Problems Based on z Transform

Problems Based on z Transform

Problem 1:

Obtain the z transform of δ(n)

Solution:

Problem 2:

Obtain the z transform of u(n)

Unit step sequence u(n) = 

Solution:

Figure shows the ROC of u(n)

Problem 3:

Obtain z transform of x(n) = aⁿ u(n)

Solution:

By definition of z transform, X(z) = 

Following figure shows the ROC of aⁿ u(n)

Hint: ROC of the right hand sided sequence (causal sequence) is outside the circle.

Problem 4:

Obtain z transform of x(n) = -aⁿ u(-n-1)

Solution:

By definition of z transform.

Here |a^-1 z| < 1 is equal to |z| < |a| ROC is the area that lies inside the unit circle of radius |a|

Problem 5:

Determine z transform of (i) x(n) = aⁿ/n! for n ≥ 0 (ii) x(n) = n u(n) for n ≥ 0 Dec 10-Marks 10

Solution:

(i) x(n) = aⁿ/n!

By definition X(z) = 

(ii) Solution:

Problem 6:

Determine z transform of x(n) = n² u(n)

Solution:

Problem 7:

Find the z transform and its ROC of

x(n) = 1, n ≥ 0

= 3ⁿ, n < 0 May 05-8 Marks

Solution:

and

Problem 8:

Determine the z transform and ROC of the signal.

Solution:

(1) is Equivalent to

Problem 9:

z transform of [u(n) - u(n-10)]

Solution:

Problem 10:

Find the z transform of the sequence

Solution:

Problem 11:

Determine z transform of the following sequence a ^{| n |}, 0 < |α| < 1

Solution:

Problem 12:

Find z transform of x(n) = a ^{- | n |}, 0 < |α| < 1

Solution:

Problem 13:

Determine z transform of

Solution:

Problem 14:

Obtain the z transform of x(n) = cos nθ u(n)

Solution:

Problem 15:

Obtain the z transform of x(n) = cos (βnt) Dec 03-3 Marks

Solution:

We know that

Problem 16:

Determine the z transform of

Solution:

From above equations (1) & (2)

Replacing z by z/a in X₁(z), X(z) becomes

Solution:

from above equations (1) & (2)

Replacing z by z/a in X¹(z), X(z) becomes

Problem 17:

Determine z transform of following sequences:

(i) x(n) = {1, 2, 3, 4, 5, 0, 7}, (ii) x(n) = {1, 3, 5, 7, 5, 0, 6}

Solution:

(i) x(n) = {1, 2, 3, 4, 5, 0, 7}

x(0) = 1, x(1) = 2, x(2) = 3, x(3) = 4, x(4) = 5, x(5) = 0, x(6) = 7

By definition

Putting for x(n) in above equation

Result:

X(z) = ∞, for z = 0, (i.e) X(z) is convergent for all values of z, except z = 0. Hence ROC is entire z plane except z = 0.

Putting for x(n) in above equation.

Result:

X(z) = ∞ for z = 0 and z = ∞. Hence ROC is entire z plane except z = 0 and z = ∞.