6.5 Characterizing Precise Oscillators: Frequency Domain
6.5.3 I 2 Power Spectral Density
When making the connection between the time domain measure of noise I2 and the frequency domain, power spectral density, we had to take the limit asτ →0 ofI2(τ). In
‡‡This is made possible bySy(f)≥0∀f. Another useful property is thatSy(f) =Sy(−f), so that, as is often done, we may focus on positivef only, by defining the one-sided density.
§§With spectrum meaning range of frequencies, as used by Newton when referring to the ‘ghostly ap- parition’ (spectre) of colours splitting from white light through a prism [2].
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6.5 Characterizing Precise Oscillators: Frequency Domain
practice this is not possible since there is a minimum non-zeroτ that can not practically be made smaller. It would be helpful if the power spectral density would take the τ averaging into account. Indeed, it can as we now describe.
We start with the variance, or power in the τa-averaged noise signal, I2(τa) =
y2tk,τa .
Here we have replaced the originalτ withτa(a for ‘average’ over) to distinguish it from autocorrelationτ, which we now label τs (s for ‘shift’).
We follow the same power spectral density construction for y2tk,τa
as we did for the ensemble averaged instantaneous power,
y2 . We first define the autocorrelation function,
Ryτa(τs)≡
yt0,τayt0−τs,τa .
We then define the power spectral density (in terms of the Fourier transform, but if necessary by the Wiener-Khinchin theorem we do not need the Fourier transform),
Syτa(f)≡ F(Ryτa)(f).
We write the autocorrelation as the inverse Fourier(-like) transform, Ryτa(τs) =
Z ∞
−∞
Syτa(f)ei2πf τsdf.
We make the connection to I2 by evaluating the autocorrelation at τs= 0; giving us the power spectral density of the ensemble average of the τa-averaged noise,
I2(τa) = y2t0,τa
=Ryτa(0) = Z ∞
−∞
Syτa(f)df.
Now, since y(t) is our fundamental building block, about which we make basic assump- tions and build our noise model around, we should connect Syτa(f) to Sy(f).
This is done by explicitly writing outyτa in terms ofyand a moving##averaging window, hτa, and then following the detailed calculations of taking the Fourier transform of the autocorrelation function.
##The window is a ‘continuously moving’ rectangular function. This may seem incompatible with adja- cent window averaging. It is however not a problem. We could, if we wished to model the process
6 Atomic Clock Background
To take advantage of some of the properties of the Fourier transform, we rewrite the process of averaging as a convolution. In our case, theτa-averaging process is convolution with a 1/τa-weighted and shiftedkk rectangular window,hτa(t), of lengthτa,
hτa(t) = τ1
aΠτa(t+τ2a).
The Fourier transform of the window is proportional to the sinc1/τa function,
|Hτa(f)|=|F(hτa)(f)|=|sinc1/τa(f)|.
The window, hτa, is called the impulse response∗ ∗ ∗ and its Fourier transform, Hτa is called thetransfer function† † †.
Usinghτa we have,
yt0,τ = 1 τ
Z t0+τ t0
y(t)dt
= Z ∞
−∞
hτa(t0−t)y(t)dt
= (hτa∗y)(t0).
of using only adjacent windows, still first take the continuous moving average convolution, and then sample the result by multiplying with a Dirac Comb. Here it is not necessary, because we are only using the autocorrelation of the moving average at zero lag and then ensemble averaging by collect- ing statistics. That is, for now we are not interested in how neighbouring windows correlate, we are rather interested in how one ensemble averaged window’s noise is distributed. We do use the valid, but practically inaccessible, theory of true continuous moving averages and its autocorrelation to introduce the idea of the power spectral density. Later on in the thesis we actually do make use of sampling the moving average to gain insight into aliasing.
kkThe flip and shift to the beginning (t=t0) of the noise integration is taken care of by the convolution.
The τ2a shift to the left concerns whicht-value is assigned the value of the integration. Normally the convolution with the rectangular function assigns the integral’s value to the center (t0+τa/2) of the rectangular function. In the Allan variance, it has been arbitrarily chosen to be the beginning, t0, which is not a problem, a mere exponential factor,e2πif τa/2, does the translation in the frequency domain. The convolution is ‘a-causal’ in either case because futurey(t) values are used in their pasts.
Later in the thesis we insist on causal windows.
∗ ∗ ∗
The reason for the name is that, if the input were an impulse (the Dirac Delta), the output, after convolution with the window, is the window itself.
† † †
The reason for this name is that, in the frequency domain, the Fourier transform of the window called the transfer function, ‘takes’ or ‘transfers’ the input to the output by simple multiplication (convolution has been transformed into multiplication).
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6.5 Characterizing Precise Oscillators: Frequency Domain
Now we can write out the previously defined autocorrelation of the noise in full, Ryτa(τs)≡
yt0,τayt0−τs,τa
= Z ∞
−∞
hτa(t0−t1)y(t1)dt1
Z ∞
−∞
hτa(t0−τs−t2)y(t2)dt2
= Z ∞
−∞
Z ∞
−∞
hτa(w)hτa(z)y(t0−w)y(t0−τs−z)dw dz
= Z ∞
−∞
Z ∞
−∞
hτa(w)hτa(z)D
y(t0−w)y(t0−τs−z)E dw dz
= Z ∞
−∞
Z ∞
−∞
hτa(w)hτa(z)Ry(τs+z−w)dw dz ,
where we have introduced new variables (w=t0−t1 and z=t0−τs−t2) and changed the variables of integration. By linearity of the integration we took the expectation in.
Furthermore by stationarity of the random process we rewrote the product of the y’s as an autocorrelation at the corresponding lag.
We are now ready to take the Fourier transform of the autocorrelation with respect to τs, as in the definition. We also introduce another new variable, to record the lag (x=τs+z−w=t1−t2), and change the relevant variable of integration,
Syτa(f)≡ F(Ryτa)(f)
= Z ∞
−∞
Z ∞
−∞
Z ∞
−∞
hτa(w)hτa(z)Ry(τs+z−w)e−2πif τsdw dz dτs
= Z Z Z
hτa(w)hτa(z)Ry(x)e−2πif(x−z+w)dw dz dx
= Z
hτa(w)e−2πif wdw Z
hτa(z)e2πif zdz Z
Ry(x)e−2πif xdx
=|Hτa(f)|2Sy(f)
= sinc2
1/τa(f)Sy(f)
= sin(πτ2(πτaf)
af)2 Sy(f).
Thus we have related the averaged noise’s power spectral density directly to the noise’s power spectral density. The effect of the averaging window is to attenuate the true noise’s high frequency components, acting as an oscillating low pass filter.
6 Atomic Clock Background