Z-Transform Study Notes for GATE & Other Electrical Engineering Exams

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

,  where z = r.e^jω

• The discrete-time Fourier Transform (DTFT) is obtained by evaluating Z-Transform at z = e^jω
• 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)} ; X₁ (z) = Z {x_l (n)} ; X₂(z) = z{x₂ (n)}; Y(z) =z (y (n))

Summary of Properties of z- Transform:

Impulse Response and Location of Poles

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