)

∣∣∣∣∣2

≈LM

∣∣∣∫ 1

0 w(x)dx∣∣∣2∫ 1

0 w2(x)dx.

To conclude this section, it is important to emphasize that,unlike formula (36) of Theorem 2, formula (42) of Theorem 3may not produce reliable results when the probability of falsealarm Pfa becomes very small, as the impact of the statisti-cal correlation between adjacent overlapped input data blocksmay no longer be negligible for very small values of Pfa.More sophisticated mathematical tools are required to deriveformulas such as (26) to provide more reliable informationon the processing gain in the case of overlapped input data.Nevertheless, formula (42) is expected to provide a reasonableestimate for the processing gain for relatively large Pfa whenthe input data overlap ratio γ = 0.5.

IV. NUMERICAL COMPUTATION OF PROCESSING GAIN

Here, numerical computation of the processing gain of thePFFT detector over the FFT summation detector is discussed.

Fig. 6. Correction term in (36) (Blackman and Hanning windows).

Fig. 7. Correction term in (36) (Blackman and Nuttall windows).

An inspection of formulas (29), (36), and (42) immediatelyreveals that the performance gain can be decomposed into thesum of two terms. The first term, which is equal to eitherDL or D2L−1 and called the main term here, depends onlyon Pfa, Pd, and L. The second term, which is referred to asthe correction term here, depends on the choice of the windowand the prototype filter, the parameters L and M , and the nor-malized frequency offset fo = M(f − n/M) of the complexsinusoid (16).

The correction term can be easily calculated by invokingcomputer routines for evaluating polynomials and integrals.Fig. 6 plots the correction term 10 log 10[κL(H, w,M, fo)]defined by (36) for fo relatively close to 0, for the L-blocksummation detector employing the Blackman window withM = 1024 and L = 8, and for the PFFT detector employingthe Hanning window of length LM as the prototype filter.Fig. 7 presents the correction term in (36) for the sameL-block summation detector in Fig. 6 but for a different PFFTdetector that employs the Nuttall window as the prototype filter.Experimentation with various designs of the prototype FIRfilters H of (35) and various classical windows shows that the


Fig. 8. DL as a function of L.

correction terms in (29), (36), and (42) all behave in a similarway and can take relatively small positive values for fo close to0 but become negative with large absolute values as fo movesaway from 0.

The computation of the main term in the processing gain ismuch less straightforward, as it requires the computation of theinverse function GL(z, y) defined by (19). Let DL be definedby (30) and let Do

L be defined by

DoL = 10 log 10

G2L−1 (Pd, T2L−1(Pfa))G1 (Pd, T1(Pfa))

=D2L−1, L ≥ 1. (45)

DL and DoL are the main terms in (36) and (42), respectively.

They respectively represent the performance gain of the PFFTdetector over the FFT summation detector for 0% (γ = 0) and50% (γ = 0.5) overlap ratios when the frequency f of thecomplex sinusoid (16) coincides with the channel center fre-quency and the window and the prototype filter in the FFT andPFFT filter banks are both derived from the same continuousprototype window function. Since L is often a power of 2 inpractice, we also define dn and do

n by

dn = DL don = Do

L, L = 2n. (46)

Computer programs have been written to compute DL and D0L

for any given Pd, Pfa, and L. To give an idea of the behavior ofDL for increasing L, DL is plotted as a function of L in Fig. 8for Pd = 0.9 and Pfa = 10−2, 10−4, 10−6, 1 ≤ L ≤ 512. WhenL is relatively small, for example, 2 ≤ L ≤ 16, DL increasesroughly from 1 to 3.5 dB, as can be seen from Fig. 9. Notethat the curves for Do

L are similar, and DL and DoL are always

nonnegative.The behavior of the dependencies of DL and Do

L on L canbe explored by plotting dn and do

n as a function of n, for givenvalues of Pfa and Pd. Typical results are shown in Figs. 10–13.Once the window function and the prototype FIR filter areselected, the theoretical performance gain of the PFFT detectorover the FFT summation detector can be obtained by computingthe main term and then adding the applicable correction term.To give an example, assume that Pd = 0.9, Pfa = 0.01, L = 64,

Fig. 9. DL as a function of L (1 ≤ L ≤ 30).

Fig. 10. dn as a function of n, with Pd = 0.8.

and γ = 0 (no data overlapping); assume that the window inthe L-block summation detector is the Blackman window, andthe prototype FIR filter in the PFFT detector is the Hanningwindow. Then, log2(L) = n = 6, and the correction term DL

computed at the given Pd and Pfa is equal to dn, which is foundto be 6 dB from Fig. 12. On the other hand, the correction termat fo = 0 is computed to be 0.6 dB. Hence, the processing gainfor the given parameters is obtained as follows:

gL =DL + correction term

= dn + correction term = 6 + 0.6 = 6.6 (dB).

The processing gain may take negative values when thenormalized frequency offset fo is relatively close to the channelboundaries fo = ±0.5. This is a manifestation of fundamentaltradeoffs that exist between the FFT summation and PFFTdetectors. The analysis developed in this paper provides a pow-erful tool for precisely quantifying their relative advantages.


Fig. 11. don as a function of n, with Pd = 0.8.

Fig. 12. dn as a function of n, with Pd = 0.9.

V. COMPUTER SIMULATIONS

Preliminary computer simulations have been performed todemonstrate the application of the theoretical results derived inthis paper to the performance comparison of FFT summationand PFFT detectors. Let Pfa = 0.01, γ = 0 (no input dataoverlapping), M = 1024, and L = 8. Assume that the windowof length M = 1024 employed in the FFT summation detectoris Blackman and the prototype filter in the PFFT filter bankis the Hanning window of length LM = 8 × 1024 = 8192.In the simulations, each input data record of length LMconsists of LM consecutive samples of the sinusoidal signal(16) whose frequency f is randomly selected from the inter-

Fig. 13. don as a function of n, with Pd = 0.9.

Fig. 14. Simulated Pd plotted as a function of the SNR |A|2/σ2 for thetwo eight-block summation detectors employing the Blackman and Hanningwindows, respectively, and the PFFT detector with eight taps in each polyphasebranch employing the Hanning window as the prototype filter.

val [−0.1/(LM) + n/M, 0.1/(LM) + n/M ] for some fixedinteger 0 ≤ n ≤ M − 1, or, equivalently, the normalized fre-quency offset fo = M(f − n/M) is uniformly distributed inthe interval [−0.1/L, 0.1/L] = [−0.0125, 0.0125]. For thesegiven parameters, the simulated probability of detection Pd isplotted as a function of the input SNR |A|2/σ2 in Fig. 14,where three detectors are compared: 1) the eight-block 1024-channel summation detector defined by the Blackman windowof length M = 1024; 2) the eight-block 1024-channel sum-mation detector defined by the Hanning window of lengthM = 1024; and 3) the 1024-channel PFFT detector definedby the Hanning window of length M × L = 8192. These de-tectors are represented as the Blackman summation, Hanningsummation, and Hanning polyphase in the legend in Fig. 14,respectively. It can be observed that, to achieve a value of Pd =0.9, the Hanning polyphase detector outperforms the Black-man summation detector by about 3 dB, whereas the Hanning


summation detector outperforms the Blackman summation de-tector by about 0.5 dB. The same information can also beobtained using formula (36) without conducting simulations.Since L = 8, we have n = log2(L) = 3, and for Pfa = 0.01and Pd = 0.9, from Fig. 12, it can be seen that dn ≈ 2.5 dB.From Fig. 6, it can be seen that for fo ∈ [−0.0125, 0.0125],the correction term 10 log 10[κL(H, w,M, fo)] defined by (36)takes values roughly in the range from 0 to 0.5 dB. Hence, theprocessing gain gL of the Hanning polyphase detector over theBlackman summation detector ranges from about 2.5 to about3.0 dB. This is largely consistent with the curves in Fig. 14.Hence, without going through computer simulations, one canbe certain that the Hanning polyphase detector outperformsthe Blackman summation detector by about 2.5–3 dB for thesinusoid signal (16) with normalized frequency offset randomlyvarying relatively close to 0.

VI. WINDOW ENERGY NORMALIZATION

Consider the basic FFT detector shown in Fig. 1. If a singleinput data block containing a length-M segment of the complexsinusoid defined by rk = e2πjkf , k = 0, 1, . . ., is processed,the channel spectral power of the nth channel is equal tothe squared magnitude response of the rectangular windowevaluated at the frequency offset f − n/M . In general, if thewindow W = {wm}M−1

m=0 is applied to the input data blocks,the channel spectral power of the nth channel is equal tothe squared magnitude response of the window W evaluatedat the frequency offset f − n/M . Traditionally, comparativeperformance analysis of the basic FFT detector in Fig. 1 withdata windowing is confined to the analysis of the magnituderesponses of windows. This has culminated in the work [13]where a comprehensive analysis is presented for various win-dows. When comparing the magnitude responses of windowsin [13], the window W is dc-normalized with

∑M−1m=0 wm = 1.

This practice is widely accepted and used in digital filterdesigns and is perfectly relevant in the FIR filter design context.However, within the framework of CFAR detection, when com-paring performance for more sophisticated detection schemesusing spectral data averaging and the polyphase structure, thistraditional ad hoc approach is not completely satisfactory, and aquantitative approach is necessary. The insights into the relativeperformance of the FFT summation and PFFT detectors yieldedby the quantitative framework provided by (29), (36), and (42)are not easily observable through ad hoc approaches.

Let us look again at formulas (36) and (37), which arerespectively reproduced for easy reference as follows:

gL = DL + 10 log 10[κL(H, w,M, fo)] (47)

κL(H, w,M, fo) =

∣∣∣∫ 1



×

∣∣∣∣LM−1∑m=0

hme−2πjm foM

∣∣∣∣2

∣∣∣∣LM−1∑m=0


∣∣∣∣2 (48)

Fig. 15. Magnitude responses (dc-normalized windows).

where DL is defined by (30), and fo = M(f − n/M) is thenormalized frequency offset of the complex sinusoid defined by(16). If L = 1, the PFFT detector and the L-block summationdetector are both reduced to the simple basic FFT detectordefined by the length-M windows H and WM , respectively,and, in this case, since DL = 0, the processing gain gL ofthe window H over the window WM is now defined andcomputed by

gL = 10 log 10

∣∣∣∣M−1∑m=0

hme−2πjm foM

∣∣∣∣2

∣∣∣∣M−1∑m=0

wM (m)e−2πjm foM

∣∣∣∣2 . (49)

The right-hand side of (49), as a function of fo, is nothingbut the difference (in decibels) between the squared magni-tude responses of H and WM evaluated at fo/M . This islargely consistent with traditional intuitions. However, H andWM are now both energy-normalized, not dc-normalized, i.e.,∑M−1

m=0 h2m =

∑M1m=0 w2

m = 1. What this implies is that, withinthe framework of CFAR detection, one should compare themagnitude responses of energy-normalized windows (or FIRfilters) and not those of dc-normalized windows. As an illustra-tion, the magnitude responses of the dc-normalized Blackman,Hanning, Hamming, Nuttall, and rectangular windows areplotted as a function of fo in Fig. 15, whereas the magni-tude responses of the energy-normalized Blackman, Hanning,Hamming, Nuttall, and rectangular windows are plotted as afunction of fo in Fig. 16. Over a large portion of the FFTchannel defined by fo ∈ [−0.5, 0.5], the rectangular windowoutperforms all the other windows. Furthermore, the Hanningwindow provides a positive processing gain relative to theBlackman window and is preferable from the detection per-spective. Simulations have confirmed these observations. Incomparison, in Fig. 15, the curve for the squared magnituderesponse of the rectangular window is located below those of allthe other windows. This may lead to the misconception that the


Fig. 16. Magnitude responses (energy-normalized windows).

rectangular window is inferior to the other windows in detectionperformance, which is completely incorrect.

In general, for L ≥ 1, formula (48) provides a very conve-nient mathematical tool to analyze the relative performance forany window in the FFT summation detector and any proto-type filter in the PFFT detector. Graphs of 10 log 10[κL(H, w,M, fo)] make it relatively easy to determine the frequencybands within the FFT channel, where the processing gain gL ispositive and negative, respectively. If a priori information aboutthe bandwidths of signals to be detected is available, the designof the prototype filter H in the PFFT detector can be optimized.Formula (43) plays the same role for the case of overlappedinput data blocks.

VII. CONCLUSION

We have set up a simple framework for comparing the perfor-mance of the PFFT and FFT summation detectors. Within theframework of CFAR detection, simple mathematical formulashave been derived that can be used for analyzing the relativeperformance of these detectors given the assumption that theamount of input data is the same. It has been found that,when comparing windows and prototype filters, contrary to theaccepted practice in digital filter designs, it is more meaningfulto compare the magnitude responses of energy normalized, notdc-normalized, windows and prototype filters. The numericalresults provided for typical parameters for the main term inthe processing gain are very useful for making appropriatetradeoffs in the design of solutions for practical signal detectionproblems.

ACKNOWLEDGMENT

The authors would like to thank the reviewers and the as-sociate editor for their constructive comments and suggestionsthat have helped to improve the clarity and the readability ofthis paper.

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Sichun Wang (M’06–SM’06) received the B.Sc. andM.Sc. degrees in mathematics from Nankai Univer-sity, Tianjin, China, in 1983 and 1989, respectively,and the Ph.D. degree in mathematics from McMasterUniversity, Hamilton, ON, Canada, in 1996.

From 1983 to 1986, he was an Assistant Lecturerwith Nankai University. From 1989 to 1991, he was aLecturer with Tianjin University, Tianjin. From 1992to 1996, he was a Teaching Assistant with McMasterUniversity, where, from 1996 to 1997, he was aPostdoctoral Fellow with the Communications Re-

search Laboratory. From 1997 to 2007, he was either a Research Scientist or aResearch Consultant with one of the following organizations: Telexis, Intrinsix,Calian, Communications Research Centre Canada (CRC), Ottawa, ON, andDefence Research and Development Canada (DRDC), Ottawa. Since 2007, hehas been a Research Scientist with CRC and seconded to DRDC Ottawa. Hehas published on various topics in mathematics, signal processing, and digitalcommunications, such as maximal function estimates for the free Schrödingerequation, chaotic signal estimation, blind channel equalization, forward errorcontrol codes (turbo codes), and FFT filter bank-based CFAR detection. Hisrecent research has focused on detection and estimation algorithms in directionfinding, geolocation, and spectrum sensing for cognitive radios.

Dr. Wang has served as a reviewer for several IEEE transactions andconferences.

Robert Inkol (M’73–SM’86) received the B.Sc. andM.A.Sc. degrees in applied physics and electrical en-gineering from the University of Waterloo, Waterloo,ON, Canada, in 1976 and 1978, respectively.

Since 1978, he has been with the Defence Re-search and Development Canada, Ottawa, ON, wherehe is currently a Senior Scientist. He is also anAdjunct Professor with the Royal Military Collegeof Canada, Kingston, ON. He has been responsiblefor extensive contributions to the application of verylarge scale integrated circuit technology and digital

signal processing techniques to electronic warfare systems. In addition tohaving produced numerous publications, he is the holder of four patents.

Mr. Inkol has served as a reviewer for various publications and as a technicalprogram committee member for several IEEE conferences.

Sreeraman Rajan (M’90–SM’06) received the B.E.degree in electronics and communications fromBharathiyar University, Coimbatore, India, in 1987,the M.Sc. degree in electrical engineering fromTulane University, New Orleans, LA, in 1990, andthe Ph.D. degree in electrical and computer en-gineering from the University of New Brunswick,Fredericton, NB, Canada, in 2004.

From 1986 to 1990, he was a Scientific Officerwith the Reactor Control Division, Bhabha AtomicResearch Center, Bombay, India, where he was in-

volved in the control, safety, and regulation of nuclear research and powerreactors. From 1997 to 1998, he carried out research under a grant fromSiemens Corporate Research, USA. From 1999 to 2000, he worked on opticalcomponents and the development of signal processing algorithms for AdvancedFiber Optic Modules at JDS Uniphase. From 2000 to 2003, he was involvedin channel monitoring, dynamic equalization, and control of optical power foradvanced fiber optical communication systems while at Ceyba Corporation.In 2004, he developed signal processing algorithms for noninvasive medicaldevices while working at Biopeak Corporation. Since December 2004, he hasbeen a Defense Scientist with the Defence Research and Development Canada,Ottawa, ON, Canada. He is the holder of one patent and two disclosuresof invention, and has authored several journal and conference papers. Hisbroad area of research interests includes signal processing, biomedical signalprocessing, communications, and pattern classification.

Dr. Rajan is a member of the IEEE Engineering in Medicine and BiologySociety (EMBS), the Communications Society, and the Eta Kappa Nu HonorSociety. He is currently the Vice Chair of the IEEE Ottawa Section and theChair of the IEEE Ottawa EMBS Chapter. He has been involved with severalIEEE conferences in various capacities. He has served as a reviewer for severalIEEE transactions and conferences.

François Patenaude received the B.A.Sc. degreein electrical engineering from the University ofSherbrooke, Sherbrooke, QC, Canada, in 1986 andthe M.A.Sc. and Ph.D. degrees in electrical engi-neering from the University of Ottawa, Ottawa, ON,Canada, in 1990 and 1996, respectively. His graduatework was conducted in collaboration with the MobileSatellite Group of the Communications ResearchCentre (CRC), Ottawa.

In 1995, he joined CRC to work on signal process-ing applications for communications and spectrum

monitoring. His main research interests include automatic signal recognition,detection and estimation in the spectrum monitoring context, and real-timesignal processing.

detection of narrow-band signals through the fft and polyphase fft filter banks: noncoherent versus...

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