CYCLIC SPECTRAL ANALYSIS FROM THEAVERAGED CYCLIC PERIODOGRAM

Roger Boustany, Jerome Antoni

University of Technology of Compiegne60205 Compiegne, France

Abstract: The theory of cyclostationarity has emerged as a new approach tocharacterizing a certain type of nonstationary signals. Many aspects of the spectralanalysis of cyclostationary signals have been investigated but were essentiallybased on the use of the smoothed cyclic periodogram. This paper proposes acyclic spectral estimator based on the averaged cyclic periodogram which benefitsfrom better implementation properties. It shows that an unexpected but importantcondition for this estimator to be valid is to set enough overlap between adjacentsegments in order to prevent cyclic leakage. It proves that setting the percentageof overlap to 75% with a hanning window, or 50% with a half-sine window fixes theproblem. It also shows that in certain situations the cyclic leakage associated withthe averaged cyclic periodogram can be made exactly zero, in contrast with thesmoothed cyclic periodogram. Illustrative examples finally confirm the obtainedresults, where it is also demonstrated how to use them for efficiently estimatingthe Wigner-Ville spectrum. Copyright c©2005 IFAC.

Keywords: Signal processing, Spectral analysis, Averaged periodogram,Cyclostationary signals, Cyclic spectral estimation, Cyclic leakage

1. INTRODUCTION

During the last two decades, the theory of cyclo-stationarity has emerged as a new approach tocharacterizing a certain type of nonstationary sig-nals (Gardner, 1991),(Gardner, 1990),(Giannakis,1998). Cyclostationarity extends the class of sta-tionary signals to those signals whose statisticalproperties change periodically with time. Suchsignals are generally not periodic but random intheir waveform, yet they are inherently generatedby some periodic mechanism.As opposed to stationary signals, CS signals con-tain extra information due to their hidden peri-odicities. In the time domain, this extra infor-mation is carried by the temporal variations ofstatistical descriptors such as the instantaneousauto-correlation function (Gardner, 1990). Veryinterestingly, the same information relates in the

frequency domain to correlations between spectralcomponents spaced apart by specific frequencies,referred to as cyclic frequencies (Gardner, 1991).As opposed to general nonstationary signals, CSsignals also enjoy a well-understood theory whichactually generalizes all the signal processing toolshistorically developed for stationary signals. Inparticular, the very powerful spectral analysis ofstationary signals finds an interesting generaliza-tion. Actually many aspects of the spectral analy-sis of CS signals have already been investigated inthe specialized literature (Gardner, 1986), (Hurd,1989), (Roberts et al., 1991), (Dandawate andGiannakis, 1994) but they were mainly based onthe smoothed cyclic periodogram. The aim ofthis paper is to study the averaged cyclic peri-odogram as an estimator of the cyclic spectrum.The reason for this choice is that the averaged

periodogram is the most popular technique forthe spectral analysis of stationary signals due toits high computational efficiency, its robustnessagainst outlying data and its ability to remove nonstationary trends. As far as we know, the averagedcyclic periodogram has been rarely advocated inthe cyclostationary literature.The paper is organized as follows. In section 2,the basic terminology of cyclostationary signalsare reviewed. In section 3, an averaged cyclic pe-riodogram is established on the basis of a generalquadratic form for cyclic spectral estimation. Insection 4, the spectral estimator introduced insection 3 is studied in terms of bias and variance.Finally, illustrative examples are provided in sec-tion 5.

2. CYCLOSTATIONARY SIGNALS ANDTHEIR TERMINOLOGY

In the following we will consider sampled signalswith ∆ denoting their sampling period and ntheir temporal index. For simplicity we shall onlyconsider signals with zero mean and thereforerestrict our analysis to purely random signals.This assumption is without loss of generality sinceefficient techniques exist for centering non-zeromean cyclostationary signals (Antoni et al., 2004).Stricto sensu, a cyclostationary (CS) signal X[n]is a signal whose joint probability density functionis a quasi-periodic function of time. This entailsthat any typical statistical descriptor of signalX[n] is also a quasi-periodic function of timein particular the instantaneous auto-correlationfunction

R2X [n, τ ] = EX[n + βτ ]X[n− βτ ]∗

(1)

(where β + β = 1, the parameter β allowingfor a general formulation of various equivalentdefinitions) accepts a Fourier series

R2X [n, τ ] =∑

αi∈AR2X [τ ;αi]ej2παin∆ (2)

over the spectrum A = αi of cyclic frequen-cies αi associated with the non-zero Fourier co-efficients R2X [τ ;αi] known as the cyclic auto-correlation functions or cyclo-correlation func-tions of signal X. The Fourier transform of thecyclic auto-correlation function is a spectral de-scriptor of signal X and is known as the cyclicpower spectrum or cyclo-spectrum

S2X(f ; αi) = ∆∞∑

τ=−∞R2X [τ ;αi]e−j2πfτ∆ (3)

The next section will focus on estimating thecyclic power spectrum using the averaged cyclicperiodogram on the basis of a general quadraticform.

3. CYCLIC SPECTRAL ESTIMATION USINGTHE AVERAGED PERIODOGRAM

3.1 A General Quadratic Form

We claim that any non-parametric estimatorS2X(f ; α; L) of the cyclic spectrum S2X(f ;α) canbe deduced given a finite-length signal X[n]L−1

n=0

from the general quadratic form

S2X(f ; α; L) = ∆L−1∑p=0

L−1∑q=0

QL[p, q]X[p]X[q]∗

e−j2π(f+βα)p∆ej2π(f−βα)q∆ (4)

where QL is a suitably chosen positive semi-definite kernel such as to preserve the interpre-tation of S2X(f ; α;L) as a power density and inparticular S2X(f ; 0; L) ≥ 0 for α = 0. Let usdefine also QL(λ, η) the double DTFT (DiscreteTime Fourier Transform) of kernel QL which willbe used hereafter:

QL(λ, η) =

∆2L−1∑p=0

L−1∑q=0

QL[p, q]e−j2πλ(p−q)∆e−j2πηq∆ (5)

Different kernels QL result in different spectralestimators. Due to space limitations, we only con-sider here the case of the averaged cyclic peri-odogram. A more complete study will be pre-sented in another paper.Let w[n]Nw−1

n=0 be a positive and smooth Nw-long data-window and let wk[n] = w[n−kR] be itsshifted version by R samples. Then the averagedcyclic periodogram is

S(W )2X (f ;α; L) =

1K∆

K∑

k=1

X(k)Nw

(f + βα)X(k)Nw

(f − βα)∗ (6)

with X(k)Nw

(f) = ∆∑R+Nw−1

n=R wk[n]x[n]e−j2πfn∆

the short-time DTFT of the kth weighted sequencewk[n]x[n]R+Nw−1

n=R and K = b(L−Nw)/Rc + 1(where byc stands for the greatest integer smallerthan or equal to y). The averaged cyclic peri-odogram is obtained from (4) wherein

QL[p, q] =1K

K∑

k=1

wk[p]wk[q] ⇔ QL(λ, η) =

W (λ)W (λ− η)∗DR∆K (η)e−jπηR∆(K−1) (7)

with W (f) the DTFT of w[n] and DR∆K (λ) =

K−1 sin(πλR∆L)sin(πλR∆) the Dirichlet kernel. Formula (6)

can be very efficiently implemented by means ofthe FFT algorithm by imposing Nw to be a powerof 2. It remains to state which constraints mustfulfil kernel QL such as to yield a valid spectralestimate in terms of (i) minimum estimation biasand (ii) minimum estimation variance.

0 0

α αf f

a) b)

∆α ~ 1/(L∆)

Fig. 1. Illustration of the effect of cyclic leakage.

3.2 Bias Analysis

For the proposed quadratic estimator to have min-imum bias, it must hold that ES2X(f ; α;L) 'S2X(f ; α) as closely as possible. Using Eqs.(4) and(5), it comes that:

ES2X(f ; α;L) =1∆

∑

αi∈A

∫ 12∆

−12∆

QL(λ, α− αi)

S2X(f − λ + β(α− αi); αi)dλ (8)

This implies two conditions to be met, which wenow investigate.

3.2.1. Condition I (power calibration) The firstcondition is to impose that the leading termindexed by α = αi equals S2X(f ; α). This yields∑L−1

p=0 QL[p, p] = 1 which entails the well-knowncalibration ‖w‖2 =

∑n w[n]2 = 1.

3.2.2. Condition II (cyclic leakage minimization)The second condition is to impose that the

terms indexed by αi 6= α in Eq.(8) be zero.Such terms may be interpreted as interferencesresulting from the action of the smoothing kernelQL(λ, η) on the cyclic spectra positioned at cyclicfrequencies αi 6= α. The phenomena is akin toenergy leakage in the cyclic frequency(see Fig.1);we shall refer to it as cyclic leakage as firstdiscovered in reference (Gardner, 1986). It is ingeneral impossible to make these interferencesexactly zero (except in some special cases with theaveraged cyclic periodogram as shown hereafter),however it is still possible to seek kernel QL suchas to minimize them. To see this, let us considerthe situation where S2X(f ;α) = Sα

2X is a constantfunction in the f -frequency, so that condition IIbecomes that of making function

BQ(η) =1∆

∫ 12∆

−12∆

QL(λ, η)dλ (9)

as close as possible to zero for any η 6= 0.This boils down to seeking kernel QL such as tominimize the bandwidth of |BQ(η)|. In the case ofthe averaged cyclic periodogram:

|BQ(η)| = 1∆

∣∣DR∆K (η)

∣∣ · |W2(η)| (10)

where W2(f) is the DTFT of w[n]2. Although theDirichlet kernel DR∆

K (η) has a narrow central lobe

0

1

~ 1/Nw

1R∆

1KR∆ 1

KR∆η

|DR∆(η)|K|W2(η)|/∆

~ 1/K

height to be minimised

~ 1L∆

.2

.4

.6

.8

Fig. 2. Structure of function BQ(η) in the aver-aged cyclic periodogram.

of bandwidth 1/(KR∆) ∼ 1/(L∆), it also hasseveral other lobes with period 1/(R∆) - see Fig.2.It is therefore important that function W2(η)decreases fast enough in order to keep the effect ofthose secondary lobes as small as possible. Sincethe bandwidth of W2(η) is on the order of 1/Nw,the condition is that R ¿ Nw, i.e. importantoverlap must be placed on adjacent data-windows.In practice, R = Nw/3 (67% overlap) will begood enough with a Hanning or a Hammingdata-window. With a half-sine data-window R =Nw/2 (50% overlap) will produce an excellentcyclic leakage minimization because the secondarylobes of DR∆

K (η) happen to fall on the nulls ofW2(η). These results are illustrated in Fig.3 andsummarized in the below table:

Secondary lobes roll-off for typical data-window

Hamming Hanning half-sine

height of 2nd lobe -2.0 dB -1.7 dB -3.0 dB

height of 3rd lobe -8.8 dB -7.8 dB −∞height of 4th lobe −∞ −∞ −∞

Therefore the following rule:

Proposition 1. For the averaged cyclicperiodogram, Condition II requires that the

acquisition time be long enough so that L∆ issignificantly larger than the inverse of the

minimum α-spacing - say ∆minα - to be resolved,

that is L∆ À 1/∆minα and either R ≤ Nw/3 with

a Hanning and a Hamming data-window orR ≤ Nw/2 with a half-sine data-window.

Last but not least, it is noteworthy that the av-eraged cyclic periodogram can be made exactlyunbiased in α provided that the cyclic spectrumA is known. The idea is to choose R∆ such that∆min

α happens to coincide exactly with a zero ofDR∆

K (η). For instance, one case of practical impor-tance is when the signal of interest is periodically-correlated with a known cycle N , such that wehave ∆min

α = 1/(N∆). Then the following ruleapplies:

0-50

-40

-30

-20

-10

0

0

0 0

0 0

-50

-40

-30

-20

-10

0

-50

-40

-30

-20

-10

0

-50

-40

-30

-20

-10

0

-50

-40

-30

-20

-10

0

-50

-40

-30

-20

-10

0

R = Nw

R = Nw/2

R = Nw/3

dBdB

dB

Hanning data window half-sine data window

η

η

η

1/(R∆) 2/(R∆)

1/(R∆) 2/(R∆)

1/(R∆)

3/(R∆)

η 1/(R∆)

η 1/(R∆) 2/(R∆)

η1/(R∆) 2/(R∆) 3/(R∆)

a)

b)

c)

d)

e)

f)

Fig. 3. Illustration of the effect of segmentsoverlapping on cyclic leakage in the av-eraged cyclic periodogram: full lines =function |BQ(η)|; dotted lines = functions|DR∆

K (η)|/∆ and |W2(η)|. Hanning window:a) no overlap, b) 50% overlap, c) 67% over-lap. Half-sine window: d) no overlap, e) 50%overlap, f) 67% overlap.

Proposition 2. For a periodically-correlatedsignal with cycle N , setting R = kN/K, (k ∈ INbut not a multiple of N) completely cancels the

effect of cyclic leakage.

3.2.3. General bias evaluation Conditions I andII together guaranty that the proposed quadraticspectral estimator of the cyclic spectrum is unbi-ased in the case of CS white signals. The sameconclusion obviously does not hold true in thegeneral case of non-white signals.

Proposition 3. Provided that conditions I and IIare met, the proposed quadratic spectral

estimator has bias:

ES2X(f ; α;L) − S2X(f ;α) 'IQ

2∂2

∂f2S2X(f ; α) +O(1/L)

with IQ = ∆−1∫ 1/2∆

−1/2∆λ2QL(λ, 0)dλ the inertia

of kernel QL(λ, 0).

The proof follows from a trivial Taylor expansionof Eq.(8). Proposition 3 generalises the bias for-mula known in the spectral analysis of stationarysignals. However, care should be taken to meetcondition II which protects against the effect ofcyclic leakage, otherwise Eq.(3) becomes muchmore intricate. To our knowledge this point hasrarely been addressed before.Finally, provided that conditions I and II aremet, the resolution in the α-frequency essentially

depends on the acquisition time L∆ , i.e. ∆α ∼1/(L∆).

3.3 Variance Analysis

It finally remains to prove that the proposedquadratic estimator is consistent. We start withthe following general result (the proof is not givenhere due to limited space):

Proposition 4. For large L:

V arS2X(f ; α; L)'EQ1 · S2X(f + βα; 0)S2X(f − βα; 0)

+ EQ2 · SXX∗(f ; α)SXX∗(f ;α)∗

where EQ1 = 1∆2

∫∫ 1/2∆

−1/2∆|QL(λ, η)|2 dλdη and

EQ2 = 1∆2

∫∫ 1/2∆

−1/2∆QL(λ, η)QL(λ− η,−η)∗dλdη

The above formula reveals the existence of twoterms carried by EQ1 and EQ2, respectively. How-ever it happens that in many instances the sec-ond term is non-zero only on a countable set offrequencies in particular when the processes ofinterest are real. Hence the following result:

Proposition 5. For large L, the variance ofS2X(f ;α; L) is

V arS2X(f ;α; L)'EQ1 · S2X(f + βα; 0)S2X(f − βα; 0)

almost everywhere.

This very simple formula can be checked to gen-eralize the result obtained for stationary signals(α = 0) and also accepts the results of refer-ences (Hurd, 1989) and (Dandawate and Gian-nakis, 1994) as particular cases. It states thatthe variance of S2X(f ; α; L) is proportional to thecross-spectra S2X(f+βα; 0) and S2X(f−βα; 0) atfrequencies f + βα and f − βα, and to the energyEQ1 = ‖QL‖2 of kernel QL.The particular expression taken by EQ1 in the caseof the averaged cyclic periodogram becomes aftersome manipulations:

EQ1 =K−1∑

k=−K+1

Rw[kR]2 · K − |k|K2

(11)

where Rw[k] =∑

n w[n − k]w[n] is the auto-correlation function of the data-window w[n]. Thisresult is valid for any value of R. Assessmentof formula (11) with classical data-windows (e.g.Hanning, Hamming, half-sine) shows that EQ1 is adecreasing function of R/Nw where the minimumis virtually reached as soon as R ≤ Nw/3. Forlarge L we have found that the minimum of EQ1

tends rapidly to ‖Rw‖2 /L.

00.1

0.20.3

0.4

> 0

.1

.2

.3

.4

.5

00.1

0.20.3

0.40.5.01

.02

.03

.04

.05

00.1

0.20.3

0.4.002

.004

.006

.008

.01

00.1

0.20.3

0.40.5.002

.004

.006

.008

0.01

1N 2

N 3N 4

N

> 01N 2

N 3N 4

N

> 01N 2

N 3N 4

N

> 01N 2

N 3N 4

N

α

f

α

f

α

f

α

f

a) b)

c) d)

Fig. 4. Illustration of the effect of cyclic leakageon the estimated cyclic coherence function.The cyclic coherence function is that of astationary signal and is theoretically zeroeverywhere (α 6= 0). A Hanning data-windowis used with a) R = Nw, b) R = Nw/2, c)R = Nw/3, and d) R = Nw/4.

4. ILLUSTRATIVE EXPERIMENTS

The first experiment has for objective to illustratethe necessity of properly setting the percentageof overlaps in the averaged cyclic periodogram. Astationary signal was synthesized (∆ = 1s) and itscyclic coherence function (β = 1/2) - theoreticallynil at α 6= 0 - was computed for several valuesof alpha in the range [0.001; 0.08] Hz in order tocheck for the presence of cyclostationarity:

γ2X(f ; α; L) =

S2X(f ; α; L)[S2X(f + βα; 0; L)S2X(f − βα; 0; L)

]1/2

We used the averaged cyclic periodogram withthe following settings: Nw = 60, a Hanning data-window, and FFT sizes of 128 samples; this pro-vided a 1% level of significance of 0.002 to thecyclic coherence function. Different values of theincrement parameter R were tested, i.e. R = Nw,R = Nw/2, R = Nw/3, and R = Nw/4. Resultsare displayed in Fig.4 for the four tested cases.As expected from the analysis of section 3.2, thecase R = Nw - Fig.4.a - produces significantcyclic leakage: instead of being statistically zeroas expected from the theory, γ2X(f ; α) carriesa significant amount of ”energy” at α = 1/Nw

leaking from α = 0. Similarly, a small amountof leakage - but still statistically significant (!) -is present at α = 2/Nw. The same phenomenaoccurs at α = 2/Nw when R = Nw/2 - Fig.4.b.However as soon as R ≥ Nw/3, it is checked thatcyclic leakage virtually disappears as establishedin section 3.2 - Fig.4.c-d. Although not shown herevery similar observations were obtained with ahalf-sine data-window except that cyclic leakagethen disappeared for R ≥ Nw/2 in accordancewith the discussion of section 3.2. This experimentinsists once again on the importance of correctly

0 0.04 0.08 0.12 0.16 0.2-100

-50

0

50

100Time signal

Time [s]

0 5 10 15 20 2515

20

25

30

35 Power spectrum (dB)

Frequency [kHz]

Fig. 5. a) Time signal and b) corresponding powerspectrum computed from the averaged pe-riodogram with a half-sine window of 1024samples and 67% overlap. The frequency res-olution is ∆f ∼ 0.1 kHz. The 6-σ Confidenceinterval is ' 2.6 dB.

0

5

10

15

20

25

0

100

200

300

400

500

0

.05

.1

.15

.2

.25

.3

f [kHz]

α [Hz]

Fig. 6. Cyclic coherence function, computed fromthe averaged cyclic periodogram with a half-sine window of 256 samples and 67% overlap.The frequency resolution is ∆f ∼ 0.3 kHz.The cyclic frequency resolution is ∆α ∼ 1 Hz.The 1% level of significance is 0.017.

setting R in the averaged cyclic periodogram, arequirement that is truly specific to cyclic spectralanalysis.The second experiment illustrates the use of

cyclic spectral analysis on an industrial system.The system of interest is a hydraulic pump rotat-ing at Ω ' 3000 rpm to be monitored by meansof vibration analysis. The vibration signal is cap-tured by an accelerometer mounted on the pumpcasing and is sampled at a rate of 50 kHz. The firstobjective is to assess the origin of the pump vibra-tions, and secondly to characterize the ”mechani-cal signature” of these vibrations. Figures 5.a and5.b display 0.2 seconds of the vibration signal andthe corresponding power spectrum, respectively.Inspection of these plots suggests that the signalessentially has a random structure: no clear har-monic structure is revealed in the spectrum, buttwo wide resonance peaks are found around 4.5kHz and 10.5 kHz, the origin of which is to beinvestigated. To this purpose, the cyclic coherencefunction γ2X(f ; α) (β = 1/2) is computed fordifferent values of α ranging from 1 Hz to 500 Hzwith cyclic resolution ∆α = 1 Hz. The result isdisplayed in Fig.6. There is clearly a region withhigh cyclic coherence at α = 391 Hz, i.e. at eight

Freq

uenc

y [k

Hz]

Time [ms]

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

(one rotation)

Time [ms]

0 2 4 6 8 10 12 14 16 18 20

a) Wigner-Ville spectrum

b) Wigner-Ville distribution

Freq

uenc

y [k

Hz]

0

5

10

15

20

25

Fig. 7. a) Wigner-Ville spectrum from the ana-lytic signal, synchronized on the blade pass-ing frequency and displayed over one rota-tion. The frequency resolution is ∆f ∼ 0.3kHz. The finest time resolution is ∼ 1 ms.Only values above 3 standard deviations ofthe stationary hypothesis are displayed. b)Smoothed pseudo-Wigner-Ville distributiondisplayed over one rotation. Frequency res-olution ∆f ∼ 0.4 kHz. The finest time reso-lution ∼ 1 ms.

times the pump rotation speed. Note that thisharmonic is not present in the power spectrum,so it does not relate to periodic vibrations butto some kind of periodic modulation of randomcarriers. The origin of this modulation is actuallyrecognized as the passing frequency of the eightblades of the pump propeller, i.e. 8×Ω = 391 Hz,with Ω = 48.9 Hz. Because the cyclic coherencefunction at α = 391 Hz is above the 1% level ofsignificance, it confirms that the vibration signaleffectively exhibits a certain amount of cyclosta-tionarity at this cyclic frequency. The final stepis to characterize more finely the structure of thiscyclostationary source by means of the Wigner-Ville spectrum

W2X [n, f) = FR2X [n, τ ]=

∑

αi∈AFR2X [τ ; αi] =

∑

αi∈AS2X(f ; αi).

where F· stands for the Fourier transform. TheWigner-Ville spectrum is computed by using αi’sequal to all multiples of 391 Hz so that theblade time-frequency signature can be extractedfrom the remaining signal. The result is dis-played in Fig.7.a, revealing that the two resonancepeaks initially detected in the power spectrumare linked to the blade activity; more specificallythey are two random carriers which are periodi-cally amplitude modulated with the blade pass-ing frequency. Incidentally, Fig.7.b displays the

smoothed pseudo-Wigner-Ville distribution com-puted on one rotation of the pump with similartime-frequency resolution in order to stress thefact that the proposed estimator of the Wigner-Ville spectrum drastically helps to suppress in-terference terms and consequently to improve thequality of the time-frequency lecture.

5. CONCLUSION

The purpose of this paper was to provide a cyclicspectral estimator based on the averaged cyclicperiodogram - surprisingly rarely suggested inprevious works. The conditions for this kernelto yield minimum bias and consistent estimatorshave been addressed in details. It was shownthat by setting 50% overlap with a half-sine datawindow, or 67% overlap with a Hanning or aHamming data-window, the averaged cyclic peri-odogram offers excellent cyclic leakage rejection;at the same time this precaution guarantees anearly minimum estimation variance. Moreover,by providing the relevant conditions in designingkernel QL, the material presented herein leavesopen a wide scope of possibilities for designingnew and original cyclic spectral estimators.

REFERENCES

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Gardner, W. (1990). Introduction to Random Pro-cesses. 2nd ed.. McGraw-Hill. Chap.12.

Gardner, W. (1991). Exploitation of spectral re-dundancy in cyclostationary signals. IEEESignal Processing Magazine pp. 14–36.

Giannakis, G. (1998). Cyclostationary signal anal-ysis. in The Digital Signal Processing Hand-book, IEEE Press. Chap. 5.

Hurd, H. (1989). Nonparametric time series analy-sis for periodically correlated processes. IEEEtrans. on Information theory 35(2), 350–359.

Roberts, R., W. Brown and H. Loomis (1991).Computationally efficient algorithms forcyclic spectral analysis. IEEE Signal Process-ing Magazine pp. 38–49.