IEEE T R A N S A C T I O N S O N I N S T R U M E N T A T I O N A N D M E A S U R E M E N T , VOL. 37. N O . 2 , J U N E 1988
213
The Geometric Mean of Power (Amplitude) Spectra
has a Much Smaller Bias Than the Classical
Arithmetic (RMS) Averaging
RIK PINTELON. J . SCHOUKENS,
AND
J. RENNEBOOG
"v
Abstract-The statistical properties of the geometric mean of power
(amplitude) spectra resulting from a discrete Fourier transform (DFT),
are compared with those of the arithmetic (RMS) averaging. It is shown
that the asymptotic bias of the geometric mean is a function of the
fourth-order moments of the measurement noise.
6
Xm
I. PRELIMINARY
CONSIDERATIONS
ONSIDER TWO time series x, y and a linear model
which relates one time series to another (Fig. 1). Both
series are disturbed with zero mean noise, denoted by n,
and nY, respectively. These contaminating noise sources
are jointly uncorrelated, and are each uncorrelated with
x, and y .
It is well known that the power spectra of time series
can be calculated by means of a discrete Fourier Transform (DFT). For two wide-sense ergodic time series x, y
satisfying (1.1):
C
Fig. 1. Linear relationship between two time series
cross spectra of x, y. In practice both x and y are measured. Hence, supposing that no leakage nor aliasing effects occur in the DFT algorithm, (1.2) becomes after
classical averaging of the DFT spectra over independent
experiments:
S3f)
=
IH(f)j2s, +
s;(f)
=
H(f>Sn
NYY
k= +m
the cross power spectrum calculated via a DFT converges
uniformly in O( 1/ T o ) to the exact value S,,( f ) , as the
time domain window width To increases to infinity ([3, p.
249, th. 7.4. I]). The same is true for the auto power spectrum ([3, p. 123, theorem 5.2.21). In other words the
leakage errors on the power spectra obtained via a DFT,
vanish in 1/ T o .
In both deterministic and stochastic cases the following
properties are valid for the model shown in Fig. 1, in the
absence of disturbing noise ( n , = 0, n,. = 0 )
SJf)
=
(H(f)/2s,,(f)
SJf)
=
H(f)L(f)
where S,,, Syy,Sxyare, respectively, the auto power and
Manuscript received January 28, 1988. This work was supported in part
by the National Fund for Scientific Research, Belgium.
J. Schoukens and R. Pintelon are with the National Fund for Scientific
Research, Vrije Universiteit, 1050 Brussels, Belgium.
J . Renneboog is with the Electrical Measurement Department, Vrije
Universiteit, 1050 Brussels, Belgium.
IEEE Log Number 8820509.
S X f ) = S,(f) + N ,
(1.3)
where N,,, NYYare the autopower spectra of n,, n,.
The cross and auto power spectra are used to calculate
the transfer function of a Device Under Test (DUT) (1.4)
or the coherence function CxY(f ) of two time series (1.5),
[6]. Clearly HXY(f ) and HY,( f ) are biased estimates of
H(f),which give, respectively, a too small and a too
large approximations of the true value. The complex coherence function quantifies the bidirectional linear association existent between x and y. I t allows the verification
of the linearity assumption, which is important in such
applications as system identification and source bearing
estimation in acoustics [l]. I t there exists a perfect linear
relationship between the two time series (e.g., model of
Fig. 1 with n, = 0 and nY = 0), then the magnitude of the
coherence function equals 1 over the whole frequency
band. In the presence of disturbing noise or nonlinearities, the modulus of the coherence function will be less
than 1 [ l ] , [2]. So, if one wishes to detect the nonlinearity
of a DUT, the influence of the contaminating (measurement) noise should be eliminated
om
0018-9456/88/0600-0213$01.OO @ 1988 IEEE
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"
214
I h t i t : ‘ T R A N S A C T I O N S O N I N S T R U M t N I A 1 1 0 N A N D M E A S U R E M E N 1. VOL.
It can be concluded that an unbiased estimate of the
auto power spectrum is required in a lot of measurement
applications. The classical RMS averaging does not fulfill
this requirement.
77. N O . 1. J U N E I Y X X
stationary, and weakly correlated ([3], p. 94, theorem
4.4.1). Moreover, Schoukens [4] showed by means of a
realistic example that these asymptotic properties (a)-(c)
are reached for practical values of the number of time domain samples N . In Lemma 4 of the Appendix it is demonstrated that the p.d.f. of stationary noise is transformed
by the DFT into an even p.d.f.. This property is valid for
a finite number of time domain samples (divisible by
four).
A . Aritlztnetic (RMS) A\,erugr
11. AVERAGING
OF POWER SPECTRA
Averaging of a power spectrum is useful when averaging the Fourier spectrum is impossible (e.g., random
noise or burst random), or when there exists an uncertainty on the trigger position from one experiment to another, due to inaccurate locking of the signal generator
and the spectrum analyzer (or transient recorder). Indeed,
an error of 1 sample on the trigger position causes a fault
of 180” on the phase at the Nyquist frequency!
The aim here is to estimate the power spectrum ( A 2 )
starting from the measured real (Rm) and imaginary (Im)
parts of the Fourier coefficients, obtained via a DFT, when
averaging of these real and imaginary parts over M independent experiments is impossible. It will be supposed
that the DFT gives the correct Fourier spectrum, which is
the case for periodic band limited signals (e.g., multisine,
filtered pseudo random noise), for transients if the sample
frequency is sufficiently high (e.g., impulse, burst random), and for wide-sense ergodic random noise if the time
domain window To is sufficiently large (see Section I).
Note that the results of this section remain valid for Fourier spectra obtained in other ways than via the DFT, as
long as the noise on their real and imaginary parts satisfies
assumptions (2.2). The statistical properties of the different averaging modes will be compared in this section.
Define the measured real and imaginary parts of the
Fourier spectrum as
R,,, = R
I,,,
=
I
+r
+i
~ { r =} ~ { i =} o
(b)
E { r 2 } = E { i ’ } = u2
(c)
E{ri} = 0
(d)
r , i have an even probability
density function (p.d.f.).
l M
~i~~
= M IC
= I ( R ~ I+ Itn;)
-
(2.1.1)
which has a bias of
b ( A k M s ) = 2“’.
(2.1.2)
For small noise levels ( u ’ << A 2 ) , the bias on the RMS
estimate of the amplitude spectrum ARMS is given by
UL
b(ARMS)
=--
A
2(M- 1)A’
(2.1.3)
B. Geometric Averuge
The geometric mean of the power spectrum is
M
rr
A~~~~
2
=
(Rtnf
I=I
+~mf)
which has a bias of (Lemma 1, Appendix), ( u 2
2 u2
~(&EOM)
=
~
-
(2.2.1)
<< A ’ )
l
(2.2.2)
Proceeding in the Same way the bias of the geometric
mean of the amplitude spectrum is found ( u ’ << A ’ )
7
UL
(2.1)
where r , i are. respectively, the noise on the real ( R ) and
imaginary ( I ) parts, and with A’ = R’ + I ? . It will be
supposed that the noise on real and imaginary parts satisfies the following conditions:
(a)
The arithmetic average of the power spectrum is defined as
b(AGEOM)
=
2(M -
Retnurks:
1) In case the fourth-order moments of real and imaginary part are equal (2.2.4).
E(r4} = E { P } =
(2.2)
/.L4
(2.2.4)
and if in addition to (2.2), r and i are uncorrelated into
the fourth-order moments, then the contribution of the
fourth-order moments to the bias can be calculated. This
gives for (2.2.2) (Lemma 2, Appendix)
It is well known that assumptions (2.2) are asymptotically
valid after a DFT, and that r . i are asymptotically normal
distributed random variables for an increasing number of
time domain samples, provided the time domain noise is
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(2.2.5)
PINTELON e! d . : GEOMETRIC M E A N OF POWER SPECTRA
where ku is the Kurtosis factor ( p4 = ku u 4 ) , and for
(2.2.3)
TABLE 1
DEFINITION
OF T H E SIMULATIONS
I N T H E FREQUENCY
DOMAIN
simulation
1
. [1
-
8
3
.
2
3
4
(2.2.6)
Notice that (2.2.4) is asymptotically valid after a DFT
[3]. For Gaussian noise satisfying (2.2), condition (2.2.4)
is valid, so that the contribution of the fourth-order moments to the bias is zero in this case (ku = 3 for a Gaussian p.d.f.).
2) The logarithmic function is under assumptions (2.2)
(a)-(c), the unique function, independent of u2,that makes
the contribution of the second-order moments to the bias
of the power spectrum zero (Lemma 3, Appendix). It also
follows that it is impossible to find a function, independent of u2,that at the same time eliminates the second and
any other higher order moments.
3) The standard deviation of the geometric mean and
the RMS average is found via linearization ( u << A )
1 lo1
215
M
4 . m
1.6e5
1 .Oe6
4.0~6
S/Nrrq(dB)
2.0
1.5
4.9
7.4
11.0
17.0
0.5
.~
TABLE I1
SIMULATION
RESULTSWITH W H I T ELAPLACE
NOISE ON THE F O U R I E R
COEFFICIENTS
I
I
Laplace probability density funcuon (ku
SIMULATION
=6)
TABLE 111
RESULTSWITH WHITE UNIFORM NOISEON THE FOURIER
COEFFICIENTS
I
Uniform pmbabdity density function (h= 915)
D. Conclusion
The averaging modes only differ in their bias. Both
RMS and geometric mean estimates are nonconsistent,
however for practical signal-to-noise ratios ( A / u > 1 ),
the asymptotic bias ( M -+ 0 3 ) of the geometric mean is
much smaller than that of the RMS value. In other words,
the geometric mean of the amplitude spectrum has much
better noise properties than the arithmetic mean.
111. SIMULATIONS
Two kinds of simulations are distinguished here. The
first is a frequency domain approach: white noise satisfying (2.2) (Gaussian, Laplace, and uniform) is added to
the real ( R ) and imaginary ( I ) part of one spectral line.
It allows the verification of the asymptotic expressions
((2.1.3) and (2.2.6)) for the bias on the estimates, obtained via the arithmetic and the geometric averages, respectively. The second is a time domain approach: white
time domain noise with asymmetrical p.d.f. (Rayleigh,
exponential) is transformed via a 2048 point FFT into frequency domain noise satisfying (2.2). It allows the verification of the asymptotic normality of the frequency domain noise, after a DFT.
Four different frequency domain simulations are considered. They are defined in Table I as functions of the
number of experiments M and the signal-to-noise ratio
S I N , , in decibels. The time domain simulations of Section 111-B are also defined with respect to Table I.
A. Frequency Domain Simulations
The following values are chosen for the real and imaginary parts of the spectral line: R = 3 , I = 4 ( A = 5 ) .
Hence, the asymptotic bias of the arithmetic and geomet-
TABLE 1V
SIMULATION
RESULTSWITH WHITE G A U S S I ANOISE
N
ON THE FOURltR
COEFFICIENTS
I
I
Gaussian probability density function (kn = 3)
-I
ARMS)
- 17c-2
- 6.01 4
i7c-3
0 80
0 45
0 20
0 05
___
ric means become
(3.1)
(3.2)
The theoretical values of the bias (3.1), (3.2) are compared in Tables 11-IV with the values (ARMs - A ) ,
( A G E O M - A ) , obtained via the simulations defined in Table I for three different p.d.f.’s. Clearly the asymptotic
bias of the geometric mean is much smaller than the bias
of the corresponding RMS average. Taking into account
the standard deviation Std = u/( M - 1 ) I/’ of the estimates, it can be seen that the predicted bias of the geometric mean for Laplace and uniform noise (3.2), coincides very well with the simulations, even for very low
signal to noise ratios. According to the theory, the estimates of the amplitude via the geometric average for
Gaussian noise, equals the exact value within the 95-percent confidence interval.
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216
IEEE T R A N S A C T I O N S ON I N S T R U M E N T A T I O N A N D M E A S U R E M E N T . V O L . 3 7 . NO
SIMULATION
RESULTSWITH
TABLE V
EXPONENTIAL
TIME
DOMAIN
NOISE
WHITE
Exponential probability density funcnon
38e3
Ice3
2*4
5mp
5 m 5
5
m
86e3
5*-4
2e-5
5436
0436
0196
0050
5 I96
-
SO50
020
00s
TABLE VI
SIMULATION R t s u L r s WITH W H I T E R A Y L t l C H
TlMt DOMAIN
NOISE
Rayleigh probability density function
~
I
I
Std
\Illl"l*"O"
I
2
3
4
Ilk2
38c3
I*?
2 *-4
2. J U N k I Y X X
tude) spectrum via e.g., the maximum likelihood (ML)
method or the method of the moments. The most appropriate assumption one can make is that the frequency domain noise satisfying (2.2) is normally distributed. It is
well known that in this case the power spectrum A2 is
noncentral chi-square distributed ([7, ch. 24, p. 2431).
The disadvantages of the ML method and the method of
moments are, with respect to the geometric mean, the
much longer calculation time and their sensitivity to deviations from the Gaussian assumption. Moreover, they
require an estimate of the variance c2 of the disturbing
noise, which results for the method of moments in a larger
uncertainty upon the estimated amplitude (power) spectrum than given by (2.2.7). This is the price which must
be paid for its asymptotic unbiasedness property. Remark
also that the ML approach leads to a rather involved algorithm ([7, p. 244, equation (24.18)]).
L
~
B. Time Domain Simulations
The same values for R , I are chosen as in Section IIIA. Due to the linearity of the DFT, M approximately independent experiments can be obtained as follows. First
2048 samples of white time domain noise, with noneven
p.d.f. (exponential, Rayleigh), are transformed via an
FFT into 1022 samples (1024 - (dc Nyquist)) of complex frequency domain noise satisfying (2.2). Next, to
limit the computing time, each of these 1022 spectral lines
is added separately to R , I , which gives 1022 approximately independent experiments of R,, I,,,. The whole
procedure is repeated until the desired number M is
reached. It should be remarked that this way of constructing the independent experiments gives worst case results
for the estimates. It can be seen from Tables V, VI, that
the exact amplitude always lies in the 95-percent confidence interval of the geometric mean, just as it is the case
for Gaussian frequency domain noise (see Table IV).
Hence, after an FFT with a finite number of time domain
samples, it is justified, from a practical point of view, to
treat the frequency domain noise as being uncorrelated
and normally distributed.
In order to appreciate the noise levels in the time domain, the relationship between the signal-to-noise ratios
in the time and frequency domains is given in (3.3), for
a signal with a flat amplitude spectrum containing F spectral lines, when a DFT based on N time domain samples
is being used
+
S/Nlre,
=
p
2F S/Ntime.
V . CONCLUSION
An original averaging principle for auto power spectra,
which has a much smaller bias than the classical RMS
estimate, has been presented. Its statistical properties are
verified by means of simulations. It has been demonstrated that the geometric mean leads to the smallest bias
on the estimates of an auto power spectrum, in case the
p.d.f. and the variance of the disturbing noise are unknown. There is no increase in uncertainty with respect
to the RMS average.
For practical signal-to-noise ratios in the time domain,
the geometric mean generates almost unbiased estimates
of the auto power spectra.
APPENDIX
Lemma I : Statistical properties of the geometric
mean: The statistical properties of the geometric mean
will be studied by first taking the logarithm, and second
expanding in Taylor's series. This approach is valid for
signal to noise ratios that are not too small ( 0' << A * ) ,
so that the higher order terms can be neglected. Define
the logarithmic average as
ALOG
= log
I M
C log (Rm? + I m ; ) .
ML=l
= -
(A&OM)
(A.1)
Expansion of (A. 1 ) for M
=
1 in Taylor's series gives
(3.3)
Assume for example N = 2048, and F = 32, then the
signal-to-noise ratio in the time domain is about 15-dB
smaller than in the frequency domain.
IV. COMPARISON
W I T H OTHERMETHODS
If the p.d.f. of the disturbing noise is known, it is possible to obtain consistent estimates of the power (ampli-
with
x=
(7)
x =
(1)
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PINTELON
(’I
217
d . : GEOMETRIC M E A N OF POWER SPECTRA
and where x satisfies (2.2). Equation (A.2) leads to the
following expression for the bias on the logarithmic average (A. I ) :
b(ALOG)
=
E{ALOc(Rm, Im)
(A.ll)
ALOG(R, I ) ]
-
(A.3)
which can be written as
Q.E.D..
Lemma 2: Contribution of the Fourth-Order Moments
to the Bias of the Geometric Mean: Proceeding in the
same way as in Lemma 1 , the contribution of the fourthorder moments to the bias of the logarithmic average is
found to be
where
( A . 12)
and
d2ALOG
-=(-+2
I2
a
-1
R2
2
+ I2
( R ~
(A.6)
so that the coefficient of o2 in (A.4) is zero. Since the
p.d.f. of the noise on the Fourier coefficients is even, the
expected value of the third-order moments is zero, and
the bias on (A. 1) is at most a linear function of the fourthorder moments
0 4 , E { r 4 } ,E { i 4 } .
(A.7)
Because the sample mean of A,, (Rm, Im) over M independent experiments converges to its expected value. it
can be concluded that (A. 1) is of the form
ALoG = log ( A 2 ) +
After a few calculations we get the asymptotic bias ( M
03) of the geometric mean (cfr. (A.8) and (A.9))
-+
which demonstrates the proof.
Lemma 3: The Logarithmic Function gives the Unique
Averaging Mode that Eliminates the Second-Order Moments in the Bias of the Power Spectrum: We are looking
forafunctiong(A), s o t h a t E { g ( A m ) - g ( A ) } doesnot
contain second-order moments. Expansion of g ( A m ) in
Taylor’s series gives
c+b
In (10)
( A . 14)
~
where X , x are defined in Lemma 1. The expected value
of the second-order term in (A. 14) is set to zero
with
R
I
c=27r‘+2-if
A
A*
[;
=o
E -xT-
(A.15)
so that a partial differential equation for g ( A ) is obtained
a 2 d 4 + a 72 d =4
8 R2
ar
which can be written as (A’ = R2
Finally an expression for AGEOM
is found
0
(A. 16)
+ I?)
(A.17)
Equation (A. 17) has the general solution ( C , , C-,are constants)
with
6=c+b.
g ( A ) = C , In ( A ) + C?.
Equation (A.9) allows the calculation of the standard deviation and the bias of the geometric mean
One could remark that proceeding in the same way, the
expected value of the sum of the second- and the fourthorder terms in (A.14) could be set to zero. This would
however lead to a function that depends upon the standard
deviation u of the disturbing noise.
std(AiE0,)
= A2 std(t) =
2 Ao
~
JM-I
(A. 10)
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(A.18)
218
IEEE T R A N S A C T I O N S ON I N S T R U M E N T A T I O N A N D M E A S U R E M E N T . VOL
Lemma 4: A Nonsymmetric P. D, F, is Transformed by
the DFT into a Symmetric P. D. F. (P.D. F. of the DC
Component Excepted): This property has already been
proven for an infinite number of time domain samples [3].
It will be demonstrated here for a finite number N (divisible by four).
dence is not required) is symmetric. Put z = x - y , where
x and y are identically distributed random variables. The
p.d.f. f ; of z is given by [ 5 ]
+m
-m
can be written under the form ( A .19). Division of ( A .19)
in the even and the odd spectral lines
N/2 - I
u/(N) =
C
-m
+ k CN / 2 nkW;
nkW!
c +m
=
(WN = exp ( - j
=
g))
C
k=O
k=O
(nk
+
(-1)‘nk+N/2)
wi
(A.19)
C
k=O
&(u,
z
+ u ) du
(u = w -
z)
(A.22)
= fv,,(
y,
.)’
REFERENCES
(nk - n k + N / 2 )
k(2/+ I)
WN
(a)
N/4 -I
a2/=
-m
since x and y are identically distributed: & ( x , y )
N/2 -I
a2/+1 =
)
=f,(z>
N/2- I
=
+m
N- I
k=O
37. N O . ?. J U N E 19XX
[(nk
+
nk+N/2)
wk2’
(b) (A.20)
gives ( A . 2 0 ) . Since { n k } is stationary, the p.d.f. of nk
equals the p.d.f. of n k + N / 2 , and the p.d.f. of (nk +
n k + ~ ~ j 4 S) O. it can
n k + j q / 2 ) equals the p.d.f. ( n k + ~ / 4
be cdncluded from ( A . 2 0 )that &e p.d.f. ifthe frequency
domain noise is symmetric, if the p.d.f. Of the difference
of two identically distributed random variables (indepen- ( n k + N / 4 -k
%+3N/4)]
[I] J . A. Cadzow and 0. M. Solomon, “Linear modeling and the coherence function,” IEEE Trans. Acousr., Speech, Signal Proc., vol.
ASSP-35, no. I . pp. 19-28, Jan. 1987.
[2] G. Clifford Carter, “Coherence and time delay estimation,” Proc.
IEEE, vol. 75, no. 2, pp. 236-255, Feb. 1987.
[3] D. R. Brillinger, Time Series, Data Analysis, and Theory. New York:
Holt, Rinehart, and Winston, Inc., 1975.
141 J . Schoukens, and J . Renneboog, “Modeling the noise influence on
the Fourier coefficients after a discrete Fourier transform,” lEEE Trans.
Instrum. Meas., vol. IM-35, no. 3, pp. 278-286, Sept. 1986.
[5] A . Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw Hill, 1972.
[6] J . S. Bendat and A. G . Piersol, Engineering Applications of Correlation and Spectral Analysis. New York: Wiiey & Sons, 1980.
[7] M . Kendall and A . Stuart, The Advanced Theory of Statistics, vol. 2 .
London: Statistical Inference and Statistical Relationship, fourth edition, Charles Griffin, 1979.
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