Question: 1 -

What is the ROC of the signal x(n)=δ(n-k), k>0?

z=∞

Entire z-plane, except at z=∞

Entire z-plane, except at z=0

z=0

Answer:

Entire z-plane, except at z=0

Solution:

Given x(n)=δ(n-k)=1 at n=k
=> X(z)=z^-k
From the above equation, X(z) is defined at all values of z except at z=0 for k>0.
So ROC is defined as Entire z-plane, except at z=0.

Question: 2 -

What is the z-transform of the signal x(n)=(0.5)ⁿu(n)?

1/(1+0.5z⁻¹); ROC |z|<0.5

1/(1+0.5z⁻¹); ROC |z|>0.5

1/(1−0.5z⁻¹); ROC |z|>0.5

1/(1−0.5z⁻¹); ROC |z|<0.5

Answer:

1/(1−0.5z⁻¹); ROC |z|>0.5

Solution:

Given x(n)=(0.5)ⁿu(n)=(0.5)ⁿ for n≥0


So, X(z)=∑_n=0 ^∞ 0.5ⁿ z⁻ⁿ=∑_n=0 ^∞ (0.5z⁻¹)ⁿ


This is an infinite GP whose sum is given as


X(z)=1/(1−0.5z⁻¹) under the condition that |0.5z^-1|<1


=> X(z)=1/(1−0.5z⁻¹) and ROC is |z|>0.5.

Question: 3 -

What is the z-transform of the following finite duration signal?

x(n) = {2, 4, 5_↑, 7, 0, 1}

2 + 4z + 5z² + 7z³ + z⁵

2 + 4z^-1 + 5z^-2 + 7z^-3 + z^-5

2z² + 4z + 5 +7z^-1 + z^-3

2 + 4z + 5z² + 7z³ + z⁴

Answer:

2z² + 4z + 5 +7z^-1 + z^-3

Solution:

Substitute the values of n from -2 to 3 and the corresponding signal values in the z-transform formula
We get, X(z) = 2z² + 4z + 5 + 7z^-1 + z^-3.

Question: 4 -

Consider a discrete-time signal given by

x[n] = (-0.25)ⁿ u[n] + (0.5)ⁿ u[-n-1]

The region of convergence of its Z-transform would be

the entire z-plane

the region inside the circle of radius 0.5 and centered at origin

the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.

the region outside the circle of radius 0.25 and centered at origin

Answer:

the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.

Solution not available.

Question: 5 -

The Z-Transform X(z) of a discrete time signal x(n) is defined as ____________

∑ _n=−∞^∞ x(n) zⁿ

∑ _n=0^∞ x(n) zⁿ

∑ _n=−∞^∞ x(n) z^-n

None of these

Answer:

∑ _n=−∞^∞ x(n) z^-n

Solution not available.