Predicting the variance of a measurement with 1/f noise
aa r X i v : . [ phy s i c s . d a t a - a n ] M a y Predicting the variance of a measurement with 1/f noise
Benjamin Lenoir a a Onera – The French Aerospace Lab, 29 avenue de la Division Leclerc, F-92322 Châtillon, France
Published in
Fluctuation and Noise Letters
17 May 2013
Abstract
Measurement devices always add noise to the signal of interest and it is necessary to evaluate thevariance of the results. This article focuses on stationary random processes whose Power SpectrumDensity is a power law of frequency. For flicker noise, behaving as /f and which is present in manydifferent phenomena, the usual way to compute the variance leads to infinite values. This articleproposes an alternative definition of the variance which takes into account the fact that measurementdevises need to be calibrated. This new variance, which depends on the calibration duration, themeasurement duration and the duration between the calibration and the measurement, allows avoidinginfinite values when computing the variance of a measurement. Keywords
This article deals with stationary random processes whose power spectrum density (PSD) S ( f ) behavesas ∀ f ∈ R , S ( f ) = h | f | α (1)with α ∈ ] −
3; + ∞ [ . The condition on α allows covering “flicker noise” ( α = − ), which is the focus ofthis article, as well as white noise ( α = 0 ). The fact that flicker noise describes correctly experimentalphenomena in many different fields has been verified [3, 8, 7, 9].Because of the infinite value taken by S ( f ) when f → , discussions focus on the extend of the /f spectrum at low frequencies. However, the yet imperfectly understood mechanisms of the /f noise do notgive reasons to expect a low-bound on the frequency range [2]. The problem becomes more acute whenone tries to compute the variance of a measurement made with an instrument having this type of noise,since it leads to infinite values. One way to tackle this problem is to use a “conditional spectrum” [6].The idea is to take into account the fact that the physical phenomenon leading to the /f noise has beenobserved for a finite amount of time called t ∗ . If, during the measurement process, the signal is averagedover a period called T , the variance of the measured average scales as log( t ∗ /T ) [4, 5]. This approachleads to a finite value for the variance. But it introduces the parameter t ∗ , which may be completelyarbitrary when one is interested by the average over the period T , and adds the unknown offset log( t ∗ ) .This article is not concerned with the derivation of | f | α noises from physical laws, nor with the ideaof giving a lower bound to the frequency range of /f noise. Instead, the goal is to provide an efficienttool to compute the variance of a measurement made with flicker noise and more generally with a PSDleading to infinite variance due to its behavior close to zero. To do so, a new definition of the variance isintroduced. It takes into account the measurement process, which is composed of the measurement itselfbut also of the calibration of the instrument. This quantity is first introduced for flicker noise. Someproperties are given in this case and generalizations are made to other type of PSD.1 . Lenoir, 2013 2/4 Let us consider a stationary stochastic process n whose PSD is S ( f ) = 1 / | f | . This noise is added to adeterministic signal called s . The aim of the measurement process considered here is to know the meanof the signal s over a duration T and to characterize the precision of this measurement. The usual wayto proceed is to compute the variance V ( T ) using the transfer function h T of the variance operator V ( T ) = 2 Z ∞ S ( f ) | H T ( f ) | df, (2)where H T ( f ) = sinc ( πT f ) is the Fourier transform of h T . V ( T ) is equal to + ∞ because of the integrandbeing equivalent to /f when f → + .The idea to get around this problem is inspired by the methodology used to make measurements. Atsome point in the measurement process, the instrument is calibrated, i.e. its bias is measured so as to beremoved from the measurements of interest. The measurement equation of the instrument is m = s + b + n, (3)where m is the measurement, s the signal of interest, b the bias of the instrument and n the noise.The calibration process consists in measuring subsequently the mean of s and the mean of − s , eachmeasurement lasting τ / . Assuming that s and b are constant over τ , it gives two quantities m c and m c such as b = m c + m c − h n i τ , (4)where h·i τ is the mean over the duration τ . As a result, when making a measurement of the signal s overa period T , the variance of the measurement depends on the measurement itself and the calibration ofthe bias, i.e. on the quantity − τ Z τ/ − τ/ n ( t ) dt + 1 T Z τ/ Tτ/ n ( t ) dt, (5)with ∆ the duration between the end of the calibration and the beginning of the measurement. Thisleads to consider the following new definition for the variance U ( τ, T, ∆) = E (cid:20)(cid:16) − I τ − τ/ + I Tτ/ (cid:17) (cid:21) = E (cid:2) ( n ∗ h τ,T, ∆ ) ( t ) (cid:3) , (6)where I ba = b R a + ba n ( t ) dt , E is the expectation operator and h τ,T, ∆ is defined in figure 1. U ( τ, T, ∆) doesnot depend on t , which is an arbitrary time, because n is a stationary stochastic process. (cid:1)(cid:2)(cid:3) (cid:2)(cid:3) (cid:2)(cid:3) (cid:4)(cid:5) (cid:2)(cid:3) (cid:4)(cid:5)(cid:4) T (cid:3) T (cid:1) (cid:3) (cid:2) t h (cid:2) ,T , (cid:5) | H τ , T , ∆ | Figure 1: Transfer function of the variance U defined in equation (6). Left:
In the temporal domain.
Right:
In the frequency domain with T = 10 s, τ = 1 s and ∆ = 100 s.The transfer function of the variance U , represented in figure 1, is equal in the Fourier domain to H τ,T, ∆ ( f ) = − sinc ( τ πf ) + exp (cid:20) − i πf (cid:18) ∆ + T + τ (cid:19)(cid:21) sinc ( T πf ) . (7) . Lenoir, 2013 3/4 This expression generalizes Allan variance [1] with uneven sizes for the samples and discontinuity betweenthe samples. Since | H τ,T, ∆ ( f ) | is equivalent to f when f → + , the integral U α ( τ, T, ∆) = 2 Z ∞ f α | H τ,T, ∆ ( f ) | df, (8)is finite for α ∈ ] −
3; 1[ . For a flicker noise ( α = − ), computing the integral leads to the followingexpression: U − ( τ, T, ∆) = − πτ ) − πT ) − T ) τ T ln [2 π (∆ + T )] − τ ) τ T ln [2 π (∆ + τ )] + 2∆ τ T ln [2 π ∆]+ 2(∆ + τ + T ) τ T ln [2 π (∆ + τ + T )] . (9)Obviously, the following property is verified: U − ( τ, T, ∆) = U − ( T, τ, ∆) . It means that in term ofvariance the duration of the calibration and the measurement are equivalent. Figure 2 shows, for given τ and ∆ , the value of the integration time T for which the variance reaches a minimum. −4 −2 Duration between calibration and measurement ∆ (s) O p t i m a l m ea s u r e m en t du r a t i on ( s ) τ = 1 s τ = 10 s τ = 100 s τ = 1000 s Figure 2:
Left:
Value of U − , defined by equation (9), as a function of the measurement duration T , andthe duration between the calibration and the measurement ∆ , for a calibration duration τ = 1 s. Right:
Value of the measurement duration T for which U − is minimum. The method presented in the previous section can be applied to the other value of the coefficient α . Anexact result can be obtained for a white noise ( α = 0 ): U ( τ, T, ∆) = 1 τ + 1 T . (10)In this case, the variance does not depend on the quantity ∆ . This is a characteristic of white noise becauseof the absence of correlation in the noise. Similarly, one can obtain analytically the value equation (8)for a brown noise ( α = − ): U − ( τ, T, ∆) = 4 π (cid:18) ∆ + T τ (cid:19) . (11)In this case, the standard deviation, which is the square root of the variance, scales as √ ∆ . This is acharacteristic of random walks which are described by brown noise.Finally, for values of α larger than , it is necessary to take into account the cut-off frequency f c ofthe measurement device: the quantity ˜ U α ( τ, T, ∆ , f c ) is the same as U α but with an integration up to f c in equation (8). Therefore U α ( τ, T, ∆) = ˜ U α ( τ, T, ∆ , + ∞ ) . This allows to solve the divergence of U α for α ≥ . Figure 3 shows the behavior of ˜ U α for a white noise ( α = 0 ) and for different values of f c . Theintegration time T for which the value of ˜ U is different from the value of U are those smaller than /f c . . Lenoir, 2013 4/4 −4 −3 −2 −1 Measurement duration T (s) ~ U ( s − ) f c = 1 Hzf c = 10 Hzf c = 100 Hzf c = Inf Figure 3: Value of ˜ U as a function of the integration time T for τ = 1 s and ∆ = 100 s. The plots areindexed by the cut-off frequency of the measurement device. This article aims at answering the following question while avoiding infinite values: what is the varianceof a signal mean value measurement when the measurement noise is a flicker noise? To do so, a newdefinition of the variance was introduced. It was inspired by the usual measurement methodology whichconsists in calibrating the device before making measurements. This led to a variance which dependson three quantities : the calibration duration, the measurement duration and the duration between thecalibration and the measurement. And it can be generalized for Power Spectrum Density of the form | f | α with α > − . Acknowledgments
The author is grateful to CNES (Centre National d’Études Spatiales, France) for its financial support.
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