The Z-Transform
The z-transform converts a discrete-time signal into a complex frequency domain representation.
- $ X(z) = \sum_{n=-\infty}^\infty (x[n]z^{-n}) $
Some Properties:
Linearity:
- $ ax1[n]+bx2[n] = aX1(z)+bX2(z) $
Time-Shifting:
- $ x[n-k] = z^{-k}X(z) $
Scaling in Z domain:
- $ a^{n}Y(z) = X(a^{-1}Z) $
Time Reversal:
- $ x[-n] = X(z^{-1}) $
Convolution:
- $ x1[n]* x2[n] = X1(z)X2(z) $
Inverse Z-Transform
Returns a complex variable representation back into a discrete-time signal.
- $ x[n] = Z^{-1}[X(z)] = \int X(z)z^{n-1}\ $
in this case the integral is around a counter-clockwise clothed path encircling the origin of the complex plane and entirely inside the R.O.C.
Absolute Convergence
A series $ \sum_{n=-\infty}^\infty (An) $ is said to absolutely converge if $ \sum_{n=-\infty}^\infty |(An)| $ converges
The R.O.C. (Region of convergence, absolute convergence in this case) is the set of points in the complex plane for which the summation of the Z-Transform converges.
For example:
$ X(z)=\sum{n=-\infty}^\infty x[n]z^{-n} $ converges if the absolute value $ |x[n]z^{-n}| < 1 $.
But the norm of $ |z^{-n}| = 1 $, so the series converges if $ |x[n]| < 1 $.
