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**Z-Transform**

- Computation of the Z-transform for discrete-time signals.
- Enables analysis of the signal in the frequency domain.
- Z-Transform takes the form of a polynomial.
- Enables interpretation of the signal in terms of the roots of the polynomial.
- z
^{−1}corresponds to a delay of one unit in the signal.

The Z - Transform of a discrete time signal x[n] is deﬁned as

- The discrete-time Fourier Transform (DTFT) is obtained by evaluating Z-Transform at z = ejω.
- The z-transform defined above has both sided summation. It is called bilateral or both sided Z-transform.

**Unilateral (one-sided) z-transform**

- The unilateral z-transform of a sequence x[n] is defined as

**Region of Convergence (ROC):**

ROC is the region where z-transform converges. It is clear that z-transform is an infinite power series. The series is not convergent for all values of z.**Significance of ROC**

- ROC gives an idea about values of z for which z-transform can be calculated.
- ROC can be used to determine causality of the system.
- ROC can be used to determine stability of the system.

**Summary of ROC of Discrete Time Signals for the sequences**

**Characteristic Families of Signals and Corresponding ROC**

**Note:** X(z) = z{x(n)} ; X_{1} (z) = Z {x_{l} (n)} ; X_{2}(z) = z{x_{2} (n)}; Y(z) = z (y (n))

Summary of Properties of z- Transform:**Impulse Response and Location of Poles**

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39 videos|7 docs|19 tests

### Initial Value Theorem of Z transform

- Video | 06:23 min
### Final Value Theorem of Z transform

- Video | 10:50 min
### Inverse Z transform by Partial Fraction

- Video | 11:07 min
### Inverse Z transform by Long Division Method

- Video | 10:17 min

- Test: Laplace Part - 1
- Test | 10 ques | 30 min
- Test: Laplace Part - 2
- Test | 10 ques | 30 min