for carrying out antenna transfer function measurements in the frequency domain, thus borrowing intact many highly refined calibration procedures associated ...
PROCEEDINGS OF THE 2008 IEEE INTERNATIONAL CONFERENCE ON ULTRA-WIDEBAND (ICUWB2008), VOL. 2
UWB Antenna Characterization James S. McLean and Robert Sutton
Abstract—The characterization of UWB antennas is considered with emphasis placed on the determination of compact descriptors appropriate for UWB communications antennas. All of the descriptors can be derived from the antenna transfer function, thus providing a unified approach in which the transfer function is the basis for all of the subsequent computations. It is shown that the unambiguous determination of the antenna transfer function necessarily involves the extraction of linear phase from measured or simulated data, and that the accomplishment of this task is facilitated by the invocation of analytic function theory, specifically the BodeHilbert integral relationships. Finally, some arguments are given for carrying out antenna transfer function measurements in the frequency domain, thus borrowing intact many highly refined calibration procedures associated with characterization of 2-port networks using automatic vector network analyzers including the so-called adapter removal algorithms. Index Terms—Antenna measurements, Antenna radiation patterns, Ultra-wideband antennas.
I. INTRODUCTION
T
HE quantitative characterization of an antenna intended for UWB applications is fundamentally no different from that of any other antenna. However, a complete description of the physical system comprised by any practical antenna is complicated. For example, the complex vector frequency domain transfer function given as a function both rotation angles and frequency can completely describe the far field behavior of an antenna. Unfortunately, it is difficult to consider such a quantity of data simultaneously. Thus, compact descriptors are often sought to facilitate performance assessment and comparison between antennas. It is these compact descriptors that are typically tied to a particular application. Since the requirements for UWB antennas are so different from those of most traditional applications, it is useful to introduce descriptors specifically for UWB applications. To this end, parameters and descriptors appropriate for UWB antennas have been published [1-9]. We discuss four of them: energy gain, correlation coefficient, correlated energy gain, and differential time delay. The first three are described in some detail in [8] and the differential time delay pattern is described in [10]. All four quantities are scalar functions of both rotation directions. From these patterns, single-valued figures of merit such as mean effective energy gain can be computed using averaging over direction, possibly weighted in accordance with statistical environmental data. James McLean and Robert Sutton are with TDK R&D Corp.
978-1-4244-1827-5/08/$25.00 ©2008 IEEE
The first three of the antenna pattern descriptors can be expressed directly in terms of the antenna transfer function as described in [1,7,11,12] and thus their determination amounts to the determination of this complex antenna transfer function. The fourth descriptor, the differential time delay, is also computed using the transfer function, but in a less direct fashion. In [10], two different approaches for computing a differential delay pattern are given. The first involves determining the value of time delay that maximizes the correlation. The second involves computing the linear phase factor or delay associated with the transfer function by first generating the associated minimum phase transfer function and then computing the phase difference between the original transfer function and its minimum phase counterpart. In the hypothetical case in which the antenna does not distort a pulse, but rather the transfer function exhibits purely linear phase with frequency in all directions, the two computed time delays would be exactly the same. This is the case when an antenna that exhibits a frequency independent phase center and exhibits low distortion is rotated about a point other than the phase center. II. DETERMINATION OF ANTENNA TRANSFER FUNCTION In principle, the antenna transfer function can be determined the frequency or time domain as was shown in [7]. In practice, the Fourier transform (or inverse transform) of data measured in one domain will not result in a perfect facsimile of the data measured in the other. This is due to a number of factors, but limited frequency range and time-domain sample rate are especially significant since an antenna is not in general a low-pass system. While time-domain measurement systems have matured significantly in recent years, frequencydomain automatic vector network analyzers exploit an enormous amount of previous work in calibration and deembedding. That is, it is possible to purchase an automatic vector network analyzer (VNA) and a calibration kit with traceable standards and embedded calibration software. Such a system is conceivable for a time domain approach, but is not yet available. It is also worthwhile to note that the unavoidable presence of noise in all measurements also accounts for differences between such transfer functions measured directly in one (frequency or time) domain and transfer functions derived via transformation of data measured in the other domain. For example, the time domain response of an antenna as derived from frequency domain measurements contains at any given time instant the effects of noise at all frequencies. In some situations it might even be advantageous to blank or zero data in a frequency range for which the antenna’s response is so poor that the data obtained is clearly dominated by noise. In
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PROCEEDINGS OF THE 2008 IEEE INTERNATIONAL CONFERENCE ON ULTRA-WIDEBAND (ICUWB2008), VOL. 2 any case, even if sampling and windowing problems are overcome, agreement between such directly measured and transformed responses cannot be expected to be perfect. This argues for having the capability to make measurements in time and frequency domains. The antenna transfer function can be experimentally determined in the frequency domain using a modified 2antenna or 3-antenna method carried out with a vector network analyzer as described [7] and [11-13]. The “connectorized” antenna is, in fact, the logical extension of the term “noninsertable” device [14]. That is, in order to determine the transfer function of one antenna, it is necessary to make a measurement using another antenna (the adapter) and then to “remove” (the effects of) this adapter from the measurement. Thus, the operation is completely analogous to the so-called “adapter removal” technique developed for use with a VNA. The antenna is a 2-port device, albeit with a second port that is not only of a diverse form, but also is not accessible. The principal difference is, that in an adapter removal approach, for 2-port network measurements one still uses 1-port calibration standards for the non-insertable port [14]. Here, however, because the definition of the antenna transfer function places the other port in the far field, only progressive wave motion exists at this hypothetical port and thus the port is perfectly matched. Therefore, no one-port calibration standards are required. In fact, one can employ a sufficiently large ground plane as a one-port short circuit standard and thereby arrive at a one-antenna approach to determining the transfer function as was shown in [15]. When the 3-antenna method is used, the antenna transfer function given in terms of three port-to-port transfer scattering parameter measurements is:
H A (ω ) = where
S
AB 21
CA 2πRc0 S 21 (ω )S 21AB (ω ) jωR / c0 e jω S 21BC (ω )
(1)
H A (ω ) denotes the transfer function of antenna A and
(ω )
denotes the forward scattering parameter in the equivalent reciprocal 2-port network formed from antennas A and B. In general, the complex vector transfer function of an antenna will have two orthogonal components for any given direction. Typically, many metrology antennas such as the double-ridged horn discussed in [11], are nominally linearly polarized and the cross polarization is very small in the antenna’s principal planes. However, such an antenna usually does not exhibit negligible cross polarization in directions outside the principal planes. Thus, to completely characterize an antenna, it is necessary to measure or otherwise determine complex transfer functions for both the co-polarized and the cross-polarized components. If the polarization state cannot be assumed for any of the antennas in a three-antenna measurement, the approach becomes much more involved. It is, however, possible to use the technique given in [16] and [17] to determine the so-called complex polarization ratios of the three antennas:
Rc (θ , φ , ω ) =
Eθ ( R, θ , φ , ω ) Eφ ( R, θ , φ , ω )
=
Fθ (θ , φ , ω ) Fφ (θ , φ , ω )
.
(2)
In the far field, the complex polarization ratio is a function of direction and not of radial distance. It is the complex ratio of the two orthogonal electric field components of the radiated field. Thus, it is also the ratio of the two components of the complex vector transfer function. Once one component of the transfer function has been obtained, the other can be determined via the complex polarization ratio. Because of the known sensitivity of the feed region of the horn to mechanical tolerances, it was thought in [11] that it should not be assumed that the horns were identical. However, when two antennas can be made so precisely as to be able to be assumed to be identical, the 2-antenna approach can be used to obtain the transfer function from a single insertion loss measurement. Of course this saves effort, but it is also worthwhile to note that, given unavoidable measurement errors, the 2-antenna approach as used in [7] might in some cases be more accurate than the 3-antenna method. Since we are necessarily interested in measuring insertion phase over very broad bandwidth, absolute errors on the order of fractions of millimeters are significant. Thus, when changing antennas for a 3-antenna measurement, very small errors in positioning can be incurred resulting in significant phase error. On the other hand, the machining operations for a relatively simple antenna can be performed very precisely. Moreover, the mechanical inspection of such a device is straightforward and thus it should be possible to obtain practically identical antennas. Clearly, this argument applies even more strongly to the most general 6-measurement approach required to account for unknown polarization. If such an approach is to be effectively employed, some attention must be paid to repeatable fixturing and positioning. Regardless of whether one assumes that the two antennas are identical, or employs a 3-antenna approach, it is necessary to take a complex square root in process of determining the transfer function. Proper unwrapping of the phase function is therefore required. Some discussion of this is presented in references [7,11]. This ambiguity is, in fact, the same as that encountered in the “adapter removal” algorithm used with many vector network analyzers. The measurement of the through insertion loss in the 2-antenna approach is equivalent to measuring the return loss of a 2-port network terminated in a short. In the typical adapter removal algorithm [14] involving one-port calibration standards at the non-insertable port, the electrical length of the unknown network must be known to within 90 degrees in order to unambiguously determine the scattering matrix representation. Here, since through measurements are being made, it is necessary to know the electrical length to within 180 degrees. The unambiguous determination of antenna transfer functions depends on the definition of and the determination of the distance R “between” the antennas as shown in Eqn 1. This is not straightforward for finite-sized antennas and several very
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PROCEEDINGS OF THE 2008 IEEE INTERNATIONAL CONFERENCE ON ULTRA-WIDEBAND (ICUWB2008), VOL. 2 different approaches have been used to extract the proper distance including, the extrapolation of amplitude decay [6], least-squares fitting of linear phase [11], and finally, the use of the Bode-Hilbert integral relations to determine a minimum phase function [18,19]. The fourth approach has been used to define the time-delay pattern descriptor given in [10] and is discussed in section IV of this paper. It is useful to note that for many antennas, at any given frequency, the effective phase centers computed for two orthogonal polarizations are not colocated. Thus, the frequency loci of the phase centers for the two polarizations might be markedly different. Clearly, the ambiguity involving the distance R is even greater when polarization is considered. III. ANTENNA PATTERN DESCRIPTORS As mentioned earlier, three of the four antenna pattern descriptors mentioned earlier--energy gain, correlation coefficient, and correlated energy gain--can be expressed directly in terms of the antenna transfer function. The energy gain of an antenna is the time domain analog to the frequency domain power gain:
1 GE (θ ,φ ) = 2 π c0 where
³
∞
−∞
2 G 2 H (θ , φ , ω ) a (ω ) ω 2 dω
³
∞
−∞
a (ω ) dω 2
.
(3)
a (ω ) is the incident voltage pulse as defined in [8].
It is illustrative to note that when the input impedance match is perfect, the energy gain is simply the average of the power gain, G, weighted in accordance with the input pulse:
GE
(θ , φ ) = ³
∞
−∞
2
G (θ , φ , ω ) a (ω ) dω
³
∞
−∞
2
a (ω ) dω 2
(4)
The normalized correlation coefficient provides a measure of how similar the radiated waveform in a particular direction is to that in a chosen reference direction and is given by:
ρ (θ , φ ) = max τ
2 G G ∞ 2 º Re ª ³ a ω 2 H (θ , φ , ω ) ⋅ H ∗ (θ 0 , φ0 , ω ) e − jωτ d ω º » «¬ −∞ »¼ » G G 2 2 ∞ ∞ 2 2 2 2 ³−∞ a ω H (θ , φ , ω ) dω ³−∞ a ω H (θ0 , φ0 , ω ) dω »¼ (5) where the reference direction is taken as (θ 0 , φ 0 ) . Of course
ª « « « ¬
the correlation coefficient depends on both the transfer function of the antenna and the input pulse. Finally, the correlated energy gain as defined in [8] has been shown to be the product of the correlation coefficient in (3) and the energy gain in (5). As has been noted by many researchers the correlation coefficient and the correlated energy gain are most meaningful when the maximization operation shown in (6) is included in the definition.
IV. SECONDARY PATTERN DESCRIPTORS: DIFFERENTIAL DISTANCE AND T IME DELAY PATTERNS It is not possible to assign a unique time delay to any dispersive system such as a practical antenna. Many attempts have been made to extend the concept of group delay to UWB antennas. However, group delay in the strictest sense is a narrowband quantity: it is simple the negative of the slope of
φT , of the transfer function: ∂φ (θ , φ , ω ) . τ g (θ , φ , ω ) = − T ∂ω
the phase,
(6)
Thus, the group delay is a not a single number, but rather is function of frequency as well as direction and can even take on negative values over limited frequency ranges. When one considers a limited frequency range, the group delay of an antenna might vary slowly enough over this range as to be able to be taken as constant. However, when a dispersive antenna is considered over a broad frequency range, the traditional group delay is function of frequency and cannot easily be reduced further. It is interesting to note that the group delay of a typical log-periodic dipole antenna or many other frequency-independent antennas is actually a decreasing function of frequency in the antenna’s pass band [20]. Thus, its meaning as an actual delay is dubious. Having stated this, it must be admitted that many UWB antennas are reasonably well behaved over the moderately wide 3.1-10.6 commercial UWB band and averaging of the group delay sometimes yields meaningful results [21]. However, many antennas are not and many UWB applications involve much greater fractional bandwidths than do commercial UWB communications antennas. In principle, for some antennas, the integral quantities involving the complex transfer function such as correlation coefficient could be expressed in terms of the group delay using integration by parts, but the utility of this is not apparent. In any case the entire frequency range of group delay would be involved. In the more general case, there are several reasonably consistent approaches for determining time delays associated with broadband antennas. In Eqn 5 of this paper, a maximization of the correlation integral is performed over a delay, . For each direction, a single value of is required to maximize the normalized correlation and the correlated energy gain. Thus, this quantity constitutes a uniquely defined delay. When the difference between the value of this delay for a particular direction is taken with that in a reference direction, a differential delay pattern can be obtained. A second uniquely defined delay can be obtained by extracting the linear phase from the antenna transfer function using one of the methods described in [10]. Perhaps the most rigorous approach is the application of the Bode-Hilbert [19] integral relationship to derive the minimum phase transfer function associated with the actual transfer function of an antenna:
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PROCEEDINGS OF THE 2008 IEEE INTERNATIONAL CONFERENCE ON ULTRA-WIDEBAND (ICUWB2008), VOL. 2 ∞ 2ω ln H (ω ′ ) − ln H (ω ) dω ′ . Φ m (ω ) = π ³0 ω ′2 − ω 2
(7) REFERENCES
The linear phase is thus the difference between the phase of the actual transfer function and the associated minimum phase transfer function:
Φ linear (θ , φ , ω ) = ∠H (θ , φ , ω ) − Φ m (θ , φ , ω )
where
(8)
Φ m (θ , φ , ω ) is the associated minimum phase
transfer function. linear phase by:
τ d (θ , φ ) =
The time delay is given in terms of the
Φ linear (θ , φ , ω )
ω
.
(9)
Extracting this linear phase is equivalent to shifting the time domain response to begin exactly at zero time. Again, taking the difference between the delay computed for a particular direction and that for the reference direction provides a differential delay pattern that can be compared to the one derived from the maximization of the correlation integral. If the time domain radiated waveform or more strictly the impulse response of the antenna in a given direction is a timeshifted replica of that in the reference direction, the two differential delay patterns would be identical. Of course, the correlation would also be unity. For many practical antennas, the patterns are very similar. However, when the off axis time domain response bears little resemblance to the on axis response, these patterns are markedly different. V. CONCLUSIONS The characterization of UWB antennas has been briefly addressed with emphasis on frequency domain measurements of the antenna transfer function. Frequency domain measurements are preferred, but only because of the availability of the highly refined calibration techniques associated with vector network analyzers. From the transfer function three primary descriptors, the energy gain, the correlated energy gain, and the correlation coefficient can be derived. In the definition of correlation coefficient, it is meaningful to perform maximization over a time delay. The time delay required to maximize the correlation and hence the correlated gain is thus in itself a secondary descriptor. When this time delay is considered relative to its value on the principal axis the differential time delay pattern is obtained. The time delay pattern is not uniquely determined and a second time delay, derived from the difference between the phase of the antenna transfer function and its associated minimum phase counterpart is also presented. Only under strict limiting conditions would these two quantities be identical.
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