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[1] The previously published linearized continuum model of a linear (one-dimensional) array of mutually injection locked oscillators is generalized to include the effects of time delay in the interoscillator coupling. This generalization is motivated by a desire for causal solutions describing the dynamics of the phase distribution in the aperture of a phase array antenna driven by such an array of oscillators. The solutions for the phase dynamics will, in general, be noncausal unless the coupling delay is taken into account. In the present formulation the coupling delay is represented by an exponential factor introduced in the Laplace transform of the partial differential equation arising in the continuum model. As a result of this, the transform of the phase distribution exhibits an infinite set of branch points in the complex frequency plane and the inverse transform is computed as the sum of the integrals around the branch cuts.

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[2] Some years ago, Pogorzelski et al. [1999] introduced a linearized continuum analysis of the dynamic behavior of the phase distribution in an array of mutually injection locked oscillators. Such arrays are useful in providing excitations for the elements of a phased array antenna as shown in Figure 1. The most attractive feature of such arrays is that beam agility is easily implemented using the scheme proposed by Liao and York [1993]. They showed that by antisymmetric detuning of the end oscillators of the array, one may produce linear phase distributions across the array and thus steer the radiated beam. The continuum analysis of Pogorzelski et al. [1999] was based on modeling the phase distribution as a continuous function of time and position in the array. This continuous function was shown to satisfy a partial differential equation of diffusion type wherein the excitation was a function describing the detuning of the oscillators from the ensemble frequency. Solutions of the equation were easily obtained analytically via Laplace transformation and the results provided considerable insight into the behavior of such arrays in both one and two dimensions in analogy with heat conduction. However, as in the previous discrete formalism [Pogorzelski, 2008], the solutions were not causal in that the response at any point in the array began at the same time as the excitation no matter how far apart were the excitation and response points. There was no provision in the theory for the finite speed of propagation in the medium; that is, no provision for the time delay in the interoscillator coupling lines. The present paper describes an effort to account for such time delay in the analysis of the array within the linear continuum approximation used in the previous analysis that neglected delay.

2. Continuum Model With Coupling Delay

[3] The continuum model expresses the phase distribution across the array as a continuous function taking on the value of the phase of each oscillator as the independent variable takes on the value corresponding to the location of that oscillator in the array. Thus adapting the solution from the linearized discrete model, one may view the ideal solution for the Laplace transform of the dynamic phase distribution across the array in terms of the continuum model as the continuous function,

where s (measured in locking ranges) is the transform variable conjugate to time (measured in inverse locking ranges), x is the point at which the phase is measured and y is the independent variable of the tuning distribution over the array [see Pogorzelski, 2008, equation (25)]. The function _{tune}(y) is the Laplace transform of the difference between the free running frequency of the oscillator at position y and the steady state ensemble frequency of the array prior to detuning which is taken as the reference frequency, measured in locking ranges. The problem to be addressed here is the derivation of an approximate differential equation for this continuous phase distribution in the manner presented by Pogorzelski et al. [1999] in the absence of coupling delay, this time accounting for the delay.

[4] We begin with the discrete equation,

that describes the behavior of the array with coupling delay, d, measured in inverse locking ranges [see Pogorzelski, 2008, equation (6)]. Viewing the index i, as a continuous variable, x, and j as a continuous variable, y, this equation may be rewritten in the form,

or, approximating to second order in Δx with Δx = Δi = 1,

Setting,

representing step detuning at time zero of the oscillator at location y, the Green's function g(s,x,y) is immediately apparent. It is,

We note in passing that if d is set to zero, g becomes,

which agrees with the result obtained in previous work [Pogorzelski et al., 1999] and has the well known inverse Laplace transform,

However, returning now to the nonzero d case, numerical computation of the inverse Laplace transform of (6) reveals that the influence of the nearest neighboring oscillators of a detuned one is evident at a time only one delay time, d, after the detuning begins. Because the round trip delay between two neighboring oscillator is 2d, not d, this violates causality.

[5] The key to resolving this causality dilemma may be found in comparing (1) with (6) with x = y. Bearing in mind that (1) is the exact solution in this linear approximation, we note that the denominator of (1) is,

These denominators introduce branch points where the quantity under each radical is zero. That is, (9) has branch points where (s + 2) = 2 e^{−sd} and where (s + 2) = −2 e^{−sd} whereas (10) only has branch points where (s + 2) = 2 e^{−sd}. Now, envisioning inversion of the Laplace transforms (1) and (6) by deformation of the Bromwich contour around the branch points and their associated cuts, we conclude that (6) will lead to a solution that is missing the contributions from the branch points where (s + 2) = −2 e^{−sd}. It will be shown that this is the root of the causality difficulty. Before resolving that issue however, it will be important to first understand the meaning of the branch cut contributions to the inversion integral and the reason half of them are absent from (6).

[6]Figure 2 is a plot of the branch point locations in the complex s plane for d = 2 inverse locking ranges. The dots denote the branch points associated with and the circles denote those associated with . (See Appendix A.) We will take all of the branch cuts to extend horizontally to the left from each branch point and deform the Bromwich contour to the left around each cut. The inversion integral will thus be converted to a sum of the contributions from each branch cut integration. Let us focus first on the cuts associated with . The continuum model represented by equation (4) is a good approximation when the phase is spatially slowly varying so that spatial derivatives of higher order than the second can be neglected. In the case of the Green's function, the expression on the right side of (4) is zero everywhere except at x = y. When x = y, the second derivative term cancels the right hand side. Therefore, for all values of x and y, the second spatial derivative of g will be small when (s + 2) is nearly equal to 2 e^{−sd}. Thus, the continuum model represented by (4) is most accurate near this subset of the branch points, the solid dots in Figure 2. It is not accurate far from the curve on which the branch points lie. In particular, it is not accurate in the right half plane. In Laplace transform theory, we associate the time at which the time function switches on with the time after which it is no longer permitted to close the inversion contour in the right half plane where the transform is presumably analytic. However, as pointed out above, the approximations leading to (6) fail in the right half plane so any conclusions drawn from (4) based on its behavior there are not valid.

[7] We now investigate the meaning of the portion of the solution associated with the circles in Figure 2. Clearly, one may transition from the dots to the circles by merely replacing e^{−sd} with −e^{−sd}. If this is done in equation (2) it becomes immediately evident that the circles in Figure 2 are associated with phase behavior that alternates in sign from one oscillator to the next in the array. Note that this is not to say that the oscillations of adjacent oscillators are 180 degrees out of phase. Rather, the variations of the phase from one oscillator to the next alternate in sign. The form of the partial differential equation corresponding to (4) may be obtained by reversing the sign of the exponential delay factor in (4) yielding,

in which (s, x, y) is a slowly varying function describing the phase distribution in an array with coupling that includes a sign reversal from one oscillator to the next. However, as mentioned above, from (2) is clear that a distribution f_{i} = −1 _{i} would be supported by a similar array without the sign reversal in the coupling. That is,

is an admissible phase distribution for the original array with coupling delay d and no sign reversals. Solving (11) and using (12) one obtains the solution corresponding to the circle branch points in Figure 2 and the analog of (6) is,

where i here is the square root of minus one not to be confused with the index I that has been replaced by x. The branch of the square root has been chosen to make the imaginary part negative so that the solution decreases with |x – y|. This choice makes the real part of the square root positive above the cut. It should be noted that this solution could not have been obtained directly from (4) because (4) embodies the assumption of small phase variation from one oscillator to its nearest neighbors to justify replacement of the finite differences with the second derivative. This same assumption is embodied in (11) to obtain the second derivative of (s, x, y) but would not be valid for f(s, x, y). This explains the absence of half of the branch points in (6) and the other half in (13).

[8] It remains to determine how much of each solution, (6) and (13), should be used to obtain the correct result. That is, we have

and we wish to determine the constants A and B with the condition A + B = 1 to produce the correct right hand side of the differential equation. Setting x = y, equation (14) becomes,

In the limit of large s, the inverse Laplace transforms of the two terms in (15) become Fourier-like series by virtue of the branch cut locations. That is,

These expressions are termed “Fourier-like” because of the τ dependence of the coefficients. Recall that they are accurate for large values of m and n; i.e., high frequencies. The high frequency portions of the spectrum are responsible for any discontinuities in the function or its derivatives. Based on these approximations, the expression for the time derivative of the phase distribution, which is the inverse Laplace transform of sg(s,y,y), becomes,

where,

Now, the temporal discontinuities in (19) or its derivatives are representative of the interactions among the oscillators in the array. For example, a discontinuity in the phase of the detuned oscillator as a function of time is expected at time 2d when the effect of detuning has had time to propagate from the detuned oscillator to its nearest neighbors and return. No such discontinuity is expected at time d. The Fourier-like series in (19) can form a slope discontinuity when τ = d due to the 1/n^{2} dependence of the coefficients. However, if A = B, the discontinuity generated by the two series will cancel by virtue of the pt/(2d) in the arguments of the trigonometric functions. When τ = 2d, a discontinuity in the second derivative will be generated by each series and these will add. However, such a discontinuity at τ = 2d is expected due to coupling with nearest neighbors. Similarly, the discontinuities cancel for τ equal to any odd multiple of d and add for any even multiple but, of course, appear in increasingly higher order derivatives as τ increases. Thus, we have established that for the discontinuities to appear at the correct times in the dynamic evolution of the detuned oscillator, A must be equal to B. Since we have already established that A + B = 1, we have that,

3. Discussion of a Numerical Example of the Continuum Model Solution

[9] We select as an example the infinite array with one oscillator detuned at t = 0 and, without loss of generality, set y = 0. That is, we set the origin of coordinates at the single detuned oscillator. As shown by Pogorzelski [2008], the finite array solution can be obtained using image theory. As above, it is found that analysis of the errors in the continuum approximation is facilitated if one plots not the phase but the time derivative of the phase of each oscillator. Recall that this is obtained via multiplication of the transforms by s. (The initial values are all zero.) That is, it is the oscillator phase variation for an impulse excitation rather than a unit step in time. Evaluating the inverse transform of (22), multiplied by s, for each x as a sum of integrals around the branch cuts we obtain the solution plotted in Figures 2a–2e as solid lines. (The ripple shown in the insert of Figure 3a is due to truncation of the branch cut integral series.) Note that this approximates the corresponding exact solution (1), multiplied by s, shown in Figures 3a3b3c3d–3e as dashed lines. However, some error is evident, particularly the noncausal contributions to the solution. In an effort to improve the accuracy of the solution by comparing (22) with (1), it was found that the approximation of the arcsech function with the square root can be improved by multiplication of the square root by a constant. That is,

with C = 1 is a good approximation near the branch points but poor along the branch cuts far from the branch points leading to a poor result for early times even to the extent of violating causality. Along the branch cuts far from the branch points, (23) becomes,

so that far from the branch points, yields a better approximation than the previous one corresponding to C = 1. Moreover, it does not deteriorate the approximation very badly near the branch points. Thus, the solution with C = is better overall particularly in that it is, to a very good approximation, causal.

[10] The above circumstance actually arises from the approximation of the finite differences in (3) with the spatial second derivative in (4) and implies that a more accurate version of (4) would be,

The corresponding enhanced version of (22) would then be,

Evaluating this for the improved value of C, again multiplied by s, as a sum of branch cut integrals results in Figures 4a4b4c4d–4eas solid lines which, when compared with the exact result in Figures 4a–4e as dashed lines, is clearly more accurate than the previous continuum result in Figures 3a–3e, certainly more accurate than the result of the original continuum model neglecting coupling delay shown in Figure 5. Nevertheless, some small noncausal contributions remain, increasing as one progresses to oscillators further from the detuned one. It is further noted that, upon integration of these results to obtain the response to step detuning, the small causality errors will also be integrated and will therefore persist throughout the temporal solution.

4. Concluding Summary

[11] The previously published continuum model for coupled oscillator arrays has been generalized to account for delay in the coupling lines. Straightforward introduction of delay factors in the Laplace transformed partial differential equation of the continuum model results in a solution that violates causality. Here it was shown that by including a second type of solution the causality violation is reduced but not eliminated. The root of this difficulty was shown to be related to the approximation of the finite differences in the discrete theory with derivatives in the continuum theory. Improvement of this aspect of the continuum model significantly mitigated the causality difficulties. The improved continuum model appears to be adequate for application in a study of the beamsteering dynamics of coupled oscillator based phased arrays including the effects of coupling delay.

Appendix A:: Discussion of the Branch Points

[12] The locations of the branch points of the Laplace transform shown in Figure A1 can be found by solving the equation,

In Figure 1 the dots correspond to the plus sign and the circles to the minus sign. This equation can be solved in terms of the Lambert W function (implemented in Mathematica® as “ProductLog”) which is defined to be the solution, w(z), of,

That is, the ProductLog function is the inverse of the we^{w} function, z(w), so that,

In terms of this function, the solution of (A1) is,

Because the W function is not single valued, as there are multiple solutions to (A1), an index n is used to denote the particular solution in use to express a particular root, s_{n}, of (A1).

[13] For large values of s, the roots lie on a simple curve for which an equation can be obtained as follows. First, let s = σ + iω. Then, beginning with (A1) and (A5), it is easy to see that,

and that,

Now for large values of ω,

and we have,

an equation for a curve passing through the branch points for large s. Moreover, from (A5) it is clear that the branch points are separated in ω by 2π/d for each selection of sign and by π/d from one sign to the other. Figure 2 shows this curve and the roots of (A1) for d = 2. As d decreases the branch points move away from the origin and increase in separation. Interestingly, there is a critical value of d below which some of the circle branch points move onto the real axis. This critical value is 0.139232271 for which the smallest circle branch points merge on the real axis at σ = −9.18224297. Figure A1 shows the branch cut locations for d = 0.12; i.e., less that the critical value.

[14] Finally, we note that as d approaches zero, all but two of the branch points move to infinity. There remain a branch point at the origin, as one might expect from the previously reported work on the continuum model, and one of the circle type at σ = −4. This result was unexpected and indicates that there is a missing contribution in the earlier result. Consider the previously obtained result for the infinite array with the oscillator at y step detuned at t = 0 [Pogorzelski et al., 1999]

This has the branch point at s = 0. However, the result above in the limit as d approaches zero is,

which has the additional branch point at s = −4. These two solutions agree in the limit as s approaches zero indicating that they will yield similar time function behavior for large t. However, they differ in their large s behavior so one can expect differing small t behavior in the time functions. This may be demonstrated explicitly for x = y whence the inverse transforms may be obtained analytically. They are,

and,

where the I's are modified Bessel functions. These two functions are plotted ((A11) dashed, (A12) solid) along with their difference in Figure A2 where it may be seen that, as expected, they differ primarily for early times and even there the difference is small. Thus, we conclude that the previously obtained result, while strictly speaking incorrect, still provides a reasonable approximation to the correct behavior.

Acknowledgments

[15] The work reported here was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Partial support from the JPL Mars Technology Program, the Interplanetary Network Directorate Technology Program, and the JPL Advanced Concepts Office of the NASA Office of Space Exploration are gratefully acknowledged. Thanks are due to Vahraz Jamnejad of JPL for helpful theoretical discussions concerning this work. Much of the computation and graphing associated with this work was done using Mathematica® by Wolfram Research. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not constitute or imply its endorsement by the United States Government or the Jet Propulsion Laboratory, California Institute of Technology.