Z transform

              Z transform permit the processing of digital signals and discrete systeme.

1. Definition
The Z transform is defined by : [Image]

2. Stability in Z

A discrete system whose transfer function is expressed according to Z is stable if all the poles have a module lower than 1.

3. Propriety

3.1. Linearity


3.2. Delay Theorem
[Image] It is noticed that to delay a causal signal of N samples is equivalent to multiply its Z transform by[Image].

3.3. Product of convolution


3.4. Multiplication by[Image]


3.5. Multiplication by [Image]


3.5. Multiplication by[Image]


4. Z transform of tests signals
Dirac : [Image]. Step: [Image].

5. End value and initial value theorem
Initial value theorem: [Image] End value theorem : [Image]

6. Research of original[Image]

        It's necessary to arrange the initial expression to obtain forms in tables by carrying out operations such as the decomposition in simple elements