The z-transform is a mathematical tool used in signal processing and control systems to analyze discrete-time signals and systems. It transforms a discrete-time signal, which is a sequence of numbers, into a complex frequency-domain representation. The z-transform is particularly useful for analyzing linear time-invariant (LTI) systems and solving difference equations.

Definition

• For a discrete-time signalx[n], where n is an integer (typically representing time samples), the z-transform is defined as:

where:

• X(z) is the z-transform of the signal x[n],
• z is a complex variable, typically expressed as z=re ^jω
• The summation is over all integer values of n.

• The region in the complex plane where the summation converges is called the region of convergence (ROC).

Types of Z-Transform

1. Unilateral (One-Sided) z-Transform:
   
   • Considers only non-negative time indices (n≥0):

• Used for causal signals and systems (signals that are zero for n<0).

2. Bilateral (Two-Sided) z-Transform:

• Considers all time indices ( n=−∞ to ∞ ).
• Used for non-causal signals or systems.

Key Properties of the z-Transform

The z-transform has several properties that make it a powerful tool for analyzing discrete-time systems. Below are the most important ones:

1.Find the z transfer of signal x(n) ={2 3 4 5 6}

2.Find the z-transform of signal x(n) ={1 2 5 7 0 1}

3.Find the z-transform of signal x(n)= [3(2)ⁿ -4(3)ⁿ]u(n)

4.Find the z-transfer of signal x(x)= coswn u(n)

Compile by Er. Basant kumar yadav