Frequency Resolution Effects on FRF Estimation:
Cyclic Averaging vs. Large Block Size
Allyn W. Phillips, PhD
Research Assistant Professor
Andrew T. Zucker
Research Assistant
Randall J. Allemang, PhD
Professor
Structural Dynamics Research Laboratory
Department of Mechanical, Industrial and Nuclear Engineering
University of Cincinnati
Cincinnati, Ohio 45221-0072
U. S. A.
ABSTRACT
It is always desirable to use acquired data in the most effective way
possible. This paper explores acquiring several contiguous blocks
of data and then processing the data in various ways. The first
method will be the traditional approach of utilizing each contiguous block, in whole, as one RMS spectral average. Other approaches will involve dividing the contiguous block up into subsets
which will then be cyclic averaged. The advantages and disadvantages of each approach are discussed in terms of total test time,
processing time, data quality, and memory usage which becomes
significant for moderate to large (N i xN o ) test situations. Experimental results are shown for typical structural cases.
Nomenclature
N = Time blocksize.
N s = Frequency blocksize.
N avg = Number of RMS spectral averages.
N c = Number of cyclic averages.
N i = Number of inputs.
N o = Number of outputs.
Δ f = Frequency resolution (Hz.).
1. Introduction
The averaging of signals is normally viewed as a summation or
weighted summation process where each sample function has a
common abscissa. Normally, the designation of history is given to
sample functions with the abscissa of absolute time and the
designation of spectra is given to sample functions with the
abscissa of absolute frequency. The spectra are normally generated
by Fourier transforming the corresponding history. In order to
generalize and consolidate the concept of signal averaging as much
as possible, the case of relative time can also be considered. In this
way relative history can be discussed with units of the appropriate
event rather than seconds and a relative spectrum will be the
corresponding Fourier transform with units of cycles per event.
This concept of signal averaging is used widely in structural
signature analysis where the event is a revolution. This kind of
approach simplifies the application of many other concepts of
signal relationships such as Shannon’s sampling theorem and
Rayleigh’s criterion of frequency resolution.
2. Background
The process of signal averaging as it applies to frequency response
functions is simplified greatly by the intrinsic uniqueness of the
frequency response function. Since the frequency response
function can be expressed in terms of system properties of mass,
stiffness, and damping, it is reasonable to conclude that in most
realistic structures, the frequency response functions are
considered to be constants just like mass, stiffness, and damping.
This concept means that when formulating the frequency response
function using H 1 , H 2 , or H v algorithms, the estimate of frequency
response is intrinsically unique, as long as the system is linear and
the noise can be minimized or eliminated. In general, the auto- and
cross-power spectrums are statistically unique only if the input is
stationary and sufficient averages have been taken. Generally, this
is never the case. Nevertheless, the estimate of frequency response
is valid whether the input is stationary, non-stationary, or
deterministic.
The concept of the intrinsic uniqueness of the frequency response
function also permits a greater freedom in the testing procedure.
Each function can be derived as a result of a separate test or as the
result of different portions of the same continuous test situation. In
either case, the estimate of frequency response function will be the
same as long as the time history data is acquired simultaneously
for the auto- and cross-power spectrums that are utilized in any
computation for frequency response or coherence function.
Generally, averaging is utilized primarily as a method to reduce the
error in the estimate of the frequency response function(s). This
error can be broadly considered as noise on either the input and/or
the output and can be considered to the sum of random and bias
components. Random errors can be effectively minimized through
the common approach to averaging. Bias errors, generally, cannot
be effectively minimized through the common approach to
averaging. Cyclic averaging, however, does minimize the bias
error caused by the truncation of the time domain signals in
conjunction with the use of a discrete Fourier transform (DFT).
This error is commonly known as the leakage error.
The approaches to signal averaging for these different situations
vary only in the relationship between each sample function used.
Since the Fourier transform is a linear function, there is no
theoretical difference between the use of histories or spectra.
(Practically, though, there are precision considerations.) With this
in mind, the signal averaging useful to frequency response function
measurements can be divided into three classifications:
•
Asynchronous
•
Synchronous
•
Cyclic
These three classifications refer to the trigger and sampling
relationships between sample functions.
2.1 Asynchronous Signal Averaging
The classification of asynchronous signal averaging refers to the
case where no known relationship exists between individual
sample functions. The FRF is estimated solely on the basis of the
intrinsic uniqueness of the frequency response function. In this
case, the power spectra (least squares) approach to the estimate of
frequency response must be used since no other way of preserving
phase and improving the estimate is available. In this situation, the
trigger to initiate digitization (sampling and quantization) takes
place in a random fashion dependent only upon the equipment
availability. The triggering is said to be in a free-run mode.
2.2 Synchronous Signal Averaging
The synchronous classification of signal averaging adds the
additional constraint that each sample function must be initiated
with respect to a specific trigger condition (often the magnitude
and slope of the excitation). This means that the frequency
response function can be formed as a summation of ratios of X(ω )
divided by F(ω ) since phase is preserved. Even so, the power
spectra (least squares) approach is still the preferred FRF
estimation method due to the reduction of variance and the
usefulness of the ordinary coherence function. The ability to
synchronize the initiation of digitization allows for use of nonstationary or deterministic inputs with a resulting increased signal
to noise ratio and reduced leakage. Both of these improvements in
the frequency response function estimate are due to more of the
input and output being observable in the limited time window.
The synchronization takes place as a function of a trigger signal
occurring in the input (internally) or in some event related to the
input (externally). An example of an internal trigger would be the
case where an impulsive input is used to estimate the frequency
response. All sample functions would be initiated when the input
reached a certain amplitude and slope. A similar example of an
external trigger would be the case where the impulsive excitation
to a speaker is used to trigger the estimate of frequency response
between two microphones in the sound field. Again, all sample
functions would be initiated when the trigger signal reached a
certain amplitude and scope.
2.3 Cyclic Signal Averaging
The cyclic classification of signal averaging involves the added
constraint that the digitization is coherent between sample
functions. This means that the exact time between each sample
function is used to enhance the signal averaging process. Rather
than trying to keep track of elapsed time between sample
functions, the normal procedure is to allow no time to elapse
between successive sample functions. This process can be
described as a comb digital filter in the frequency domain with the
teeth of the comb at frequency increments dependent upon the
periodic nature of the sampling with respect to the event measured.
The result is an attenuation of the spectrum between the teeth not
possible with other forms of averaging.
This form of signal averaging is very useful for filtering periodic
components from a noisy signal since the teeth of the filter are
positioned at harmonics of the frequency of the sampling reference
signal. This is of particular importance in applications where it is
desirable to extract signals connected with various rotating
members. This same form of signal averaging is particularly
useful for reducing leakage during frequency response
measurements and also has been used for evoked response
measurements in biomedical studies.
A very common application of cyclic signal averaging is in the
area of analysis of rotating structures. In such an application, the
peaks of the comb filter are positioned to match the fundamental
and harmonic frequencies of a particular rotating shaft or
component. This is particularly powerful, since in one
measurement it is possible to enhance all of the possible
frequencies generated by the rotating member from a given data
signal. With a zoom Fourier transform type of approach, one shaft
frequency at a time can be examined depending upon the zoom
power necessary to extract the shaft frequencies from the
surrounding noise.
The application of cyclic averaging to the estimation of frequency
response functions can be easily observed by noting the effects of
cyclic averaging on a single frequency sinusoid. Figures 1 and 2
represent the cyclic averaging of a sinusoid that is periodic with
respect to the observation time period T . Figures 3 and 4 represent
the cyclic averaging of a sinusoid that is aperiodic with respect to
the observation time period T . By comparing Figure 2 to Figure 4,
the attenuation of the nonperiodic signal can be clearly observed.
3. Theory of Cyclic Averaging
In the application of cyclic averaging to frequency response
function estimates, the corresponding fundamental and harmonic
frequencies that are enhanced are the frequencies that occur at the
integer multiples of Δ f . In this case, the spectra between each
Δ f is reduced with an associated reduction of the bias error called
leakage.
fundamental frequency resolution Δ f of the spectra via the
1
Rayleigh Criteria (Δ f = ).
T
1
0.8
0.6
The Fourier transform of a history is given by:
0.4
0.2
0
+∞
−0.2
X(ω ) =
−0.4
∫∞ x(t) e
− jω t
(1)
dt
−
−0.6
−0.8
−1
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
Figure 1. Contiguous Time Records (Periodic Signal)
Using the time shift theorem of the Fourier transform, the Fourier
transform of the same history that has been shifted in time by an
amount t 0 is:
+∞
∫∞ x(t + t ) e
X(ω ) e− j ω t0 =
1
0.8
0
− jω t
dt
(2)
−
0.6
0.4
For the case of a discrete Fourier transform, each frequency in the
spectra is assumed to be an integer multiple of the fundamental
1
frequency Δ f = . Making this substitution in Equation (2)
T
2π
(ω = n
with n as an integer) yields:
T
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
1
X(ω ) e
Figure 2. Averaged Time Records (Periodic Signal)
−j n
2π
t
T 0
+∞
=
∫∞ x(t + t ) e
0
− jω t
dt
(3)
−
Note that in Equation (3), the correction for the cases where
t 0 = N T with N is an integer will be a unit magnitude with zero
phase. Therefore, if each history that is cyclic averaged occurs at a
time shift, with respect to the initial average, that is an integer
multiple of the observation period T , then the correction due to the
time shift does not effect the frequency domain characteristics of
the averaged result. All further discussion will assume that the
time shift t 0 will be an integer multiple of the basic observation
period T .
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
The signal averaging algorithm for histories averaged with a
boxcar or uniform window is:
Figure 3. Contiguous Time Records (Non-Periodic Signal)
1
x(t) =
0.8
1
Nc
Nc − 1
Σ
i=0
x i (t)
(4)
0.6
0.4
where:
0.2
0
•
x i (t) = Time history, average i
•
N c = Number of cyclic averages
•
x(t) = Cyclic averaged time history
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
1
Figure 4. Averaged Time Records (Non-Periodic Signal)
The first observation to be noted is the relationship between the
Fourier transform of a history and the Fourier transform of a time
shifted history. In the averaging case, each history will be of some
finite time length T which is the observation period of the data.
Note that this time period of observation T determines the
For the case where x(t) is continuous over the time period N c T ,
the complex Fourier coefficients of the cyclic averaged time history
become:
Ck =
1
T
T
∫ x(t) e
0
− jω t
dt
(5)
Ck =
1
T
T
∫
0
1
Nc
Nc − 1
Σ
i=0
x i (t) e−
jω t
0
(6)
dt
10
−1
Finally:
Ck =
1
Nc T
T
Nc − 1
∫ Σ
0
x i (t) e−
jω t
(7)
dt
i=0
Since x(t) is a continuous function, the sum of the integrals can be
replaced with an integral evaluated from 0 to N c T over the
original function x(t). Therefore:
Normalized Amplitude
10
−2
10
−3
10
−4
10
1
Nc T
∫
x(t) e−
jω t
(8)
dt
−5
10
0
The above equation indicates that the Fourier coefficents of the
1
cyclic averaged history (which are spaced at Δ f = ) are the same
T
Fourier coefficients from the original history (which are spaced at
Δ f = N c T ).
Figures 5 through 8 show the cyclic averaging effect in the
frequency domain for the cases of 1, 2 and 4 averages with a
uniform window applied to the data. Figures 9 through 12 show
the cyclic averaging effect in the frequency domain for the cases of
1, 2 and 4 averages with a Hanning window applied to the original
contiguous data.. These figures demonstrate the effectiveness of
cyclic averaging in rejecting nonharmonic frequencies. Practically,
these figures also demonstrate that, based upon effectiveness or the
limitations of the dynamic range of the measured data, a maximum
of 16 to 32 averages is recommended. Realistically, 4 to 8 cyclic
averages together with a Hanning window provides a dramatic
improvement in the FRF estimate.
0
1
2
3
4
5
Frequency Resolution: (Delta f)
6
7
8
Figure 6. Cyclic Averaged (N c = 2) Window Characteristics,
Uniform Window
0
10
−1
10
Normalized Amplitude
Ck =
Nc T
−2
10
−3
10
−4
10
−5
10
0
10
Normalized Amplitude
−2
10
−3
10
−4
10
−5
0
1
2
3
4
5
Frequency Resolution: (Delta f)
6
7
8
Figure 7. Cyclic Averaged (N c = 4) Window Characteristics,
Uniform Window
−1
10
10
0
1
2
3
4
5
Frequency Resolution: (Delta f)
6
7
8
Figure 5. Cyclic Averaged (N c = 1) Window Characteristics,
Uniform Window
0
0
10
10
−1
10
−1
10
Normalized Amplitude
Normalized Amplitude
−2
−2
10
−3
10
10
−3
10
−4
10
−4
10
−5
10
−5
10
−4
−6
−3
−2
−1
0
1
Frequency Resolution: (Delta f)
2
3
10
4
Figure 8. Cyclic Averaged (N c = 4) Window Characteristics,
Uniform Window
0
1
2
3
4
5
Frequency Resolution: (Delta f)
6
7
8
Figure 10. Cyclic Averaged (N c = 2) Window Characteristics,
Hanning Window
0
0
10
10
−1
10
−1
10
−2
Normalized Amplitude
Normalized Amplitude
10
−2
10
−3
10
−3
10
−4
10
−5
10
−6
10
−4
10
−7
10
−5
10
0
−8
1
2
3
4
5
Frequency Resolution: (Delta f)
6
7
8
Figure 9. Cyclic Averaged (N c = 1) Window Characteristics,
Hanning Window
10
0
1
2
3
4
5
Frequency Resolution: (Delta f)
6
7
8
Figure 11. Cyclic Averaged (N c = 4) Window Characteristics,
Hanning Window
0.08
0
10
0.06
0.04
−1
10
0.02
0
−0.02
Normalized Amplitude
−2
10
−0.04
−0.06
−3
10
−0.08
−0.1
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
−4
10
Figure 15. Contiguous Time Records with Hanning Window
(Random Signal)
−5
10
0.03
−6
10
−4
−3
−2
−1
0
1
Frequency Resolution: (Delta f)
2
3
4
Figure 12. Cyclic Averaged (N c = 4) Window Characteristics,
Hanning Window
0.02
0.01
0
−0.01
The results of cyclic averaging of a general random signal with the
application of a uniform window are shown in Figures 13 and 14.
Likewise, the results of cyclic averaging of a general random signal
with the application of a Hanning window are shown in Figures 15
and 16.
−0.02
−0.03
−0.04
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
1
Figure 16. Averaged Time Records with Hanning Window
(Random Signal)
0.1
4. Practical Example
0.05
0
−0.05
−0.1
−0.15
0
0.5
1
1.5
2
2.5
Time Periods (T)
3
3.5
4
Figure 13. Contiguous Time Records (Random Signal)
0.05
0.04
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
Time Periods (T)
0.7
0.8
0.9
Figure 14. Averaged Time Records (Random Signal)
1
Examples from two different lightly damped structures will be
used to discuss the points of this paper. The two structures tested
consist of a circular plate and a body-in-white. Both stuctures
were tested with a random excitation via two shaker inputs and the
responses were measured with accelerometers. Each structure was
tested under two different digital signal processing techniques.
The first technique used cyclic averaging with N c cyclic averages,
N avg spectral averages, and N time lines. In comparison, the
second method used the same N avg spectral averages, no cyclic
averages, and increased the number of time lines to N c xN so that a
large blocksize resulted and the total test time for both sets of
measurements were the same. As a result of the larger blocksize
the frequency resolution was increased for this measurement set as
compared to the first measurement set by a factor of N c . As stated
the total test time for both techniques was the same since each case
required the collection of NxN c xN avg time lines. The question to
be answered is whether cyclic averaging provides better results
than the same measurement with a finer frequency spacing and
what are the advantages/disadvantages of one technique over the
other. When the frequency axis of the cyclic averaged test is
extracted from the finer frequency/larger blocksize run and
overlaid the results are very comparable (a plot is not shown since
the two different FRF’s can not even be distinguished). This is a
result of the comb filtering of cyclic averaging and in the abscence
of other measurement noise it is expected that the two cases would
exactly overlay one another. An analytical study will be conducted
to verify if this is indeed the result. If all the frequency lines of the
finer frequency/larger blocksize run are kept and the results of the
Phase (Deg)
Input:
1 Output: 3
180
0
0
10
1
0.9
0.8
−1
10
0.7
Magnitude
two techniques are compared, as shown in Figures 17 and 18 for
the circular plate and Figures 19 and 20 for the body-in-white, the
effects of cyclic averaging on leakage can be observed. A dramatic
improvement in coherence as a result of cyclic averaging
eliminating leakage is seen in the comparison of the circular plate
data. An improvement, though minimal, is also present in the
body-in-white data. Had cyclic averaging not been employed, the
results would have looked like those presented in Figures 21 and
22, for the circular plate and body in white respectivily, which are
the results from a pure random excitation in which the same
amount of data was taken but cyclic averaging was not
implemented.
0.6
−2
10
0.5
0.4
0.3
−3
10
Phase (Deg)
Input: 1 Output 4
0.2
180
0.1
0
−4
10
3
10
0
1
10
20
30
Frequency (Hertz)
50
60
0
Figure 19. Body in White
2
10
0.8
10
0.6
0
10
0.4
Phase (Deg)
Input:
1
Magnitude
40
1 Output: 3
180
0
0
10
1
0.9
−1
10
0.2
0.8
−1
10
0
50
100
150
200
250
Frequency (Hertz)
300
350
400
0
450
Figure 17. Lightly Damped Circular Plate
Magnitude
0.7
−2
10
0.6
−2
10
0.5
0.4
0.3
−3
10
Phase (Deg)
Input: 1 Output 4
0.2
180
0.1
0
−4
10
3
10
0
1
10
20
30
Frequency (Hertz)
40
50
60
0
Figure 20. Body in White
2
10
0.8
0
10
0.4
−1
10
0
50
100
150
200
250
Frequency (Hertz)
300
350
Figure 18. Lightly Damped Circular Plate
400
0
450
0
3
1
2
0.2
−2
10
180
10
10
0.8
1
Magnitude
Magnitude
0.6
Phase (Deg)
Input: 1 Output 4
1
10
10
0.6
0
10
0.4
−1
10
0.2
−2
10
0
50
100
150
200
250
Frequency (Hertz)
300
350
Figure 21. Lightly Damped Circular Plate
400
0
450
Phase (Deg)
Input:
dramatically reduces both the memory requirements and the
computational effort. This reduction may be trivial for small test,
but substantial for moderate to large tests.
1 Output: 3
180
0
0
10
1
0.9
0.8
−1
10
Magnitude
0.7
0.6
−2
10
0.5
0.4
0.3
−3
10
0.2
It is also important to remember that if the results are going to be
used for the estimation of modal parameters, it is much more
important to have frequency response functions accurate at the
measured frequencies than to have a large number of frequencies
calculated. For large tests using current polyreference modal
parameter estimation techiniques, the additional frequency lines do
not contribute significantly the the results. In fact, the
measurements are often decimated before inclusion. However, if
there is leakage at the resonances in the large block FRFs that has
been eliminated in the cyclic averaged data, the results of using the
large block FRFs will bias the solution of the modal parameters.
0.1
−4
10
0
10
20
30
Frequency (Hertz)
40
50
60
0
Figure 22. Body in White
Besides improved data quality through the minimization of
leakage, cyclic averaging offers two other noteworthy advantages.
In comparison to the large blocksize case, cyclic averaged data
requires less processing time. This is a direct result of the number
of frequency lines being a factor of N c smaller when compared to
the large blocksize case. For larger N i xN o cases this translates into
N c xN i xN o lines which need not be processed when calculating
FRF’s. The second advantage, amount of memory required to save
results, is also a direct result of this factor of N c differnece in the
number of frequency lines. Since cyclic averaged results do
contain a factor of N c less frequency lines when compared to the
large blocksize case it requires less space to store the results. For
N
large N i xN o tests this means a savings of N i xN o x
x(N c − 1)
2
complex values.
Assuming standard data storage (1 complex number requires 16
btyes), a moderately sized test (N i = 4, N o = 256, N s = 1024,
N = 2048) would require approximately 35 Megabytes of storage.
If the blocksize is increased by a factor of 4 (N i = 4, N o = 256,
N s = 4096, N = 8192), the in memory required storage increases
proportionately to almost 140 Megabytes. However, in contrast,
the use of 4 cyclic averages (N i = 4, N o = 256, N s = 1024,
N = 8192) requires only the additional storage for the raw time
series, the spectral storage requirements remain the same (about 38
Megabytes). Note also that the computational time also increases
proportionately.
One advantage of the larger blocksize is that through its smaller
Δ f spacing a more accurate reading of the resonance frequncies
maybe obtained. Also closely spaced modes may become more
visible. However, if the data is going to be utilized in an
appropriate parameter estimation routine, the routine should be
able to pinpoint the resonance frequencies and closely spaced
modes without this increased frequency resolution.
5. Conclusions
The results of this study indicate that while cyclic averaging and
increased frequency produce essentially the same result for the
frequencies in common, however, the use of cyclic averaging
6. References
[1]
Bendat, J.S.; Piersol, A.G., Random Data: Analysis and
Measurement Procedures, John Wiley and Sons, Inc.,
New York, 1971, 407 pp.
[2]
Bendat, J. S., Piersol, A. G., Engineering Applications of
Correlation and Spectral Analysis, John Wiley and Sons,
Inc., New York, 1980, 302 pp.
[3]
Otnes, R.K., Enochson, L., Digital Time Series Analysis,
John Wiley and Sons, Inc., New York, 1972, 467 pp.
[4]
Hsu, H.P., Fourier Analysis, Simon and Schuster, 1970,
274 pp.
[5] Allemang, R. J., "Investigation of Some Multiple
Input/Output Frequency Response Function Experimental
Modal Analysis Techniques", Doctor of Philosophy
Dissertation, University of Cincinnati, Mechanical
Engineering Department, 1980, 358 pp.
[6]
Potter, R.W., "Compilation of Time Windows and Line
Shapes for Fourier Analysis", Hewlett-Packard Company,
1972, 26 pp.
[7]
Allemang, R.J., Phillips, A.W., "Cyclic Averaging for
Frequency Response Function Estimation", Proceedings,
International Modal Analysis Conference, 1996, 8pp.