# Patent

## Title
Ensemble oscillator and related methods

## Abstract

An ensemble clock comprises: an input for receiving a reference signal; multiple free-running oscillators each configured to generate a corresponding free-running frequency; an output oscillator configured to generate a controlled frequency having a frequency responsive to a control signal; a differencer unit configured to derive difference measurements indicative of time and frequency-based errors associated with each of (i) the controlled frequency, and (ii) the free-running frequency, relative to the reference signal; and a controller configured to generate the control signal responsive to the difference measurements.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application claims priority to U.S. Provisional Application No. 60/489,396, filed Jul. 23, 2003, and is related to U.S. Provisional Application No. 60/488,894, filed Jul. 21, 2003, each of which is incorporated herein in its entirety by reference.
This application is also related to U.S. Non-Provisional application No. 10/898,418, now U.S. Pat. No. 6,958,951, filed Jul. 23, 2004, entitled “Adaptive Kalman Filter Process for Controlling an Ensemble Clock,” by Duven, which is incorporated herein in its entirety by reference.


STATEMENT OF GOVERNMENTAL INTEREST
This invention was made with U.S. Government Support under SPAWAR Contract Nos. N00039-01-01-C-2233 and N00039-02-C-2217. The U.S. Government has certain rights in this invention.


BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates generally to multiple oscillator systems.
2. Description of the Related Art
Conventional systems, such as satellite systems, disadvantageously use complex and expensive atomic oscillators, such as rubidium or cesium oscillators, to provide precise and stable reference frequencies. In such systems, there is a need to provide reference frequencies approaching the accuracies available from atomic standards, without using the atomic standards.
SUMMARY OF THE INVENTION
An embodiment of the present invention includes an ensemble clock or oscillator that provides an accurate output frequency. The ensemble clock comprises: an input for receiving a reference signal; multiple free-running oscillators each configured to generate a corresponding free-running frequency; an output oscillator configured to generate a controlled frequency having a frequency responsive to a control signal; a differencer unit configured to derive difference measurements indicative of time and frequency-based errors associated with each of (i) the controlled frequency, and (ii) the free-running frequency, relative to the reference signal; and a controller configured to generate the control signal responsive to the difference measurements.
Further apparatus and method embodiments of the present invention will be apparent from the ensuing description.



BRIEF DESCRIPTION OF THE DRAWINGS
Various embodiments are described below with reference to the drawings
FIG. 1 is a block diagram of an example ensemble oscillator system constructed and operated in accordance with the principles of the present invention.
FIG. 2 is a block diagram of an example arrangement of a Time Measurement Unit (TMU) of the system of FIG. 1
FIG. 3 is a block diagram of an example arrangement of a Signal Generation Unit (SGU) of the system of FIG. 1.
FIG. 4 is a block diagram of an example Adaptive Kalman Filter Processor (AKFP) of the system of FIG. 1 and oscillator error models used by the AKFP.
FIG. 5 is a flow chart of an example method executed by the AKFP.
FIG. 6 is a is a flow chart of another example summary method executed by the AKFP.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT(S)
Overview
FIG. 1 is a block diagram of an example ensemble oscillator system 100, including an oscillator ensemble unit (OEU) 102, a time measurement unit (TMU) 104, a central processor unit (CPU) or controller 106, optional environmental sensors 108, a signal generation unit (SGU) or output oscillator 110 (also referred to as a frequency controlled oscillator), and an input/output unit (IOU) 112.
OEU 102 includes multiple (m) ensemble frequency sources or oscillators 1141-114m that generate corresponding ensemble free-running frequency signals 1161-116m (also referred to as frequencies 1161 . . . 116m). The ensemble oscillators 114 generate frequencies 116 concurrently. Each ensemble oscillator is a stand-alone or free-running oscillator in that system 100 does not control the phase and frequency of the free-running oscillator. OEU 102 provides ensemble free-running frequency signals 116 to respective inputs of TMU 104.
Oscillators 114 can include any type or a mix of types of oscillators, including quartz oscillators, temperature-compensated quartz oscillators, ovenized quartz oscillators, micro-mechanical oscillators, atomic clocks, and so on. Oscillators 114 may be co-located with one another and with the other modules of ensemble oscillator system 100. Alternatively, one or more of oscillators 114 may be located at a remote location with respect to the other oscillators and connected to OEU 102 by either wired or wireless links, and/or a telecommunication network.
Each of free-running frequency signals 116 may be represented as any type of waveform, such as a sine wave, a square wave, a trapezoid or triangle wave, or a pulsed waveform, for example. In an embodiment, each of frequency signals 116 is at approximately the same frequency as the other frequency signals, such as at or near 10 MHz. In another embodiment, the frequencies of frequency signals 116 are substantially different from one another.
SGU 110 generates a controlled frequency signal 120 (also referred to as a controlled frequency 120 or an output signal 120) having a frequency and a phase responsive to a control signal 122, and provides the controlled frequency signal to IOU 112. In an embodiment, the frequency of controlled frequency signal 120 is approximately matched to the frequencies of ensemble frequency signals 116. SGU 110 also generates a feedback signal 124 and provides the feedback signal to an input of TMU 104. Like controlled frequency signal 120, the frequency and phase of feedback signal 124 are responsive to control signal 122. In an embodiment, feedback signal 124 and controlled frequency signal 120 are the same signal. Thus, feedback signal is also referred to herein as controlled frequency signal 124 or controlled frequency signal 124.
IOU 112 receives a master time reference signal 126 (also referred to as a precise time signal 126 or a reference signal 126) from an external source (not shown), in addition to controlled frequency signal 120. Examples of such precise time include GPS time, including, for example, a GPS derived IPPS signal, a JTIDS/ETR signal, and a Have Quick signal. IOU 112 passes precise time signal 126 to TMU 104. From controlled frequency signal 120 and precise time signal 126, IOU 112 produces a plurality of output signals 130. IOU 112 includes logic to produce output signals 130 such as 1/1M/5M PPS signals (where M=106), 1/10/1M/10M PPMinute signals, Have Quick, IRIG-B or other time codes, and 5 MHz/10 MHz/other clocks, etc.
TMU 104 (also referred to as differencer unit 104) measures differences, such as time differences or intervals, for example, between controlled frequency signal 124 and each of (i) time reference signal 126, and (ii) ensemble frequency signals 116, and passes the measured time differences as signals 132 to controller 106. Measured differences (or difference measurements) 132 are indicative of frequency and phase differences between controlled frequency signal 124 and each of (i) time reference signal 126, and (ii) ensemble signals 116. Specifically, measured differences 132 indicate time and frequency-based errors associated with each of (i) controlled frequency 124, and (ii) each or free-running frequencies 116, relative to reference signal 126.
CPU 106 controls ensemble oscillator system 100. CPU 106 includes an Adaptive Kalman Filter Processor (AKFP) 134 (also referred to as an AKFprocessor 134). AKFP 134 processes time interval date 132 to produce control signal 122 for controlling the phase and/or frequency of controlled frequency signal 120 and feedback signal 124. CPU 106 and AKFP are computer based controllers including memory, input/output modules and ports, software or computer program modules, and any other digital and/or analog logic as is know in the art to support all of the functionality required of CPU 106 and AKFP 134 described herein.
CPU 106 also includes an optional sensor module 135 configured to operate as described below.
Environmental sensors 108 include one or more vibration sensors for sensing a level of vibration associated with one or more corresponding oscillators 114. Environmental sensors 108 may also include a temperature sensor for sensing the temperature of oscillators 114. Environmental sensors 108 provide sensed data 136, including vibration levels and temperature, to sensor module 135 of CPU 106. Sensor module 136 passes difference measurements 132 from differencer unit 104 to AKFP 134 only when the sensed vibration levels indicated in signal 136 are below a predetermined level.
TMU
FIG. 2 is a block diagram of an example arrangement of TMU 104. TMU 104 includes a frequency down-converter module 202 and a differencer 208 following module 202. Module 202 frequency down-converts ensemble frequency signals 1161-116m to corresponding lower-frequency ensemble signals or ensemble free-running time signals 212a-212n (also referred to as free-running times 212), such as periodic pulsed signals. Also, module 202 frequency down-converts controlled frequency signal 124 to a lower-frequency controlled time signal 214 (also referred to as a controlled time 214), such as a periodic pulsed signal. In and embodiment, module 202 includes dividers to divide the frequency f of each ensemble frequency signal 116i, and controlled frequency signal 124, to a lower-frequency f/N. In an example, frequency f is 10 MHz and N is 107. Time signals 212 and 214 may each be a 1PPS signal.
Differencer 208 measures time differences between controlled time signal 214 and (i) each of ensemble time signals 212, and (ii) precise time signal 126, to produce time difference signals 132. TMU 104 provides time difference signals 132 to AKFP 134 in the form of a vector including the time differences 132.
SGU
FIG. 3 is a block diagram of an example arrangement of SGU 110. SGU 110 includes a digital-to-analog-converter (DAC) 302 to convert control signal 122 from a digital format into an analog signal 304. A low pass filter (LPF) 306 filters signal 304, to produce a signal 308. A voltage controlled oscillator (VCO) 312 generates controlled frequency signal 120, having its frequency and phase controlled responsive to control signal 308.
Adaptive Kalman Filter Processor (AKFP)
As is known in the art, a frequency source or oscillator generates a frequency having a frequency error y(t) (or “y”). The frequency error tends to wander or drift randomly at a rate. This error is referred to as frequency drift or rate error d(t) (or “d”). The oscillator frequency may be divided-down to produce a time signal. The time signal has a time error x(t) (or “x”). An oscillator error model or set of dynamic relationships, i.e., equations, may be used to quantify the aforementioned oscillator time, frequency and frequency drift errors. In the present invention, AKFP 134 uses or assumes such oscillator error models or dynamic relationships of oscillators in system 100 to transform time interval data 132 into control signal 122.
FIG. 4 is a block diagram of an embodiment of AKFP 134 and a set of oscillator error models 402, 404 and 406 corresponding respectively to oscillators or frequency sources 114, 110 (312) and 112 of system 100, used by the AKFP. Ensemble oscillator error models 4021-402m represent errors associated with ensemble oscillators 1141-114m. Specifically, oscillator error models 4021-402m respectively represent or characterize time errors x1 . . . xm (or X1 . . . m) of respective time signals 2121 . . . 212m, frequency errors y1 . . . ym of respective ensemble signals 1161 . . . 116m, and frequency drift errors d1 . . . dm associated with respective frequency errors y1 . . . ym.
Oscillator error model 404 represents errors associated with VCO 312. Specifically, oscillator error model 404 represents time error x0 of time signal 214, frequency error y0 of signal 124, and frequency drift error d0 associated with frequency error y0.
IOU 112 is modeled as a reference time source that generates a reference frequency, from which is derived precise time signal 126. The reference time source is represented as oscillator error model 406. Oscillator error model 406 represents time error xg of time signal 126.
Models 402 and 404 include unit random noise inputs a, b and c weighted with respective weighting parameters alpha “α,” beta “β,” and gamma “γ,” to derive the time, frequency and frequency drift errors of the respective oscillators. In the embodiment depicted in FIG. 4:
(i) the frequency errors y0, 1 . . . m are represented as first and second integrals of the product of weighting parameters beta0, 1 . . . m and alpha1 . . . m (also referred to herein as random process weighting (RPW) parameters) with unit variance white noise random processes, and
(ii) the time errors x0, 1 . . . m are represented as first integrals of the sum of the fractional frequency errors and the product of weighting parameters gamma0, 1 . . . m with additional unit variance white noise random processes.
Thus, generally, the time, frequency, and frequency drift errors are functions of both the corresponding alpha α, beta β, and gamma γ parameters, and the noise processes. The alpha and beta parameters determine the magnitudes and rates of growth of the variances and covariances of the time, frequency and frequency rate errors.
As mentioned above, TMU 104 generates time differences (i.e., signals 132) between time signal 214 and (i) each of time signals 212, and (ii) time signal 126. Accordingly, also depicted in FIG. 4 are subtractive adders 410 and 412, fed by the respective outputs of error oscillator models 402-406, that model the time differencing action of TMU 104 (i.e., differencer 208) with respect to time errors associated with the oscillators. Adders 410, 412 yield time difference errors z (specifically, errors Z1 . . . zm and zg) as functions of time errors x and error perturbation signals v. These errors are associated with time interval data 132.
AKFprocessor 134 receives time interval data 132. AKFprocessor 134 uses time interval data 132 together with the oscillator error models or dynamic relationships depicted in FIG. 4 to:
(a) compute running estimates of the time, frequency, and frequency drift errors of each of ensemble oscillators 114 and VCO 312 relative to the reference time scale or external time reference provided, e.g., by a GPS timing receiver;
(b) control VCO 312 such that its time and frequency errors are maintained close to zero relative to the reference time scale; and
(c) adaptively estimate the alpha and beta parameters corresponding to each of the ensemble oscillators 114 and VCO 312.
AKFprocessor 134 uses the estimates of the time x, frequency y, and frequency drift d errors of each of ensemble oscillators 114 and VCO 312 as state variables in a Kalman filter process. AKFprocessor 134 also uses the alpha and beta parameters in the Kalman filter process. The estimated errors and alpha and beta parameters are used to update the oscillator error models.
The significance of the alpha and beta parameters is that they determine the rate of error growth in the time and frequency errors of each oscillator. Adaptive estimation of these parameters in the present invention permits the VCO control aspect of AKFprocessor 134 to select weighting parameters such that ensemble oscillators having relatively lower noise (that is, lower time, frequency and frequency drift errors) exert more influence on the frequency control of VCO 312 than do ensemble oscillators associated with relatively higher noise. Also, maintaining time, frequency, and frequency drift estimates on ensemble oscillators 114 during periods of precise time signal (e.g., GPS) availability allows AKFprocessor 134 to maintain tighter control on VCO error growth during precise time signal (e.g., GPS) outages than is possible when there is only one free-running oscillator.
As depicted in FIG. 4, AKFprocessor 134 includes an adaptive Kalman filter 420 that 420 receives time interval measurements 132 and produces (i) state variable estimates x, y and z, and (ii) estimates of the alpha, beta and gamma parameters. AKFprocessor 134 also includes a State Space Controller (SSC) 424 that produces control signal 122 based on the state variable estimates from Kalman filter 420.
FIG. 5 is a flow chart of an example method 500 executed by AKFprocessor 134. Method 500 includes eight (8) main method steps, described below. Process flow proceeds from Step 1 to Step 8 as depicted in FIG. 5. Adaptive Kalman filter 420 executes Steps 1-5, 7 and 8 of method 500, while SSC executes Step 6. Steps 1-8 depicted in FIG. 5 represents both method steps and software modules of AKFP 134.
Listed first below are the overall inputs to and outputs from AKFprocessor 134 and method 500. Then, a description of each of the Steps 1-8 of method 500 ensues. In the embodiment described below, AKFprocessor 134 operates on time, fractional frequency, and fractional frequency rate errors. “Fractional frequency error” is equal to frequency error divided by the nominal center frequency of a given frequency signal, and “fractional frequency rate error” is equal to frequency rate error divided by the nominal center frequency. Frequency error, fractional frequency error, frequency rate error and fractional frequency rate error are generally referred to herein collectively and individually as frequency-based error(s).
Inputs













Z
(m + 1)-dimensional vector of time difference measurements



corresponding to time interval measurements 132



[x1 − x0, . . . , xm − x0, xg − x0]T


SigmaV
(m + 1)-dimensional vector of measurement error σ's


Tcompute
Number of seconds needed to compute the VCO control



values


Tcrank
Number of seconds between input measurement sets to the



Kalman filter


Ncranks
Number of input measurement sets between re-evaluations



of the α, β estimates


kValues
(m + 1,3)-dimensional matrix of initial α, β, γ values



for the m free-running oscillators and the VCO


kMin
(m + 1,2)-dimensional matrix of minimum allowable values



for the α, β estimates


kMax
(m + 1,2)-dimensional matrix of maximum allowable values



for the α, β estimates









Outputs














FracFreqRate
Digital values to be used by the SGU to generate a



VCO control voltage/signal that drives the VCO time and



frequency errors to zero


kEstimate
Current value of the estimated α and β parameters


Out2
Current value of the a posteriori estimates of the time,



frequency, and frequency drift errors of the VCO and m



free-running oscillators


Out3
Square-root of the diagonal of the a posteriori covariance



matrix









Step 1—Initialize Parameters.
Purpose: When the AKFprocessor 134 receives control for the first time, in step numerous parameters and matrices needed by the algorithm, i.e., method 500, are initialized to appropriate values. The parameters initialized and their significance are as follows:













Uk
Current VCO fractional frequency rate control


ControlPoles
Location in the z-plane of the VCO controller's poles


m
Number of free-running oscillators


n
Number of state variables to be estimated


Xm
Current a priori estimates of the state variables



(estimates of x, y and d before incorporating current



measurements)


Xp
Current a posteriori estimates of the state variables



(estimates of x, y and d after incorporating current



measurements)


Pm
Current a priori covariance matrix


Pp
Current a posteriori covariance matrix


Hmatrix
Measurement matrix


dXmdTh
Partials of Xm with respect to α2 and β2


dXpdTh
Partials of Xp with respect to α2 and β2


dPmdTh
Partials of Pm with respect to α2 and β2


dPpdTh
Partials of Pp with respect to α2 and β2


dJdTh
Partials of J with respect to α2 and β2


Twait
Tcrank − Tcompute


PhiVCO
VCO transition matrix with Δt = Tcrank


GammaVCO
VCO input matrix with Δt = Tcrank


PhiCompute
VCO transition matrix with Δt = Tcompute


GammaCompute
VCO input matrix with Δt = Tcompute


PhiWait
VCO transition matrix with Δt = Twait


GammaWait
VCO input matrix with Δt = Twait


PhiMatrix
Transition matrix for all n states, Δt = Tcrank









Inputs: None.

Outputs: The parameters listed above.

Step 2—Save Input Parameters.
Purpose: Since parameters Tcompute, Tcrank, Ncranks, kValues, kMin, and kMax are usually constant over many calls to the AKFprocessor 134, these values are saved in persistent memory when supplied, and the saved values are used when the input parameters are not supplied. In addition, a few parameters that are derived from the input parameters are also computed and saved in persistent memory.
Inputs: Tcompute, Tcrank, Ncranks, kValues, kMin, kMax, m
Outputs: Tc=Saved value of Tcompute
dT=Saved value of Tcrank
Nc=Saved value of Ncranks
Theta=[β12, . . . , β(m+1)2, α12, . . . , αm2]T 
Finv=diag(Theta2)
dF=(2m+1,2m+1) matrix of 0's
FinvDiv=[(m+1) 1's, (m) 10's]
ThMin=kmin2 
ThMax=kMax2 
Step 3—Kalman Measurement Update.
Purpose: Adaptive processing starts with a standard Kalman Measurement Update as described by the equations shown below. This results in an optimal estimate of the state vector up to and including the current set of measurements. It also provides the matrices needed in Steps 4 and 5 for adaptive estimation of the α and β parameters.

    
    
        I=Vector of indexes for which SigmaVi>0
        H=Ith rows of Hmatrix
        Rmatrix=diag(SigmaVi2)
        B=H*Pm*HT+Rmatrix
        Binv=B−1 
        Kmatrix=Pm*HT*B−1 
        ImKH=I−Kmatrix*H
        Nu=Zi−H*Xm
        Xp=Xm+Kmatrix*Nu
        Pp=ImKH*Pm

Inputs: SigmaV, Hmatrix, Pm, Xm, Theta

Outputs: H=Measurement matrix for the measurements being used

    
    


Binv=Inverse of the innovations covariance matrix
ImKH=a priori to a posteriori gain matrix
Nu=Current measurement residuals (innovations)
Xp=a posteriori state variable estimates
Pp=a posteriori covariance matrix
kEstimate=Current α and β estimates (optional AKFprocessor 134 output)
Out2=Current Xp estimates (optional AKFprocessor 134 output)
Out3=sqrt(diag(Pp))=Current state uncertainty levels (optional output)
Step 4—Adaptive Measurement Update.
Purpose: Together, Steps 4 and 5 adaptively estimate the α and β parameters. These parameters determine the rate of growth of a clock/oscillator's time and frequency uncertainty. In a given class of oscillator (e.g., ovenized crystal oscillators) these parameters can vary by factors of 2-10, and between different classes of oscillators (e.g., crystal vs. atomic oscillators) these parameters can vary by several orders of magnitude. In an ordinary (non-adaptive) Kalman filter, these parameters are typically specified by a human operator/analyst. This is frequently done by a separate (i.e., side) analysis of time sequences of clock/oscillators that are similar to those being processed, and usually requires an iterative trial-and-error process in which the performance of the Kalman filter with a certain set of parameter values is observed and then modified until acceptable performance is achieved. The AKFprocessor 134 of the present invention uses the principles of advanced statistical estimation theory to estimate the parameter values from data provided by the actual oscillators being processed, including the ensemble of free-running oscillators 114. The general principles for adaptive Kalman filter processing are given in References [1], [2] and [3], each of which is incorporated herein by reference in its entirety. Step 4 computes matrices that are used in Step 5 to estimate the α and β parameter estimates. The equations used in Step 4 to compute the matrices are as follows:

dNudTh(i)=−H*dXmdTh(i)

dBdTh(i)=H*dPmdTh(i)*HT

dKdTh(i)=ImKH*dPmdTh(i)*HT*Binv

dXpdTh(i)=ImKH*dXmdTh(i)+dKdTh(i)*Nu

dPpdTh(i)=ImKH*dPmdTh(i)*ImKHT





  
    
      
        
          dJdTh
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          +
          
            
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          -
        
      
    
  
  
    
      
        
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          where
          ⁢
          
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Inputs: H, dXmdTh, dPmdTh, ImKH, Binv, Nu, dJdTh, dF
Outputs: djdTh=Partials of J with respect to α2 and β2, where J is a conditional information matrix that represents a measurement of the quality of the alpha and beta estimates. That is, J represents how close to reality the alpha and beta estimates are. Therefore, J is a quantity to be minimized, e.g., when estimates of alpha and beta are correct, J is zero.
dF=Information added to the Fisher information matrix over the current Ncranks accumulation interval
Step 5—Compute New α and β Estimates.
Purpose: New Maximum Likelihood (ML) estimates of α2 and β2 are computed in this step on a schedule of one new set of estimates per Ncranks calls to the AKFprocessor 134. The equations used to calculate these estimates are as follows:

diag(Finv)=(1+1/FinvDiv)*diag(Finv)

Finv=inv[inv(Finv)+dF]

Theta=max{ThMin, min[ThMax, Theta−Finv*dJdTh)]}

dXmdTh=0 dPmdTh=0

dXpdTh=0 dPpdTh=0

dJdTh=0 dF=0

p=0

Inputs: Finv, FinvDiv, dF, dJdTh, Theta, ThMin, and ThMax

Outputs: Theta=(2m+1)-dimensional vector of α2 and β2 estimates

dXmdTh=Partials of Xm with respect to α2 and β2 
dXpdTh=Partials of Xp with respect to α2 and β2 
dPmdTh=Partials of Pm with respect to α2 and β2 
dPpdTh=Partials of Pp with respect to α2 and β2 
dJdTh=Partials of J with respect to α2 and β2 
dF=Partial sum of Fisher information matrices
p=Crank counter (number of AKFprocessor 134 calls since last step-5 update)
Step 6—Compute VCO Control Value.
Purpose: This component of the AKFprocessor 134 uses estimates of the current time and fractional frequency error of the VCO/clock 312 produced by the adaptive Kalman filter to compute what the rate of change of fractional frequency error (Uk=dyVCO/dt) must be so that both the time and fractional frequency errors of the VCO/clock are quickly driven to zero. These control values are converted into a time-varying control voltage V(t) in the Signal Generation Unit (SGU) and are applied as input to the VCO to achieve the desired time and frequency control for this oscillator. The control values are calculated by the equation

Uk=−Kcontrol *xyVCO,

    
    
        where xyVCO is a 2-element column vector containing the predicted time and fractional frequency error of the VCO/clock at time (tk+tCompute), where tk represents current time and tCompute represents the amount of time needed by the CPU computer to process the measurements taken at time tk. xyVCO is calculated from the a posteriori state estimate Xp(tk) produced by the adaptive Kalman filter according to the equation

xyVCO=PhiCompute*Xp(1:2)+GammaCompute*Uk.

        Kcontrol is a 2-element row vector of control gains produced by the Ackerman design procedure described in Ref. [3]:

Kcontrol=nth row of P=C−1αC(Φ) where

C=[Γ, ΦΓ, . . . , Φn−1Γ]

αC(z)=Desired characteristic polynomial for VCO control

Φ=VCO transition matrix (PhiVCO) and

Γ=VCO control input matrix (GammaVCO).

    
    
    Inputs: PhiVCO, GammaVCO, ControlPoles, PhiCompute, GammaCompute, Xp, Uk
    Outputs: xyVCO=Estimated VCO time/frequency error at time (tk+tCompute)
    
        Uk=VCO fractional frequency rate control value
        FracFreqRate=Uk
    
    


Step 7—Kalman Time Update.
Purpose: This step produces the expected a priori state vector (Xm) and associated covariance matrix (Pm) for the next measurement time, tk+1=tk+Tcrank. The equations used for computing these quantities are as follows:




  
    
      
        
          Xm
          ⁡
          
            (
            
              1
              ⁢
              
                :
              
              ⁢
              2
            
            )
          
        
        =
        
          
            PhiWait
            *
            xyVCO
          
          +
          
            GammaWait
            *
            Uk
          
        
      
    
  
  
    
      
        
          Xm
          ⁡
          
            (
            
              3
              ⁢
              
                :
              
              ⁢
              n
            
            )
          
        
        =
        
          
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            ⁡
            
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                  n
                
                ,
                
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        =
        
          
            
              
                γ
                ^
              
              i
              2
            
            *
            dT
          
          +
          
            
              (
              
                1
                /
                3
              
              )
            
            ⁢
            
              
                β
                ^
              
              i
              2
            
            *
            
              dT
              3
            
          
          +
          
            
              (
              
                1
                /
                20
              
              )
            
            ⁢
            
              
                α
                ^
              
              i
              2
            
            *
            
              dT
              5
            
          
        
      
    
  
  
    
      
        
          
            q
            12
          
          ⁡
          
            (
            i
            )
          
        
        =
        
          
            
              (
              
                1
                /
                2
              
              )
            
            ⁢
            
              
                β
                ^
              
              i
              2
            
            *
            
              dT
              2
            
          
          +
          
            
              (
              
                1
                /
                8
              
              )
            
            ⁢
            
              
                α
                ^
              
              i
              2
            
            *
            
              dT
              4
            
          
        
      
    
  
  
    
      
        
          
            q
            13
          
          ⁡
          
            (
            i
            )
          
        
        =
        
          
            (
            
              1
              /
              6
            
            )
          
          ⁢
          
            
              α
              ^
            
            i
            2
          
          *
          
            dT
            3
          
        
      
    
  
  
    
      
        
          
            q
            22
          
          ⁡
          
            (
            i
            )
          
        
        =
        
          
            
              
                β
                ^
              
              i
              2
            
            *
            dT
          
          +
          
            
              (
              
                1
                /
                3
              
              )
            
            ⁢
            
              
                α
                ^
              
              i
              2
            
            *
            
              dT
              3
            
          
        
      
    
  
  
    
      
        
          
            q
            23
          
          ⁡
          
            (
            i
            )
          
        
        =
        
          
            (
            
              1
              /
              2
            
            )
          
          ⁢
          
            
              α
              ^
            
            i
            2
          
          *
          
            dT
            2
          
        
      
    
  
  
    
      
        
          
            q
            33
          
          ⁡
          
            (
            i
            )
          
        
        =
        
          
            
              α
              ^
            
            i
            2
          
          *
          dT
        
      
    
  
  
    
      
        
          Qmatrix
          =
          
            diag
            ⁡
            
              (
              
                
                  Q
                  0
                
                ,
                
                  Q
                  1
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  Q
                  m
                
              
              )
            
          
        
        ,
        where
      
    
  
  
    
      
        
          Q
          0
        
        =
        
          
            
              [
              
                
                  
                    
                      
                        q
                        22
                      
                      ⁡
                      
                        (
                        0
                        )
                      
                    
                  
                  
                    
                      
                        q
                        23
                      
                      ⁡
                      
                        (
                        0
                        )
                      
                    
                  
                
                
                  
                    
                      
                        q
                        23
                      
                      ⁡
                      
                        (
                        0
                        )
                      
                    
                  
                  
                    
                      
                        q
                        33
                      
                      ⁡
                      
                        (
                        0
                        )
                      
                    
                  
                
              
              ]
            
            ⁢
            
              
            
            ⁢
            and
            ⁢
            
              
            
            ⁢
            
              Q
              i
            
          
          =
          
            ⌊
            
              
                
                  
                    
                      q
                      11
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
                
                  
                    
                      q
                      12
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
                
                  
                    
                      q
                      13
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
              
              
                
                  
                    
                      q
                      12
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
                
                  
                    
                      q
                      22
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
                
                  
                    
                      q
                      23
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
              
              
                
                  
                    
                      q
                      13
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
                
                  
                    
                      q
                      23
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
                
                  
                    
                      q
                      33
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                
              
            
            ⌋
          
        
      
    
  
  
    
      
        Pm
        =
        
          
            PhiMatrix
            *
            Pp
            *
            
              PhiMatrix
              T
            
          
          +
          Qmatrix
        
      
    
  




Inputs: PhiWait, GammaWait, xyVCO, Uk, PhiMatrix, Theta, dT, Xp, Pp

Outputs: Xm=a priori state vector estimate for next measurement time

Pm=a priori covariance matrix for next measurement time
Step 8—Adaptive Time Update.
Purpose: Calculations needed to complete the adaptive estimation of the α and β parameters are executed in this step. In this case the partials of Xm and Pm at the next measurement time, tk+1=tk+tCrank, are computed from the partials dXpdTh and dPpdTh generated in Step 4 for time tk. The equations to be used for these calculations are as follows:




  
    
      
        dXmdTh
        =
        
          PhiMatrix
          *
          dXpdTh
        
      
    
    
      
        (
        a
        )
      
    
  
  
    
      
        
          
            dPmdTh
            ⁡
            
              (
              i
              )
            
          
          =
          
            PhiMatrix
            *
            
              dPpdTh
              ⁡
              
                (
                i
                )
              
            
            *
            
              PhiMatrix
              T
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              dPmdTh
              
                j
                ,
                j
              
            
            ⁡
            
              (
              i
              )
            
          
          =
          
            
              
                dPmdTh
                
                  j
                  ,
                  j
                
              
              ⁡
              
                (
                i
                )
              
            
            +
            
              ⌊
              
                
                  
                    
                      
                        (
                        
                          1
                          /
                          3
                        
                        )
                      
                      ⁢
                      
                        dT
                        3
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          2
                        
                        )
                      
                      ⁢
                      
                        dT
                        2
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          1
                          /
                          2
                        
                        )
                      
                      ⁢
                      
                        dT
                        2
                      
                    
                  
                  
                    dT
                  
                
              
              ⌋
            
          
        
        ⁢
        
          

        
        ⁢
        
          (
          
            
              where
              ⁢
              
                
              
              ⁢
              i
            
            =
            
              
                0
                ⁢
                
                  
                
                ⁢
                and
                ⁢
                
                  
                
                ⁢
                j
              
              =
              
                [
                
                  1
                  ,
                  2
                
                ]
              
            
          
          )
        
      
    
    
      
        (
        b
        )
      
    
  
  
    
      
        
          
            dPmdTh
            ⁡
            
              (
              i
              )
            
          
          =
          
            PhiMatrix
            *
            
              dPpdTh
              ⁡
              
                (
                i
                )
              
            
            *
            
              PhiMatrix
              T
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              dPmdTh
              
                j
                ,
                j
              
            
            ⁡
            
              (
              i
              )
            
          
          =
          
            
              
                dPmdTh
                
                  j
                  ,
                  j
                
              
              ⁡
              
                (
                i
                )
              
            
            +
            
              ⌊
              
                
                  
                    
                      
                        (
                        
                          1
                          /
                          3
                        
                        )
                      
                      ⁢
                      
                        dT
                        3
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          2
                        
                        )
                      
                      ⁢
                      
                        dT
                        2
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          1
                          /
                          2
                        
                        )
                      
                      ⁢
                      
                        dT
                        2
                      
                    
                  
                  
                    dT
                  
                
              
              ⌋
            
          
        
        ⁢
        
          

        
        ⁢
        
          (
          
            
              where
              ⁢
              
                
              
              ⁢
              i
            
            =
            
              
                
                  [
                  
                    1
                    ⁢
                    
                      :
                    
                    ⁢
                    m
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                and
                ⁢
                
                  
                
                ⁢
                j
              
              =
              
                [
                
                  
                    3
                    ⁢
                    i
                  
                  ,
                  
                    
                      3
                      ⁢
                      i
                    
                    +
                    1
                  
                
                ]
              
            
          
          )
        
      
    
    
      
        (
        c
        )
      
    
  
  
    
      
        
          
            dPmdTh
            ⁡
            
              (
              i
              )
            
          
          =
          
            PhiMatrix
            *
            
              dPmdTh
              ⁡
              
                (
                i
                )
              
            
            *
            
              PhiMatrix
              T
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              dPmdTh
              
                j
                ,
                j
              
            
            ⁡
            
              (
              i
              )
            
          
          =
          
            
              
                dPmdTh
                
                  j
                  ,
                  j
                
              
              ⁡
              
                (
                i
                )
              
            
            +
            
              ⌊
              
                
                  
                    
                      
                        (
                        
                          1
                          /
                          20
                        
                        )
                      
                      ⁢
                      
                        dT
                        5
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          8
                        
                        )
                      
                      ⁢
                      
                        dT
                        4
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          6
                        
                        )
                      
                      ⁢
                      
                        dT
                        3
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          1
                          /
                          8
                        
                        )
                      
                      ⁢
                      
                        dT
                        4
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          3
                        
                        )
                      
                      ⁢
                      
                        dT
                        3
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          3
                        
                        )
                      
                      ⁢
                      
                        dT
                        2
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          1
                          /
                          6
                        
                        )
                      
                      ⁢
                      
                        dT
                        3
                      
                    
                  
                  
                    
                      
                        (
                        
                          1
                          /
                          2
                        
                        )
                      
                      ⁢
                      
                        dT
                        2
                      
                    
                  
                  
                    dT
                  
                
              
              ⌋
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            (
            
              
                where
                ⁢
                
                  
                
                ⁢
                i
              
              =
              
                
                  
                    [
                    
                      m
                      +
                      
                        1
                        ⁢
                        
                          :
                        
                        ⁢
                        2
                        ⁢
                        m
                      
                    
                    ]
                  
                  ⁢
                  
                    
                  
                  ⁢
                  and
                  ⁢
                  
                    
                  
                  ⁢
                  j
                
                =
                
                  [
                  
                    
                      3
                      ⁢
                      i
                    
                    ,
                    
                      
                        3
                        ⁢
                        i
                      
                      +
                      1
                    
                  
                  ]
                
              
            
            )
          
          .
        
      
    
    
      
        (
        d
        )
      
    
  




Inputs: PhiMatrix, dXpdTh, dPpdTh, and dT.

Outputs: dXmdTh=Partials of Xm(tk+1) with respect to α2 and β2 

dPmdTh=Partials of Pm(tk+1) with respect to α2 and β2.
FIG. 6 is a flow chart of an example method 600 summarizing the flow and some of the important features of Steps 1-8 from method 500. Entering method 600 it is assumed that AKFP 134 has already passed through initializing Steps 1 and 2, and is now operating in a steady-state condition wherein system 100 continuously passes difference measurements 132 to AKFP 134, and AKFP 134 has already been cyclically generating various inputs to and outputs from the Steps 1-8.
It is also assumed that AKFP 134 uses the dynamic relationships depicted in FIG. 4 to represent time errors and frequency-based errors (e.g., fractional frequency and fractional frequency rate errors) associated with oscillators 114 and 110. The errors are relative to the reference signal 126. The alpha and beta parameters mentioned above are random process weighting (RPW) parameters that determine the magnitudes and rates of growth of the variances and covariances of the time and frequency-based errors.
A first step 602 of method 600 includes adaptively and optimally estimating the RPW parameters (i.e., the alpha and beta parameters) associated with each free-running frequency (e.g., of free-running frequency signals 116). These RPW estimates effectively indicate relative noise levels associated with each free-running oscillator or its free-running frequency.
A next step 604 includes generating, using the RPW parameter (i.e., alpha and beta) estimates from step 602, variances and covariances associated with each free-running frequency.
A next step 606 includes generating, using the variances and covariances from step 604, error weights associated with each free-running frequency.
A next step 608 includes optimally estimating weighted time and frequency-based errors associated with each free-running frequency, wherein the weighted error estimates are weighted with the corresponding error weights from step 606.
Steps 604, 606 and 608 collectively represent a step of deriving, based on the RPW parameters from the first module, weighted time and frequency-based error estimates associated with each free-running frequency, wherein the weighted errors estimates are weighted to favor less noisy free-running frequencies over more noisy free-running frequencies.
A next step 610 includes optimally estimating, based on the weighted error estimates from step 608, time and frequency-based errors associated with the controlled frequency (e.g., of controlled frequency signal 124). Steps 602-610 can be performed by adaptive Kalman filter 420.
A next step 612 includes generating a frequency control signal (e.g., frequency control signal 122) based on the error estimates from step 610. State Space Controller 424 can perform step 612. In steps 604-612, lower noise oscillators are used more heavily than higher noise oscillators in generating the frequency controlled signal. Specifically, the weighted error estimates from step 608 favor the lower noise oscillators, as indicated by the RPW parameters.
CONCLUSION
While the above description contains many specifics, these specifics should not be construed as limitations of the invention, but merely as exemplifications of preferred embodiments thereof. Those skilled in the art will envision many other embodiments within the scope and spirit of the invention as defined by the claims appended hereto.
REFERENCES 


    [1] M. D. Shuster, D. W. Porter, and B. D. Gibbs: “Maximum Likelihood Estimation Techniques for Trident Missile System Parameters”, Business and Technological Systems, Inc. report number BTS10-82-64/rb 1037, September 1983.
    [2] P. S. Maybeck: “Stochastic Models, Estimation, and Control”, vol. 2, Academic Press, New York, 1982.
    [3] G. F. Franklin, J. D. Powell, and M. Workman: “Digital Control of Dynamic Systems”, 3rd Ed., Addison-Wesley, Menlo Park, Calif., 1998.

## Claims

1. An ensemble clock, comprising:
an input for receiving a reference signal;
multiple free-running oscillators each configured to generate a corresponding free-running frequency;
an output oscillator configured to generate a controlled frequency having a frequency responsive to a control signal;
a differencer unit configured to derive difference measurements indicative of time and frequency-based errors associated with each of (i) the controlled frequency, and (ii) the free-running frequency, relative to the reference signal; and
a controller configured to generate the control signal responsive to the difference measurements.

2. The ensemble clock of claim 1, wherein the differencer unit comprises:
first circuitry configured to derive from each free-running frequency a corresponding free-running time;
second circuitry configured to derive from the controlled frequency a controlled time; and
third circuitry configured to measure time differences between the controlled time and each of
(i) the reference signal, and
(ii) the free-running times,
wherein the difference measurements include the time differences.

3. The ensemble clock of claim 2, wherein the reference signal is a pulsed signal and each free-running time is a pulsed signal.

4. The ensemble clock of claim 2, wherein:
the first circuitry is configured to divide-down each free-running frequency to produce each free-running time; and
the second circuitry is configured to divide-down the controlled frequency to produce the controlled time.

5. The ensemble clock of claim 1, wherein the controller includes an adaptive Kalman filter processor configured to generate the control signal based on the measured differences so that control signal drives toward zero the time and frequency-based errors associated with the controlled frequency.

6. The ensemble clock of claim 5, wherein the adaptive Kalman filter processor includes:
an adaptive Kalman filter configured to estimate, based on the difference measurements, the time and frequency-based errors associated with the controlled frequency and each free-running frequency; and
a state space controller that generates the control signal based on the error estimates from the adaptive Kalman filter.

7. The ensemble clock of claim 5, wherein the Adaptive Kalman Filter Processor comprises:
(a) dynamic relationships associated with each free-running frequency and the controlled frequency to represent the time and frequency-based errors corresponding to variances and covariances having magnitudes and rates of growth determined by random process weighting (RPW) parameters, whereby the RPW parameters are indicative of relative noise levels associated with the corresponding frequencies; and
(b) an adaptive Kalman Filter including the following modules that, when operated in a steady-state condition, use the dynamic relationships and the difference measurements to produce their respective results:
(i) a first module configured to adaptively estimate the RPW parameters associated with each free-running frequency;
(ii) a second module configured to derive, based on the RPW parameters from the first module, weighted time and frequency-based error estimates associated with each free-running frequency, wherein the weighted errors estimates are weighted to favor less noisy free-running frequencies over more noisy free-running frequencies; and
(iii) a third module configured to estimate, based on the weighted error estimates from the second module, time and frequency-based errors associated with the controlled frequency; and

(c) a state space controller configured to generate the control signal based on the error estimates from the third module such that the control signal causes the time and frequency-based errors associated with the controlled frequency to be driven toward, and maintained over time near, zero with respect to the reference frequency.

8. The ensemble clock of claim 7, wherein the second module of the adaptive Kalman filter includes means for performing the following processing steps:
(I) generate, based on the RPW parameter estimates from the first module of the adaptive Kalman filter, variances and covariances associated with each free-running frequency;
(II) generate, using the variances and covariances from step (I), error weights associated with each free-running frequency; and
(III) estimate the weighted time and frequency-based errors associated with each free-running frequency, wherein the weighted error estimates are weighted with the corresponding error weights from step (II).

9. The ensemble clock of claim 5, further comprising:
a vibration sensor, coupled with the controller, for sensing a level of vibration of at least one of the multiple ensemble oscillators, wherein the controller includes a sensor module configured to pass the difference measurements from the differencer unit to the adaptive Kalman filter only when the sensed vibration level is below a predetermined level.

10. The ensemble clock of claim 1, wherein each of the free-running oscillators includes a crystal oscillator.

11. The ensemble clock of claim 1, wherein the multiple free-running oscillators includes at least three free-running oscillators.

12. The ensemble clock of claim 1, wherein the output oscillator includes a voltage controlled oscillator responsive to the control signal.

13. The ensemble clock of claim 1, wherein each free-running frequency is a continuous wave frequency.

14. The ensemble clock of claim 1, wherein the frequency-based errors include fractional frequency errors and fractional frequency rate errors.

15. The ensemble clock of claim 1, wherein the reference signal is a GPS time signal.

16. An ensemble clock, comprising:
an input for receiving a reference signal;
multiple free-running oscillators each configured to generate a corresponding free-running frequency;
an output oscillator configured to generate a controlled frequency having a frequency responsive to a control signal;
a differencer unit configured to derive, at least indirectly, time difference measurements between the controlled frequency and each of
(i) the reference signal, and
(ii) the free-running frequencies; and

a controller including a adaptive Kalman filter processor configured to generate the control signal responsive to the time difference measurements.

17. A method of precision timekeeping, comprising:
(a) receiving a reference signal;
(b) generating multiple free-running frequencies;
(c) generating a controlled frequency having a frequency responsive to a control signal;
(d) deriving difference measurements indicative of time and frequency-based errors associated with each of (i) the controlled frequency, and (ii) each free-running frequency, relative to the reference signal; and
(e) generating the control signal responsive to the difference measurements.

18. The method of claim 17, wherein step (d) comprises:
deriving from each free-running frequency a corresponding free-running time;
deriving from the controlled frequency a controlled time; and
measuring time differences between the controlled time and each of
(i) the reference signal, and
(ii) the free-running times,
wherein the difference measurements include the time differences.

19. The method of claim 18, wherein:
deriving each free-running time includes dividing-down each free-running frequency to produce the corresponding free-running time; and
deriving the controlled time includes dividing-down the controlled frequency to produce the controlled time.

20. The method of claim 17, wherein step (e) comprises performing an adaptive Kalman filter process over the measured differences so as to generate the control such that the control signal drives toward zero the time and frequency-based errors associated with the controlled frequency.

21. The method of claim 20, wherein step (e) comprises:
(e)(i) performing an adaptive Kalman filter process over the difference measurements to estimate the time and frequency based-errors associated with the controlled frequency and each free-running frequency; and
(e)(ii) generating the control signal based on at least some of the error estimates from step (e)(i).

22. The method of claim 20, wherein step (e) further comprises:
(e)(i) using dynamic relationships associated with each free-running frequency and the controlled frequency to represent time and frequency-based errors corresponding to variances and covariances having magnitudes and rates of growth determined by random process weighting (RPW) parameters associated with each of the frequencies, whereby the RPW parameters are indicative of relative noise levels associated with the corresponding frequencies;
(e)(ii) the following adaptive Kalman filtering steps that, when executed in a steady-state condition, use the dynamic relationships and the difference measurements to produce their respective results:
adaptively estimating the RPW parameters associated with each free-running frequency;
deriving, based on the RPW parameter estimates, weighted time and frequency-based error estimates associated with each free-running frequency, wherein the weighted errors estimates are weighted to favor less noisy free-running frequencies over more noisy free-running frequencies; and
estimating, based on the weighted error estimates from the deriving step, time and frequency-based errors associated with the controlled frequency; and

(e)(iii) generating the control signal based on the error estimates associated with the controlled frequency.

23. The method of claim 22, wherein the deriving step of step (e)(ii) comprises:
generating, based on the RPW parameter estimates, variances and covariances associated with each free-running frequency;
generating, using the variances and covariances, error weights associated with each free-running frequency; and
estimating the weighted time and frequency-based errors associated with each free-running frequency, wherein the weighted error estimates are weighted with the corresponding error weights from the generating error weights step.

24. The method of claim 20, further comprising:
prior to step (e), sensing a level of mechanical vibration associated with said generating of the multiple frequency signals in step (b); and
passing the difference measurements to the adaptive Kalman filter process only when the sensed vibration level is below a predetermined level.

25. The method of claim 17, wherein the reference signal is a GPS time signal.

26. The method of claim 17, wherein the frequency-based errors include fractional frequency errors and fractional frequency rate errors.