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The Minimum Phase Nature of the Transfer Function of the
Impulse Radiating Antenna
J. S. McLean1 , R. Sutton1 , and H. Foltz2
1
2
TDK R&D Corp., USA
University of Texas — Pan American, USA
Abstract— The Impulse Radiating Antenna, a reflector antenna employing a TEM feed, has
been shown to provide excellent time-domain pulse reproduction on its principal axis. We examine the on-axis response of a representative, well-designed Impulse Radiating Antenna in the
frequency- and time-domains and show that it is the existence of the pre-pulse in the time-domain
response that causes the frequency-domain transfer function to be non-minimum phase in nature. Moreover, the hypothetical time-domain response corresponding to the minimum phase
frequency domain transfer function derived in turn from the magnitude of the on-axis response
of the impulse radiating antenna is essentially identical to the actual time domain response except that the pre-pulse occurs after the main pulse. That is, the time domain response associated
with the minimum phase transfer function appears almost as a mirror image of the actual time
domain response with the symmetry occurring around the center of the main pulse.
1. INTRODUCTION
The Impulse Radiating Antenna (IRA), a reflector antenna employing a TEM feed structure [1],
has been shown to provide excellent time-domain pulse reproduction on its principal axis; more
specifically as stated in [1], “a step-like signal into the antenna gives an approximate delta-function
response in the far field.” While this statement succinctly describes the ideal time domain behavior,
the frequency domain counterpart is that perfect time domain pulse reproduction requires satisfaction of the distortionless transfer function criterion in the frequency domain. Distortionless transfer
functions, in turn, are a small subset of a more general group, minimum-phase transfer functions.
When a minimum-phase network exhibits a transfer function magnitude which is nearly constant
with frequency, its associated phase function necessarily satisfies the distortionless transfer function
criterion; however, many broadband antennas and other physical systems exhibiting flat or nearly
flat transfer functions are not minimum-phase. An example of such a broadband, non-minimum
phase antenna is the Log-Periodic Dipole Antenna (LPDA), the transfer function of which exhibits
a nominally flat magnitude, but deviates greatly from minimum phase behavior [2]. It is well known
that this antenna exhibits poor time domain pulse fidelity.
2. FREQUENCY DOMAIN TRANSFER FUNCTION
~ gives the far field radiated
The frequency domain complex vector antenna transfer function H
~
electric field E in terms of the power-normalized incident voltage a at the antenna input port [3]:
−jβR
~ (R, θ, φ, ω)
E
~ (θ, φ, ω) e
= −jω H
a (ω) .
√
η0
2πRc0
(1)
This definition explicitly includes a differentiation of the transmitted signal through the jω factor1 .
Thus, a hypothetical antenna with a unity transfer function for all frequency would still differentiate
the incident signal in the sense that the radiated electric field would resemble the time derivative
of the input voltage. Thus, the inverse Fourier transform of the antenna transfer function may
be called antenna impulse response, but it actually gives the far field electric field due to a step
incident voltage. Note that the definition for antenna transfer function given in Eq. (1) has units
of meters and is identical to the frequency domain counterpart of the normalized impulse response
defined in [4, 5].
1 Strictly speaking, the antenna transfer function does not directly yield the electric field due to the incident voltage; therefore
the “impulse response” derived from it does not give the electric field due to a voltage impulse at the input. The definition
was arrived at by several different groups of researchers and is a compromise that provides some symmetry between forms for
reception and transmission [3]. However, the authors note that there are other conventions that have been proposed.
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However, the differentiation in Eq. (1) is distinct from the intrinsic high-pass nature of the
impedance matching and power transfer characteristics of any finite-sized antenna. A finite size
antenna driven by a step input cannot sustain a DC electric field that decays as 1/R in the far-field;
~ must have at least one zero at ω = 0. This can also be
therefore, one can conclude from (1) that H
seen in the relationship between the complex vector effective length ~heff and the antenna transfer
function:
√
η0 Z0 ~
~ (θ, φ, ω) =
H
heff (ω)
(2)
Z0 + ZA (ω)
where Z0 is the normalizing impedance for the antenna input port and ZA is the antenna input
impedance. The frequency domain transfer function must exhibit at least one zero at DC. Some
antennas (for example, a short monopole) have an effective height that is constant and finite in
the limit as ω → 0; however, such antennas have an input impedance ZA that becomes infinite
at DC. The authors are unaware of any antenna that has both nonzero effective height and finite
impedance at DC. The conclusion is that the relationship between electric field and input voltage
must always have at least two zeroes at ω = 0, and therefore the response to a step input must be
zero average in the time domain.
The Impulse Radiating Antenna is primarily a reflector antenna. However, the TEM feed
structure itself also radiates and behaves essentially as a P × M antenna. It can be shown that one
type of canonical P × M antenna sometimes referred as a balanced transmission line wave (BTW)
sensor [6], a backward-radiating, terminated uniform transmission line, has an asymptotic slope of
12 dB/octave in its gain, and thus has an asymptotic slope of 6 dB/octave in its transfer function,
indicating a single zero at ω = 0. The Impulse Radiating Antenna can also be shown to exhibit a
single zero at DC in its effective length and transfer function. Both the balanced transmission line
wave sensor and the IRA are very well matched due to the internal loads and the complex voltage
division term in (2) is essentially frequency independent. Thus, when relating the incident input
voltage to the far field radiated electric field the high-pass differentiation appears twice: once in
the transfer function (and thus the effective height), and again due to the jω factor in Eq. (1) for
the electric field.
An antenna with a frequency-domain transfer function (as defined in [3]) of unity for all frequency
would generate an electric field impulse in the far field in response to a step input, and is said to
satisfy the distortionless transfer function criterion. The IRA nearly satisfies the distortionless
transfer function criterion and thus is nearly minimum phase in the frequency domain. However,
the small departure of its transfer function from minimum phase is quite interesting.
Close examination of the time domain response of the IRA reveals a pre-pulse as well as a long
shallow tail following the main pulse [1]. The pre-pulse has been shown to be step-like in the time
domain. It radiates from the TEM feed structure and necessarily precedes the main pulse from the
Figure 1: Schematic drawing of an impulse radiating antenna. The reflector employed by the Farr IRA-3
has a diameter of D = 46 cm and focal length of F = 23 cm and thus F/D = 0.5.
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reflector. It can be shown that having two radiation mechanisms (in this case direct feed radiation
and the reflector radiation) with different delay times leads to non-minimum phase behavior if the
smaller signal is the first to arrive in the far-field. The lack of minimum phase behavior in turn
implies that the distortionless criterion cannot be satisfied, and furthermore, that it cannot be
corrected through passive equalization.
As discussed above, Eq. (1) relating the electric field to the incident voltage must contain at
least two zeroes at DC, and the entire time domain response must average to zero. In [1] some
discussion is given concerning how if the area of the pre-pulse could be made to equal that under
the main impulse (thus giving zero DC average) then the tail of the response would be small. The
area of the pre-pulse is tailored by adjusting the characteristic impedance of the TEM feed of the
IRA. Thus, the canonical or ideal response of the Impulse Radiating Antenna is a step followed
immediately by an impulse of equal area. The particular IRA characterized here, the Farr Research
IRA-3, exhibits a very good on-axis, time-domain response with a very minimal tail.
3. MEASUREMENT
The port-to-port forward transfer scattering parameter of combination of a Farr Research IRA-3
impulse radiating antenna and a Farr Research TEM-1 TEM horn was measured in a fully anechoic
chamber using an automatic vector network analyzer. The analyzer was calibrated using a so-called
transmission calibration; that is, a full two-port calibration was not used due to the very long coaxial
cables connecting the network analyzer, which was located outside the chamber, to the antennas.
The magnitude of the measured transfer scattering parameter is shown in Fig. 3.
Figure 2: Solid model of impulse radiating antenna. A bifurcating ground plane located in the x-z plane has
been omitted for clarity.
-20
S21 Magnitude (dB)
-30
-40
-50
-60
-70
-80
-90
-100
0
5
10
Frequency (GHz)
15
20
Figure 3: The magnitude of the measured transfer scattering parameter S21 of the 2-port network comprised
by a Farr Research IRA-3 impulse radiating antenna and a Farr Research TEM-1 TEM horn situated in an
anechoic chamber with principal axes aligned and with 3.5 meters separation.
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Transfer Function Magnitude (dB re 1m)
0
-10
-20
-30
-40
-50
0
5
10
Frequency (GHz)
15
20
Figure 4: Magnitude of the frequency domain transfer function of the Farr IRA-3 impulse radiating antennas
as derived measured port-to-port insertion loss data.
0
Farr IRA-3
Canonical PxM
-10
20*log10|H|
-20
-30
-40
-50
-60 1
10
2
10
Frequency (MHz)
3
10
Figure 5: Magnitude on-axis transfer function of a canonical P × M antenna (length 23 cm) and a numerical
model of the Farr Research IRA-3 impulse radiating antenna. As expected the slope of the transfer function
is asymptotically = 6 dB/octave at the low end of the frequency range.
The TEM horn had been previously characterized as described in [8]. From the transfer scattering parameter measurement, the transfer function of IRA was then determined. The magnitude
of the frequency domain transfer function is shown in Fig. 4. The transfer function data given here
agrees reasonably well with the data provided by the manufacturer.
4. LOW FREQUENCY EXTRAPOLATION OF MEASURED DATA
The IRA transfer function magnitude shown in Fig. 4 appears to reach an approximately constant
value at low frequency. However, as discussed in Section 2, it is known that there is at least one
zero at DC, and therefore one can conclude that there is also a pole at a low but finite frequency.
S21 in a two-antenna measurement has at least three zeroes at DC (one in each of the transfer
functions plus the jω factor in Eq. (1)), and thus the measured signal decreases very rapidly at low
frequency. In the data presented here, unavoidable noise overcomes the measured signal before the
pole in the IRA transfer function can be seen.
To study the minimum phase behavior of the IRA, a Hilbert transform is performed to derive
the minimum phase function from the transfer function magnitude, which requires in principle
integration over all frequency. Explicit or implicit (simple truncation) extrapolation of the finite
frequency range is necessary. The value of the minimum phase function as ω → 0 is determined by
the limiting slope at low frequency. Due to the noise problem described above, the low frequency
behavior must be extrapolated.
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100
80
Phase of H (degrees)
60
40
20
0
-20
-40
-60
-80
-100 1
10
2
10
Frequency (MHz)
3
10
Figure 6: Simulated phase of far-field on-axis electric field of Farr IRA-3. A factor of exp(−jkR) has been
removed from the data. As can be seen the phase of the transmitted electric field with respect to the source
asymptotically approaches 90◦ at low frequencies. This is consistent with there being one zero at DC in the
transfer function.
The operation of the IRA in the lowest registers of its operating frequency range is similar to
that of the so-called P × M antenna or Balanced Transmission Line Wave (BTW) sensor [6]. The
antenna exhibits a pattern which is a cardioid of revolution about the principal axis and has an
effective length and transfer function the magnitudes of which increase 6 dB/octave asymptotically.
The power gain increases 12 dB per octave asymptotically. Fig. 5 shows data for the Farr IRA3 generated using a commercial FEM simulator, Ansoft’s HFSS software, with comparison to a
canonical P ×M antenna. In this numerical simulation the equal-delay balun was not modeled. The
canonical P × M antenna is a terminated two-wire transmission line with characteristic impedance
of 450 Ohms and terminated in a matched load. The length of the line is 246 mm corresponding
to the focal length and hence feed dimension of the Farr IRA-3. The data was obtained using a
numerical model implemented using the Numerical Electromagnetics Code (NEC). Note that both
antennas show 6 dB/decade slope in the magnitude of their transfer functions in low portion of the
frequency range. Since both antennas are very nearly perfectly matched at the low ends of their
respective operating frequency ranges, it can be shown that the antenna transfer function as well
as the traditional effective height rolls off with 6 dB/octave. Thus, the extrapolation of the transfer
function to DC requires that there be exactly one zero at the origin.
5. MINIMUM PHASE TRANSFER FUNCTION
The minimum-phase quality of the transfer function of an antenna is associated with the propagation of energy through the system. Having a single path through the network or system is a
sufficient condition to have minimum-phase behavior. It was shown in [7] that the broadband,
double-ridged horn is very nearly minimum phase on its principal axis, but deviates from this
condition off-axis. It was surmised that this was due primarily to interference between the direct
radiation from the horn’s aperture and fields diffracted by the edge of the horn. In [8], it was shown
that an asymmetric or half TEM horn such as the Farr Research TEM-3 exhibits a minimum phase
response on its principal axis as well as off axis in the E-plane for angles below the ground plane,
but is not minimum phase off-axis above the ground plane. This is because below the ground plane
the radiation is essentially entirely due to a single mechanism, diffraction from the edge of the
ground plane.
The question in this case is whether a similar multiple path effect applies to the IRA. In Fig. 7
we show the measured transfer function phase for the IRA in comparison with the minimum phase
function for the antenna. The minimum phase function was computed using the Bode-Hilbert
transform of the measured magnitude data, but with the data truncated at 300 MHz and replaced
with extrapolated points computed using the 6 dB/octave slope as discussed in Section 4. It can
be seen that there is good agreement above 5 GHz, but increasing non-minimum phase behavior
below 5 GHz. This is evidence that there are multiple radiation mechanisms.
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Transfer Function Phase (degrees)
90
Black- Measured
Blue - Hilbert
0
-90
-180
0
5
10
Frequency (GHz)
15
20
Figure 7: Measured phase of transfer function and minimum phase function computed from the magnitude
of the measured transfer function.
6.0E9
Black - Measured
Blue - Hilbert
Impulse Response (m/s)
5.0E9
4.0E9
3.0E9
2.0E9
1.0E9
0.0E0
-1.0E9
-5
0
5
10
15
20
Time (ns)
Figure 8: The impulse response of the IRA as derived from frequency-domain data measured in an anechoic
chamber (black). For comparison, the impulse response derived from a hypothetical transfer function with
the same magnitude but Hilbert minimum phase (blue).
2.0E9
Black - Measured
Impulse Response (m/s)
1.5E9
Blue - Hilbert
1.0E9
5.0E8
0.0E0
-5.0E8
-1.0E9
-4
-2
0
Time (ns)
2
4
Figure 9: Same data as Figure 8, on an expanded scale. Note that the rectangular pre-pulse that precedes
the main pulse in the measured data. If the transfer function is modified to be Hilbert minimum phase, the
pre-pulse is transposed such that it follows the main pulse in the response derived from the minimum phase
transfer function.
Figures 8 and 9 show, for the curves labeled “measured” data, the inverse Fourier transform of
the measured transfer function. For these curves no extrapolation or windowing was used. The
shapes correspond to the electric field that would be produced by a step input, which is the intended
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mode of operation for the IRA. The negative going pre-pulse is clearly evident starting 1.6 ns prior
to the main pulse. The step-like appearance of the pre-pulse leads one to believe that it is associated
with a pole at low frequency. Since the deviation from minimum phase behavior is at low frequency,
one might guess that the pre-pulse is the most important cause of the deviation.
As further confirmation, one can carry out a “thought experiment”, in which a new hypothetical
transfer function is generated, consisting of the actual magnitude from measurements, with the
minimum phase function derived from the Hilbert transform. This hypothetical transfer function
is then used to generate a time response via an inverse Fourier transform. The same procedure was
used in [8] to analyze a TEM horn, and it was seen that in directions where the measured transfer
function was not minimum phase, the new hypothetical time-domain response differed from the
actual response in that the precursors to the main pulse in the actual data were transposed to the
opposite (later) side of the main pulse, thus providing a clean onset to the main pulse. Thus the
precursors could be seen to be directly responsible for the deviations from minimum phase.
A similar computation was carried out for the IRA. The results are the blue curves labeled as
“Hilbert” in Figs. 8 and 9 It can be seen that in the time-domain response computed from the
minimum phase function, the pre-pulse is transposed about the main impulse such that it later.
This behavior can be qualitatively explained in terms of a simple model based on rays representing
major sources of radiation. In general, the frequency-domain minimum-phase criterion will be
satisfied only if the time-domain field from the strongest radiation source is the first to arrive at
the observation point. Enforcing the minimum phase condition in the frequency domain re-orders
the time-domain response such that the main impulse is first.
6. CONCLUSIONS
The impulse radiating antenna exhibits a transfer function that is essentially minimum phase at
high frequency, but has a small but significant deviation from minimum phase at low frequency. By
comparing the actual impulse response to a hypothetical impulse response for a transfer function
that is modified to be minimum phase, it can be seen that the pre-pulse due to direct radiation
from the feed is the most prominent cause of the non-minimum phase behavior.
ACKNOWLEDGMENT
This work was supported in part by the Army Research Office through grant W911NF-06-1-0420.
REFERENCES
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