文档介绍：Chapter6 Z-Transform
Definition
ROC (Region of Converges)
z-Transform Properties
Transfer Function
Z-Transform
In continuous signal system, we use S-Transform and FT as the tools to process problems in the transform domain; so in discrete signal system, we use Z-Transform and DFT.
Z-Transform can make the solution for discrete time systems very simple.
Z-Transform
The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems
Because of the convergence condition, in many cases, the DTFT of a sequence may not exist.
Definition and Properties
DTFT defined by:
leads to the z-transform。
z-transform may exist for many sequences for which the DTFT does not exist。
Definition and Properties
For a given sequence g[n], its z-transform G(z) is defined as:
where z= Re(z) + jIm(z) is plex variable.
Definition and Properties
If we let z=rej, then the z-transform reduces to:
For r = 1 (., |z| = 1), z-transform reduces to its DTFT, provided the latter exists。The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle。
Definition and Properties
Like the DTFT, there are conditions on the convergence of the infinite series:
For a given sequence, the set R of values of z for which its z-transform converges is called the region of convergence (ROC)
Definition and Properties
From our earlier discussion on the uniform convergence of the DTFT, it follows that the series:
converges if {g[n]r-n} is absolutely summable, ., if:
Definition and Properties
In general, the ROC of a z-transform of a sequence g[n] is an annular region of the z-plane:
where
Note: The z-transform is a form of a Laurent series and is an analytic function at every point in the ROC。
Definition and Properties
Example - Determine the z-transform X(z) of the causal sequence x[n]=n[n] and its ROC.
Now
The above power series converges to:
ROC is the annular region |z| > |α|.