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| − | The Z-Transform | + | ===The Z-Transform=== |
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. | ||
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| − | Inverse Z-Transform | + | ===Inverse Z-Transform=== |
Returns a complex variable representation back into a discrete-time signal. | Returns a complex variable representation back into a discrete-time signal. | ||
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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. | 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. | ||
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| + | ===Absolute Convergence=== | ||
| + | |||
| + | A series <math> \sum_{n=-\infty}^\infty (An) </math> is said to absolutely converge if <math> \sum_{n=-\infty}^\infty |(An)| </math> 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. | ||
Revision as of 10:23, 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.
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.
