Transcript Your Title Your Title Your Title Your Name, Your Department Faculty

Extrapolation of Wideband Responses from Resonant Antennas
using Early-Time and Low-Frequency Data
J. Michael Frye and Dr. Anthony Q. Martin
Abstract
Response Representation
Obtaining a wideband response from a resonant
antenna using traditional electromagnetic numerical
methods is often very computationally demanding. A
technique is presented which accurately extrapolates
the complete response using only early-time and lowfrequency data, which can be obtained relatively easily.
The response is represented as a summation of N
orthogonal polynomials and M damped sinusoids.
Damped sinusoids are utilized to efficiently represent
the effects of resonances in the response. A genetic
algorithm (GA) is used to select all required
parameters for stable and accurate results.
Numerical Results
 Time and freq. response represented by two summations:
E-Shaped Patch Antenna
 N weighted orthogonal polynomials: n  t  , n  f 
 Associate Hermite polynomials used
 M damped sinusoids: gm  t  , Gm  f 
 Represent “resonant” behavior of response
 Amplitude, decay factor, and resonant freq. define each pair
N 1
M 1
n 0
m0
x  t   xˆ  t    a n n  t / l1    g m  t 
N 1
M 1
n 0
m 0
Extrapolation of Driving-Point Current Response
X  f   Xˆ  f    a n  n  f / l2    Gm  f 
Applications
 Rapid numerical analysis of resonant antennas over
an ultra-wide frequency band
 Tool for the design and analysis of wideband, multiband, or switched antennas
 xˆ  t  , Xˆ  f
Extrapolation Approach
 Early-time and low-freq. data used to extrapolate
complete time & freq. response (late-time, high-freq.)

are the extrapolated versions of x  t  , X  f

 Single set of N polynomial weighting coefficients: a n
 Determined from matrix eq. using only early-time/low-freq. data
 Damped sinusoid parameters found from response data
 Aid in optimization of antenna designs which meet
performance goals for multiple applications
GA-Based Parameter Selection
 Represents possible parameter combinations as chromosomes
x  t  : time - domain response
 Finds optimal solution by evaluating ‘fitness’ of potential solutions
 Minimize difference between actual and extrapolated responses:
ˆ


X

X
ˆ
1 xx

E 

 ,
2
x
X



X  f  : freq.- domain response
 Partition into early/late-time and low/high-freq data:

X  f    X LF , X HF

 Highly resonant: cavity
modes, slot and wire
resonances
 Differentiated Gaussian
pulse voltage excitation
up to 15 GHz
 Extrapolated response:
driving-point current
due to pulse
 Genetic algorithm (GA) optimization is utilized
 Low-freq. data contains late-time information
x 

xi
Data to left of vertical dashed line is known (early-time/low-freq.)
Data to right is extrapolated (late-time/high-freq.)
Accurate extrapolation with 6% of time data and 25% of freq. data
N=32 (orthogonal polynomials), M=9 (damped sinusoids)
Cavity-Backed Slot Antenna with Monopole
 Parameters must be carefully selected for stable/accurate results
 Global search technique inspired by evolutionary biology concepts
 Computationally intensive data (late-time, high –freq.)
is extrapolated rather than directly computed.




 Accurately characterize resonances of response
 Early-time data contains high-freq. information
x  t    xET , xLT
 Dual-band (1.9 GHz & 2.4 GHz)
 Designed for wireless
communications applications
 Differentiated Gaussian pulse
voltage excitation up to 4 GHz
 Extrapolated response:
driving-point current due
to pulse
 Represent “transient” behavior of response
2
i
Extrapolation of Driving-Point Current Response
 E cannot be directly minimized in practice because it requires
complete knowledge of the response (which must be extrapolated)
 Approximate E using only early-time/low-freq data:

1
ˆ
E 
2

 
xET  xˆ ET  IFFT Xˆ
xET
LT
 xˆ LT

X LF
 Xˆ LF  FFT  xˆ HF  Xˆ HF
X LF
 By minimizing Ê , all necessary extrapolation parameters can be
reliably selected to yield an accurate extrapolation





 Vertical dash line indicates point of extrapolation
 Accurate extrapolation with 2.5% of time data and 25% of freq. data
 N=484 (orthogonal polynomials), M=44 (damped sinusoids)