Time-Domain Analysis and z Transform - Introduction to Digital Signal Processing and Filter Design - page 58

By use of matrix algebra, we can now find any one or all three unknown functions
Y 1 (z) , Y 2 (z) ,and Y 3 (z) , when the input X(z) is zero—their inverse z transforms
yield zero input responses. We can find them when all the initial states are
zero—their inverse z transform will yield zero state responses. Of course we
can find the total responses y 1 (n) , y 2 (n) ,and y 3 (n) , under the given initial states
and the input function x(n) . This outlines a powerful algebraic method for the
analysis of discrete-time systems described by any large number of equations in
either the discrete-time domain or the z -transform domain. We use this method
to find the zero input response and the zero state response and their sum, which
is the total response denoted as y 1 (n) , y 2 (n) ,and y 3 (n) .

2.3.2 Natural Response and Forced Response

It is to be pointed out that the total response can also be expressed as the sum
of the natural response and forced response of the system. First let us make
it clear that the natural response is not the same as the zero input response
of the system. The natural response is defined as the component of the total
response, which consists of all terms displaying the natural frequencies of the
system. Natural frequencies are also known as the “characteristic roots” of the
system, eigenvalues of the system determinant, and poles of the transfer

function.
A few methods are used to find the natural frequencies of a system. Suppose
that the system is described by its single input-output relationship as a 0 y(n)

+

a 1 y(n − 1 ) + a 2 y(n − 2 ) +···+ a N y(n − N) = b 0 x(n) + b 1 x(n − 1 ) +···+

b M x(n M) .

If we assume the solution to the homogeneous equation to be of the form
y c (n)

−

A(c) n , and substitute it as well as its delayed sequences, we get the
following characteristic equation:

=

A(c) n [ a 0 + a 1 (c) − 1 a 2 (c) − 2 a N (c) − N ]

+ +···+

A(c) n − N [ a 0 (c) N a 1 (c) N − 1 = + +···+ a N − 1 (c) + a N ] = 0 .

Let the N roots of the characteristic polynomial

[ a 0 (c) N a 1 (c) N − 1 + +···+ a N − 1 (c) + a N ]

be denoted by (c 1 ), (c 2 ),...,(c N ) , which are the natural frequencies. Assuming
that all the roots are distinct and separate, the natural response assumes the form

A 1 (c 1 ) n A 2 (c 2 ) n A N (c N ) n y c (n) = + +···+

which in classical literature is known as the complementary function or comple-
mentary solution .