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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: | ||
| + | |||
| + | Linearity: | ||
| + | |||
| + | * <math> ax1[n]+bx2[n] = aX1(z)+bX2(z) </math> | ||
| + | |||
| + | Time-Shifting: | ||
| + | |||
| + | * <math> x[n-k] = z^{-k}X(z) </math> | ||
| + | |||
| + | Scaling in Z domain: | ||
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| + | * <math> a^{n}Y(z) = X(a^{-1}Z) </math> | ||
| + | |||
| + | Time Reversal: | ||
| + | |||
| + | * <math> x[-n] = X(z^{-1}) </math> | ||
| + | |||
| + | Convolution: | ||
| + | |||
| + | * x1[n]* x2[n] = X1(z)X2(z) | ||
Revision as of 11:05, 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)
