Better narrowband spectral measurements

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Mechanical Systemsand

Signal Processing

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doi:10.1016/j.ym

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Mechanical Systems and Signal Processing 21 (2007) 2474–2495

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Better narrowband spectral measurements

Jason F. Dahla,�, Craig C. Smithb

aIntegrated Pro, Lehi, UT, USAbDepartment of Mechanical Engineering, Brigham Young University, Provo, UT, USA

Received 16 February 2005; received in revised form 16 December 2006; accepted 14 January 2007

Available online 5 March 2007

Abstract

Spectrum analysis based on traditional FFT techniques is a widely available, commonly understood and general purpose

tool for use with mechanical systems. In applications such as system identification, modal analysis and rotating machinery

signature analysis, relevant information content in signals is greatest near narrowband phenomena in the frequency

domain. Effort should be focused on reducing bias error in these regions in preference to reducing variance error. Time

aliasing methods of spectrum analysis can result in superior bias error reduction, producing an accurately sampled spectral

estimate and permitting window function characteristics to be decoupled from the DFT resolution, or frequency line

spacing. Windows created with time aliasing can be optimized by adjusting the amount of time or frequency localization,

according to the type of spectrum being measured. Assuming conventional criteria, time aliased windows can achieve

better mainlobe and sidelobe characteristics than other windows. An innovative approach, appropriate with continuous

spectra, is to use the window as a discrete sampling function in frequency to eliminate bias error inherent in the effective

convolution between window and signal transform. Both approaches have been successfully implemented and results are

presented using time aliased spectral estimates showing more accurate measurements of narrowband phenomena when

used in applications of rotating machinery vibration and system identification.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Spectrum analysis; Time aliasing; System identification; Modal analysis; Rotating machinery fault diagnosis; Narrowband

signals; Bias error; Spectral leakage

1. Introduction

Spectrum analysis has long found useful application with mechanical systems. System identification, modalanalysis and rotating machinery signature analysis are just some of the uses for spectrum analysis. Many ofthese applications involve signals exhibiting narrowband characteristics in the frequency domain, such asresonances, anti-resonances, or even discrete sinusoidal components. Since the relevant information content insuch signals is greatest near these narrowband phenomena, effort should be focused on reducing error in theseregions to enhance the overall measurement.

e front matter r 2007 Elsevier Ltd. All rights reserved.

ssp.2007.01.009

ing author. Tel.: +1 801 369 4670.

ess: [email protected] (J.F. Dahl).

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dx.doi.org/10.1016/j.ymssp.2007.01.009

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Nomenclature

a Kaiser–Bessel function indexDFT, FFT discrete Fourier transform, fast Fourier transformDfN N point DFT resolution (interval between k and k+1 frequency lines)f frequency (units as stated)H( f ), H[k], H(z) frequency response function in frequency, discrete frequency, Z domaink discrete frequency line indexn, N discrete time index, time aliasing period or block length (also DFT size)m, M time aliasing period index, total number of time aliased periods (blocks)p, q, r auxiliary summation variablest time (units as stated)T sample periodT0 time data segment length (NT ¼ T0)x(t), X( f ) continuous time signal, Fourier transform of x(t)x[n], X[k] discrete time signal, DFT of x[n]x(a)(t), X(a)( f ) continuous time aliased signal, Fourier transform of x(a)(t)x(a)[n], X(a)[k] discrete time aliased signal, DFT of x(a)[n]Xc Band-limited X

v(t), v[n], V[k] windowed time signal, discrete windowed time signal, DFT of v[n]o, ok frequency, discrete frequency (rad/s)w(t), w[n], W[k] window function, discrete window function, DFT of w[n]

J.F. Dahl, C.C. Smith / Mechanical Systems and Signal Processing 21 (2007) 2474–2495 2475

In current practice of spectrum analysis as applied to mechanical systems, traditional FFT-based analysis isa widely available, commonly understood and general purpose technique. While indeed useful, development ofFFT methods1 more often has been successful at reducing the variance errors associated with broadbandmeasurements, whereas many mechanical phenomena are more narrowband in nature. Reduction of bias errorin the vicinity of a resonant frequency, for example, significantly increases the signal-to-noise ratio, allowingimproved identification using spectrum-based methods.

The goal of this paper is to demonstrate the use of, and recent improvements to, time aliasing methods ofspectrum analysis [1–4] that have been developed but are not in general use. These methods can substantially

reduce and even virtually eliminate bias error in the form of spectral leakage (magnitude transferred betweenfrequency lines by non-zero sidelobes of the window transform) relative to the resolution required for anygiven problem. As will be illustrated in the examples, these methods are particularly suited to enhancingnarrowband phenomena, such as resonances and anti-resonances, or rotating machinery phenomena, but theyare also effective across the entire spectral estimate, not just in a local region, so they can be used with all typesof spectra, including broadband phenomena.

The key idea of these methods is the uncoupling of effective window length in time from line spacing infrequency by relaxing the constraint of avoiding aliasing in the time domain, when in fact; one is often primarily

interested in frequency domain information (which is correctly preserved). This creates the potential toaccomplish the objectives of windowing to remove bias error much more effectively, beyond the capabilities ofconventional windowing methods.

Our presentation will: (1) discuss uncertainty sources in spectral estimates, tradeoffs between reducing biasand variance error and motivations for improved narrowband estimates; (2) review the method of timealiasing as applied to creating a spectral estimate; (3) present an analysis of leakage bias reduction methods;

1By FFT-based spectrum analysis, we are referring here to the most common class of methods, known as directly computed, non-

parametric techniques. Although parametric techniques have been used with narrowband spectra, they are more specialized, require

assumptions or a priori knowledge about the phenomena involved, and are not as generally applicable, nor as commonly available.

ARTICLE IN PRESSJ.F. Dahl, C.C. Smith / Mechanical Systems and Signal Processing 21 (2007) 2474–24952476

(4) explain innovative bias error reduction methods afforded by time aliasing; and (5) demonstrate theeffectiveness of the innovative time aliasing methods of spectrum analysis by means of two examples.

2. Uncertainty types and associated tradeoffs in spectral estimates

Spectra are commonly estimated by using a discrete Fourier transform (DFT) (or more commonly, a fastFourier transform (FFT) of the input signal to directly compute a periodogram estimate of the autospectrumor cross-spectrum [5–8]. As with any measurement system, these estimates include a certain amount of error. Ifbias and variance errors are assumed independent, then the mean squared error of the spectral estimate at anysingle frequency line, k, can be expressed [9] as shown in Eq. (1). Accurate spectral estimates can beaccomplished by reducing either bias error or the variance error but there is a trade-off: given a finite length ofthe data record, neither one can be reduced without an increase in the other:

Mean squared errork ¼ Bias2k þ Vark. (1)

Variance uncertainty is introduced by the nature of the signal itself (as in the case of the purely randombroadband signal), and also by noise or other measurement errors. The usual approach to reducing variance inthe spectrum is the averaging of spectra in the frequency domain. By averaging an increasing number ofperiodograms, variance error can be successfully reduced. Given a finite length data record, however,increasing the number of averages requires that each periodogram be computed from a shorter length segmentof time data, causing increased bias error because of truncation end effects.

Bias uncertainty in the spectrum is primarily a function of two factors evident in the frequency domain:window transform shape and frequency resolution. The window transform is the Fourier transform of theweighting function applied in the time domain when the data is truncated into blocks of N data points, prior toapplying a DFT. Frequency resolution refers both to DFT resolution, DfN, or the spacing of frequency lines ina discrete spectrum, and also to effective frequency resolution, or the ability to resolve features in the spectrumat any given DFT resolution, which is strongly influenced by the window transform shape. Effective frequency

resolution can be a relative factor at any given DFT resolution. Changing the window transform shape canimprove effective frequency resolution and also reduce bias error by reducing leakage, or magnitudetransferred between frequency lines by non-zero sidelobes of the window transform.

Although bias error can generally be reduced by improving DFT resolution because the window transformscales with it, larger transforms incur higher computational cost, or may only be effective in a localized regionof the spectrum, as in the case of the commonly used zoom transform. We should point out that the use of thetime aliasing methods, to be described in Section 3, does not preclude the use of a larger transform forimproved DFT resolution or even a zoom transform, while still improving window transform characteristicsand effective frequency resolution across the entire spectrum. On the other hand, one of the advantages oftime aliasing methods is that, very often, acceptable spectral estimates can be achieved at a lower DFTresolution than conventional methods, as will be demonstrated in Section 6.1.

Considering the tradeoff between variance error reduction and bias error reduction as it relates tobroadband and narrowband types of spectra, it is possible to conclude the following two points:

1.
variance errors have the greatest effect in the flatter regions of a spectrum and so spectral averaging willhave a more positive effect on broadband phenomena and
2.
bias errors have the greatest effect in the regions of narrowband phenomena, such as poles and zeros, andbias error should be reduced in preference to variance error.
In justification of these conclusions, consider that for primarily broadband spectra, leakage from onefrequency line is replaced by a nearly equal amount of magnitude from adjacent frequency lines and so the netresult is less susceptible to window transform shape. For a finite amount of broadband type data, it can makesense to use a shorter DFT length and average more spectra together, maximizing variance reduction.

In the vicinity of narrowband phenomena, the situation is reversed. Sidelobe leakage can be very significantfrom one frequency line to the next because of differing magnitudes. Good effective frequency resolution isrequired to resolve important features of the spectrum, so that they are not simply ‘‘lumped together’’ into

ARTICLE IN PRESSJ.F. Dahl, C.C. Smith / Mechanical Systems and Signal Processing 21 (2007) 2474–2495 2477

larger peaks or valleys. Because significant signal energy is concentrated in a narrow band of frequencies, thesignal-to-noise ratio is likely quite good, so variance error introduced by noise on the signal is often low.Assuming adequate basic DFT resolution, the spectrum estimation method should be optimized to reducebias error by improving the effective frequency resolution and lessening sidelobe leakage. This requires alonger effective window in the time domain to reduce truncation end effects. In order to achieve longereffective windows and therefore reduced bias error, we recommend the method of time aliasing, describedin Section 3.

3. Brief review of the method of time aliasing

The method of time aliasing [1–4] creates a periodic time domain signal from the original signal, yielding adiscretely sampled frequency domain representation of the original time signal. Because the time aliased signalis a frequency sampled version of the original signal, its transform equals the transform of the original signalat the frequency line spacing in the frequency domain (see Appendix A).

Before continuing, we offer some relevant words regarding aliasing error: When attempting to reconstruct atime signal from frequency domain information (i.e., through an inverse Fourier transform), it is desirable toavoid aliasing in the time domain. Likewise, frequency domain aliasing is a concept that is well understoodand the Nyquist criterion defines how to avoid confounding of frequency domain information resulting fromsampling too coarsely in the time domain. In both cases, however, the potential for confounding between theoriginal and aliased signals is a result of implementation, not the mathematical process. When applying timealiasing to spectrum analysis of time domain signals, it is important to remember that the frequency domain is

the domain of interest and that although the time signal is modified, the frequency domain is sampled exactly,producing a periodogram estimate which is correct at every frequency line when compared with the unsampledperiodogram. As long as the basic DFT resolution is adequate, time aliasing can be used to further improvethe spectrum estimate through reduction of bias error.

3.1. Producing a time aliased spectral estimate

We first present in Eqs. (2a) and (2b). the discrete time signal, x[n], and its Fourier integral transform(assuming that x[n] is absolutely summable over �NotoN), which equivalent to a band-limited function infrequency, Xc, periodic over �NoroN, and may be sampled at discrete ok by a DFT:

x½n� ¼ xðtÞjt¼nT , (2a)

X ðejoT Þ ¼ TX1

n¼�1

x½n�e�jonT ¼X1

r¼�1

X c oþ2pr

T

� �; X c½k� ¼ X ðejokT Þjok¼2pk=T0

. (2b)

Now, define the discrete time aliased signal, x(a)[n], shown in Eq. (3a). Because it is a periodic function intime (as denoted by the term, qN), x(a)[n] is not absolutely summable, but the Fourier transform of one periodin time, of length N samples, of x(a)[n] produces periodically repeated Fourier coefficients at harmonicfrequencies, ok; in other words, a Discrete Fourier Series given in Eq. (3b), also equivalent over one period infrequency to a sampled version of the Fourier integral transform of the original signal (reference Appendix A):

xðaÞ½nþ qN� ¼X1

m¼�1

x½ðnþ qNÞ þmN�, (3a)

X ðaÞðejokT Þ ¼1

N

XN�1n¼0

xðaÞ½n�e�jokT ; X ðaÞ½K � ¼ X ðaÞðejokT Þjok¼2pk=T0. (3b)

For spectrum estimation, time aliasing is accomplished by first modulating the entire time signal, x(t) with atruncating window function, w(t) as in a conventional estimate (Fig. 1). This is followed by overlapping andsumming contiguous blocks of the windowed discrete time signal (the order of the windowing andoverlapping/summing steps are not interchangeable). Fig. 2 shows a graphical representation of overlapping/

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Fig. 1. Window function w[n] of MN point length (consisting of M contiguous blocks of N weighting points) is used to truncate discrete

signal x[n] to finite length discrete signal v[n].

Fig. 2. Diagram graphically illustrating time aliasing of discrete time signal x[n], which is MN points in length (consisting of M contiguous

blocks of N data points) to N point periodic signal x(a)[n]. One period or block of data from x(a)[n] is to be transformed to frequency

domain using a DFT (FFT).

J.F. Dahl, C.C. Smith / Mechanical Systems and Signal Processing 21 (2007) 2474–24952478

summing a discrete signal, x[n], divided into M segments of length N (shown in general case, but since thesummation over m is necessarily finite, this is equivalent to v[n]). The time aliased signal, x(a)(t), in its discretetime form, x(a)[n], is a repeating signal with period N samples that results from summing x[n+mN] over m inthe contiguous blocks of x[n]. After the time aliased signal is formed, one period (letting q ¼ 0) of the resultingperiodic, time aliased signal can be transformed using an FFT.

An important note regarding the summation process: the result is not normalized by a factor of 1/M—to doso would implement the well-known synchronous time segment averaging process [10,11] (except in the specialcase that the signal consists only of pure sinusoidal, or discrete components, which can only be represented bya Fourier series because they are not of finite length). This can be a useful technique for the special purpose ofextracting purely discrete, harmonically related sinusoidal components from a noisy signal, however, it is oflimited scope; whereas a generalized, time aliasing process is applicable to all spectra, including sinusoids, andespecially to narrowband, harmonic and non-harmonic spectral components.


3.2. Time aliasing decouples window function length from DFT resolution

The N point transform, X(a)[k], of the time aliased signal would equal, at every Mth frequency line, thetransform of an MN point transform of the original signal, had it been performed. Because transform size isreduced by a factor of M, time aliasing has been viewed as a useful tool for compressing transform size whileincluding a large amount of time data in multi-rate digital signal processing systems [12].

More importantly for our discussion, this compression allows the transform of one period of the timealiased signal to preserve the attributes of the original, MN point length window function that was appliedto the signal, versus a window function that was just a single period, N in length. Re-visiting Fig. 1, considerthat if a single block of N samples of x[n] is transformed using an FFT, then the window function is only N

points long. Likewise, if MN samples of x[n] are transformed using a single, MN length FFT, then thetime window function is MN points long, which would reduce leakage bias, but a higher cost of computation is

incurred in the transform. In contrast, first windowing the MN point long signal and time aliasing to anN point periodic signal allows an N point FFT to be used to transform one period and the transform

of the much longer, MN point window, will be precisely preserved in the frequency domain representation at theN point DFT resolution, but without the overhead of performing the larger transform.

The use of a longer effective window with a smaller transform introduces a unique flexibility whentransforming signals to the frequency domain. It allows windows to be created having optimal propertiesfor a particular type of analysis by adjusting the amount of localization in either time or frequency relativeto the DFT resolution. It can also allow a smaller FFT to be computed for an equivalent length of timedata, if desired, to reduce computational overhead, assuming processing and/or data storage resourcesare limited. Section Five will illustrate two approaches to window design, one traditional, and oneinnovative, as used with time aliasing to reduce leakage and therefore minimize bias error in spectralestimates.

4. Two approaches to reducing leakage bias error through window design

We will now discuss two approaches to reducing bias error and achieving improved narrowband spectralestimates: one, a commonly used conventional approach which we will call the Ideal Uniform Windowmethod and the other, an innovative approach, the Ideal Sampling Window method. In Section 5, it will beshown how both approaches can be better implemented through time aliasing.

4.1. Leakage bias error reduction approach number one: an Ideal Uniform Window

The implicit standard in most conventional window design is to use truncation weighting functions havingFourier transforms that approximate, as closely as possible, an Ideal Uniform Window transform. This conceptrepresents an optimal window which might be obtained assuming the desire to correctly estimate an average ofspectral energy in an interval DfN wide, centered at any given frequency line, without admitting or transmittingany energy between adjacent lines.

As shown in Fig. 3, the magnitude of the Ideal Uniform Window transform is unity across its width ofexactly DfN. Viewed as a filter, centered at the kth frequency line, the mainlobe of the ideal uniform windowtransform has unity gain and perfect cutoff (zero sidelobe magnitude). The effect of this filter assigns theaverage magnitude across the interval DfN to the kth frequency line and no others.

Especially when used with narrowband signals, mainlobe bandwidth of any real-life window transform mustbe narrow enough to resolve features in the spectrum, i.e. ensure adequate effective frequency resolution. Forexample, in rotating machinery diagnostics, if two sinusoidal components in the signal are too closely spacedrelative to the window mainlobe bandwidth, even though they may be more than DfN apart, they cannot beresolved unless DFT resolution is improved, which will scale the mainlobe narrower.

In the time domain, we recognize the inverse transform of the Ideal Uniform Window as the sinc, or sin(pt/N)/(pt/N) function, NotoN. Because practical window functions must necessarily be of finite length, the idealuniform window transform cannot exist in practice, of course. Furthermore, since the length of conventionalwindow functions is constrained to the length of the N point segment to be transformed, significant

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Fig. 3. Ideal Uniform Window transform magnitude as a function of DfN.


compromises result in traditional window function design. As discussed in Section 5, however, timealiasing allows the window function length to extend to MN points for very good approximations to an IdealUniform Window.

4.2. Leakage bias error reduction approach number two: an Ideal Sampling Window

While the Ideal Uniform Window approach is useful for many signals, effective convolution betweenwindow and signal transforms results in an average across DfN, which can be a source of bias error if the signalmagnitude varies significantly with frequency relative to a particular DFT resolution. This might occur,for example, in the vicinity of a resonance, or magnitude peak, of a signal that can be represented continuouslyin frequency.

This situation and the associated bias error are illustrated in Fig. 4, where the transform of the IdealUniform Window is overlaid on an example signal having a Fourier transform that can be representedcontinuously in frequency. The represented magnitude of the signal, produced through effective convolutionwith the window transform, resulting in magnitude averaging over DfN, is shown by a point with a cross, or‘X’. The point which would be represented by instead sampling the signal at the frequency line is shown by apoint with a circle or ‘O’. The difference between the average and sampled values is a bias error.

Another type of idealized window, the Ideal Sampling Window, may be more accurate in this situation(assuming the signal transform is a continuous function of frequency). The transform of this windowis theoretically an impulse sampler, or Dirac delta function, illustrated in Fig. 4 by a vertical line at thesample point. The effective convolution between the Ideal Sampling Window transform and the signaltransform would yield sampled values of the signal transform at the DFT resolution spacing, eliminating thebias error.

Producing a true impulse sampler in frequency would require an infinite length function in time. The nearestconventional approximation is the rectangular (box-car) window, but its effectiveness is hindered by thepenalty in sidelobe leakage if the time window function length equals the transform length, N. Like theIdeal Uniform Window in discussed in Section 4.1, however, the Ideal Sampling Window can also beapproximated very well using the method of time aliasing because the window function length can be extendedarbitrarily in time over MN samples, allowing the mainlobe of the window transform to become highlylocalized in frequency, or in other words, arbitrarily narrow relative to DfN, and approaching an impulsesampler.

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Fig. 4. With frequency domain function, X(f), a continuous spectrum with magnitude that changes significantly over DfN, use of Ideal

Uniform Window produces an average value (point X) across DfN having bias error, whereas Ideal Sampling Window results in a sampled

value (point O) and no bias.


5. Using time aliasing to reduce bias error by approaching Optimal Windows

Turning attention to better implementing the two leakage bias error approaches in Section 4, we will showthat there is considerable flexibility in achieving this goal by implementing window designs which arerealizable only by time aliasing the signal prior to transforming it to the frequency domain.

5.1. Approaching an Ideal Uniform Window by time aliasing

The smoothed sinc, or ssinc window has been presented for use with time aliasing methods [2,4]. Subsequentmodifications to this window have been made and will be presented here; however the original ssinc windowfunction remains effective and simple to implement. Used with time aliasing to extend the effective length intime, ssinc window-based transforms can become very close approximations to the ideal uniform window,significantly or even virtually eliminating leakage bias error.

The ssinc window function, Fig. 5, has a high degree of localization in time for optimal frequency domaincharacteristics across Df and is especially applicable to repetitive signals, such as those found in rotatingmachinery. It is derived from the multiplication of an inner sinc, or sin(x)/x function with an outer smoothingfunction. In Eqs. (4a)–(4c), the inner sinc function has sinusoidal frequency, p/N, where N is the length of thediscrete time aliased signal, x(a)[n] in Section 2. The outer function is also a sinc function, of sinusoidalfrequency 2p/MN, where MN is the length of the original truncated signal x[n]:

wssinc½n� ¼ wouter sinc½n��winner sinc½n� (4a)

winner sinc½n� ¼

sin pn

N�

M

2

� ��

pn

N�

M

2

� � ; 0pnoMN

2;

1; n ¼MN

2;

winner sinc½MN � n�;MN

2pnoMN � 1;

8>>>>>>>>>>><>>>>>>>>>>>:

(4b)

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Fig. 5. Overlay of ssinc window functions in time according to number of time aliased blocks, M.


wouter sinc½n� ¼

sin p2n

MN� 1

� ��

p2n

MN� 1

� � ; 0pnoMN

2;

1; n ¼MN

2;

wouter sinc½MN � n�;MN

2pnoMN � 1;

8>>>>>>>>>>><>>>>>>>>>>>:

(4c)

where M is the number of time aliased blocks of N samples each.The frequency domain magnitude of the ssinc window transform is shown in Fig. 6. The exact shape

of its transform depends on the number of time aliased blocks, M, but the window half-power width, orwidth of the mainlobe at half magnitude, is always DfN. Adding more blocks tends to ‘‘square up’’ thessinc window transform, making it approach unity across the top of the mainlobe and reducing sidelobeheight.

Modifying the outer smoothing function produces other windows with slightly modified characteristics fromthe original ssinc window [3]. For example, substituting time weighting functions of length MN: Hanningwindow [13], Nuttal window [13], or Kaiser–Bessel (a ¼ 2) window [13, Appendix B], yield respectively, thehsinc, nsinc or ksinc, window transforms shown in Figs. 7–9, which are slightly more localized in frequency,but have reduced mainlobe ripple and reduced sidelobe magnitudes.

Table 1 shows comparisons between ssinc-based window transforms and common conventional windowtransforms based on the same DFT resolution, DfN. These comparisons use figures of merit as definedin [13]. The table and figures show that, using the Ideal Uniform Window criteria, the ssinc-based windowsperform very well compared to conventional windows. Having a magnitude and bandwidth of nearlyunity and DfN, respectively, these time aliased windows achieve nearly optimal mainlobe characteristicswhile maintaining excellent to superior side-lobe leakage reduction. Because the mainlobe bandwidth ismore localized than conventional window transforms, these windows exhibit improved effective frequency

resolution at the same DFT resolution as conventional windows.

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Fig. 6. Overlay of time aliased ssinc window transform magnitudes as a function of frequency line spacing, DfN as M, the number of

blocks in the time aliasing summation is increased.

Fig. 7. Overlay of transform magnitudes of time aliased, ssinc-based windows (M ¼ 16) as a function of frequency line spacing, DfN.


5.2. Scaling of time aliased Windows to adjust time and frequency localization

Time aliasing provides a way to approach nearly optimal mainlobe window transform bandwidth accordingto the Ideal Uniform Window criteria. For some applications, however, it may be helpful to adjust frequencydomain localization to have an arbitrary mainlobe width relative to DfN. As an example, Eq. (5) shows amodification [3] of the inner sinc window previously given as Eq. (4), now including a time scaling factor, S,which results in a transform having narrower mainlobe bandwidth. This window might be termed the scaled

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Fig. 8. Overlay of transform magnitudes (decibel scale) of time aliased, ssinc-based windows (M ¼ 16) as a function of frequency line

spacing, DfN to show sidelobe characteristics.

Fig. 9. Zoomed overlay of time aliased, nsinc and ssinc window transform magnitudes (M ¼ 64) as a function of frequency line spacing,

DfN to compare mainlobe flatness.


ssinc, or possibly sssinc, window:

winner sinc½n� ¼

sin pSn

N�

M

2

� ��

pSn

N�

M

2

� � ; 0pnoMN

2;

1; n ¼MN

2;

winner sinc½MN � n�;MN

2pnoMN � 1;

8>>>>>>>>>>><>>>>>>>>>>>:

(5)

where S is the desired mainlobe width as a fraction of DfN.

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Table 1

Conventional and time aliased, smooth sinc-based Window transform comparisons

Window function Mainlobe characteristics Sidelobe characteristics

Equivalent noise

bandwidth (DfN)

�3 dB Bandwidth

(DfN)

�6 dB Bandwidth

(DfN)

Max sidelobe height

(dB)

Sidelobe roll-off

(dB/Octave)

Rectangular 1.00 0.89 1.21 �13 �6

Hanning 1.23 1.2 1.65 �32 �18

Kaiser–Bessel (a ¼ 2) 1.49 1.43 1.99 �46 �6

ssinc (M ¼ 8) 0.92 0.91 1.00 �41 �15

ssinc (M ¼ 16) 0.96 0.95 1.00 �42 �15.5

ssinc (M ¼ 32) 0.98 0.98 1.00 �41.5 �20.6

ksinc (M ¼ 8) 0.9 0.89 1.00 �47 �15.8

ksinc (M ¼ 16) 0.96 0.95 1.00 �48 �17.25

Fig. 10. Overlay of scaled ssinc window transform magnitudes (decibel scale) as a function of frequency line spacing, DfN to compare

mainlobe width (M ¼ 16 for all windows).


Some examples of possible window transforms are shown in Fig. 10.2 By scaling the inner sinc windowfunction, the degree of localization in the frequency domain can be changed so the window transform has anarbitrary mainlobe width. In the time domain, scaling the window function prior to time aliasing also allowsone to choose the degree of time localization and the influence of the window function on the signal meansquared value in the time domain.

2All of the window transforms shown in the figure have been normalized by the window RMS value in order to maintain constant

amplitude for the sake of presentation, but this is not a general restriction.


It is important to realize that conventional windows may also be time scaled; however, the time length of aconventional window, N, always equals the transform length, N, so it is not decoupled in the same way as atime aliased window, which has length M times N. The mainlobe bandwidth of a time aliased windowtransform is flexible because the effective length of the time window function can be arbitrary.

The ability to decouple the scaled ssinc window from the frequency line spacing illustrates the degree offlexibility afforded with time aliasing. Windows can be designed independently of DfN. Because the windowtransform shape is decoupled from the frequency line spacing, the window width, flatness, sidelobe amplitude,etc. all now become subject to modification as desired to achieve optimal characteristics.

5.3. Approaching an Ideal Sampling Window by time aliasing

To approximate an Ideal Sampling Window described in Section 4.2, a frequency domain windowtransform with highly localized mainlobe is desired. Such a transform is obtained by minimizing thelocalization of the time window function. An appropriate choice might be a rectangular window, but allowingit to extend in time to cover multiple time aliased blocks of data. Referring to Table 1, we see that themainlobe bandwidth of the conventional rectangular window function, expressed as a fraction of DfN, is 1.00.With time aliasing, however, rectangular window mainlobe bandwidth and magnitude are inversely related tothe number of blocks of data, M, summed during time aliasing. As M increases, mainlobe bandwidth, 1/Mdecreases, and the magnitude, also M, of a time aliased rectangular window more and more closelyapproximates a Dirac function, or an Ideal Sampling Window. The rectangular window function is not theonly window that can be used for frequency sampling, but it is convenient and sampling characteristicsimprove rapidly with increasing M. It is also worthy of note that this might be a natural choice, given thefundamental relationship between sampling in one domain and aliasing in the other [1].

6. Example results obtained using time aliased spectral estimates

Using time aliasing methods of spectrum estimation we realize an improved capability for reducing biaserror in spectra and improving measurements of narrowband spectra, such as those found in mechanicalsystems. Following are two examples that will illustrate how using time aliasing to design optimal windowscan be used to improve spectral measurements and avoid these errors.

6.1. Example one: resolving bearing harmonics and structural resonances in rotating machinery by using time

aliasing to approximate an Ideal Uniform Window

The application is the detection of structural resonances and critical bearing harmonics in a rotatingmachinery system. Such phenomena show up as narrowband peaks in the spectrum, related to the runningspeed. The expected frequencies associated with bearing harmonics can be predicted based on knowledge ofthe bearing geometry and other factors. Peak magnitudes generally increase with decreased health of thebearing, but they can be difficult to detect without sufficient effective frequency resolution.

In these two procedures, improved effective frequency resolution and reduced leakage are demonstratedwith a time aliased spectrum estimate compared to a conventional estimate. Time aliased windows,approaching an Ideal Uniform Window, are used and bias error is reduced in preference to variance error.

For both time aliased and non-time aliased methods, the same overall amount of data was used,transformed in blocks of N samples yielding the same DFT resolution, DfN in both cases. In the conventionalcase, blocks of N samples were Hanning-windowed, transformed and autospectra computed, and all thespectral estimates at DfN were averaged together in the frequency domain for maximum variance reduction. Inthe time aliased case, M contiguous blocks of N samples were first ssinc-windowed, then time aliased togetherinto a single block of N points. This block was transformed, auto-spectrum computed at DfN, and the resultingspectral estimate averaged with the spectral estimate of the next, ssinc-windowed, time aliased M blocks of N

data points. This procedure necessarily yielded fewer spectral averages and therefore improved bias reductionat the expense of variance reduction. In practice, however, the resulting number of spectral averages, between


2 and 5, was sufficient to reduce the actual noise error and the greater reduction of bias error was shown toimprove the overall measurement.

The test subject was a machinery fault simulator3 (MFS), further described in Appendix C, which is ahardware platform consisting of various rotating and reciprocating elements (unused) such as are often foundin mechanical machinery, but possibly on a different scale. The MFS is designed for repeatable configurationso that signatures of common machinery faults can be produced and measured for demonstration, teachingand research. The sensors used in this experiment were a set of industrial grade, piezo accelerometers locatedon triaxial mounts on the inboard and outboard bearing mounts and an LED tachometer was used to measureshaft rotational speed.

In the first procedure, a signal was acquired from an accelerometer on the outboard bearing mount.A purposely faulted bearing, supplied by the MFS manufacturer, was placed in the outboard mount. Thisparticular bearing had a BPFO (ball-pass frequency for outer) raceway defect, detectable through harmonicsat multiples of 3.052 times the running speed. Defects of this type can be difficult to detect because theygenerate low magnitude harmonics that occur at near integers of the running speed, (3,6,9, etc. in this case)and often in the near vicinity of other peaks in the spectrum.

In Figs. 11–13, spectra from the horizontal outboard accelerometer are shown overlaid at the same DFTresolution for both the time-aliased and conventional methods, as the time segment length, N, and accordinglythe transform size, was varied from 1024, 2048 and 4096 points. The time domain sample rate (2 kHz) wasunchanged, so DfN is finer as N is increased. In Fig. 11, a small imbalance, misalignment or both is detectableby the low magnitude (compare with next procedure, Figs. 14–17) harmonics (indicated by 1X, 2X, 3X etc.) atmultiples of the running speed (approx 2000 RPM). The BPFO harmonic is also visible at 3.052X. Bycomparing Figs. 11 and 12 it can be seen that the BPFO harmonic, visible in the time-aliased spectra atN ¼ 1024 points, is obscured or partially leaked by the wider Hanning window mainlobe in the conventionalmethod until DfN is improved by increasing N in Fig. 13. This illustrates that with this narrowband data,the effective frequency resolution is better than with the conventional method and the leakage bias lower,at a relatively coarse DFT resolution, when using a time aliased ssinc window to approximate an IdealUniform Window.

It is important to note that because only a finite amount of data was available in this experiment, thenumber of time aliased segments, M, was reduced from 16, 8 and 4 through Figs. 11–13, respectively, tomaintain some spectral averaging for the time-aliased spectra. Therefore the maximum bias reduction is visiblein Fig. 11 for the time-aliased spectrum with transform size 1024. As the DFT length is increased in Figs. 12and 13, the effective frequency resolutions of the two methods begin to converge as the shape of the time-aliased window is less optimal and the DFT resolution improves. With more data available, M could havebeen maintained at 16 as N increased, producing improved bias error reduction.

In the second procedure, a signal was acquired from the accelerometers on the inboard (channel X) andoutboard (channel Y) bearing mounts in the vertical direction. An imbalance was purposely imposed byplacing small weights on two of the rotors. The faulted bearing was replaced by a good bearing. The motorspeed was varied until the imbalance condition excited multiple structural resonances in the MFS. Spectralestimates were computed from N ¼ 1024 point transforms of data sampled at 5 kHz using three methods: theconventional Hanning windowed and time aliased ssinc-windowed methods of the previous procedure and anadditional conventional method using rectangular windowing, for illustration purposes.

In Fig. 14, conventional autospectra obtained using the Hanning window and rectangular window areoverlaid on the results obtained using a time aliased ssinc window. The rotor imbalance producedaccelerations at the bearing mounts manifest as a series of strong harmonics at multiples of the running speed(about 2700RPM) as indicated by 1X, 2X, etc. Several additional resonance peaks are obscured by therelatively wide mainlobe of the Hanning window, even though the sidelobe leakage, visible as the magnitudelevel in between peaks, is reduced. With the narrower mainlobe of the conventional, rectangular window,although the resonances are detectable, the magnitudes are incorrect, evident as much lower resonance peaks,because of sidelobe leakage in between peaks, also visible. In contrast, all the narrowband phenomena are

3Machinery Fault SimulatorTM is produced by SpectraQuest, Inc. (http://www.spectraquest.com).

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Fig. 11. MFS 1750RPM, sampling frequency 2 kHz, N ¼ 1024, autospectrum conventional Hanning window and 20 spectral averages,

compared to time aliased (M ¼ 16) ssinc window and five spectral averages. BPFO 3.052X frequency harmonic clearly visible in time

aliased, obscured in conventional.

Fig. 12. MFS 1750RPM, sampling frequency 2 kHz, N ¼ 2048, autospectrum conventional Hanning window and 10 spectral averages,

compared to time aliased (M ¼ 8) ssinc window and three spectral averages. BPFO 3.052X frequency harmonic clearly visible in time

aliased, leaked in conventional.


visible, at correct magnitudes, with the time aliased, ssinc-windowed spectrum and the lower sidelobe leakagein between peaks is evident.

6.2. Example two: approximating an Ideal Sampling Window

This is a demonstration of the effect of approximating an Ideal Sampling Window to improvemeasurements with continuously represented frequency domain functions. This experiment was performedusing numerical simulation in Matlabs to measure the frequency response function of the two filter functionsgiven in Eqs. (6) and (7). The first is a narrowband filter, H2(z) with a strong resonant pole and the second is anarrowband filter, H3(z), which has a strong zero located near the resonance. The theoretical FRF magnitude

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Fig. 13. MFS 1750RPM, sampling frequency 2 kHz, N ¼ 4096, autospectrum conventional Hanning window and five spectral averages,

compared to time aliased (M ¼ 4) ssinc window and two spectral averages. BPFO 3.052X frequency harmonic visible in conventional and

time aliased.

Fig. 14. MFS 2700RPM, sampling frequency 5 kHz, N ¼ 1024, autospectra of conventional Hanning or rectangular windows with 20

spectral averages compared to autospectrum of time aliased (M ¼ 32) ssinc window with five spectral averages.


and phase plots for the two filters are shown in Figs. 15 and 16. The sample frequency, 1/T for this experimentwas assumed to be one:

H2ðzÞ ¼1

1þ 0:03125z�1 þ 0:99z�2, (6)

H3ðzÞ ¼1þ 0:125z�1 þ 0:9999z�2

1þ 0:0156z�1 þ 0:9999z�2. (7)

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Fig. 15. Theoretical frequency response function of filter H2.

Fig. 16. Theoretical frequency response function of filter H3.


The filters were implemented using difference equations in the time domain and the input to the filter, x[n],was composed of samples from a pseudorandom noise generator. An uncorrelated noise source was used toadd a controlled amount of variance uncertainty (25% noise amplitude) to the output of the filter, y[n] for amagnitude S/N ratio of about 8 in these experiments. For each test run of the simulation, four blocks of theinput were passed through the filters before any data was collected to eliminate the possibility of transientresponse and then the input and output of the filter, x[n] and y[n], were collected in windowed segments andtransformed with the same DFT resolution to X[k] and Y[k] in the frequency domain.

Weighted (or Windowed) overlapped segment averaging (WOSA), [14] was used to accomplish spectralaveraging of the transforms. The exact same total amount of data was used for both the conventional and timealiased spectra. On each run, the estimated FRF magnitude, H[k], was calculated according to Eq. (8), fromspectrally averaged X[k] and Y[k]. The conventional spectra were windowed in 50% overlapped segments of N

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Fig. 17. Cumulative overlays of FRF magnitude measurements from 40 separate runs with H2 filter. Top two plots show conventional

method with Hanning and Kaiser–Bessel (a ¼ 3) windows; bottom two plots show time aliasing with ssinc (M ¼ 32) and rectangular

window (M ¼ 32) used to approach an Ideal Sampling Window, resulting in reduced variance and correct magnitude.


samples, yielding 64 spectral averages. Time aliased windows were overlapped by N samples, yielding 33spectral averages for each test run.

The time aliased spectra were windowed in two ways: a ssinc window with M ¼ 32 time aliased blocks toapproximate an ideal uniform window as discussed in Section 5.1, and a rectangular window with M ¼ 32time aliased blocks to approximate an ideal sampling window as discussed in Section 5.3:

H½k� ¼Y ½k�X ½k��avg

X ½k�X ½k��avg, (8)

where X[k]* is the complex conjugate of X[k], etc.Fig. 17 contains zoomed overlay plots of FRF magnitude measurements received for 40 separate test runs

with filter H2. The top two plots show all 40 test runs for the conventional method using a Hanning windowand a Kaiser–Bessel (a ¼ 3) window; and the bottom two plots shows all 40 test runs for time aliasing withssinc windowing and also with rectangular windowing. The trend of each plot represents the mean of the 40FRF magnitude measurements and the spread shows the variance.


Comparing Fig. 17 to the theoretical magnitude shown in Fig. 15 shows two things: First, in general, thatthe results in the narrowband region received using time aliasing windows were superior to conventionalmethods, with less leakage error and smaller spread, notwithstanding the steepness of the resonance peak;second, that the use of a frequency scaled rectangular window (bottom plot) as a frequency sampling functionapproaching an Ideal Sampling Window produced the most correct and repeatable FRF magnitudemeasurements, as predicted in Section 4.2.

Figs. 18 and 19 show cumulative results for filter H3, displayed in a different way, where the mean FRFmagnitude from all test runs is overlaid with the theoretical or expected FRF magnitude and the variance of

Fig. 18. Cumulative mean and variance estimating H3 filter FRF magnitude using conventional method with Hanning, Kaiser–Bessel

(a ¼ 3) and rectangular windows (40 test runs each).

Fig. 19. Cumulative mean and variance for 120 test runs estimating H3 filter FRF magnitude using time aliasing and a rectangular

(M ¼ 32) window to approach Ideal Sampling Window.


the FRF magnitude is plotted separately. Cumulative averages of 40 test runs for conventional methods usinga Hanning window, rectangular window, and a Kaiser–Bessel (a ¼ 3) window are shown overlaid in Fig. 18.Cumulative averages of 120 test runs using time aliasing and a rectangular window, approaching an IdealSampling Window are overlaid in Fig. 19. Again, time aliasing performance in Fig. 19 in the narrowbandregion is superior to conventional methods, evident by the agreement with the theoretical magnitude andreduced variance (note scale difference between figures). It is important to see that this is true both for the poleand for the zero, showing improvement with both types of narrowband phenomena when bias error is reducedby using a time aliased window as a frequency sampling function.

7. Conclusion

The purpose of this paper has been to demonstrate that the use of time aliasing for spectrum analysis asapplied to mechanical systems produces an accurately sampled spectral estimate while permitting thecharacteristics of the window function to be decoupled from the DFT resolution, or frequency linespacing. This decoupling allows the realization of windows not possible otherwise because the effective timelength of the window can extend longer than the segment used for DFT transformation. Windows usedwith time aliasing can be optimized by adjusting the amount of time or frequency localization, accordingto the type of spectrum being measured and can be used in conjunction with conventional signalprocessing methods presently used in vibration analysis, experimental modal analysis, and rotating machineryanalysis.

Assuming the conventional, or Ideal Uniform Window design criterion, time aliased windows, including thessinc window and new variations presented in this paper, have been shown to achieve better characteristicsthan conventional windows, including Hanning, Kaiser–Bessel, and others. Improved mainlobe shape allowsfor more precise magnitude measurement and improved effective frequency resolution, while reduced sidelobemagnitude controls leakage to an arbitrary degree, including virtual elimination of bias errors from leakage.When time aliased windows were used in an application involving rotating machinery vibration, bearingharmonics were detected at much coarser frequency line spacing than with conventional methods andstructural resonance magnitudes were more accurately estimated.

A new approach described in this paper is the design of windows according to an Ideal Sampling Windowcriterion, for use with continuous spectra, where the mainlobe of a time-aliased window is scaled toapproximate a discrete sampling function. This type of window was shown to produce superior frequencyresponse function measurements in a system identification application where a lightly damped resonance andanti-resonance were closely spaced in frequency.

When measuring narrowband spectra, such as those associated with many mechanical systems, bias error isgenerally of more concern than variance error. Using time aliasing in spectrum analysis has been shown tosignificantly reduce bias error, making direct spectral estimates of narrowband phenomena more accurate andapplicable than ever for mechanical system spectrum analysis.

Appendix A

The transform of one period of a discrete, time aliased signal is equal to a frequency sampled version of thetransform of the original discrete signal: form one period of the N point time aliased signal, x(a)[n] and its N

point transform, X(a)[k] and re-arrange order of summation:

X ðaÞðejokT Þ ¼1

N

XN�1n¼0

xðaÞ½n�e�jð2pkn=NÞ ¼1

N

XN�1n¼0

XM�1m¼0

x½nþmN�e�jð2pkn=NÞ

¼1

N

XM�1m¼0

XN�1n¼0

x½nþmN�e�jð2pkn=NÞe�jð2pkmN=NÞ ¼1

N

XM�1m¼0

XNþmN�1

q¼mN

x½q�e�jð2pqk=NÞ.


Recognize the periodic nature of the exponential and substitute frequency line indices:

¼1

N

XMN�1

q¼0

x½q�e�jð2pqk=NÞ ¼ X c½Mp�MN ¼ X ðe�jMopT Þ; op ¼2pp

MN¼

ok

M,

‘X ðaÞðejokT ÞN ¼ X ðejopT ÞMN jop¼ok.

Appendix B

Kaiser–Bessel Window weighting function from [13] (time shift by –MN/2 when forming ksinc window)

wKaiser2Bessel½n� ¼

I0 pa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:0�

n

MN=2

� �2s2

435

I0 pað Þ; 0pjnjp

MN

2, (B.1)

I0ðnÞ ¼X1k¼0

n

2

� �k

k!

264

3752

, (B.2)

where M is the number of time aliased blocks of N samples each.

Appendix C

A picture of the machinery fault simulator (see footnote 3) (MFS) is shown in Fig. C1 with the rotorassembly top right, motor, drive and tachometer on the left and reciprocating elements (unused) in the front.Accelerometers are mounted triaxially on the left (inboard) rotor bearing mount and vertically on the right(outboard) bearing mount. A plot of acceleration for inboard bearing mount vertical and horizontalaccelerations, channels X and Y, with no intentional faults introduced is shown in Fig. C2. There was no gainused on any of the accelerometer channels and good signal-to-noise ratio with noise floor at approximately�120 dB. One segment of time data are shown in the left plots and averaged autospectra over 20 segments (nottime aliased) in the right plots. The sample frequency was 2048Hz. Very low amplitude harmonics of the

Fig. C1. Machinery fault simulator (MFS) (see footnote 3).

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Fig. C2. MFS accelerations and magnitudes with no faults imposed (Fs ¼ 2048Hz).


running speed are visible, from an inevitable very slight shaft misalignment. Data were acquired and analyzedusing programs written in LabVIEWs (http://www.ni.com) and Matlabs (http://www.mathworks.com).

References

[1] C.C. Smith, J.F. Dahl, R.J. Thornhill, The dual nature of leakage and aliasing in digital signal processing, Active Control of

Vibration and Noise Symposium, International Mechanical Engineering Congress, November 1994.

[2] C.C. Smith, R.H. Holmes, R.J. Thornhill, Time aliasing to reduce frequency leakage when digitally processing acoustic, shock,

and vibration signals, in: Third International Congress on Air and Structure-Borne Sound and Vibration, Montreal, Canada, 1994,

pp. 986–987.

[3] J.F. Dahl, Time aliasing methods of spectrum estimation, Ph.D. Thesis, Brigham Young University, Provo, UT, USA, 2003.

[4] C.C. Smith, J.F. Dahl, R.J. Thornhill, The duality of leakage and aliasing and improved digital spectral analysis techniques, ASME

Journal of Dynamic Systems, Measurement, and Control 118 (December 1996) 741–747.

[5] P.D. Welch, A direct digital method of power spectrum estimation, IBM Journal of Research and Development, April (1961)

141–156.

[6] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, second ed., Prentice-Hall, Upper Saddle River, NJ,

USA, 1999.

[7] J.G. Proakis, et al., Advanced Digital Signal Processing, Macmillan Publishing Company, New York, NY, USA, 1992.

[8] D.E. Newland, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, third ed., Longman Scientific & Technical,

Essex, UK, 1993.

[9] G.M. Jenkins, D.G. Watts, Spectral Analysis and its Applications, Holden-Day, San Francisco, CA, USA, 1968.

[10] S. Braun, The extraction of periodic waveforms by time domain averaging, Acoustica 32 (1975) 69–77.

[11] C.M. Harris (Ed.) Shock and Vibration Handbook, fourth ed., McGraw-Hill, New York, NY, USA, 1996.

[12] R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, USA, 1983.

[13] F.J. Harris, On the use of windows for harmonic analysis with the discrete fourier transform, Proceedings of the IEEE 66 (1) (1978)

51–83.

[14] P.D. Welch, The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short

modified periodograms, IEEE Transactions on Audio and Electroacoustics AU-15 (2) (1967) 270–273.

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