Discrete-Time Signals and Systems - Digital Signal Processing Laboratory - page 18

Linear Constant Coefficient Difference Equations

The difference equation is another fundamental relation between the input
and output of LTI systems. It describes the discrete-time process executed
by the system, such as a high-pass or a low-pass filtering. The difference
equation takes the following form:

() + ( ) +… ( ) = () + ( ) +… bbxn M ( )

ay n

ay n − 1 a y n N − bx n bx n − 1 −

0 1 N 0 1 M

or compactly as follows:

N M

∑ ∑

ayn k ( −= ) bxn k ( − ) (2.3)

k k

k = 0 k = 0

Important properties of the difference equations are:

• The order of the difference equation is N , where N ≥ M .

• The coefficients a i , i = are constant,

real numbers and define the properties of the system. For example,
one set of coefficients could generate a low-pass filter, while a dif-
ferent set of coefficients could generate a highpass filter. The latter
aspect shows the great simplicity of digital systems.

0, 1, 2, … N , and b i , i = 0, 1, 2, … M

2.1.1 Introduction to Z-Transforms and the System Function H ( z )

In the previous section, we introduced two fundamental properties of LTI
systems, linear convolution and difference equation. The link between these
two fundamental aspects of LTI systems is provided by the Z-transform. The
Z-transform is a very important time-to-frequency domain transformation
of the basic discrete-time signal x ( n ). The Z-transform is defined by the
equation:

∞
∑

xnz n

−

Xz () = () (2.4)

n =−∞

where the variable z is complex and the function X ( z ) is defined in the
complex plane.

A concise list of Z-transform properties is given in Table 2.1 . Utilizing

property (2) from Table 2.1, in Equation 2.2, we get the transformed equation:

Y ( z ) = X ( z ) H ( z ) (2.5)