- Email: [email protected]
A noise-variance estimator in electroencephalography
COMPUTERS
AND
BIOMEDICAL
5, 108-113 (1972)
RESEARCH
A Noise-Variance
Estimator
in Electroencephatography J. J. REY
WILLIAM Manufacture
Beige de Lampes et de Mate’riel Electronique-M.B.L.E., Brussels, Belgium
S.A.,
Received June 18, 1971 It is hardly possible in the electroencephalographic field to distinguish the signal and the noise. What they are is for both of them so vague that very opposite views can be defended with success. Let us mention as extremes, on the one hand, the idea that a continuous electroencephalographic potential is noise due to the random-like cephalic activity only and, on the other hand, the position of many who believe that no noise remains in an average evoked potential when the summation period is long enough. The present paper describes a method for the estimation of the noise variance based upon the following definition: the noise is the observation component behaving as a random variable. We are not postulating any special distribution, nor particular property of the random component. The estimator we derive assumes implicitly some stationarity for the noise source, although this requirement has no practical importance. The application field of the method is mainly in the analysis of average evoked potential records where it is of interest to estimate the remaining noise component. GENERAL DEVELOPMENT
The object of this note is a method for estimating the noise superimposedwith a signal known by periodically sampledvalues. We assumethat the original signal can be adequately represented by an analytical development. Let x1 be the sampled value, sI be the real value of the signal at time ti, and l i be the noise present at time Ii. Then the model is given by the equation Xi = St+ E*
(1)
with the constraints Si = S(ti)p ti = to + iT,
(2)
= analytical function oft, E*= random variable with zero mean.
s(t)
Hereinafter, we establishan estimator of 0<*, the variance of the noisecomponent. 0 1972 by Academic
Press, Inc.
108
ESTIMATION
OF SUPERIMPOSED
NOISE
109
Our noise estimator is derived from the comparison of several variance equations obtained for different groupings of the sampled values. We group the Xi terms k by k and investigate their difference behaviour, that is to say, the random variable Yk,j
By similarity
=
$,
Xjkti
-
5 i-1
(3)
XCi-l)kti.
with (l), the y variable can be expressed as Yk,J
=
sk,i
+
7k.J
(4)
(9
with Sk2
=
i i=l
Esjk+i
-
S(j-l)ktil
?lk,.i
=
,i
LElk+i
-
E(j-l)kti
and
1*
(6)
We will see later that the S component is, in essence, a high-pass filtered version of the s signal. We now consider the variance of they variable. Due to the independence between its two components, we have u,‘,, = as2 + u,,*,
(7)
where us2 has the usual statistical meaning despite its analytical nature. The two right-side terms can be expressed as functions of the signal and noise as they appear in (1). The relation (5) between the S and s variables will give an expression for us?. Denoting by D the following differential operator proportional to the sampling period T, D = (772) (44, we have s(t + IT) = e21Ds(t) or, in a more condensed form, sitl = e*‘D&Q. In this formal way, we transform (5) in Sk j = V”
- e-kD)2 Sjkt ,,2 ,D - ,-D
(8)
or, expanding, D2 t O(D4) Sjk+l;Z . I
(9)
110
REY
Under the assumption that the higher-order derivatives do not contribute significantly to the S variance, we derive the relation us2 z k4 V. (10) Let us note that the previous assumption is weaker than assuming a truncation of the series appearing in (9). The V constant is approximately proportional to the variance of the s first derivative. The incidence of the value of k on V will be shown later on a special case. The CJ,,~term of (7) can be replaced, after (6), by 5 2 = 2kaC2.
(11)
Introducing this result, as well as (lo), in (7) we obtain a relation between cr:,, which we can directly evaluate, the unknown quantity V, and uC2which is to be estimated u;,, = k4 V + 2ko,?.
(12)
An estimator of crC2is obtained by eliminating V between two Eq. (12) established for different values of k. For k equals 1 and 2, the corresponding relations yield q2 = (16~: y - a;,,)/28 which is our random-noise-variance
(13)
estimator. CASE STUDY
Let us consider the special case where the signal s(t) presents some periodicity. If it is so, it can be expanded in its sine and cosine Fourier components. As they are coorthogonal, each of them contribute independently to the us2, and this case can be reduced to the simpler case where just a single sinewave is taken into account. We consider hereinafter the case of a sinewave and a random noise superimposed. The angular frequency is denoted by w for the sinewave, and its amplitude is unitary. In this simple case, we can derive the exact expression of Skj, by (8) and the corresponding us2. We get 2-1 l-coskwT 2 (14) us -5 sinwT/2 -L I ’ A comparison between (14) and (10) yields V = (wz T2/2) + O(w4T4),
It appears that our assumption of V being constant with respect to k value is justified as long as wT is moderately inferior to the unity, or in other words, that the Eq. (13) is a reliable estimator when the sinewave period is sampled at least four times. Let us now investigate more thoroughly what is the incidence on the uC2estimator given by (13). Denoting by GE2the estimated value and by uC2 the real value, we obtain after (13), (7), (14) and (11) “2 = u, ‘+A u’( (15)
ESTIMATION
OF SUPERIMPOSED
NOISE
111
with A,i
4l;cos~T~ sm UT/2 ] -[' 56 f
slnCZY~~T~]
(16)
or, expanding in wT A =+w4T4
+O(co6T6).
The A term in (15) results from the signal presence. It is always positive, however generally quite small. Correction of the noise estimator (13) for this bias is not felt justified. We indicate in the following table the maximum signal-to-noise ratio we can admit for a given precision of the noise estimator. Limiting A by the condition and applying Eq. (16) and signal/noise == I /20,*, we obtain, at the level CL= 0.1, UT
A
0.314 0.785 1.571 2.356 2.827 3.142
0.001 0.045 0.429 0.955 1.11 1.14
Maximum
signal/noise
36.98 or $16 dB 1.10 + OdB 0.12 - 9dB 0.05 -13 dB 0.04 -13 dB 0.04 -14 dB.
It is hardly possible to evaluate the estimation error when the expected signal is unknown, for it is a function of its frequency spectrum. Denoting byf(o) the amplitude of the signal component at the w frequency, the general expression of the signalto-noise ratio, such that the inequality (17) is satisfied, is given by signal/noise~(~~.i(w)2~~)/(21j:/Jw)’Adw). where A is the function (16). For a typical electroencephalographic following results at the CL= 0.1 level: T 0.020 set 0.010 0.004 0.002 0.001
(18)
frequency spectrum, we have obtained the Maximum
signal/noise
0.097 or -10 dB 0.293 - 5dB 4.26 + 6dB 54.8 $17 dB 824 +29 dB.
112
REY
They indicate that the noise-variance estimator (13) is satisfactory in very adverse conditions where the signal component dominates the noise component by a considerable factor, as well as in more ordinary situations. The spectral data we have used are those published by D. 0. Walter (I) for the electroencephalogram of a man in the resting state. For obvious reasons the data have been completed to cover the whole frequency spectrum. They are given in the following table : w/2n
f(w)
intensity
1 see-’ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 50 100
16.01 17.69 19.54 21.59 23.85 30.36 27.86 30.72 40.69 46.21 32.12 30.37 32.89 41.79 39.20 36.77 34.49 32.36 30.35 28.47 15.02 4. I8 0.17
128.16” 156.44 190.95 233.08 284.50 460.90 388.00 472.00 828.00 1067.90 516.00 461.10 541.00 873.00 768.20 675.98 594.83 523.43 460.59 405.30 112.82 8.74 0.01
y In units proportional
to “pV*/(c/sec)“.
CONCLUSION
We have proposed a method for estimating the variance of the noise remaining on average evoked potential records. This method appears to be fairly insensitive to the low-frequency components of the records, but tends to overestimate the noise in presence of frequency components higher than a fourth of the sampling frequency used for digitizing the records.
ESTIMATION
OF SUPERIMPOSED
NOISE
113
The proposed method can be applied in any field where noise estimation is required in presence of a dominant signal, this signal presenting some continuity. REFERENCE D. 0. In “Advances in EEG Analysis” (D. 0. Walter and M. A. B. Brazier, Eds.), p. 16. Electroenceph. Clin. Neurophysiol., Suppl. 27, 1969.
1. WALTER,
AND
BIOMEDICAL
5, 108-113 (1972)
RESEARCH
A Noise-Variance
Estimator
in Electroencephatography J. J. REY
WILLIAM Manufacture
Beige de Lampes et de Mate’riel Electronique-M.B.L.E., Brussels, Belgium
S.A.,
Received June 18, 1971 It is hardly possible in the electroencephalographic field to distinguish the signal and the noise. What they are is for both of them so vague that very opposite views can be defended with success. Let us mention as extremes, on the one hand, the idea that a continuous electroencephalographic potential is noise due to the random-like cephalic activity only and, on the other hand, the position of many who believe that no noise remains in an average evoked potential when the summation period is long enough. The present paper describes a method for the estimation of the noise variance based upon the following definition: the noise is the observation component behaving as a random variable. We are not postulating any special distribution, nor particular property of the random component. The estimator we derive assumes implicitly some stationarity for the noise source, although this requirement has no practical importance. The application field of the method is mainly in the analysis of average evoked potential records where it is of interest to estimate the remaining noise component. GENERAL DEVELOPMENT
The object of this note is a method for estimating the noise superimposedwith a signal known by periodically sampledvalues. We assumethat the original signal can be adequately represented by an analytical development. Let x1 be the sampled value, sI be the real value of the signal at time ti, and l i be the noise present at time Ii. Then the model is given by the equation Xi = St+ E*
(1)
with the constraints Si = S(ti)p ti = to + iT,
(2)
= analytical function oft, E*= random variable with zero mean.
s(t)
Hereinafter, we establishan estimator of 0<*, the variance of the noisecomponent. 0 1972 by Academic
Press, Inc.
108
ESTIMATION
OF SUPERIMPOSED
NOISE
109
Our noise estimator is derived from the comparison of several variance equations obtained for different groupings of the sampled values. We group the Xi terms k by k and investigate their difference behaviour, that is to say, the random variable Yk,j
By similarity
=
$,
Xjkti
-
5 i-1
(3)
XCi-l)kti.
with (l), the y variable can be expressed as Yk,J
=
sk,i
+
7k.J
(4)
(9
with Sk2
=
i i=l
Esjk+i
-
S(j-l)ktil
?lk,.i
=
,i
LElk+i
-
E(j-l)kti
and
1*
(6)
We will see later that the S component is, in essence, a high-pass filtered version of the s signal. We now consider the variance of they variable. Due to the independence between its two components, we have u,‘,, = as2 + u,,*,
(7)
where us2 has the usual statistical meaning despite its analytical nature. The two right-side terms can be expressed as functions of the signal and noise as they appear in (1). The relation (5) between the S and s variables will give an expression for us?. Denoting by D the following differential operator proportional to the sampling period T, D = (772) (44, we have s(t + IT) = e21Ds(t) or, in a more condensed form, sitl = e*‘D&Q. In this formal way, we transform (5) in Sk j = V”
- e-kD)2 Sjkt ,,2 ,D - ,-D
(8)
or, expanding, D2 t O(D4) Sjk+l;Z . I
(9)
110
REY
Under the assumption that the higher-order derivatives do not contribute significantly to the S variance, we derive the relation us2 z k4 V. (10) Let us note that the previous assumption is weaker than assuming a truncation of the series appearing in (9). The V constant is approximately proportional to the variance of the s first derivative. The incidence of the value of k on V will be shown later on a special case. The CJ,,~term of (7) can be replaced, after (6), by 5 2 = 2kaC2.
(11)
Introducing this result, as well as (lo), in (7) we obtain a relation between cr:,, which we can directly evaluate, the unknown quantity V, and uC2which is to be estimated u;,, = k4 V + 2ko,?.
(12)
An estimator of crC2is obtained by eliminating V between two Eq. (12) established for different values of k. For k equals 1 and 2, the corresponding relations yield q2 = (16~: y - a;,,)/28 which is our random-noise-variance
(13)
estimator. CASE STUDY
Let us consider the special case where the signal s(t) presents some periodicity. If it is so, it can be expanded in its sine and cosine Fourier components. As they are coorthogonal, each of them contribute independently to the us2, and this case can be reduced to the simpler case where just a single sinewave is taken into account. We consider hereinafter the case of a sinewave and a random noise superimposed. The angular frequency is denoted by w for the sinewave, and its amplitude is unitary. In this simple case, we can derive the exact expression of Skj, by (8) and the corresponding us2. We get 2-1 l-coskwT 2 (14) us -5 sinwT/2 -L I ’ A comparison between (14) and (10) yields V = (wz T2/2) + O(w4T4),
It appears that our assumption of V being constant with respect to k value is justified as long as wT is moderately inferior to the unity, or in other words, that the Eq. (13) is a reliable estimator when the sinewave period is sampled at least four times. Let us now investigate more thoroughly what is the incidence on the uC2estimator given by (13). Denoting by GE2the estimated value and by uC2 the real value, we obtain after (13), (7), (14) and (11) “2 = u, ‘+A u’( (15)
ESTIMATION
OF SUPERIMPOSED
NOISE
111
with A,i
4l;cos~T~ sm UT/2 ] -[' 56 f
slnCZY~~T~]
(16)
or, expanding in wT A =+w4T4
+O(co6T6).
The A term in (15) results from the signal presence. It is always positive, however generally quite small. Correction of the noise estimator (13) for this bias is not felt justified. We indicate in the following table the maximum signal-to-noise ratio we can admit for a given precision of the noise estimator. Limiting A by the condition and applying Eq. (16) and signal/noise == I /20,*, we obtain, at the level CL= 0.1, UT
A
0.314 0.785 1.571 2.356 2.827 3.142
0.001 0.045 0.429 0.955 1.11 1.14
Maximum
signal/noise
36.98 or $16 dB 1.10 + OdB 0.12 - 9dB 0.05 -13 dB 0.04 -13 dB 0.04 -14 dB.
It is hardly possible to evaluate the estimation error when the expected signal is unknown, for it is a function of its frequency spectrum. Denoting byf(o) the amplitude of the signal component at the w frequency, the general expression of the signalto-noise ratio, such that the inequality (17) is satisfied, is given by signal/noise~(~~.i(w)2~~)/(21j:/Jw)’Adw). where A is the function (16). For a typical electroencephalographic following results at the CL= 0.1 level: T 0.020 set 0.010 0.004 0.002 0.001
(18)
frequency spectrum, we have obtained the Maximum
signal/noise
0.097 or -10 dB 0.293 - 5dB 4.26 + 6dB 54.8 $17 dB 824 +29 dB.
112
REY
They indicate that the noise-variance estimator (13) is satisfactory in very adverse conditions where the signal component dominates the noise component by a considerable factor, as well as in more ordinary situations. The spectral data we have used are those published by D. 0. Walter (I) for the electroencephalogram of a man in the resting state. For obvious reasons the data have been completed to cover the whole frequency spectrum. They are given in the following table : w/2n
f(w)
intensity
1 see-’ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 50 100
16.01 17.69 19.54 21.59 23.85 30.36 27.86 30.72 40.69 46.21 32.12 30.37 32.89 41.79 39.20 36.77 34.49 32.36 30.35 28.47 15.02 4. I8 0.17
128.16” 156.44 190.95 233.08 284.50 460.90 388.00 472.00 828.00 1067.90 516.00 461.10 541.00 873.00 768.20 675.98 594.83 523.43 460.59 405.30 112.82 8.74 0.01
y In units proportional
to “pV*/(c/sec)“.
CONCLUSION
We have proposed a method for estimating the variance of the noise remaining on average evoked potential records. This method appears to be fairly insensitive to the low-frequency components of the records, but tends to overestimate the noise in presence of frequency components higher than a fourth of the sampling frequency used for digitizing the records.
ESTIMATION
OF SUPERIMPOSED
NOISE
113
The proposed method can be applied in any field where noise estimation is required in presence of a dominant signal, this signal presenting some continuity. REFERENCE D. 0. In “Advances in EEG Analysis” (D. 0. Walter and M. A. B. Brazier, Eds.), p. 16. Electroenceph. Clin. Neurophysiol., Suppl. 27, 1969.
1. WALTER,