Basic System Properties

1. System with and without Memory 

A system is considered as memoryless  if its output at a given time is  depended on the input at the same time  alone . Otherwise , that system has memory .  The system specified by equation (1)  is memoryless , while the one specified by equation (2) is a system with memory . 

2. Causality 

A system is causal if the output at any time depends only on the values of the input at the present and the previous time . The system defined by equation (3) is causal , but the system given by equation (4) is not . 

For now , we can conclude that all systems without memory are casual. 

3.Time Invariance 

A system is regarded as time invariant only if the characteristics of the system are fixed all the time .  If we conduct a time shift on the input values, the output values are also shifted by the same time units . This is illustrated the following example. 

An example of a time invariant system :

if   y[n] = a x[n]   

then 

y [n-n0] = a x[n-n0]

4.  Linearity 

A system is linear if the input is the weighted sum of several signals , and consequently , the out put is also the weighted sum of the responses of the system to each of those signals . 

More specifically , considering  the responses of x1(t) and x2(t) are y1(t) and y2(t) respectively, then the system has the linearity when 

1) the response to x1(t) + x2(t)  is y1(t) + y2(t)               (  the additivity property   )

2) the response to  ax1(t) is ay1(t)                                  ( scaling or homogeneity property )