Transforms and its Applications: UNIT 2: Z Transform

Z - transform of standard functions

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Z Transform: Definition and Example Important Solved Problems with formula, steps, derivation and answer based on Z‒transform of some basic functions.

Z ‒ transform of standard functions, elementary properties of Z ‒ transform

Z transform

Definition: Z ‒ transform (two‒sided or bilateral)

Let {x (n)} be a sequence defined for all integers then its Z‒transform is defined to be

Z {x (n)} = X (z) = ^∞Σ_n=-_∞ x (n) z^‒n

where z is an arbitrary complex number.

Definition: Z ‒ transform (one‒sided or unilateral)

Let {x (n)} be a sequence defined for n = 0, 1, 2, … and x (n) = 0 for n < 0, then its Z ‒ transform is defined to be

where z is an arbitrary complex number.

Definition: Z ‒ transform for discrete values of t

If f(t) is a function defined for discrete values of t where t = nT, n = 0, 1, 2, 3, …. T being the sampling period, then Z - transform of f(t) is defined as

Note:

1. Mostly we study one‒sided Z‒transform

2. If f(t) given, then replace 't' by 'n T'

3. The double braces {} are used for a sequence.

Sometimes we use [ ] or ().

Problems based on Z‒transform of some basic functions

Find the Z‒transform of the following functions:

1. 1

2. aⁿ

3. n

4. 1/n

5. 1/ n+1

6. 1/ n‒1

7. 1/ n!

8. 1/(n+1)!

9. (n+2)

10. 1/(n+2)

Transforms and its Applications: UNIT 2: Z Transform : Tag: Engineering mathematics, Maths : - Z - transform of standard functions

Home | Department: EEE Department | Ist Year | Subject: Transforms and its Applications |

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Transforms and its Applications: UNIT 2: Z Transform

Applications of Z transform

Z - transform of standard functions

Z Transform Linearity property

Z Transform: Differentiation in the Z-Domain

Z Transform: First Shifting theorem (Frequency shifting)

Z Transform: Second shifting theorem (Time Shifting)

Z Transform: Unit impulse sequence and unit step sequence

Z Transform: Initial value theorem and final value theorem

Z Transform: Differentiation

Inverse Z Transform: Partial Fractions Methods

Inverse Z Transform: Inverse integral method (Cauchy's residue theorem)

Z Transform: Convolution Theorem

Z Transform: Formation of difference equations

Application: Solution of difference equations using Z transform

Z Transform: Exercise Problems

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