Download Lecture 7 - Ohio University

EE 616
Computer Aided Analysis of Electronic Networks
Lecture 7
Instructor: Dr. J. A. Starzyk, Professor
School of EECS
Ohio University
Athens, OH, 45701
1
Outline
2

Computer Generation of Network Functions

Unit circle polynomial interpolation

Condition numbers for interpolation
Computer Generation of Network Functions
We want to obtain a network function in the semi-symbolic form
Ns 
Fs  
Ds 
This will be useful for direct evaluation of F(s) at different frequencies
(instead of using n3 / 3 operations for LU factorization each time) or
inverse Laplace transform to obtain time domain solution.
Assume we know LU factorization at frequency si and assume that
T = LU has all entries with s in the numerator. Determinant D(si)
can be found as
n
Ds i   det Ts i   det Ls i    L ij
j1
3
Computer Generation of Network Functions (cont’d)
Solving LUX =  we can get X and then F(si).
The numerator N(si) can be found as
N si   Dsi F si 
Changing si we get sets of pairs (si, N(si)) and (si, D(si)),
and we get the coefficients of N and D. This is a wellknown problem of polynomial interpolation.
4
Unit circle polynomial interpolation
Assume we have (n + 1) distinct points (xi, f(xi) = yi). We want to
find coefficients of the polynomial
n
f (x)   a j x i
j0
We have
a 0  a 1 x i  a 2 x i2  ...  a n x in  y i
or in the matrix form
or
Xa = y
1 x 0

1 x 1
 

1 x n

x 02
x 12

x 2n
 x 0n 

 x 1n 


 x nn 
y0 
a 0 
y 
a 
 1 =  1
 

 
 
a
y n 
 n
a = X-1y
The most numerically stable selection of
5
i  0,1,2,..., n
where
xi
 
X  x ij
is on the unit circle.
Unit circle polynomial interpolation (cont’d)
Let us define
and
xk = k
  exp
so
2  1
n 1
   
X  x ij  ij
1
X 
X*
n 1
Therefore the solution is a  X 1 y  1 X * y  1
and
1
n 1
or simply
6

 y
n 1
ij
1 n
aj 
y k   jk

n  1 k 0
Using this equation all coefficients of the approximating
polynomial can be explicitly calculated. Finding each coefficient
requires an addition of n + 1 complex numbers - each one easily
obtained from the approximated function values y k . In addition
the polynomial interpolation on the unit circle is the most
numerically stable algorithm.
Condition numbers for interpolation
Condition number
K X 
~

for the matrix
X
is
~
K x   max /  min
~
where max, min are the largest and smallest eigenvalues of XX*.
K(X) is a measure of perturbations in y.
In our case
XX *  XX 1 n  1  n  1I
So all the eigenvalues of XX* are equal to (n + 1) and K(x) = 1.
Note that K(x)  1 always. Therefore, the selection of all points on
the unit circle yields the best possible accuracy of the polynomial
approximation problem.
7
Condition numbers for interpolation - Example
Example: Numerical value of the numerator and denominator are
N s0  t1  3
Ds0  1  11
N s p  1  3
Ds p  1  7
Find transfer function (first order polynomials N(s) and D(s)).
n=1
since
so
8
n+1=2
a 0  a 1s
T
b 0  b1s
  exp
2  1
 1
2
Condition numbers for interpolation - Example
1 1
1
0k
a 0    1 Ns k   3  3  0
2 k 0
2
1 1
1
1k
a 1    1 Ns k   3  3  3
2 k 0
2
1 1
1
0k
b 0    1 Ds k   11  7   2
2 k 0
2
1 1
1
1k
b1    1 Ds k   11  7   9
2 k 0
2
And, we obtain
9
3s
T
2  9s
Condition numbers for various interpolations
10
Growth of error versus degree of the polynomial n
11