10.6 Other Topics of Signal Analysis

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In this section, we will briefly address some other important topics of signal analysis. We will start by

discussing bandwidth in different contexts then we will present several practically useful analysis

procedures and results on vibration signals.

10.6.1 Bandwidth

Bandwidth has different meanings depending on the particular context and application. For example,

when studying the response of a dynamic system, the bandwidth relates to the fundamental resonant

frequency, and correspondingly to the speed of response for a given excitation. In band-pass filters, the

bandwidth refers to the frequency band within which the frequency components of the signal are allowed

through the filter, the frequency components outside the band being rejected by it. With respect to

measuring instruments, bandwidth refers to the range frequencies within which the instrument measures

a signal accurately. Note that these various interpretations of bandwidth are somewhat related. For

example, if a signal passes through a band-pass filter, then we know that its frequency content is within

the bandwidth of the filter; but we cannot determine the actual frequency content of the signal through

(a)

|G( f )|

0 Frequency f

(b)

|G( f )|2

0 f

G2

r /2

G2

r

Be

Bp

Reference

Level

Equivalent

Ideal Filter

Actual

Filter

FIGURE 10.18 Characteristics of (a) an ideal band-pass filter; (b) a practical band-pass filter.

10-26 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

such an observation. In this context, the bandwidth appears to represent a frequency uncertainty in the

observation (i.e., the larger the bandwidth of the filter, the less certain the actual frequency content of a

signal that is allowed through the filter).

10.6.2 Transmission Level of a Band-Pass Filter

Practical filters can be interpreted as dynamic systems. In fact all physical, dynamic systems (e.g.,

mechanical structures) are analog filters. It follows that the filter characteristic can be represented by the

frequency-transfer function Gð f Þ of the filter. A magnitude squared plot of such a filter transfer function

is shown in Figure 10.18. In a logarithmic plot the magnitude-squared curve is obtained by simply

doubling the corresponding magnitude (Bode plot) curve. Note that the actual filter transfer function

(Figure 10.18(b)) is not flat like the ideal filter shown in Figure 10.18(a). The reference level Gr is the

average value of the transfer function magnitude in the neighborhood of its peak.

10.6.3 Effective Noise Bandwidth

Effective noise bandwidth of a filter is equal to the bandwidth of an ideal filter that has the same reference

level and that transmits the same amount of power from a white noise source. Note that white noise has a

constant (flat) PSD. Hence, for a noise source of unity PSD, the power transmitted by the practical filter is

given by

ð1

0

lGð f Þl2df

which, by definition, is equal to the power G2r

Be transmitted by the equivalent ideal filter. Hence, the

effective noise bandwidth Be is given by

Be ¼

ð1

0

lGð f Þl2df =G2r

ð10:56Þ

10.6.4 Half-Power (or 3 dB) Bandwidth

Half of the power from a unity-PSD noise source,

as transmitted by an ideal filter, is G2r

Br

􀀋

2: Hence,

Gr

􀀋 ffiffi

2 p is referred to as the half-power level. This is

alsffiffio known as a 3 dB level because 20 log10 􀀐

2 p ¼ 10 log10 2 ¼ 3 dB: (Note: 3 dB refers to a

power ratio of 2 or an amplitude ratio of

ffiffi

2 p :

Furthermore, 20 dB corresponds to an amplitude

ratio of 10 or a power ratio of 100). The 3 dB (or

half-power) bandwidth corresponds to the width

of the filter transfer function at the half-power

level. This is denoted by Bp in Figure 10.18(b).

Note that Be and Bp are different in general.

However, in an ideal case where the magnitudesquared

filter characteristic has linear rise and falloff

segments, these two bandwidths are equal (see

Figure 10.19).

10.6.5 Fourier Analysis Bandwidth

In Fourier analysis, bandwidth is interpreted, again, as the frequency uncertainty in the spectral results. In

analytical FIT results, which assume that the entire signal is available for analysis, the spectrum is

0

Be= Bp

Frequency f

|G( f )|2

G2

r / 2

Gr

2

FIGURE 10.19 An idealized filter with linear segments.

Vibration Signal Analysis 10-27

© 2005 by Taylor & Francis Group, LLC

continuously defined over the entire frequency range ½21; 1􀀉 and the frequency increment df is

infinitesimally small ðdf ! 0Þ: There is no frequency uncertainty in this case, and the analysis bandwidth

is infinitesimally narrow.

In digital Fourier transform, the discrete spectral lines are generated at frequency intervals of DF: This

finite frequency increment DF; which is the frequency uncertainty, is therefore the analysis bandwidth B

for this analysis. Note that DF ¼ 1=T; where T is the record length (or window length for a rectangular

window). It follows also that the minimum frequency that has a meaningful accuracy is the bandwidth.

This interpretation for analysis bandwidth is confirmed by noting the fact that harmonic components of

frequency less than DF (or period greater than T) cannot be studied by observing a signal record of length

less than T: Analysis bandwidth carries information regarding distinguishable minimum frequency

separation in computed results. In this sense bandwidth is directly related to the frequency resolution of

analyzed results. The accuracy of analysis increases by increasing the record length T (or decreasing the

analysis bandwidth B).

When a time window other than the rectangular window is used to truncate a measured vibration

signal, then reshaping of data occurs according to the shape of the window. This reshaping reduces leakage

due to suppression of side lobes of the Fourier spectrum of the window. At the same time, however, an

error is introduced due to the information lost through data reshaping. This error is proportional to the

bandwidth of the window itself. The effective noise bandwidth of a rectangular window is only slightly less

than 1=T; because the main lobe of its Fourier spectrum is nearly rectangular. Hence, for all practical

purposes, the effective noise bandwidth can be taken as the analysis bandwidth. Note that data truncation

(multiplication in the time domain) is equivalent to convolution of the Fourier spectrum (in the

frequency domain). The main lobe of the spectrum uniformly affects all spectral lines in the discrete

spectrum of the data signal. It follows that a window main lobe having a broader bandwidth (effective

noise bandwidth) introduces a larger error into the spectral results. Hence, in digital Fourier analysis,

bandwidth is taken as the effective noise bandwidth of the time window that is employed.

10.6.6 Resolution in Digital Fourier Results

Resolution is the frequency separation between spectral lines in digital Fourier-analysis results. For a data

record of length T; the resolution is DF ¼ 1=T irrespective of the type of window used. There is a

noteworthy distinction between analysis bandwidth and resolution. Suppose that we have a data record

of length T: If we double the length by augmenting it with trailing zeros, digital Fourier analysis of the

resulting record of length 2T will yield a spectral line separation of 1=ð2TÞ: Thus, the resolution is halved.

But, unless the true signal value is also zero in the second time interval t½T; 2T􀀉; no new information is

presented in the augmented record of duration ½0; 2T􀀉 in comparison to the original record of duration

½0; T􀀉: So, the analysis bandwidth (a measure of accuracy) will remain unchanged. If, on the other hand,

the signal itself was sampled over ½0; 2T􀀉 and the resulting 2N data points were used in digital Fourier

analysis, the bandwidth as well as the resolution would be halved.

Some relations that are useful in the digital computation of spectral results for signals are summarized

in Box 10.3.

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