| Line 3: | Line 3: | ||
The z-transform converts a discrete-time signal into a complex frequency domain representation. | The z-transform converts a discrete-time signal into a complex frequency domain representation. | ||
| − | * <math> X(z) = \sum_{n=-\infty}^\infty (x[n]z^{-n})</math> | + | * <math> X(z) = \sum_{n=-\infty}^\infty (x[n]z^{-n}) </math> |
Some Properties: | Some Properties: | ||
| Line 25: | Line 25: | ||
Convolution: | Convolution: | ||
| − | * x1[n]* x2[n] = X1(z)X2(z) | + | * <math> x1[n]* x2[n] = X1(z)X2(z) </math> |
Revision as of 11:18, 8 September 2009
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.
