A1.1 Signals and Systems

We consider discrete-time signals and systems.

1. Linear systems. Consider an operator H[ ] which denotes a system. The output of the system H[ ] given input x _l is denoted by H[ x _l]. Then, a system is linear if

   
   

   where a and b are constants and x _l and y _l are input signals.

2. Impulse response. The impulse response of a linear system is the output of the linear system when the input is a unit impluse:

   
   

   where ? _l denotes the unit impulse defined as

   
   

3. Causal. A linear system is called causal if

   
   

4. Convolution. The convolution of x _l and y _l is defined as follows

   
   

   where * denotes the convolution (operation).

5. Output of linear systems. If a linear system has impulse response { h _m}, given input x _l, the output is given by

A1.2 -transform

1. Definition. The -transform of signal x _l is defined as follows:

   
   

   where z is a complex variable.

2. Convolution property. If y _l = x _l * h _l, then

   
   

   where Y ( z), X( z), and H( z) stand for the -transforms of y _l, x _l, and h _l, respectively.

3. Shift property. If y _l = x _{l ? m}, then

   
   

4. Parseval s theorem. Let X