Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 45

Fig. 4.5 Discrete system and
signals and their relations in
time and frequency domains

x ( n ) h () y ( n ) = x ( n ) ∗ h ( n )

X ( Z ) H ( ) Y ( ) = X ( ) H ( ) Z Z Z

Z

X ∗ ( j ) H ∗ ( j ) Y ∗ ( j )

ω ω ω

Solution From ( 4.27 ) the signal samples are:

¼ 1 ;

X

x ð n Þ¼ 1
N

jnk 2p
N

N exp

for k ¼ 0

which means that the signal is equal to unity for all n (step signal).

4.8 Description of Discrete Dynamic Systems
in Time and Frequency Domains

4.8.1 Description of Discrete Systems in Time Domain

Discrete systems can be defined and described either in time or in frequency
domains. These two domains are closely related according to adequate Fourier
transform. To describe given system is then enough to define it in one of the
domains [ 7 ].
Discrete dynamic system is defined in time domain by mathematical operation
realized on its input series x(n) to get the output series y(n) = F{x(n)}. This
process is illustrated in the block scheme in Fig. 4.5 .
Examples of such relationship could be the following equations:

y ð n Þ¼ 0 : 5 ½ x ð n Þþ x ð n 2 Þ

or

y ð n Þ¼½ x ð n Þ 0 : 99y ð n 1 Þ:

Actually, there are certain simplifications in equations above and further, not
important from the viewpoint of reaching their solutions. In these equations the
sampling period T S was omitted, thus instead of x(nT S ) and y(nT S ) the simplified
x(n) and y(n) could be used. It may also be interpreted as a system or equation with
sampling period T S = 1. Such notation is usually much simpler and does not lead
to misunderstanding in time domain.