Brüel Kjær Technical Review, No. 2, 1984, p.29

Brüel Kjær Technical Review, No. 2, 1984, p.29
PREVIOUSLY ISSUED NUMBERS OF
BRÜEL & KJÆR TECHNICAL REVIEW
1-1984 Dual Channel FFT Analysis (Part I)
4-1983 Sound Level Meters - The Atlantic Divide
Design principles for Integrating Sound Level Meters
3-1983 Fourier Analysis of Surface Roughness
2-1983 System Analysis and Time Delay Spectrometry (Part II)
1-1983 System Analysis and Time Delay Spectrometry (Part I)
4-1982 Sound Intensity (Part II Instrumentation and Applications)
Flutter Compensation of Tape Recorded Signals for Narrow Band
Analysis
3-1982 Sound Intensity (Part I Theory).
2-1982 Thermal Comfort.
1-1982 Human Body Vibration Exposure and its Measurement.
4-1981 Low Frequency Calibration of Acoustical Measurement Systems.
Calibration and Standards. Vibration and Shock Measurements.
3-1981 Cepstrum Analysis.
2-1981 Acoustic Emission Source Location in Theory and in Practice.
1-1981 The Fundamentals of Industrial Balancing Machines and their
Applications.
4-1980 Selection and Use of Microphones for Engine and Aircraft Noise
Measurements.
3-1980 Power Based Measurements of Sound Insulation.
Acoustical Measurement of Auditory Tube Opening.
2-1980 Zoom-FFT.
1-1980 Luminance Contrast Measurement.
4-1979 Prepolarized Condenser Microphones for Measurement
Purposes.
Impulse Analysis using a Real-Time Digital Filter Analyzer.
3-1979 The Rationale of Dynamic Balancing by Vibration Measurements.
Interfacing Level Recorder Type 2306 to a Digital Computer.
2-1979 Acoustic Emission.
1-1979 The Discrete Fourier Transform and FFT Analyzers.
4-1978 Reverberation Process at Low Frequencies.
3-1978 The Enigma of Sound Power Measurements at Low Frequencies.
2-1978 The Application of the Narrow Band Spectrum Analyzer Type
2031 to the Analysis of Transient and Cyclic Phenomena.
Measurement of Effective Bandwidth of Filters.
1-1978 Digital Filters and FFT Technique in Real-time Analysis.
4-1977 General Accuracy of Sound Level Meter Measurements.
Low Impedance Microphone Calibrator and its Advantages.
(Continued on cover page 3)
TECHNICAL REVIEW
No. 2 — 1984
Contents
Dual Channel FFT Analysis (Part II)
H. Herlufsen...................................................................................... 3
News from the Factory…..................................................................... 46
DUAL CHANNEL FFT ANALYSIS (PART II)
by
H. Herlufsen, (M.Sc.)
ABSTRACT
In the first part of this article the basic dual channel FFT measurement was
introduced and Frequency Response Function estimates and excitation techniques were discussed.
In the second part of this article the time domain functions, Impulse Response
Function, Autocorrelation and Cross Correlation and their physical interpretation
is dealt with in some detail. The implementation of the Hilbert Transform on these
time domain functions, to compute the corresponding complex analytical signals,
is introduced, and the advantages of using the magnitude in the presentation of
these
functions
in
some
practical
situations
are
illustrated.
Calculation of sound intensity from a dual channel measurement of the sound
pressure signals from two closely spaced microphones is discussed in terms of
advantages and disadvantages.
Formulae for random errors on some of the functions derived from a dual channel
measurement on random data are also given.
SOMMAIRE
La première partie de cet article donnait les notions de base des mesures FFT en
bi-voie, et discutait des estimations des réponses en frequence et des techniques
d'excitation.
Cette deuxième partie traite plus en detail des fonctions du domaine temporel et
de réponse impulsionnelle, de I'auto-corrélation, de l'intercorrelation et de leur
interpretation physique. Elle introduit aussi l'application de la transformée de
Hilbert à ces fonctions temporelles pour calculer les signaux analytiques complexes correspondants, et illustre par quelques cas pratiques les avantages de
I'utilisation
de
I'amplitude
pour
la
representation
de
ces
fonctions.
Le calcul de I'intensite acoustique à partir de la mesure bi-voie de la pression
sonore de deux microphones rapprochés est traité par comparaison des pours et
des contres.
Des formules d'erreurs aléatoires sur quelques unes des mesures en bi-voie sur
des données aléatoires sont également données.
3
ZUSAMMENFASSUNG
Der erste Teil dieses Artikels war eine Einführung in die Zwei-Kanal-FFT-Meßtechnik. Es wurden Übertragungsfunktionsnäherungen und Anregungstechniken
diskutiert.
Der zweite Teil beschäftigt sich detailierter mit Zeitbereichsfunktionen, Impulsantwort, Autokorrelation und Kreuzkorrelation sowie ihrer physikalischen Interpretation. Die Anwendung der Hilbert-Transformation auf diese Zeitfunktionen zur
Berechnung des komplexen analytischen Signals wird vorgestellt und der Vorteil
der Darstellung des Betrags dieser Funktionen wird an einigen praktischen
Beispielen illustriert.
Die Vor- und Nachteile der Schallintensitätsmessung mit einem Zweikanalsystem
und zwei dicht nebeneinander liegenden Druckmikrofonen wird diskutiert.
Zusätzlich werden Formein fur die Berechnung der statistischen Fehler der aus
der Messung stochastischer Daten mit dem Zweikanal-System abgeleiteten Funktionen gegeben.
6. Impulse Response Function
The ideal system defined in Section 4 and shown in Fig.14 can be
described by its Frequency Response Function H{f) as discussed in
Section 4. The corresponding time domain description of the system is
given by the Impulse Response Function h(τ) which can be calculated
from the Frequency Response Function by an inverse Fourier Transform.
(6.1)
Physically the Impulse Response Function h(τ) is the response signal
from the system caused by an unit impulse input signal at time 0.
Mathematically the unit impulse signal is defined by the socalled Dirac
delta function δ(t) (see Ref. [3] and [4] for instance).
The Dirac delta function δ(t) is an impulse at t = 0, which is infinitely
short in time and infinitely high in amplitude, such that its time integral is
unity, i.e.:
The Fourier transform of δ(t) is unity at all frequencies i.e.
F { δ(t) } = 1,
-∞<f<∞
which means that all frequencies are excited to the same level if an input
signal of a (t) = δ(t) is applied to the system.
4
According to eq. (1.2) (Part I of this article) we therefore have B(f) =
H(f), since A(f) = 1 for a (t) = h (t). Thus
B(t) = F-1{ H(f) } = h(t) for a(t) = δ(t)
(6.2)
Let us now consider a general input signal a(t) to the system and find
the output signal b(t) from a time domain description. Any time signal
a(t) can be written as a sum of weighted and time shifted delta functions
i.e.:
Each of these weighted and time shifted delta functions a(τ) δ(t-τ) at
the input will at the output give a signal which is the Impulse Response
Function, weighted with a(τ) and shifted τ in time i.e.: a(τ) h(t - τ). The
output signal b(t) from the system, when excited with a(t) at the input, is
therefore the superposition of these weighted and time shifted Impulse
Response Functions, assuming linearity, giving:
(6.3)
b(t) is said to be the convolution of a(t) and h(t). Since a(t) * h(t) =
h(t) * a(t), eq.(6.3) is the same as eq. (1.1) (Part I of the article).
It can also be said that eq. (6.3) (or eq.(1.1)) is a direct consequence of
the relation: B(f) = H(f) · A(f) (eq.(1.2)) and the Convolution Theorem
for the Fourier Transform, which states that multiplication in one domain
(here the frequency domain) corresponds to convolution in the other
domain (the time domain).
From equation (6.3) we can see that h(τ) acts as a "memory function" of
the system. The output b(to) at time to not only depends upon the input
value at time to, a(to), but also depends upon the previous values in a(t)
weighted with the reversed Impulse Response Function h( to - t).
Let us first consider a single degree of freedom system shown in Fig.35,
which consists of a mass m supported by a spring, with a spring
constant k, and a viscous damper with damping coefficient c. The mass
is allowed to move in only one direction x. The input could be the force
acting on the mass f(t), while the output could be the displacement x(t).
The force caused by the spring is -kx(t) and the damping force is
-cx(t), where x(t) is the velocity (time derivative of x(t)).
5
Fig. 35. Mechanical single degree of freedom system
Having the force as the input and the displacement as the output, the
Frequency Response Function of the system is the socalled compliance.
It is for this simple system given by:
H (ω) =
X (ω)
1/m
=
F(ω)
-ω2 + jω c + k
m m
=
A
jω – (jωd-σ)
+
A*
(6.4)
jω-(-jωd - σ)
where ω = 2 π f and X (ω) and F(ω) are the Fourier Transforms of x(t) and
f(t) respectively.
A is a constant and mathematically called the residue for the resonance.
It is here imaginary and given by
A=
1
j2mωd
(6.5)
In the literature the residue is sometimes defined by R = j2A, which is
real valued, and (6.4) becomes
-
R
j2 [jω - (- jω- (jωd-σ)]
The constants ωd and σ are given by
k - ——
c
ωd= √ ——
m
4m2
and
σ =
H(ω) =
6
R
j2 [jω- (jωd-σ)]
2
c
2m
(6.6)
(6.7)
(6.8)
The corresponding Impulse Response Function h(t) is found by an
inverse Fourier Transform of (6.4) or (6.6) and is given by
h(t) = 2 | A | e-σ t sin (ωdt)
or
-σ t
h(t) = R e
sin (ωdt)
(6.9)
(6.10)
Fig.36 shows an example of log | H(f) | and phase Φ(f) of H(f) and the
corresponding h(t) for the single degree of freedom system.
ωd is called the damped natural frequency and is the frequency (in
radians per second) of the oscillations in h(t) (see Fig.36). σ is the decay
rate determining the exponential decay of h(t), shown as the dotted
envelope curve in Fig.36 (lower graph).
Notice that the higher the damping is in the system the faster is the
decay of the Impulse Response Function h(t) and the shorter will the
effective length of h(t) be.
The socalled undamped natural frequency ω0 is defined by
ω0 =
√ω
2
d
2
+σ =
√ km
(6.11)
If the system had no damping i.e. c = 0, the mass would be oscillating at
a frequency of coo after an initial impulse excitation.
The damping is also often given by the damping ratio ζ which is defined
by
ζ=
σ
ω0
(6.12)
For ζ = 1 the system is said to be critically damped since in that
situation ωd = 0 and there will be no oscillations in h(t).
The Impulse Response Function can thus be used for determination of
the damped natural frequency ωd, the decay rate σ and the constant A
(or R) for a single degree of freedom system. A practical example of
determination of decay rate σ will be given later.
In some acoustical and electroacoustical applications, where an input
can be defined and measured, the Impulse Response Function is used
7
Fig. 36. Logarithmic amplitude log H(f) and phase φ(f) of the Frequency Response Function and corresponding Impulse Response
Function of a single degree of freedom system
for determination and recognition of reflections, their time delays and
individual magnitudes. For non-dispersive media where no filtering oc8
curs in the frequency band of interest, reflections will show up as narrow
peaks in the Impulse Response Function at the points in time corresponding to the delay times. The Impulse Response Function can therefore also be used for identification of transmission paths
This
technique has some advantages compared to the use of Cross Correlation Function, which is discussed in Section 7. Since the Impulse Response Function describes the system independent of the signals
involved, the propagation paths can be recognized even though the input
signal is shaped in frequency by some filtering. Filtering of the signals
will cause the Cross Correlation Function to be smeared and can
therfore make the transmission paths identification difficult. This will be
illustrated in Section 7. In order to be able to calculate the Impulse
Response Function, however, there has to be some input signal at all the
frequencies of consideration, since the Frequency Response Function
will be undefined at those frequencies where no input signal exists.
Traditionally the Impulse Response Function hasn't been used very much
compared to the Frequency Response Function. h(t) is a real valued
function and is presented on a linear amplitude scale. The dynamic
range of a linear presentation is very limited and oscillations in h(t) due
to filtering of the signal in the system often complicates the interpretation of the results.
With the implementation of the Hilbert Transform in the time domain, all
the time domain functions including the Impulse Response Function can
be made analytical (complex), i.e. they will have a real and an imaginary
part. They can therefore be presented in terms of magnitude and phase
as well. The Hilbert Transform, which is defined as a convolution
integral, is used to compute the imaginary part jã(t) of the analytical
signal from the real valued time signal a(t). It is denoted by H and we
have
(6.13)
and the socalled analytical time signal a(t) is then defined as
(t) = a(t) + jã(t)
(6.14)
The theory behind the Hilbert Transform is defined and discussed in Ref.
[12] and [13]. It is found that the Hilbert Transform of a time function can
be computed much simpler in the frequency domain compared to a
direct calculation in the time domain. The spectrum of the time signal
a(t) is shifted -90° for positive frequencies and +90° for negative
9
frequencies. An inverse Fourier Transform of this modified spectrum will
result in the time signal a(f), which Is the Hllbert Transform of a(t). This
technique is used in the B&K Analyzers Type 2032/2034.
Having the complex (analytical) Impulse Response Function
(6.15)
the magnitude is defined as
(6.16)
while the phase is given by
(6.17)
The magnitude | h(t) | can be considered as being the envelope of h(t)
and this presentation is often much more useful than just the real valued
h(t). Firstly, the magnitude does not feature the oscillations often seen in
h(t), but gives a smooth curve where identification of reflections, for
instance, is much easier and where the peak amplitudes will be correct
and consistent. Secondly, the magnitude can be shown on a logarithmic
amplitude scale giving more dynamic range in the presentation of the
data compared to the conventional linear scale. Fig.37 illustrates this for
an acoustical system, where both the input and the output is measured
by use of microphones. The Impulse Response Function in this and all
the subsequent examples is computed as an inverse Fourier Transform
of the Frequency Response Function estimate H1(f) (see Section 4, Part
I of this article). In the upper graph the real valued h(t) is shown, while
the lower graph shows the magnitude of the complex h(t) with a
logarithmic amplitude scale. The magnitude of h(t) reveals more details
in the Impulse Response Function, and the correct levels and time delays
for the different reflections are easily found from the magnitude curve.
In this example the input spectrum is band limited but the Impulse
Response Function still detects the different reflections as narrow peaks
at the different time delays. The Cross Correlation Function for the same
signals does not resolve the reflections as clearly as the Impulse Response Function, due to the frequency band limitation. The Cross Correlation Function for the same signals will be shown in Section 7, (Fig.47).
10
Fig. 37. Impulse Response Function of an acoustical system. In the upper
graph the real valued h(t) is shown. In the lower graph the
magnitude of the complex Impulse Response Function h(t) is
plotted on a logarithmic amplitude scale
Computation of the magnitude of the Impulse Response Function was
implemented already several years ago by use of Time Delay Spectrometry (TDS). The magnitude of h(t) is then often referred to as the Energy
Time Curve (ETC). For many acoustical and electroacoustical applications the ETC created new interest and possibilities for the time (delay)
domain description of systems (Ref. [13]).
Let us return to the application of the Impulse Response Function for
determination of damping, given by the exponential decay in h(t) (6.9)
for a single degree of freedom system. When a measurement is performed over a frequency range where the system has several resonances, the Impulse Response Function will be complicated and more
difficult to interpret.
Assuming that the system is linear, it can be considered as a combination of single degree of freedom systems (Fig.35) and is called a multi
11
degree of freedom system. The Frequency Response Function is therefore a linear combination of single degree of freedom Frequency Response Functions of the form (6.4), shown in Fig.36. Each resonance is
described by the resonance frequency ωd, the decay rate σ and the
constant A (or R) called the residue.
The Impulse Response Function will be the corresponding linear combination of single degree of freedom Impulse Response Functions of the
form (6.9), shown in Fig.36 (lower graph), and can therefore be written as
or
where r is the resonance number (mode number).
Fig. 38. Frequency Response Function (driving point accelerance of a
mechanical system) revealing five resonances (upper graph).
Pseudo-random force excitation is used. In the lower graph the
first resonance is extracted by use of the weighting function
shown in Fig.39.
12
If the different resonances are well separated in frequency, and the
system is lightly damped, each resonance can be treated separately
after the measurement, by application of a weighting function on the
Frequency Response Function as exemplified in Fig.38. The Frequency
Response Function (H1(f)) is here the driving point accelerance for a
plate and is measured using shaker excitation. The shaker is attached to
the structure via a force transducer and a push rod as shown in Fig.23
(Section 5, Part I of the article). A pseudo-random excitation signal is
used in order to avoid leakage in the analysis (see Section 4.6). The
system is almost linear and the calculated Frequency Response Function
in Fig.38 (upper graph) is therefore samples at f = k · ∆ f = k · 1/T of the
true Frequency Response Function.
In the lower graph of Fig.38 the first resonance (first mode of vibration)
at 844 Hz (ωo ≅ ωd = 5303rad/s) is extracted by application of a
rectangular weighting function, the position and width of which being
defined by the user. Some tapering (half-Hanning) is applied in the
beginning and at the end of the rectangular weighting in order to avoid
sharp edges (truncation) in the edited Frequency Response Function.
The weighting function applied in Fig.38 is shown in Fig.39.
Fig. 39. The user defined weighting function used for extracting the first
resonance, as shown in Fig.38.
13
Fig. 40. Real valued Impulse Response Function h(t) (upper graph) and
magnitude of complex Impulse Response Function #(t), shown
on a logarithmic amplitude scale (lower graph), for the first
resonance
An inverse Fourier Transform of the edited Frequency Response Function (H1(f)) in the lower graph of Fig.38 results in the Impulse Response
Function of this single resonance (first mode of vibration) and is shown
in Fig.40 (upper graph). The magnitude of the corresponding complex
(analytical) Impulse Response Function, (6.15) and (6.16), is computed
and shown in the lower graph of Fig.40, with a logarithmic amplitude
scale. The exponential envelope curve e-σ t, determined by the decay
rate σ, can now be seen as a linear decay. The time constant τ of the
exponential decay is given by τ = 1/σ. The decay corresponding to the
-1
time constant τ is given by the factor e or in dB: -20 log(e) = -8,7 dB.
By use of the reference cursor the time constant τ can be found directly
and is for this resonance τ1 = 103,6ms, as seen from the ∆x readout in
the lower right corner of Fig.40. The decay rate for this resonance is
therefore
14
1
= 9,65 rad/s
τ1
σ1
ζ1 = ω = 0,0018.
01
σ1 =
and the damping ratio is
In a similar manner the other resonances can be extracted from the
same measurement and the decay rate of these can be found. In Fig.41,
the second resonance (second mode of vibration) and the fifth resonance (fifth mode of vibration) are extracted using the same kind of
weighting function as the one shown in Fig.39 shifted to the relevant
resonance.
The magnitude of the corresponding Impulse Response Functions on
logarithmic amplitude scales are shown below the respective edited
Frequency Response Functions. The time constants τ2 and τ5 are found
to be τ2= 92,0 ms and τ5 = 48,9 ms (see ∆x in the cursor read outs). This
corresponds to decay rates of
σ2 = 10,9 rad/s and σ5 = 20,4rad/s respectively.
The undamped natural frequency ω0r, on is approximately equal to the
damped natural frequency ωdr, since the damping is light and is 1204 Hz
and 2312 Hz (i.e. 7565 rad/s and 14527 rad/s) for the two resonances.
This gives damping ratios of ζ2 = 0,0014 and ζ5 = 0,0014 respectively.
The decay rate for a resonance can also be determined from the half
power (3dB) bandwidth ∆f =
∆ω
of the resonance peak in the Frequen2π
cy Response Function, assuming that the resonance is well separated
from other resonances. In this case σ is given by σ = π f or in terms of
π∆f
∆f.
damping ratio ζ = ω = 2f For lightly damped structures it is nec0
0
essary to use a frequency resolution which is high in order to find the
half power bandwidth of the resonances. Zoom analysis will therefore
often be required and several measurements must be performed in order
to cover the frequency range of interest. Using the Impulse Response
Function calculation in combination with the editing technique as shown,
only one baseband measurement is required in order to find the decay
rate (or damping ratio) for the different resonances. Notice however that
pseudo-random excitation has to be used in order to avoid leakage.
Leakage will have the effect of broadening out the resonance peak (see
Fig.10) and the estimated decay rate will therefore be wrong (too high),
at least when H1(f) is used for calculation of the Impulse Response
Function.
15
Fig. 41. a)
b)
16
Extraction of the second resonance and the corresponding
Impulse Response Function
Extraction of the fifth resonance with the corresponding
Impulse Response Function.
A zoom analysis, using random noise excitation and a resolution of
0,25 Hz (compared to a resolution of 4 Hz in the baseband measurement)
on each of the three resonances (first, second and fifth resonance) gave
the following 3dB bandwidths:
∆f1 = 3,10 Hz, ∆f2 = 3,15 Hz and ∆f5 = 6,25 Hz
The corresponding decay rates are:
σ1 = 9,8 rad/s, σ2 = 9,9rad/s and σ5 = 19,7rad/s,
corresponding quite well with the results from the Impulse Response
Functions. Since each zoom analysis is 16 times slower than the baseband analysis (16 times higher resolution) the total analysis time for the
three resonances is 3 x 16 = 48 times longer when using zoom analysis
compared to the baseband and Impulse Response Function method.
Fig. 42. Same Frequency Response Function as in Fig.38, here measured
by use of Impact Hammer excitation. The first resonance is
extracted by use of a weighting function (upper graph) and the
corresponding Impulse Response Function (magnitude of) is
shown with logarithmic amplitude scale in the lower graph
17
Impact hammer excitation can also be used for the test, and the
baseband analysis and Impulse Response Function method can be
applied. The extra artificial damping caused by the exponential weighting function, which might be applied on the response signal in Channel
B, is known and can be corrected for as explained in Section 5.6. Fig.42
shows the edited driving point accelerance for the same structure, point
and direction as used before (Fig.38). The analysis bandwidth of 4 Hz is
also the same. The first resonance is extracted by use of the same
weighting function as used in the lower graph of Fig.38 and shown in
Fig.39. The time constant of the Impulse Response Function is found to
be τ1 = 52,12ms. The exponential weighting which was applied on the
response signal had a length (time constant) of τw = 99,97ms. Thus the
true decay rate is
σ1 = τ1 1
1
τw = 19,18rad/s -10,00rad/s = 9,18rad/s,
which is the same, within 5% accuracy, as the decay rate found in the
test using pseudo-random excitation with a shaker.
7. Autocorrelation and Cross Correlation
The mathematical definitions of the Autocorrelation Function Raa(τ) of a
signal a(t) and the Cross Correlation Function Rab(τ) between two
signals a(t) and b(t) are given by
Comparing these definitions to the definition of covariance σab between
two variables a and b given by (3.2) it is seen, apart from the subtraction
of the mean values µa and µb in σab, that the Autocorrelation Raa(τ) is
the covariance of the time signal a(t) and a replica of this signal
displaced τ in time i.e. a(t+τ)). Likewise, the Cross Correlation Rab(τ) is
the covariance of a(t) and the time signal b(t) with a displacement of τ
in time i.e. b(t+τ).
The practical interpretation of Autocorrelation Function Raa(τ) is therefore to what degree the time signal a(t) is similar with a displaced
version of itself as a function of the displacement τ. Likewise the Cross
Correlation Function Rab(τ) gives a measure of the degree of similarity
18
between the time signal a(t) and a displaced version of the time signal
b(t) as a function of the time displacement τ.
In practice it is often more convenient to work with the normalised
Correlation Functions given by:
ρ aa (τ) =
Raa(τ )
Raa(0)
and
(7.3)
(7.4)
ρaa(τ) and ρab(τ) are called the Autocorrelation Coefficient Function and
the Cross Correlation Coefficient Function respectively. Raa(0) is the
total power in the signal a(t) given by
and Rbb(0) is the total power in the signal b(t) given by
It can be shown that | ρaa (τ) | ≤ 1 and that | ρab(τ) | ≤ 1 for all τ.
ρaa (τ) and ρab (τ) correspond to the Correlation Coefficient ρxy defined
by (3.1).
From (7.1) and (7.2) it is easily shown (using the convolution theorem for
the Fourier Transform) that the Autocorrelation Function Raa(τ) and the
Cross Correlation Function Rab(τ) are related to the Autospectrum
SAA(f) and the Cross Spectrum SAB(f) via the Fourier Transform as:
and
When Raa(τ) and Rab(τ) are calculated from a dual channel FFT measurement, (7.5) and (7.6) are used as seen in Fig.2 (Section 2). The
inverse Fourier Transform performed in practice is of course the inverse
DFT (or rather inverse FFT) as discussed in Section 2.1.
19
Since SAA(f) is real and even Raa(τ) is also real and even i.e.
Raa(τ) = Raa(-τ). Rab(τ) is also real, because SAB(f) is conjugate even,
but is in general not even, because SAB(f) in general have an imaginary
part different from zero.
The obvious applications of the Autocorrelation Function are:
a) Detection of echoes (reflections) in the signal.
If an echo (reflection) with a time delay of τo exists in the signal, the
Autocorrelation Function will peak at τ = τo (apart from at T = 0) and
the value of the Autocorrelation Coefficient Function at τo, ρaa(τo),'
will give a measure of the relative strength of the echo (reflection). It
is here assumed that the signal is a broadband signal, which will
become clear shortly.
b) Detection of periodic signals buried in random noise
The Autocorrelation Function of a periodic signal is also periodic,
since a periodic signal will correlate with itself at a delay of one
period T, two periods 2 T, etc. The random background noise signal
will have an Autocorrelation Function which diminishes to zero with
increasing delay and the periodic signal can therefore be detected
after a certain delay time in the Autocorrelation Function.
Fig.43 shows the Autocorrelation Coefficient Function ρaa for a) broadband random noise, b) sinewave and c) sinewave buried in broadband
random noise.
The Autocorrelation Coefficient Function for the broadband random
noise diminishes very quickly to zero (Fig.43.a)). This is a consequence
of the socalled uncertainty principle for functions related by the Fourier
Transform. If the bandwidth of the Autospectrum GAA(f) is ∆f and the
width of the Autocorrelation Function Raa(τ) is ∆t, the uncertainty
principle states that
∆f · ∆t ≥ 1
(7.7)
In the example, Fig.43 a), the bandwidth of the random noise is ∆f≈
1
= 0,96ms.
1040 Hz and it is seen that ρxy(τ) ≈ 0 for τ = ∆ t = 1,4 ms ≥
∆f
The Autocorrelation Coefficient Function for the sinewave is a cosine
with the same period length T as shown in Fig.43.b). The sinewave
correlates 100% with itself with delays of τ = 0, τ = 1 period length, τ = 2
period lengths etc. In Fig.43.b) the period length is 1,251 ms (see cursor
readout ∆X).
20
Fig. 43. Autocorrelation Coefficient Function for different types of signals
a) Broadband random noise b) Sine wave
21
Fig. 43. Autocorrelation Coefficient Function for different types of signals
c) Sine wave buried in broadband random noise
It is thus obvious that detection of echoes (reflections) can only be
performed using the Autocorrelation Function if the signal is broadband
and with enough bandwidth that the corresponding peaks in the Autocorrelation Function become sufficiently narrow and well separated.
It should be mentioned that for this application, use of the Cepstrum is
often preferred over use of the Autocorrelation Function, since the
Cepstrum is less sensitive to the shape of the Autospectrum. A description of Cepstrum techniques is found in Ref. [14].
In Fig.43.c) it is seen that the Autocorrelation Coefficient Function
detects the periodic signal (a sinewave) which is buried in the random
background noise.
If the periodic signal is a signal containing several frequency components it might be more suitable to use the Autospectrum rather than the
Autocorrelation Function since the frequencies of the different compo-
22
nents and their levels are directly detected in the Autospectrum. Zoom
analysis might however be required if their levels are very low compared
to the background noise level.
The most common applications of Cross Correlation Function are the
following:
a) Determination of time delays. If a signal is propagating from one
point A to another point B the Cross Correlation Function Rab(τ)
between the signals a{t) and b{t) at the two points will have a peak
at the time displacement corresponding to the time delay (propagation time) τ between the two points. It is assumed that the signals
involved are broadband and that the propagation is non-dispersive
which means that propagation velocity is independent of frequency.
If the velocity depends upon the frequency the propagation is said to
be dispersive and this will complicate the situation. There are two
possibilities of getting around this problem.
One possibility is to apply a weighting function on the Cross Spectrum (similar to the weighting functions applied on the Frequency
Response Function described in Section 6), before the inverse Fourier Transform, in order to find the Cross Correlation Function for only
a limited frequency range, where the propagation velocity can be
considered as nearly constant. The peak in the Cross Correlation
Function will of course be smeared out as a consequence of the
uncertainty principle (7.7).
Another possibility is to use the phase of the Cross Spectrum
directly, since this is related to the delay time. For a single delay of
τo(f) the phase of the Cross Spectrum is ΦAB(f) = 2 π f τo(f) and the
time delay can thus be found as a function of frequency τo(f).
b) Identification of transmission paths. If there are several transmission paths for the signal from point A to point B there will be several
peaks in the Cross Correlation Function, one peak for each transmission path at a delay time τn corresponding to the propagation time
τn for the nth path. The amplitude of each peak in the Cross
Correlation Coefficient Function will give a measure of the relative
strength of each transmission path. Again it is here assumed that the
signals are broadband and that the propagation is non-dispersive. If
the propagation is dispersive, application of weighting functions on
23
the Cross Spectrum before the inverse Fourier Transform as described under a) might still make identification of transmission paths
possible.
c) Detection of signals buried in extraneous noise. If a signal s(t),
which can be either deterministic or random, is buried in extraneous
noise in both a(t) and b(t) i.e. a(t) = s(t) + n(t) and b(t) = s(t) +
m(t), where n(t) and m(t) are assumed to be uncorrelated, the Cross
Correlation Function Rab(τ) will only contain information about the
correlated part s(t) in a(t) and b(t).
This is because the uncorrelated noise terms n(t) and m(t) at each
frequency will be averaged out in the Cross Spectrum SAB(f) which
therefore will only contain contribution from the correlated parts in
a(t) and b(t) as explained in Section 2.1. In the situation considered
here the Cross Correlation Function Rab(τ) will give the Autocorrelation Function RSS(τ) of the signal s(t) without any influence from n(t)
and m(t).
Fig.44 illustrates the applications a) and b). In this acoustical experiment
the input is measured with a microphone close to a sound source and
the output is measured with a microphone at a distance of approximately 1,1m.
The source generates broadband random noise and the input Autospectrum GAA(f) (upper graph of Fig.44) is seen to be fairly flat over the
frequency range considered (0-12,8 kHz). The Cross Correlation Coefficient Function ρab(τ). shown in the middle graph, reveals a number of
propagation paths (direct sound and reflections). The peaks corresponding to the different paths are narrow, because the signals are broadband
and it is quite easy to separate the paths and find their propagation
times.
Fig.45 shows similar results when the source is fed by band-limited
random noise. The Cross Correlation Coefficient Function ρab(τ) (the
graph in the middle) again reveals the different propagation paths, but
the peaks are now very broad due to the frequency band limitation of the
signals involved. The broadening of the peaks complicates the identification of the different reflections which illustrates one of the difficulties
often encountered in practice.
24
Fig. 44. The input Autospectrum of a broadband random noise signal
from a sound source (upper graph). Real valued ρab(τ) (middle
graph). Magnitude of the complex ρab(τ) (lower graph)
25
Fig. 45. The input Autospectrum of a band limited random noise signal
from the sound source (upper graph). Real valued ρab(τ) (middle
graph). Magnitude of the complex ρab(τ') (lower graph)
26
In other words, the signals have to be so broadband that the peaks in
the Cross Correlation Function become narrow and separated, exactly
as discussed for application of Autocorrelation Function for detection of
echoes.
For the same reasons, as those mentioned for the analytical Impulse
Response Function h(t) in Section 6, it is an advantage in many situations to work with the analytical Autocorrelation Coefficient Function
ρaa(τ) and the analytical Cross Correlation Coefficient Function ρab(τ)
rather than with the real valued ρaa(τ) and ρab(τ)- These are defined by:
and
ρaa(τ) = ρaa(τ) + jρaa(τ)
(7.8)
ρab(τ) = ρab(τ) + jρab(τ)
(7.9)
where -ρaa(τ) and -ρab(τ) are the Hilbert Transforms of ρaa(τ) and
ρab(τ) respectively. The lower graphs in Fig.44 and 45 show the magnitude of the analytical Cross Correlation Coefficient Functions corresponding to the real valued Cross Correlation Coefficient Functions
Fig. 46. Same results as for Fig.44, but obtained with zoom analysis
between 512 and 13312 Hz
27
shown in the graphs in the middle. The peaks are easier to identify since
the oscillations in ρab(τ) are avoided. This is specially an advantage in
the situation with band limitation as in Fig.45. If a zoom analysis is
performed, the magnitude | ρab(τ) | should also be used for measurement of the peak amplitudes for the different reflections, since the real
part ρab(τ) might not peak where | ρab(τ) | peaks. Fig.46 illustrated this
where the exact same signals as in Fig.44 are analysed, however using a
zoom analysis from 512Hz to 13312hz. This frequency range covers all
the important information as does the baseband frequency range used in
Fig.44. In the real part ρab(τ), an amplitude for the peak at 3,3ms is
found to be 0,36, while the magnitude | ρab(τ) | gives an amplitude of
0,51 which is the same as the amplitude found in Fig.44.
In Fig.47 the magnitude of the Cross Spectrum GAB(f) and the magnitude of the Cross Correlation Coefficient Function are shown for the
same signals, as those analyzed by use of the Impulse Response
Function in Fig.37 (Section 6). The main reflections can still be seen in
the Cross Correlation Coefficient Function, but their peaks are so
Fig. 47. Magnitude of Cross Spectrum and
Magnitude of Cross Correlation Coefficient Function of the
same signals as those analyzed by Impulse Response Function in Fig.37.
28
smeared out, due to the band limitation seen in the Cross Spectrum, that
some of the smaller reflections are not detected. Fig.37 shows that the
Impulse Response Function is insensitive to this band limitation of the
input signal, since it describes the system independent of the signals
involved.
Note that there has to be some power in the input signal at all the
frequencies in the analysis bandwidth, before a well-defined Frequency
Response Function (and thereby a well-defined Impulse Response Function) can be computed.
Since we are working with the DFT (or rather FFT) which computes the
spectra at discrete frequencies, the effect of circular correlation has to
be taken into account. As a consequence of computing the spectra at
discrete frequencies k ∆ f = k1/T, the time signals are artificial period
signals ranging from -∞ to +∞ with the time record in the analyzer of
length T being the period length. The computed Correlation Functions
are the Correlation Functions of these artificial periodic time signals
displaced relative to each other, and a circular correlation effect, as
shown in Fig.48, will appear. The end of one signal (end of the period in
one signal) will overlap with the beginning of the other signal (the
beginning of the period in the other signal) and the estimated Correlation
Function will be incorrect. In Fig.48 (lower graph) the Correlation Func-
Fig. 48. Circular correlation effect. The Correlation Function is assumed
to be even in the lower graph
29
Fig. 49. Zero pad of the time records in order to avoid the effect of the
circular correlation shown in Fig.48. The bow tie correlation is
shown in the lower graph
tion between the two signals is assumed to be even, which is not the
case in general for Cross Corrleation Functions. This problem of circular
correlation is overcome by setting the signal equal to zero in the second
half of the record as illustrated in Fig.49. This is called "zero pad". The
zero pad will, however, also cause the estimated Correlation Function to
be biased, but the bias error is known and can be corrected for. Due to
the artificial nullifying of the signals in the last half of the record the
estimated Correlation Function ^R(τ) will be lower than the true Correlation Function R(τ), the extent depends upon the displacement τ. The
relation is
^ Τ) = |T/2 – τ | R(τ) ; 0 ≤ τ ≤ T/
R(
2
T /2
assuming rectangular weighting in the analysis.
30
(7.10)
|T/2 - τ |
Correction for the factor -—-——
is called the "bow tie" correction (see
T
/2
Fig.49) and can be applied as a post-processing after the measurement.
In Fig.49 the Correlation Function R(τ) is assumed to be even (as in
Fig.48), which only in general is true for Autocorrelation Functions.
In all the measurements shown in this Section, zero pad and rectangular
weighting has been used.
T
If the signals involved are transients with a duration shorter than /2, the
bow tie correction should obviously not be used.
8. Sound Intensity
→
The sound intensity is a vector quantity I describing the direction and the
amount of the net flow of acoustic energy at a given position in a sound
field.
A dual channel FFT measurement of the sound pressure signals from
two closely spaced microphones can be used for calculation of the
sound intensity.→The calculated intensity is the component Ir of the
intensity vector I in the direction given by the line joining the acoustic
centres of the two microphones.
Fig.50 shows a microphone probe where the two microphones are
placed in a face to face configuration (the B&K Sound Intensity Probe
Type 3519). This sound intensity probe features very little disturbance of
the sound field. The shadowing effect of one microphone on the other is
minimal and the effective separation between the microphones is well
Fig. 50. Two microphone sound intensity probe (B&K Type 3519) with the
microphones in face to face
→ configuration. The component Ir of
the sound intensity vector I, is measured positive in the direction
indicated by the arrow (from microphone A to microphone B)
31
defined. A side by side configuration of the microphone is also used
sometimes, for instance in situations where the space is very confined or
when measurements has to be performed as close as possible to a
surface.
It can be shown that the intensity component Ir, with the direction from
microphone A to microphone B defined as the positive direction, is given
by
1
Ir = Im {GAB(f)}
(8-1)
ωρ∆r
where Im {GAB(f)} is the imaginary part of the one-sided Cross Spectrum between the two sound pressure signals pA(t) and pB(t), ρ is the
density of air and ∆r is the spacing between the microphones. The air
density is given by the ambient pressure p (in mbar) and the air
temperature t (in °C) via the formula
ρ = 0,3485
p
[
273 + t
kg
]
m3
In the B&K Analyzers Types 2032 and 2034, the microphone spacing ∆r,
the ambient pressure p and temperature t are entered from the front
panel and the sound intensity is calculated according to (8.1) from the
basic dual channel measurement. The sound pressure Autospectra
GAA(f) and GBB(f) are therefore available as well from the same measurement. The reactivity of the sound field defined as LK = LI - LP, where
LI is the intensity level (in dB relative to 1 pW/m2) and LP is the sound
pressure level (in dB relative to 20 µ Pa), can be checked from the same
measurement. The reactivity LK should always be checked, since it is
one of the factors which determines the statistical errors in the intensity
estimates and should be within the dynamic range of the analysis. For a
plane wave propagating in free field the reactivity is zero dB (LP = LI)
and the sound field is called an active sound field. In an ideal diffuse
field the intensity is zero and the sound field is purely reactive.
The sound intensity can also be calculated directly in real time using the
formula
Ir = -
1
2ρ∆r
∫
(pA + pB) (pB – pA) dt
(8.2)
where the bar indicates time averaging and PA and pe are the sound
pressure time signals as before. This approach is implemented using 1/3
octave and 1/1 octave (and 1/12 octave) digital filters in the B&K Intensity
Analyzer Type 3360.
32
Compared to this technique one of the advantages of using the two
channel FFT approach to sound intensity is that, since the spectrum is
calculated in narrow bands with constant bandwidth, this method is very
suitable for detection and recognition of pure tones, in a sound field.
There might be several closely spaced pure tones, coming from different
directions, which can only be separated using narrow band analysis,
assuming that these frequency components are sufficiently stable. Harmonic related frequency components in the sound field can also easily
be identified using constant narrow bandwidth and a linear frequency
scale. Fig.51 shows an example where the B&K Analyzer Type 2032 is
used. The sound intensity is shown on a socalled bipolar display, where
positive intensity components point upwards in the upper half of the
display and negative intensity components point downwards in the lower
half of the display. One sound source is seen to radiate harmonic related
frequency components in a direction towards the front of the probe and
gives a resulting positive intensity level at these frequencies. Another
source, which radiates broadband random noise in the opposite direction at the probe position, is causing the net flow of acoustical energy, in
the direction of the probe, to be negative at the other frequencies.
One of the disadvantages of using the two channel FFT approach as
opposed to the digital filter method is that the analysis can only be
performed in real time in a limited frequency span. Since Hanning
weighting is used when analyzing normal continuous signals, there has
Fig. 51. Example of a narrow band sound intensity spectrum, calculated
from a two channel measurement (B&K Analyzer Type 2032 or
2034)
33
to be 75% overlap of the time records in order to get true real time
analysis, i.e. a resulting (overall) weighting function on the time signals
which is flat (see Ref.16). This means for instance that for the high speed
B & K Analyzer Type 2032 a true real time analysis can only be performed
in a frequency span which is not wider than ~1/4 x 5 kHz. The digital
filter Analyzer Type 3360 will always work in real time up to 10kHz in the
1
/3 octave and 1/1 octave mode.
Another disadvantage of the two channel FFT approach is that longer
averaging time is needed in order to get the same statistical accuracy in
the measurement as by use of the digital filter analyzer, except for very
low frequencies. The broader bandwidths of the 1/3 octave or 1/1 octave
filters give an averaging in frequency which will decrease the random
error (see Section 9). Specially if the sound field is very reactive (high
sound pressure level compared to the intensity level) as in diffuse sound
fields, the sound intensity measurement using two channel FFT can be
very time consuming in order to obtain a certain statistical accuracy of
the estimated sound intensity spectrum.
Consider an example where a number of averages nd = 500 is required
in order to get a specified statistical accuracy.
a) FFT Method:
Let us assume that we are working with a frequency span which is so
wide that the analysis time for computing one sound intensity spectrum, is longer than the record length in the analyzer. For the high
speed B&K Analyzer Type 2032, this means that the frequency span
is wider than 5kHz. One sound intensity spectrum is computed
approximately every 160msec giving a total analysis time of Ttotal ≈
500 x 160 msec = 1 min 20 sec.
b) Digital Filter Method:
nd = 500 is equivalent to a BT product (Bandwidth x Averaging Time)
1
of 500. For the /3 octave at 1 kHz, for instance, B ≈ 230 Hz and the
total analysis time is therefore TTotal = 500
≈ 2,2 sec. For the 1/3 ocB
tave filter at 10kHz, TTotal ≈ 0,22sec and for the
100 Hz, TTotal ≈ 22 sec.
1
/3 octave filter at
For many applications it is also much easier to handle and interpret the
reduced amount of data from a 1/3 octave or a 1/1 octave analysis. Since
acoustic measurements are often specified in 1/3 octave or 1/1 octaves,
an analysis using the system with the standardized filters would be
preferred.
34
Finally it should be mentioned that the digital filter system Type 3360 is a
dedicated intensity measurement system giving some practical advantages for the user, such as a remote control unit and a separate display
unit.
The basic limitations in the two microphone approach to analysis of
sound intensity are the same for both systems. The finite distance Ar
between the two microphones gives an upper bound for the useful
frequency range in the analysis, and the phase mismatch between the
two channels in the measurement system, including the microphones,
gives a lower limit on the frequency range. A discussion of this is found
in Ref.15 and 22.
The principal applications of sound intensity are: a) sound power determination, b) ranking of noise sources, c) noise source location (for this
purpose it is often useful to make an intensity mapping in terms of
contour plots or 3-D plots, made by use of further post-processing and
plotting routines), d) transmission loss determination, e) sound absorption measurements.
Examples of these applications are dealt with in the References [15, 20,
21, 27-30] and will not be discussed further in this article.
9. Statistical errors in the estimates
When the analysis is performed on random signals (time histories) the
different functions derived from a dual channel FFT measurement are
estimated with limited accuracy. It is here assumed that we have stationary, continuous or transient random signals, which are band-limited
Qaussian distributed white noise signals over the analysis bandwidth.
Averaging of the Autospectra and the Cross Spectrum is performed over
only a finite number of records nd giving the estimates SAA(f), SBB(f)
and SAB(f) of SAA(f), SBB(f) and SAB(f) by:
(9.1)
(9.2)
35
(9.3)
From these two-sided spectral density estimates, the one-sided spectral
density estimates GAA(f), GBB(f) and GAB(f) are calculated according to
(2.8) and (2.9) (Part I of the article, Section 2).
The errors can be divided into two types, bias errors b[X] and random
errors σ[X], where X is the estimate of the true value X. A bias
error b[X] is a systematic error introduced either in the measurement
or in the analysis (post-processing). In some situations a bias error can
be estimated either by performing certain measurements (of phase and
gain differences in the two instrumentation channels for instance) or
from theoretical calculations. The bias error can in such situations be
corrected for, but this is not the case in general.
The random error σ[X] is the standard deviation of the estimates X and
is due to the fact that averaging is not performed over an infinite number
of records i.e. over infinitely long time, as shown in (2.4), (2.5) and (2.7).
Averaging over the finite number of records nd causes a random scatter
in the estimates.
The bias error b[X] is defined by:
B [x] = E [x] - X
(9.4)
while the random error is defined as the standard deviation of X and
therefore given by (see (3.3) in Section 3):
σ[X] =
√E[(X
– E [X])2]
(9.5)
E[ ] means "expected value of" and X is the true value.
Often it is more practical to work with the normalized errors instead of
the absolute errors b[X] and σ[X], The normalized bias error ∈b, and
the normalized random (standard) error ∈r gives the error as a fractional
portion of the true value and are defined by:
∈b =
b [X]
X
∈r = σ[X]
X
It is assumed that X ≠ 0.
36
(9.6)
Fig. 52. Illustration of bias error and random error
Fig.52 gives an illustration of how a bias error and a random error
influences the estimates of a value.
Bias errors in the different functions can be due to several effects such
as
a) Extraneous noise at input and/or at output.
b) Other inputs to system which are correlated with measured input.
c) Leakage (resolution bias).
d) Non-linearities.
e) Propagation time delays not compensated for.
The effects of a), b) and c) were dealt with for the Frequency Response
Function estimates in Section 4 and the influence of non-linearities were
mentioned in Sections 3 and 5, and shall therefore not be discussed
further here. If there is a propagation time delay τ from input to output
of the system, this time delay has to be set between the records in the
dual channel FFT measurement (see Fig.13 Section 3). Otherwise the
Cross Spectrum will be biased with a factor of (1 - τ/T), where T is the
record length in the analysis. Thus
GAB(f) ≈ (1 – τ/T) GAB(f) , 0 ≤ τ ≤ T
(9.8)
The functions derived from the Cross Spectrum will therefore also have
a bias error.for example, the Coherence Function estimate is given by
γ2 (f) ≈ (1 - τ/T)2 γ2(f) , 0 ≤ τ ≤ T
(9.9)
The random errors will now be discussed for the Autospectral and Cross
Spectral density estimates and for the estimates of Coherence Function
and Frequency Response Functions. These formulae were developed by
Bendat and Piersol (Ref. [23] and [24]). The random errors associated
with the sound intensity measurements will also be given.
37
The distribution of the data is assumed to be Gaussian (normal). This is
a fair assumption when the random error ∈r is small. The filtering (FFT
analysis) also tends to make the distribution of the time data closer to
the Gaussian distribution.
For Gaussian distributed estimates X (without any bias error ∈b) it can
be shown that there is approximately 68% chance of the true value X to
be within the interval,
X (1 - ∈ r) ≤ X ≤ X (1 + ∈ r)
(9.10)
and approximately 95% chance of the true value to be within the interval
X (1 – 2 ∈r) ≤ X ≤ X (1 + 2 ∈r)
(9.11)
9.1. Auto Spectral and Cross Spectral density estimates
If averaging is performed over nd time records of the random signals
a{t) and b(t), the normalized random (standard) error for a frequency
component of the Autospectral density estimates GAA(f) and GBB(f) at
any frequency is given by
∈r [GAA(f)] = ∈r [GBB(f)] = 1
√nd
(9.12)
Notice that the random error is independent of frequency, nd corresponds to the socalled BT-product (product of analysis bandwidth B and
total effective time record length TTotal) for the analysis. If rectangular
weighting is used in the analysis we have B = ∆f = 1/T and TTotal = nd T,
giving B · TTotal = 1 nd T = nd. T is the record length for each analysis.
T
If a smooth weighting function such as the Hanning weighting is used,
the bandwidth B will be wider than ∆f. The effective time length Teff of
each record will, however, be correspondingly shorter than T, due to the
time weighting function. For each independent record, B · Teff will be
unity and therefore B · TTotal = B · Teff ·nd = nd.
Since the effective time length is shorter than T when Hanning weighting
is used, averaging of spectra estimates from time records having some
overlap (50% or 75% overlap for instance) will allow for a higher BTproduct i.e. a higher number nd, compared to an analysis performed with
rectangular weighting for a given amount of time data, see Ref. [16].
38
The magnitude | GAB(f) | of the Cross Spectral density GAB(f) will at any
frequency be estimated with a normalized random error of
(9.13)
where γ2(f) is the Coherence Function.
The lower the Coherence is in the measurement the more averaging is
required in order to obtain a certain statistical accuracy in the estimate.
This is intuitively clear due to the fact that the lower the Coherence is
the more random variation will there be in the individual estimates
A*i (f) · Bi(f) (see Fig.4 and 9, Section 2 and 3).
9.2. Coherence Function estimates
The Coherence Function γ2(f) will also be estimated with only a limited
accuracy from the Autospectral and Cross Spectral density estimates.
(9.14)
Fig. 53. Normalized random error ∈r[γ2(f)] of Coherence Function estimates as a function of number of averages nd, for different
values of (true) Coherence γ2(f)
39
The normalized random (standard) error is given by
(9.15)
If the number of averages nd = 1, we have that
Some averaging will therefore always have to be performed in order to
estimate the Coherence Function. Even if the Coherence is unity, averaging is required to verify this in a measurement.
Fig.53 shows the relationship between the normalized random error
∈r [γ2(f)] and the number of averages nd for different values of (true)
Coherence γ2(f).
9.3. Frequency Response Function Estimates
When the Frequency Response Function of an ideal system is estimated
from H1(f) or H2(f) and the signals are random noise signals (bandwidth-limited, Qaussian distributed white noise signals) there is a random error in both magnitude and phase. The error depends upon the
number of (independent) averages nd and the Coherence γ2(f) in the
measurement.
The normalized random error for the magnitude of the estimate
H(f)(H1(f) or H2(f)) is given by
(9.16)
In Fig.54 the normalized random error ∈r [H(f)] is plotted as a function of number of averages nd for different values of Coherence γ2{f}.
The more uncorrelated noise there is in the measurement i.e. the lower
the Coherence is, the more averages have to be performed in order to
get a certain statistical accuracy.
Fig.55 shows the estimates of H1(f) and γ2(f) from a dual channel
measurement on an electrical filter network with random noise excita-
40
tion. 100 averages are performed. The estimated Coherence γ2(f) at
176 Hz is 0,6, due to uncorrelated noise at the output. The random error
of γ2(f) is
The random error for |H1(f)| at the same frequency is
which in dB is ≈ 0,5dB.
Thus there is 68% chance of the true value |H1(f)| at 176 Hz, to be
within the interval (since we have no bias error, see Section 4.2)
0,94 · |H1(f)| ≤ |H1(f)| ≤ 1,06 |H1(f)|
Fig. 54 Normalized random error ∈r [ |H1(f)| ] of magnitude of Frequency
Response Function estimates (H1(f) or H2(f) as a function of
number of averages nd for different values of Coherence γ2(f)
41
Fig. 55. Estimates of |H1(f)| and γ2 from a dual channel measurement
on an electrical filter network, with random noise excitation and
100 averages
or in dB
-24,8 dB ≤ |H1(f)| (dB) ≤ -23,8 dB since |H1(f)| = -24,3 dB.
The random errors at the same frequency for the Autospectra GAA(f)
and GBB(f) and the magnitude of the Cross Spectrum GAB(f), which
are used for the calculations, are
and
Notice that these random errors are greater than the errors on the
derived functions γ2(f) and H1(f).
The random error for the phase angle φ(f) of H(f) can be approximated
by
42
σ [φ (f)] ≈ ∈r [H(f)]
(9.17)
when the error ∈r [H(f)] is small. Φ(f) is measured in radians. The
true value H(f) is here assumed, with 68% confidence, to be inside a
circle of radius σ [H(f)] and centered at H(f). Thus
σ [Φ(f)] ≈ sin (σ [Φ(f)]) ≈
σ [H(f)]
H(f)
giving equation (9.17), for small σ [Φ(f)] or ∈r [H(f) ].
9.4 Sound Intensity
The sound intensity Ir (component in the direction of the probe) in a
random sound field is as the other functions, estimated with a limited
statistical accuracy. The bias error basically depends on the spacing
between the microphones (high frequencies) and on the residual phase
mismatch between the two channels in the measurement system (low
frequencies). This is described in the references.
The random error is determined by the Coherence Function γ2(f) between the two sound pressure signals pA(t) and pB(t), the phase angle
φAB of the Cross Spectrum GAB(f) between the two signals and the
number of averages nd. In Ref. [25] a formula for the normalized random
error ∈r [ Îr ] is derived
(9.18)
Calculations of Coherence γ2(f) and phase angle φAB(f) in different
situations are givep in Ref. [26]. These calculations indicate that the
random error ∈r [Îr] mainly depends upon the reactivity LK of the sound
field, defined as the difference between sound intensity level LI and
sound pressure level Lp, and also depends upon the interfering sound
field in the different situations (diffuse background noise or an interfering point source either perpendicular to or behind the probe, which
points towards the source of interest). The random error, however, was
found to be practically independent of the frequency and distance
between the microphones.
It is therefore very important to check and report the reactivity of the
sound field where the intensity measurements are performed, in order to
be able to judge the validity of the results.
43
References
[12] THRANE, N.
"The Hilbert Transform", B & K Technical
Review, No.3 - 1984.
[13] BIERING H. &
PEDERSEN, O.Z.
"System Analysis and Time Delay Spec
trometry", B&K Technical Review, No.1
and 2, 1983
[14] RANDALL, R. B. &
HEE, J.
"Cepstrum Analysis", B & K Technical
Review, No.3, 1981.
[15] GADE,S.
"Sound Intensity", B&K Technical Review, No.3 and 4, 1982.
[16] THRANE, N.
"Zoom FFT", B&K Technical Review,
No.2, 1980.
[17] FAHY, F.J.
"Measurement of Acoustic Intensity Using the Cross Spectral Density of two
microphone signals.", Journal of the
Acoustical Society of America 62(4)
1057-1059, 1977.
[18] CHUNG, J.Y.
"Fundamental Aspects of the Cross
Spectrum method of measuring Acoustic
Intensity". Senlis, 1981, pp.1-10.
[19] CHUNQ, J.Y. &
POPE, J.
"Practical measurement of Acoustic Intensity - the two microphone cross spectral method" Proc. Internoise 1978,
pp.893-900.
[20] GADE, S.,et al:
"Sound Power determination using
Sound Intensity measurements" (Part I
and II) B&K Appl. Notes.
[21] CROCKER, M.J.et al.
"Application of Acoustic Intensity Measurements for the evaluation of Transmission loss of structures" Senlis 1981,
pp.161-169.
44
[22] QINN, K.B.et.al.
"Sound Power determination in highly reactive environments using sound intensity measurements", B&K Application
Notes.
[23] BENDAT, J.S. &
PIERSOL, A. G.
"Random Data: Analysis and Measurement Procedures", Wiley-lnterscience,
1971.
[24] BENDAT, J.S. &
PIERSOL, A. G.
"Engineering Applications of Correlation
and Spectral Analysis", John Wiley &
Sons, 1980.
[25] SEYBERT, A. F.
"Statistical Errors in Acoustic Intensity
Measurements", Journal of Sound and
Vibration, 1981, 75, pp.519-526.
[26] DYRLUND, O.
"A note on Statistical Errors in Acoustic
Intensity Measurements.", Letters to the
Editor, Journal of Sound and Vibration,
1983. 90, pp.585-589.
[27] NICOLAS, J.,
BENQIT, R.,
SAUVE, S.
"Noise source identification and reduction in a chocolate plant", Proceedings
of Noise-Con. '83.
[28] YANO, H.,
TACHIBANA, H.,
ISHII, K.
"Application of sound intensity measuring technique to sound insulation measurements", ICA 1983.
[29] RASMUSSEN, G.,
RASMUSSEN,P.
"Identification of energy sources and absorbers", Proceedings of the 2nd International Modal Analysis Conference,
1984.
[30] COLEMAN, R.B.,
HODGSON.T.H.
"Acoustic Intensity Measurements on a
350 kN Automatic Punch Press", Proceedings of lnter-Noise'83.
45
News from the Factory
Dual Channel Signal Analyzers Type 2032/2034
The Analyzers Type 2032 and Type 2034 combine the use of realtime,
digital filtering techniques with a 2 k (2048 points) Fourier Analysis in two
channels. Thus 801-line spectra are calculated with a chosen resolution
from 1,95 mHz to 32 Hz selected in binary steps anywhere in the frequency range from DC to 25,6kHz.
Both analyzers are extremely fast. The standard version, Type 2034, has
a real-time bandwidth of 1,6 kHz in single channel mode and 800 Hz in
dual channel mode. The high speed version, Type 2032, is approximately
six times faster.
34 different functions are available such as Frequency Response, Impulse Response, Autospectra, Cross Spectrum, Correlation Functions,
Enhanced Signals and their spectra, Distribution Functions, Coherence,
Coherent Power, Sound Intensity, Cepstra, etc., etc. Most of these
functions, including time domain functions, can be shown in 6 different
formats: Real Part, Imaginary Part, Magnitude, Phase, Nyquist Plot and
Nichols Plot. Two micro-processors are used in carrying out all the
calculations: One for signal processing and the other for displaying
(postprocessing) the data. Thus any function can be displayed at any
time in any format and scaling, not only after a measurement has been
completed, but also during a measurement.
46
The primary use of a dual channel analyzer is for system analysis, i.e.
performing Frequency Response (and Impulse Response) measurements. Two different ways of calculating the Frequency Response Function are implemented in the 2032/2034: The traditional way H1 =
GAB/GAA and the new way H2 = GBB/GBA. H1 suppresses uncorrelated
noise at the output and gives the best estimate at anti-resonances. H2
suppresses uncorrelated noise at the input and gives the best estimate
at the resonance peaks.
The analyzers are equipped with extremely flexible trigger facilities.
A Free Running, Internal, External or manual trigger source can be
selected in addition to a Generator trigger, where data collection is
synchronized to the sequence of the built-in, zooming pseudo-random or
impulse generator. Zooming random noise and a sine signal with a
selectable frequency are also available.
Each channel can be independently set to have a Flat, Manning, Transient, Exponential, Flat Top, Kaiser Bessel or User Defined weighting
function. These weighting functions can even be applied to the Frequency Response Function, Cross Spectrum and Autospectra before calculating the Impulse Response, Cross Correlation and Autocorrelation
Functions.
The analyzers have seven cursor functions. Namely, Main Cursor, Harmonic Cursor, Sideband Cursor, Delta Cursor, Mask Cursor, Reference
Cursor and even a Flex Cursor by which the x-axis can be scaled in any
arbitrarily chosen engineering unit.
In spite of the high level of complexity and sophistication, the analyzers
are very easy to use, because operation is largely self-explanatory with
all relevant control settings clearly shown on its display screen, and
because complete measurement and display setups can be stored for
later recall and use.
The analyzers have a fully instrumented front end, by which they can
accept signals from most B&K microphone preamplifiers, and via the
Line Drive Amplifier Type 2644 from most accelerometers, force transducers and the Impact Hammer Type 8202 in either of its channels. In
addition, the 2032/2034 are equipped with DC-coupled direct inputs for
the input of electrical signals.
47
Output of the data which is displayed on a 12-inch large raster screen
can be made to an X-Y recorder, a video recorder or to the B&K
Graphics Recorder Type 2313, where a complete copy of the screen
including graphs and alphanumerics are made in less than 10 seconds.
The 2032/2034 are also equipped with an extremely flexible and sophisticated IEC 625-1/IEEE Std. 488-1978/ANSI MC 1.1-1975 interface which
allows simple connection to most desk-top calculators.
Graphics Recorder Type 2313
The Brüel & Kjær Graphics Recorder Type 2313 offers fast, fully annotated, graphic plots of measured frequency spectra and time functions as
presented on the display screen of B&K Digital Frequency Analyzers, as
well as documents measurement data transmitted by other equipment
furnished with an IEC/IEEE interface.
For maximum operating flexibility, a range of interchangeable Application Packages is available. These plug-in the front of the 2313 and
especially format data to suit different application requirements of B&K
analyzing equipment. In addition, they permit a variety of special tasks
to be carried for which extra equipment would normally be required. For
example, several hundred complete measurement spectra can be stored
in the Recorder for future print out or transmission to other equipment.
Also, new types of measurement can be made using the 2313 to reformat
stored and incoming data, plus use as a system controller is possible.
Where use as a conventional alphanumeric printer/graphics recorder is
concerned, the 2313 may be operated without an Application Package.
High-resolution graphic plots are made using a 512-point print head and
as many as 128 (ISO 646 and 2022) characters may be printed. It accepts
50m rolls of electrosensitive paper (metallized type) and from just one
roll the equivalent of 160 A 4 charts can be obtained.
48
PREVIOUSLY ISSUED NUMBERS OF
BRÜEL & KJÆR TECHNICAL REVIEW
(Continued from cover page 2)
3-1977 Condenser Microphones used as Sound Sources.
2-1977 Automated Measurements of Reverberation Time using the Digital Frequency Analyzer Type 2131.
Measurement of Elastic Modulus and Loss Factor of PVC at High
Frequencies.
1-1977 Digital Filters in Acoustic Analysis Systems.
An Objective Comparison of Analog and Digital Methods of Real
Time Frequency Analysis.
4-1976 An Easy and Accurate Method of Sound Power Measurements.
Measurement of Sound Absorption of rooms using a Reference
Sound Source.
3-1976 Registration of Voice Quality.
Acoustic Response Measurements and Standards for MotionPicture Theatres.
2-1976 Free-Field Response of Sound Level Meters.
High Frequency Testing of Gramophone Cartridges using an
Accelerometer.
1-1976 Do We Measure Damaging Noise Correctly?
4-1975 On the Measurement of Frequency Response Functions.
3-1975 On the Averaging Time of RMS Measurements (continuation).
SPECIAL TECHNICAL LITERATURE
As shown on the back cover page, Brüel & Kjær publish a variety of
technical literature which can be obtained from your local B & K
representative.
The following literature is presently available:
Mechanical Vibration and Shock Measurements
(English), 2nd edition
Acoustic Noise Measurements (English), 3rd edition
Architectural Acoustics (English)
Strain Measurements (English, German)
Frequency Analysis (English)
Electroacoustic Measurements (English, German, French, Spanish)
Catalogs (several languages)
Product
Data
Sheets
(English,
German,
French,
Russian)
Furthermore, back copies of the Technical Review can be supplied as
shown in the list above. Older issues may be obtained provided they are
still in stock.
Printed in Denmark by Nærum Offset
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement