Page 131 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

3. Numerical Linear Algebra 119

From the above equation it is observed that Output (t) is bounded only
when all the eigen values are less than 0 (i.e.) negative values. This is the test
for stability of the system using eigen values.

13. POSITIVE DEFINITE MATRIX TEST FOR
MINIMA LOCATION OF THE FUNCTION
F (X1, X2, X3, … XN)

2
2
2
Consider the function 2 x 1 + 2 x 2 +2 x 3 -2 x 1 x 2 -2 x 2 x 3. To find the
minima of the function f(x), partial differentiate the function with respect to
x1, x2, x3 and equate to zero and solve for x1, x2, x3 to get (x 0, y 0, z 0). The
slope at this point (x 0, y 0, z 0) is zero. To confirm that the points so obtained
are minima, all the partial second derivative values are positive. This is
tested using matrix technique as described below.
Represent the equation as the product of three matrices

[x 1 x 2 x 3] 2 -1 0 x 1
-1 2 -1 x 2

0 -1 2 x 3

If the matrix 2 -1 0 is the positive definite matrix, then the
-1 2 -1
0 -1 2

2
2
2
function f(x) = 2 x 1 + 2 x 2 +2 x 3 -2 x 1 x 2 -2 x 2 x 3 is the minima function
and the minima occurs at the point where the slope is zero (i.e.) first
derivative is zero. If all the eigen values of the matrix are positive, then the
matrix is positive definite matrix. Thus sign of all the eigen values of the
matrix defined above decides whether the point (x 0, y 0, z 0) at which the slope
is zero is minima or not.

14. WAVELET TRANSFORMATION USING
MATRIX METHOD

The 1D signal can be analyzed by using wavelet transformation. This can be
used to compress the data and to denoise the signal. Wavelet transformation
of the 1D signal contains the information regarding the low frequency
content of the signal, which is called approximation co-efficients and the