Discrete Time Fourier Transform(DTFT) exists for energy and power signals. Z-transform also exists for neither energy nor Power (NENP) type signal, up to a certain extent only. The replacement z = ejw is used for Z-transform to DTFT conversion only for absolutely summable signal.
So, the Z-transform of the discrete time signal x(n) in a power series can be written as −
The above equation represents a two-sided Z-transform equation.
Generally, when a signal is Z-transformed, it can be represented as −
Or
If it is a continuous time signal, then Z-transforms are not needed because Laplace transformations are used. However, Discrete time signals can be analyzed through Z-transforms only.
Region of Convergence
Region of Convergence is the range of complex variable Z in the Z-plane. The Z- transformation of the signal is finite or convergent. So, ROC represents those set of values of Z, for which X(Z) has a finite value.
Properties of ROC
The Z-transform is uniquely characterized by −
Signals and their ROC
x(n) | X(Z) | ROC |
---|---|---|
δ(n) | 1 | Entire Z plane |
U(n) | 1/(1−Z−1) | Mod(Z)>1 |
anu(n) | 1/(1−aZ−1) | Mod(Z)>Mod(a) |
−anu(−n−1) | 1/(1−aZ−1) | Mod(Z)<Mod(a) |
nanu(n) | aZ−1/(1−aZ−1)2 | Mod(Z)>Mod(a) |
−anu(−n−1) | aZ−1/(1−aZ−1)2 | Mod(Z)<Mod(a) |
U(n)cosωn | (Z2−Zcosω)/(Z2−2Zcosω+1) | Mod(Z)>1 |
U(n)sinωn | (Zsinω)/(Z2−2Zcosω+1) | Mod(Z)>1 |
Example
Let us find the Z-transform and the ROC of a signal given as x(n)={7,3,4,9,5}, where origin of the series is at 3.
Solution − Applying the formula we have −
ROC is the entire Z-plane excluding Z = 0, ∞, -∞
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1. What is the Z-transform in electrical engineering? |
2. How is the Z-transform different from the Fourier transform? |
3. What are the advantages of using the Z-transform in electrical engineering? |
4. How is the Z-transform applied in signal processing? |
5. What are some common applications of the Z-transform in electrical engineering? |
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