Transmit TACAN Bearing Information with a Circular Array

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 onNaval 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 onlywithin desired azimuth ranges [2]. In addition, an array could be given an operational bandwidth covering not only the current TACAN bands of 962–1024MHz and 1025–1087MHz [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 ofVivaldi 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 15Hz and that is designed to yield sinusoidal AM components, in the pulse amplitudes at the aircraft receiver, at 15Hz and 9 × 15Hz = 135Hz. 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 15Hz modulation component relative to a zero time marked by burst reception.That coarse bearing and the phase of the 135Hz 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 andpattern 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.


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.

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  elements are derived below with the goal of providing accurate TACAN bearing measurement in receivers at arbitrary bearings.

International Journal of Antennas and Propagation
There are several steps.Formally assuming the array to be circularly symmetric and requiring its pattern sampled at  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  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.

The Array.
Center the -element array on the origin with symmetry about the vertical axis and with element indices increasing with bearing.Align element 0 with bearing 0 ∘ (any bearing can be made the new zero by changing the reference-burst timing) and interpret element indices modulo  so that the elements adjacent to element 0, for example, can be indexed with ±1 or {1,  − 1}.In the development below, each summation ∑ over index  should be read as a sum over element indices  = 0, . . .,  − 1, and each summation ∑ over index ℓ should be read as the doubly infinite sum over ℓ = −∞, . . ., ∞.
Let k designate the real wavenumber vector of a transmitted signal, and let complex vector-valued function ⃗   (k) be the origin-referenced embedded far-field complex pattern of element .We assume elements are identical in the sense that for all k of interest, where linear operation k  → Rk rotates real vector k about the vertical by 2/ to increase bearing.Identity R  = R 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.

One Weight Implies the Others. Write the time-varying far-field complex array pattern as
using array symmetry (1) on the right.A classic TACAN system's pattern rotates spatially at frequency 1/ = 15 Hz, but here we require that behavior only at  equally spaced bearings.Period  rotation over 2/ in angle is given by Substituting Rk for k in (2) and a change of index yield Likewise, applying (2) to the right side of (3) yields Substituting ( 5) and ( 6) into (3) and comparing terms then formally show that  +1 () =   ( − /) for all , so A rotating bearing-sampled pattern thus implies weight periodicity  0 () =   () =  0 ( − ).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 where  is bearing.Positive real modulation indices 2 and 2 are kept small enough that |(, )| = (, ), for simple receiver demodulation.The terms at frequencies 1/ and 9/ are, respectively, used for coarse and fine bearing measurement.
The  = 0 array pattern should be, using arbitrary scaling, 2.1.4.Determining Weight  0 ().Let wavenumber vector k imp and complex polarization unit vector ⃗  govern co-pol propagation at  = 0 at the most important elevation.Using superscripts to index coefficients, the Fourier series of associated pattern sample ⟨ ⃗ (k imp , ), ⃗ ⟩ and weight  0 () take forms The co-pol array pattern at k = k imp is, by ( 2) and (7), Fourier-series forms (10) and (11) and simple algebra then yield after defining DFT sum (periodically extended in ℓ) which allows ℎ ℓ to be computed from the embedded complex element patterns.The  = 0 pattern (9) yields coefficients  ℓ .From these ℎ ℓ and  ℓ we can obtain  ℓ using the uniqueness of Fourier series and (13), which imply for integer ℓ.Thus Fourier series (11) can be written as This and (7) specify weights that fix the co-pol array pattern for the  wavenumber vectors of form R  k imp to ideal values.The pattern in other directions/polarizations cannot be independently specified and depends on the element patterns.

The Received Signal's Overall Amplitude Modulation.
Much of the above can be generalized to arbitrary polarization unit vector ⃗  and wavenumber vector k.Generalizing Fourier series (10) along with (13) and ( 14), Using (17) for  ℓ , the nonzero Fourier coefficients are Fourier sum (18) is a complex constant times a real modulation function if each of  ±1 / 0 and  ±9 / 0 is a conjugate pair.

Simulation
We tested the approach using weights and performance measures computed from simulated vertical-polarization element patterns ⟨ ⃗   (k), ⃗ ⟩ of 45 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 k imp at the north horizon for a zero "most important elevation."Modulation indices 2 and 2 were each set to 0.2 per Shestag [2].
The embedded co-pol element pattern ⟨ ⃗   (k), ⃗ ⟩ of the Vivaldi radiator appears in Figure 2. Essentially all of the samples used in DFT (14) were significant in magnitude.
Figure 3 shows that the co-pol array pattern obtained approximates 15 Hz rotation, and the Figure 4 slice at  = 0 of that pattern hews closely to desired form (8) from Shestag [2].In both figures, gain is normalized to the  = 0 peak.Section 2.2 discussion assumed that, for  ∈ {1, 9}, the hypotenuse of a right triangle with side lengths | Δ | and | Σ | 55.9 cm 14.5 cm 15.9 cm  was essentially of the latter length because | Δ | was relatively tiny.This is verified in Figure 5. 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 0.1 ∘ in magnitude, while the magnitudes of the coarse errors never exceed 0.02 ∘ .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 10 ∘ and 2 ∘ for the coarse and fine readings, respectively [5], are breached.

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.

Figure 7 :
Figure 7: Bearing measurement errors computed from the phases of  1Σ and  9Σ for coarse (solid curve) and fine (dashed curve) bearing information, respectively.