Impulse Responses of Linear UWB Antenna Arrays and the Application to Beam Steering Werner S¨orgel, Christian Sturm, and Werner Wiesbeck Institut f¨ur H¨ochstfrequenztechnik und Elektronik (IHE) Universit¨at Karlsruhe (TH), Kaiserstr. 12, 76131 Karlsruhe, Germany Phone: +49 721 608 6255, Fax: +49 721 691 865 E-mail:
[email protected] Abstract— Ultra Wideband (UWB) systems enable the transmission of short pulses with a very high time resolution. The application of UWB arrays leads to small beam widths without sidelobes even for sparse arrays with large element spacing. However the transmitted and received signals are subject to distortions due to the dispersive properties like ringing and filtering of the antennas. These effects are analyzed for true time delay beam steering in the paper by the means of the antenna array impulse response. This is derived for the example of a linear array for the frequency range of 3.1-10.6 GHz. The theoretical results are verified by measurements.
I. I NTRODUCTION Since the United States Federal Communications Commission (FCC) has opened the spectrum from 3.1 GHz to 10.6 GHz, i.e. a bandwidth of 7.5 GHz, for unlicensed use [1] with up to -41.25 dBm/MHz effective isotropic radiated power (EIRP) numerous applications in communications and sensor areas are showing up. All these applications have in common that they spread the necessary energy over a wide frequency range in this unlicensed band in order to radiate below the limit. The resulting UWB systems are mainly limited by the strict power regulations, which protect the wireless systems assigned to frequencies within the UWB band. The usage of highly efficient UWB antennas and antenna systems with multiple antennas is an important option in order to cope with the challenging power restriction and enables features like exploitation of spatial diversity, beam-forming and direction estimation. These applications require the understanding of the characteristics and specialties of UWB antenna arrays. Frequency domain array theory by itself is not sufficient for assessing the influence of UWB antenna arrays on the system performance. Therefore the treatment of the transient radiation characteristics of UWB antenna arrays in time domain is given by deriving the antenna array’s transient response and analyzing its quality measures like dispersion and ringing. The influence of the single element’s transient response, the geometry of the array, the coupling between neighboring elements and the feeding network are discussed in the context of a typical array geometry: the linear array. The theoretical modeling is illustrated by experimental results. The paper finishes with simulation results for the application of a linear array of Vivaldi antennas (exponentially tapered slot antennas) for time domain beam steering.
II. M ODELING UWB A NTENNA A RRAYS A. Transfer Function and Impulse Response In general the electrical properties of antennas are characterized by input impedance, efficiency, gain, effective area, radiation pattern and polarization properties [2]. For narrow band applications it is possible to analyze these parameters at the center frequency of the system. For larger bandwidths all of them become more or less frequency dependent, but the straight forward evaluation of the named parameters as functions of frequency is not sufficient for the characterization of the transient radiation behavior. One proper approach to take transient phenomena into account is to model the antenna as a linear time invariant transmission system that translates the exciting voltage uT x at the terminal into the radiated electric far field erad . This system can be fully characterized by its transient response ~h( ; ). Assuming free space propagation, this can be written according eq. (2) as shown in [3]. The dimension of the antenna’s normalized transient response ~h is m/s. Together with the time integrating convolution ~h this relates to the meaning of a normalized1 effective height. uRx ~einc p = ~h( ; ) p (1) ZL0 ZF0 ~erad ( ; ) 1 r @~ uT x p = (t ) h( ; ) p (2) r c0 @t ZF0 ZL0 The effective antenna gain Ge can be readily calculated ~ from the antenna’s response in frequency domain H. 2 !2 ~ H(f; ; ) (3) c0 The total field of an antenna array is the coherent sum of the fieldstrengths ~ei of the single elements:
Ge (f; ; ) =
~etot
= =
N X i N X i
~ei 1 (t ri
ri ) c0
@ ~ hi ( ; ) @t
r
ZF0 uT x;i :(4) ZL0
The voltages uT x;i at the port of each radiator are determined by the properties of the signal source, the feed network (signal 1 Normalized means that the regarded voltages are normalized to the square p root of the antennas characteristic impedance ZL0 and the electric field is normalized to the square root of the free space characteristic impedance p ZF0 .
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distribution and matching) and the mutual coupling between the single radiators. The presence of electromagnetic coupling of the single radiators can be modeled using the self and mutual coupling impedance matrix Zc [4]. It can be transformed into a scattering parameter matrix Sc , which describes the N element array as a linear N -port, see eq. (5). If a single source signal is assumed, which is distributed by a linear feed network with given transfer functions Wi (t) on the N single radiators, the array together with the feed network acts as one single antenna with a transfer function for the radiated electric field. This transfer function can be obtained by analyzing the linear network consisting of feed network, mutual coupling network and coupling to the radiated far field. The latter is modeled as a directional and polarized far field port of the array [5]. This model includes all relevant effects due to mismatch and mutual coupling [6]. The impedance matrix can be obtained from measurements or numerical simulations. Analytic expressions are available for some special array geometries of narrow band radiators like linear arrays of dipole antennas [4]. Applying multiple sources this type of model has been used with great success for multiple input multiple output (MIMO) antenna arrays [6]. 1=2
Sc = ZL0 (Zc + ZL0 )
1
(Zc
1=2
ZL0 ) ZL0
omni-directional radiators are more likely to couple than more directive ones. In the following it is assumed that mutual coupling and antenna mismatch can be neglected in the bandwidth of interest. Then the array factor can be written as the scalar sum of the path transfer functions Wi (t) of the feeding network for each element i: far =
The assumption of identical transient responses h~i ( ; ) for all elements means that not only identical elements are used but they are also identically oriented. The transfer function ~har of the antenna array can then be written as a convolution of the single elements transient responses ~h and the time domain array factor far : ~har = ~h far : (7)
The arrays transient response ~har enables the modelling of impulse radiation independently of the applied source pulse uT x . The time domain array factor far is determined by the array topology, the feeding network and the mutual coupling of the elements. For the synthesis of ultra wideband arrays the combination of the time domain array factor and the transient response of the single element is of high importance. Therefore eq. (7) is analyzed for different parameters. The accuracy is enhanced by the introduction of measured transient responses for the single elements. This yields realistic results for the dispersive behavior of the whole array. In order to separate the influences of the different phenomena, possible simplifications and the related assumptions are shown. The mutual coupling may introduce additional ringing due to reflections and re-radiation. It can be neglected if the distance between the elements is large enough. The amount of coupling also depends on the type of the single radiators:
Wi (t)
(t
(8)
r^;i ):
i=1
The time delay r^;i refers to the relative delay for the signal propagating from the single element i to an observation point in the array’s far field zone in direction r^. The delay r^;i is the difference between the time of arrival (TOA) to the same observation point for radiation from element i and radiation from a reference point in the array structure, which is usually the center of the array. The weighting functions Wi (t) are the transfer functions of the active or passive feeding network of the array. In the case of a passive power divider network the weighting functions Wi (t) are bound to N X
(5)
In eq. (5) ZL0 denotes the diagonal matrix of the port impedances to which the scattering parameters are normalized. Exploiting the linearity of the convolution operator, assuming identical transient responses h~i for all elements and far field conditions (ri r) eq. (4) is rewritten as r N X 1 @~ ZF0 ri ~etot = h( ; ) uT x (t ) Wi (t): (6) r @t ZL0 c0 i
N X
i=1
jjWi (t)jj2
(9)
1:
In the case of an ideal dispersion free feeding network they reduce to Wi (t) = wi (t feed;i ). This leads to a simplified formulation for the time domain array factor: far =
N X
wi (t
r^;i
(10)
feed;i ):
i=1
As well as a tapered amplitude distribution is used in frequency domain in order to lower the sidelobes, the wi can be used in time domain in order to lower the ringing for the off-boresight directions. In the following calculations for an ideal feeding network an uniform amplitude distribution with 1 wi = p (11) N is assumed. B. Evaluation The transient response of the antenna array can be evaluated with respect to the peak value p and the full width at half maximum (FWHM ) w0:5 of its analytic 2 envelope j~h+ ar j [7]. The peak value indicates, what maximum output voltage of the array might be expected for a given incident field. The FWHM is a measure for the time interval where the energy of the radiated signal is concentrated for a given direction ( ; ). p( ; ) = max j~h+ ar (t; ; )j
(12)
t
FWHM ( ; ) =
276
2 j~ h+ ar j
max
j~ h+ ar (t; ; )j=p( ; )=2
t
min
j~ h+ ar (t; ; )j=p( ; )=2
denotes the complex analytical signal in time domain.
t
(13)
z
The FWHM is the interval between the last and the first crossing of j~h+ ar j the half peak value p=2. For a more detailed analysis of the antenna’s transient response see [3], [7], [8]. The FWHM can also be used to define a time domain far field criterion [9]. In the far field region r > rF F it is assumed that for the boresight direction the propagation delays are equal for all elements. The maximum error at a given distance r is the difference in propagation delay between the center of the array and the outmost element at D=2 = 12 (N 1)d is ! r r D2 t= 1+ 2 1 : (14) c0 4r
r
y x
The requirement, that this error shall be less than a fraction of the duration of the impulse response FWHM results in rFF >
8c0
D2 : FWHM
(15)
There is up to now no standardized value for available in the literature. Practical values range from = 1=8:::1=3 depending on the required accuracy. For an array with N = 4 elements, spacing d = 40 mm and single elements for the FCC frequency range with a typical FWHM = 115 ps this range of yields a rFF = 4:6:::1:8 m. In frequency domain the well known far field criterion for harmonic excitation with f = 10:6 GHz yields for the same array rFF > 2D2 = = 1:8 m. For smaller r the transient response ~har becomes more and more dependent on the distance r.
Fig. 1.
Coordinate system with a single Vivaldi element.
element i is given relative to the center element m of the array dependent on the angle . r;i
=
ri
rm c0
=
1 c0
N +1 2
(16)
i d sin
Similar expressions like eq. (16) are readily obtained for other geometries like a circular array or a rectangular array following similar geometrical considerations. d i
m
N y rN
III. U NIFORM S PACED L INEAR A RRAY In Fig. 1 the orientation of a single Vivaldi element is shown together with the employed spherical coordinate system. The xy-plane coincides with the H-plane and the xzplane coincides with the E-plane3 of the antenna. The single element Vivaldi antenna consists of an exponentially tapered slot, etched onto the metallic layer on a Duroid 5880 substrate. The narrow side of the slot is used for feeding the antenna, the opening of the taper points to the direction of the main radiation. The Vivaldi is designed with a Marchand balun feeding network, which has been developed for the frequency range of 3.1-10.6 GHz with a return loss better 10 dB. The size of this antenna is 78x75x1.57 mm3 . The radiation is supported by the travelling wave structure. The propagation velocity on the structure varies only slightly with frequency. The CW gain is calculated from the measured directional transient response h( ; ) according to eq. (3). The maximum gain of the single element is 7.9 dB at 5.0 GHz [10]. In Fig. 2 the geometry of the linear array is shown. The single elements are arranged along the y-axis and are separated with equal spacing d. An observation point in the far field region is visible with the same direction for each element. The propagation paths are parallel rays emerging from the single elements as shown in Fig. 2. In eq. (16) the delay for 3 The H-plane contains the magnetic field vector H ~ and the direction of maximum radiation. Accordingly the E-plane contains the electric field vector ~ and the direction of maximum radiation. E
rm ri r2 r1
m
x
Fig. 2.
Geometry of a uniform linear array.
A. Results for the Uniform Linear Array For the uniform linear array (ULA) the main parameters are: the number of elements N , the spacing of the elements d, the distribution of the excitation, especially a time delay between neighboring elements . In Fig. 3 the measured [7] magnitude of the transient response analytic envelope is shown for the single Vivaldi element in H-plane. The measurement frequency range is 0:4 20 GHz. Its high peak value (pmax =0.35 m/ns) and the short FWHM = 115 ps of the transient responses envelope stand for very low dispersion and ringing. For off boresight directions j j > 30 the antenna exhibits no unique center of radiation. Also the amplitude decreases in these directions. For = 180 an appreciable backward radiation occurs with p( = 180 ) = 0:08 m/ns. The transient response for this direction is more dispersive and shows a width of FWHM = 807 ps. The measured
277
0
0
−5
−5 0
−10
−20 −30
−15
3
−20
2 180 90
1
−10
|h|/(1 m/ns) in dB
−10
−10
−20 −30
−15
3
180
−25
0
−90 −180
y in degree
0
time in ns
−30
0
Fig. 3. Measured transient response jhj of a single Vivaldi element. Measurement with VNWA, frequency range 0.4-20 GHz.
The application of phased arrays is a established technology for steering the antenna beam electronically. For ultra wideband applications the steering of the antenna main beam direction is an interesting option for location aware devices in order to scan sequentially the surrounding space and to
−30
y in degree
N=1 N=2 d=4cm N=4 d=4cm N=7 d=4cm N=12 d=4cm
1.2
|hn,max| in m/ns
1
;
IV. B EAM S TEERING
−90 −180
Fig. 4. Transient response of a linear array with N = 4 Vivaldi elements with spacing d = 8 cm. Simulation based on the measured single elements response.
transient response of one single Vivaldi element is taken for the simulations of the arrays transient response according to eq. (7). Fig. 4 shows an example for the resulting array impulse response. Assuming a uniform amplitude distribution and negligible coupling, the maximum peak value of the arrays transient response p pmax ;ar = max par ( ; ) = N pmax ;i (17) p increases N as can be seen in Fig.5. In the same manner as the main peak increases, the peak p amplitude for the offboresight directions is decreased 1= N because for these directions the impulse responses of the single elements are resolved in time and the single element i is excited with p 1= N according to eq. (11). Depending on the duration and the spectral properties of the excitation this ringing can cause significant sidelobes [11]. Neglecting the mutual coupling the array gain increases linearly with the number of elements Garray = N Gi . The spacing d of the array elements has a significant impact on the transient response: the maximum dimension (N 1)d of the array determines the maximum ringing for the off-boresight directions in terms of FWHM as can be seen from Fig. 6. Larger element spacing narrows the beam with constructive interference of the single element signals whereas the peak value p( ) for the off-boresight directions depends on the number of elements. The antenna gain in the frequency domain is strongly affected by the element spacing. For element spacings with d > frequency selective grating lobes emerge. Therefore an element spacing d < 28 mm is required for a grating lobe free operation below 10.6 GHz. As seen before for short pulses these grating lobes transform into a ringing behavior.
−25
90
1
0
time in ns
−20
2
|h|/(1 m/ns) in dB
0
0.8 0.6 0.4 0.2 0
−150
−100
−50
0 50 y in degree
100
150
Fig. 5. Pattern for the peak value p( ) of the array impulse response for varying number of elements N ; d = 4.
cancel interference from other devices. In contrast to narrowband systems this is achieved by introducing a mutual delay between neighboring elements. The time domain array factor far according to eq. (10) is shown in Fig. 7 for = 80 ps. Each line in the diagram represents the angle dependent delay (t r^;i feed;i ). The directions where the lines cross yield the maximum radiation. For small shifts the crossing in backward direction j j > 90 will be attenuated by the convolution with the single element response for these directions. For increasing the two crossing points move towards each other until they coincide for = d=c0 at = 90 . This is the special case of an endfire array. If the mutual delay is larger, no crossing occurs anymore and the transient responses of the single elements are resolved in time. The directions for maximum radiation are calculated by
278
max
= arcsin
c0 d
:
(18)
2
0.5 N=1 N=2 d=4cm N=4 d=4cm N=7 d=4cm N=12 d=4cm
1.8 1.6
0
1.2
time in ns
FWHM in ns
1.4
1 0.8
−0.5
−1
0.6 0.4
−1.5
0.2 −150
−100
−50
0 50 y in degree
100
150
−180
Fig. 6. Pattern for the FWHM w0:5 ( ) of the array impulse response for varying number of elements N ; d = 4.
In Fig. 8 the peak pattern p( ) is shown for different sensitivity to changes of is obtained as @ max 180 c0 q = @ d 1
c0 2
:
0 y in degree
90
180
Fig. 7. Time domain array factor far for N = 7 elements with spacing d = 4 cm and a delay between neighboring elements of = 80 ps.
The
1 0.9 0.8
1
−90
(19)
N=1 N=7 d=4cm Dt=0ns N=7 d=4cm Dt=0.06ns N=7 d=4cm Dt=0.08ns N=7 d=4cm Dt=0.2ns
0.7 |hn,max| in m/ns
0
0.6
d
For example a linear array with spacings d = 4 cm exhibits in the main beam direction a sensitivity to a variation of of 0:43 =ps. This determines the requirements on the timing accuracy of an active, true time delay beam steered array. The simulation of an N = 4 element Vivaldi array with d = 40 mm yields a -3dB width of the peakpattern of 20 . Therefore the accuracy of the beam steering should be at least 10 , which corresponds to 23 ps. If the signals for the different elements are subject to independent delay jitter this will lead to a reduced peak value. As expected from the time domain array factor, the dispersive characteristics of the scanning array can not be better than that of the single element for the scanning direction max . It is worsening for the directions aside max , because the resolved transient responses of the array factor coincide with the main lobe of the single element. Therefore they may encounter an appreciable amplification. V. M EASUREMENT R ESULTS A test set up with N = 4 elements stacked in H-plane with a spacing of d = 40 mm has been built for validation purposes. The single elements are identical samples of the Vivaldi antenna, which has been described above. The test fixture consists of poly vinyl chloride (PVC) material with slots, in which the planar single elements are inserted. The Vivaldis are provided with standard 50 SMA connectors. Therefore a power divider is needed for splitting the input signal from a single 50 source into four 50 outputs. The test setup is shown in Fig. 9. The requirements for the power divider are ultra wideband frequency range (at least the FCC frequency range). Furthermore equal delays and minimum distortion for all branches are required. The ideal case of frequency independent operation can not be obtained due to the necessary impedance conversions and the adherent reflections within the structure. Distortions that cannot be
0.5 0.4 0.3 0.2 0.1 0
−150
−100
−50
0 50 y in degree
100
150
Fig. 8. Pattern for the peak value p( ) of the array impulse response for varying .
avoided should be very similar for all branches. This leads to a highly symmetric concept of a two stage power divider. It makes use of tapered lines which substitute the well known quarter wavelength transformers of narrowband splitters. The tapered lines have the length of l = 16 mm. The power divider was fabricated in microstrip technology on DUROID 5880 substrate with the thickness t = 0:79 mm. This provides a line width of 0.66 mm for 100 at the beginning and 2.4 mm for 50 at the end of the tapered lines. The dimensions of the whole power divider PC board are 35x130 mm2 . For all ports the measured input reflection coefficients Sii are below -15 dB in the frequency range from 3-11 GHz. The transmission coefficients Si;1 are varying between -6 dB and -8 dB. However the difference of the transmission factor magnitude between two output ports is less then 0.5 dB due to the symmetry of the concept. From the the transmission factors the frequency dependent group delay g = d! d is calculated. The group delays are constant and nearly identical g;i = 0:7 ns for all branches. The absolute maximum of the measured transient response is 0.53 m/ns. This means an implementation loss of 2.5 dB for the peak magnitude compared to the prediction
279
2 meas. simu. w. divider simu. ideal
1.8 1.6
FWHM in ns
1.4 1.2 1 0.8 0.6 0.4 0.2 0
Fig. 9. Power divider and 4 element Vivaldi antenna array for 3.1-10.6 GHz. Fig. 11. meas. simu. w. divider simu. ideal
0.6
|hn,max| in m/ns
0.4 0.3 0.2 0.1
−150
−100
−50
0 50 y in degree
100
−100
−50
0 50 y in degree
100
150
Comparison of measured and simulated FWHM w0:5 ( ).
The dispersive properties of an UWB array has to be taken into account for beam steering purposes. Also the timing accuracy of the delay network single elements has to be considered. Low loss and low dispersive feeding networks are subject to further development. The presented analysis is independent of the signal that is applied to the array. The dispersive properties of the array and their impact on a given signal can be tested in a simulative design study with the presented time domain tools.
0.5
0
−150
150
R EFERENCES
Fig. 10. Comparison of measured and simulated peak value p( ). The offset in direction for the measured result are mainly due to the used cables, which had not exactly the same length.
with eq. (8). The reasons are the mutual coupling (1 dB loss according to numerical simulations for d = 40 mm) together with the applied cables and the feeding network (1.5 dB loss). The effect of the mutual coupling decreases for larger d. The beam width of the measured transient response in Fig. 10 agrees very well with the prediction. Also the measured and the predicted F W HM agree very well as can be seen from Fig. 11 . VI. C ONCLUSION The application of UWB antenna arrays has been analyzed on the basis of the arrays transient response, which provides all relevant information about their transient radiation and reception characteristic. With respect to a given array topology simplified expressions for the time domain array factor have been introduced. Together with measurements of the transient response of a single element the parameters of linear arrays are analyzed. It is seen that the appearance of grating lobes for sparse arrays in narrow band applications translates into additional dispersiveness for off boresight directions. In time domain this means additional ringing and therefore a higher FWHM than within the main beam in addition to the well known broadening of the beam due to basis aspects. The maximum of the FWHM depends on the size of the array.
[1] Federal Communications Commission (FCC), ”Revision of Part 15 of the Commission’s Rules Regarding Ultra Wideband Transmission Systems”, First Report and Order, ET Docket 98-153, FCC 02-48; Adopted: February 14, 2002; Released: April 22, 2002. [2] C. A. Balanis, Antenna Theory, Analysis and Design, Wiley, New York, 1997. [3] C. E. Baum, ”General Properties of Antennas”, Sensor and Simulation Notes, Note 330, Air Force Research Laboratory, Directed Energy Directorate, New Mexico, 1991. [4] J. D. Kraus, Antennas, Electrical Engineering Series, McGraw-Hill International Editions, second edition, 1988. [5] W. Wiesbeck and E. Heidrich, ”Wide-band multiport antenna characteristics by polarimetric RCS measurements”, IEEE Transactions on Antennas and Propagation, Vol. 46, No. 3, pp. 314-350, 1998. [6] C. Waldschmidt S. Schulteis, W. Wiesbeck,”Complete RF system model for analysis of compact MIMO arrays”, IEEE Transactions on Vehicular Technology, Vol. 53, Iss. 3, pp. 579-586 May 2004. [7] W. S¨orgel and W. Wiesbeck, ”Influence of the Antennas on the Ultra Wideband Transmission”, accepted for JASP special issue on UWB, to be published 2005. [8] E. G. Farr, C. E. Baum, ”Extending the Definitions of Antenna Gain and Radiation Pattern into the Time Domain”, Sensor and Simulation Notes, Note 350, Air Force Research Laboratory, Directed Energy Directorate, Kirtland, New Mexico, 1992. [9] L. Bowen, E. Farr, W. Prather, ”A Collapsible Impulse Radiating Antenna”, Ultra-Wideband Short-Pulse Conference, Tel Aviv 2000. [10] W. S¨orgel, C. Waldschmidt, W. Wiesbeck, ”Transient Responses of a Vivaldi Antenna and a Logarithmic Periodic Dipole Array for Ultra Wideband Communication”, IEEE Antennas and Propagation Society International Symposium, Vol. 3, pp. 592-595, Columbus, Ohio, June 2003. [11] A. Shlivinski, E. Heyman, ”A Unified Kinematic Theory of Transient Arrays” in P. Smith, S. Cloude (eds.) Ultra-Wideband, Short-Pulse Electromagnetics 5, Kluwer Academic Press, New York, 2002. [12] W. S¨orgel, W. Waldschmidt, W. Wiesbeck, UWB Antenna Arrays in M. G. di Benedetto, W. Hirt, T. Kaiser, A. Molisch, I. Oppermann, D. Porcino (eds.), UWB Communication Systems - A Comprehensive Overview, EURASIP Book Series, Hindawi Publisher, 2005.
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