Transmit TACAN Bearing Information with a Circular Array

Abstract

Using TACAN and array fundamentals, we derive an architecture for transmitting TACAN bearing information from a circular array with time-varying weights. We evaluate performance for a simulated example array of Vivaldi elements.

Academic Editor:Toni Björninen

1, Radar Division, Naval Research Laboratory, Washington, DC 20375, USA

Received 6 August 2015; Accepted 30 September 2015; 16 December 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The Tactical Air Navigational (TACAN) system provides distance and bearing information to aircraft from ground stations and is widely used in military settings. Traditionally, a ground station's physically rotating transmit antenna creates bearing-dependent amplitude modulation from which aircraft can determine their bearings from that ground station. Where space for such a dedicated, special-purpose transmit antenna is difficult to obtain, such as on Naval vessels, sharing a multifunction array with other systems is an option. In that case the TACAN application would use time-varying array weights to approximate a rotating pattern.

Replacing the rotating antenna with a circular array would have benefits beyond facilitating the consolidation of apertures. Certainly these would include simplified maintenance [1] and the potential for elevation beam shaping and/or operation only within desired azimuth ranges [2]. In addition, an array could be given an operational bandwidth covering not only the current TACAN bands of 962-1024 MHz and 1025-1087 MHz [3] but also future TACAN bands considered likely to result from revised spectrum allocations [4].

With those motivations, this paper derives time-varying TACAN array weights for a uniform cylindrical array. While TACAN specifications [5] address both the static elevation pattern and the dynamic azimuth pattern, here we focus on the latter. Our design example assumes an array of Vivaldi elements characterized by embedded element patterns obtained through HFSS simulations. To evaluate the design, we use a bearing-error metric that falls naturally out of the derivation.

The standard TACAN ground transmitter of interest slowly amplitude-modulates a fast pulse signal with an antenna pattern that rotates at 15 Hz and that is designed to yield sinusoidal AM components, in the pulse amplitudes at the aircraft receiver, at 15 Hz and 9 × 15 Hz = 135 Hz. A reference burst transmitted as the rotating main lobe passes north enables an aircraft to obtain a coarse bearing from the transmitter as the phase of the 15 Hz modulation component relative to a zero time marked by burst reception. That coarse bearing and the phase of the 135 Hz component then together yield a fine bearing measurement. Here we focus on creating a time-varying array pattern that permits accurate bearing estimation at the receiver using this process. The fast pulse modulation and reference bursts are independent of the antenna and pattern used and are not considered further here.

This paper presents the initial study into the development of the time-harmonic weights required for transmitting the TACAN waveform from a circular array. A discussion on the theory is provided and validated using simulations.

2. Theory

The next section derives the array structure and time-varying array weights. Performance is then derived as a function of those weights and the complex embedded array patterns.

2.1. Deriving the Array

Time-varying weights for a circular array of [figure omitted; refer to PDF] elements are derived below with the goal of providing accurate TACAN bearing measurement in receivers at arbitrary bearings.

There are several steps. Formally assuming the array to be circularly symmetric and requiring its pattern sampled at [figure omitted; refer to PDF] equally spaced bearings to smoothly rotate in space with time turns out-no surprise-to formally imply that the weights must also rotate so that only one weight requires explicit design. That design follows from the desired temporal modulation of the array-pattern amplitude along a single direction. The pattern modulation between the [figure omitted; refer to PDF] bearings thus addressed explicitly takes the desired general form automatically, with only pattern magnitude and signal modulation indices free to vary modestly (given reasonable assumptions) with bearing.

2.1.1. The Array

Center the [figure omitted; refer to PDF] -element array on the origin with symmetry about the vertical axis and with element indices increasing with bearing. Align element [figure omitted; refer to PDF] with bearing [figure omitted; refer to PDF] (any bearing can be made the new zero by changing the reference-burst timing) and interpret element indices modulo [figure omitted; refer to PDF] so that the elements adjacent to element [figure omitted; refer to PDF] , for example, can be indexed with [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . In the development below, each summation [figure omitted; refer to PDF] over index [figure omitted; refer to PDF] should be read as a sum over element indices [figure omitted; refer to PDF] , and each summation [figure omitted; refer to PDF] over index [figure omitted; refer to PDF] should be read as the doubly infinite sum over [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] designate the real wavenumber vector of a transmitted signal, and let complex vector-valued function [figure omitted; refer to PDF] be the origin-referenced embedded far-field complex pattern of element [figure omitted; refer to PDF] . We assume elements are identical in the sense that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] of interest, where linear operation [figure omitted; refer to PDF] rotates real vector [figure omitted; refer to PDF] about the vertical by [figure omitted; refer to PDF] to increase bearing. Identity [figure omitted; refer to PDF] will be used freely.

In practice imperfect array construction will result in nonidentical embedded element patterns, so the transmitted TACAN waveform will vary somewhat from the ideal derived here. We have yet to study such errors but hope to eventually.

2.1.2. One Weight Implies the Others

Write the time-varying far-field complex array pattern as [figure omitted; refer to PDF] using array symmetry (1) on the right. A classic TACAN system's pattern rotates spatially at frequency [figure omitted; refer to PDF] Hz, but here we require that behavior only at [figure omitted; refer to PDF] equally spaced bearings. Period [figure omitted; refer to PDF] rotation over [figure omitted; refer to PDF] in angle is given by [figure omitted; refer to PDF] Substituting [figure omitted; refer to PDF] for [figure omitted; refer to PDF] in (2) and a change of index yield [figure omitted; refer to PDF] Likewise, applying (2) to the right side of (3) yields [figure omitted; refer to PDF] Substituting (5) and (6) into (3) and comparing terms then formally show that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] A rotating bearing-sampled pattern thus implies weight periodicity [figure omitted; refer to PDF] . This will not produce rotation for all bearings, but we will preserve property (7) for simplicity of structure and in order to obtain nearly rotating behavior.

2.1.3. Desired Modulation

The desired complex array pattern is an arbitrary constant complex amplitude modulated by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is bearing. Positive real modulation indices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are kept small enough that [figure omitted; refer to PDF] , for simple receiver demodulation. The terms at frequencies [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are, respectively, used for coarse and fine bearing measurement.

The [figure omitted; refer to PDF] array pattern should be, using arbitrary scaling, [figure omitted; refer to PDF]

2.1.4. Determining Weight [figure omitted; refer to PDF]

Let wavenumber vector [figure omitted; refer to PDF] and complex polarization unit vector [figure omitted; refer to PDF] govern co-pol propagation at [figure omitted; refer to PDF] at the most important elevation. Using superscripts to index coefficients, the Fourier series of associated pattern sample [figure omitted; refer to PDF] and weight [figure omitted; refer to PDF] take forms [figure omitted; refer to PDF] The co-pol array pattern at [figure omitted; refer to PDF] is, by (2) and (7), [figure omitted; refer to PDF] Fourier-series forms (10) and (11) and simple algebra then yield [figure omitted; refer to PDF] after defining DFT sum (periodically extended in [figure omitted; refer to PDF] ) [figure omitted; refer to PDF] which allows [figure omitted; refer to PDF] to be computed from the embedded complex element patterns. The [figure omitted; refer to PDF] pattern (9) [figure omitted; refer to PDF] yields coefficients [figure omitted; refer to PDF] . From these [figure omitted; refer to PDF] and [figure omitted; refer to PDF] we can obtain [figure omitted; refer to PDF] using the uniqueness of Fourier series and (13), which imply [figure omitted; refer to PDF] for integer [figure omitted; refer to PDF] . Thus Fourier series (11) can be written as [figure omitted; refer to PDF] This and (7) specify weights that fix the co-pol array pattern for the [figure omitted; refer to PDF] wavenumber vectors of form [figure omitted; refer to PDF] to ideal values. The pattern in other directions/polarizations cannot be independently specified and depends on the element patterns.

2.2. Performance

2.2.1. The Received Signal's Overall Amplitude Modulation

Much of the above can be generalized to arbitrary polarization unit vector [figure omitted; refer to PDF] and wavenumber vector [figure omitted; refer to PDF] . Generalizing Fourier series (10) along with (13) and (14), [figure omitted; refer to PDF] Using (17) for [figure omitted; refer to PDF] , the nonzero Fourier coefficients are [figure omitted; refer to PDF] Fourier sum (18) is a complex constant times a real modulation function if each of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a conjugate pair. To distinguish desired and undesired pair behaviors, we can define sum and difference coefficients. For each [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] Fourier sum (18) then becomes [figure omitted; refer to PDF] Combining sums and differences of conjugate pairs yields [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Ideally [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are negligibly small so that [figure omitted; refer to PDF]

In analogy to (8), magnitudes [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the modulation indices, and angles [figure omitted; refer to PDF] relate to coarse and fine bearing estimates [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

2.2.2. The Fine Bearing Measurement

The receiver can compute [figure omitted; refer to PDF] from (27) directly, but computing [figure omitted; refer to PDF] requires resolving the ninefold ambiguity in (28). The key is to let [figure omitted; refer to PDF] and assume that since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are close as angles, [figure omitted; refer to PDF] . Scaling (29) by [figure omitted; refer to PDF] and using (27) and (28), [figure omitted; refer to PDF] for some integer [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , rounding yields the first of the three needed computational steps, and (30) and (29) yield the other two: [figure omitted; refer to PDF]

2.2.3. Intrinsic Bearing Measurement Error

The measured bearing generally contains some error even when [figure omitted; refer to PDF] and when the receiver measures angles [figure omitted; refer to PDF] and [figure omitted; refer to PDF] perfectly. To derive the intrinsic residual fine bearing error relative to actual bearing [figure omitted; refer to PDF] , add [figure omitted; refer to PDF] to each side of (28) and apply angle-folding map [figure omitted; refer to PDF] . This yields [figure omitted; refer to PDF] , where the right side is unchanged because the map is an identity when [figure omitted; refer to PDF] . The intrinsic fine bearing error is therefore [figure omitted; refer to PDF] Replace 9 by unity to derive intrinsic coarse bearing error [figure omitted; refer to PDF]

3. Simulation

We tested the approach using weights and performance measures computed from simulated vertical-polarization element patterns [figure omitted; refer to PDF] of [figure omitted; refer to PDF] Vivaldi radiators embedded in the uniform circular array of Figure 1. The 1 GHz carrier frequency and 22.9 cm (11.0 in) array radius used were convenient but have no TACAN significance. HFSS array simulation with one element driven and the others terminated yielded one embedded element pattern, and (1) provided the rest. Time-varying array excitations are from (7) and (17). We aimed wavenumber vector [figure omitted; refer to PDF] at the north horizon for a zero "most important elevation." Modulation indices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] were each set to [figure omitted; refer to PDF] per Shestag [2].

Figure 1: Simulated uniform circular array of [figure omitted; refer to PDF] Vivaldi radiators.

[figure omitted; refer to PDF]

The embedded co-pol element pattern [figure omitted; refer to PDF] of the Vivaldi radiator appears in Figure 2. Essentially all of the samples used in DFT (14) were significant in magnitude.

Figure 2: Real (solid) and imaginary (knobby) parts and magnitude (dashed) of embedded co-pol pattern [figure omitted; refer to PDF] of element zero, on a linear scale, with [figure omitted; refer to PDF] at bearing [figure omitted; refer to PDF] and the elevation of [figure omitted; refer to PDF] . The elements are identical, so the knobs every [figure omitted; refer to PDF] mark those [figure omitted; refer to PDF] where the curves also yield the [figure omitted; refer to PDF] for [figure omitted; refer to PDF] used in DFT (14).

[figure omitted; refer to PDF]

Figure 3 shows that the co-pol array pattern obtained approximates 15 Hz rotation, and the Figure 4 slice at [figure omitted; refer to PDF] of that pattern hews closely to desired form (8) from Shestag [2]. In both figures, gain is normalized to the [figure omitted; refer to PDF] peak.

Figure 3: Co-pol array gain as a function of bearing and time over period [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 4: This paper's synthesized [figure omitted; refer to PDF] co-pol array gain plotted over the desired pattern of (8) and [2].

[figure omitted; refer to PDF]

Section 2.2 discussion assumed that, for [figure omitted; refer to PDF] , the hypotenuse of a right triangle with side lengths [figure omitted; refer to PDF] and [figure omitted; refer to PDF] was essentially of the latter length because [figure omitted; refer to PDF] was relatively tiny. This is verified in Figure 5.

Figure 5: Approximating (25) by (26) is validated by the small error ratio [figure omitted; refer to PDF] shown here for [figure omitted; refer to PDF] (solid curve) and [figure omitted; refer to PDF] (dashed curve).

[figure omitted; refer to PDF]

Figure 6 shows that time-average array gain [figure omitted; refer to PDF] and modulation indices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] vary little with bearing. Average gain is consistent with Figure 4, and the modulation indices approximate the 20% desired value.

Figure 6: Time-averaged co-pol array gain (dashed curve using same normalization as Figures 3 and 4) and modulation indices for the [figure omitted; refer to PDF] Hz (solid curve) and [figure omitted; refer to PDF] Hz (knobby curve) components of the AM signal.

[figure omitted; refer to PDF]

The most important quantities computed in this system simulation are undoubtedly the intrinsic errors (32) and (33) in the coarse and fine bearing measurements, respectively, intrinsic because they assume noise-free reception at the aircraft. Those are shown in Figure 7. The intrinsic errors in the fine bearing measurement never exceed [figure omitted; refer to PDF] in magnitude, while the magnitudes of the coarse errors never exceed [figure omitted; refer to PDF] . While this appears to suggest that coarse measurement is more accurate, this is somewhat illusory, as the error component due to signal noise, not included here, will generally dominate and be substantially greater for the coarse measurement than for the fine measurement. Certainly the Figure 7 numbers leave plenty of room for those noise-related errors before the TACAN system error limits of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for the coarse and fine readings, respectively [5], are breached.

Figure 7: Bearing measurement errors computed from the phases of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for coarse (solid curve) and fine (dashed curve) bearing information, respectively.

[figure omitted; refer to PDF]

4. Conclusions

In this preliminary study, we developed time-harmonic weights to allow a uniform circular array to support TACAN transmission of bearing information. We have shown how those time-varying weights can be determined from the embedded element pattern. Design and error calculations for an example circular array of Vivaldi elements suggest that acceptable accuracy is feasible with reasonable arrays.

Appropriate future work to expand upon these beginnings includes examining performance over an appropriate elevation interval, considering other array dimensions and numbers of elements, exploring other element geometries, and, of course, validating the theoretical development via measurements. Probably most important, however, is to explore the effects of imperfect knowledge of the embedded element patterns.

Acknowledgment

This work is supported by the InTop program of the Office of Naval Research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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[2] L. N. Shestag, "A cylindrical array for the TACAN system," IEEE Transactions on Antennas and Propagation , vol. 22, no. 1, pp. 17-25, 1974.

[3] A. Casabona, "Antenna for the AN/URN-3 Tacan beacon," Electrical Communications , vol. 33, pp. 35-59, 1956.

[4] G. W. Hein, J.-A. Avila-Rodriguez, S. Wallner, B. Eissfeller, M. Irsigler, J.-L. Issler, "A vision on new frequencies, signals and concepts for future GNSS systems," in Proceedings of the 20th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS '07), pp. 517-534, Fort Worth, Tex, USA, September 2007.

[5] S. J. Foti, M. W. Shelley, R. Cahill, T. MacNamara, K. C. Shi, I. Collings, H. K. Wong, "An intelligent electronically spinning TACAN antenna," in Proceedings of the 19th European Microwave Conference, pp. 959-965, IEEE, London, UK, September 1989.