Properties of Z Transform MCQ Quiz - Objective Question with Answer for Properties of Z Transform - Download Free PDF

Last updated on Jun 12, 2025

Latest Properties of Z Transform MCQ Objective Questions

Properties of Z Transform Question 1:

An LTI system is minimum phase if

  1. All poles are inside unit circle
  2. All zeros are inside unit circle
  3. All poles are outside unit circle
  4. All poles and zeros are inside unit circle

Answer (Detailed Solution Below)

Option 4 : All poles and zeros are inside unit circle

Properties of Z Transform Question 1 Detailed Solution

Explanation:

Minimum Phase LTI System

Definition: A Linear Time-Invariant (LTI) system is considered to be a minimum phase system if all its poles and zeros are located within the unit circle in the Z-plane. The unit circle is defined as the set of points in the complex plane where the magnitude of the complex number is equal to 1. For a minimum phase system, both stability and minimum phase conditions are satisfied.

Key Characteristics of a Minimum Phase System:

  • All poles of the system must lie strictly inside the unit circle. This ensures the system is stable.
  • All zeros of the system must also lie strictly inside the unit circle. This ensures the system has the minimum phase property.
  • The system's phase response is minimized, which is crucial in many control and signal processing applications where phase linearity or minimal phase shift is required.

Explanation of the Correct Option:

The correct option is:

Option 4: All poles and zeros are inside the unit circle.

This is the correct condition for a minimum phase LTI system. For a system to be classified as minimum phase, it is not sufficient for just the poles or just the zeros to be inside the unit circle—both must satisfy this condition. This ensures that the system is stable and has the minimum phase property.

Properties of Z Transform Question 2:

Two systems h[n] = A(b1)n u[n] and h2[n] = A(b2)n u[n] are cascaded. If the effective , find a possible value of (A, b1, b2)

  1. A = 1/2, b1 = 1/2, b2 = -1/2
  2. A = 2, b1 = 2, b2 = -2
  3. A = 1, b1 = 1/2, b2 = -1/2
  4. A = 1, b1 = 1, b2 = -1

Answer (Detailed Solution Below)

Option 3 : A = 1, b1 = 1/2, b2 = -1/2

Properties of Z Transform Question 2 Detailed Solution

Explanation:

Analysis of the Given Problem:

The problem involves cascading two systems with impulse responses h1[n] = A(b1)nu[n] and h2[n] = A(b2)nu[n]. The overall transfer function of the cascaded system is given as:

H(z) = 4 / (4 - z-2).

We need to determine the possible values of A, b1, and b2 that satisfy the given transfer function. Let us solve step-by-step:

Step 1: Analyze the impulse response and transfer function of individual systems

The impulse response of the first system is:

h1[n] = A(b1)nu[n],

where u[n] is the unit step function. The Z-transform of h1[n] is:

H1(z) = A / (1 - b1z-1) for |z| > |b1|.

Similarly, the impulse response of the second system is:

h2[n] = A(b2)nu[n].

The Z-transform of h2[n] is:

H2(z) = A / (1 - b2z-1) for |z| > |b2|.

Step 2: Combine the two systems

When the two systems are cascaded, the overall transfer function is the product of the transfer functions of the individual systems:

H(z) = H1(z) × H2(z).

Substituting the expressions for H1(z) and H2(z):

H(z) = [A / (1 - b1z-1)] × [A / (1 - b2z-1)].

H(z) = A2 / [(1 - b1z-1)(1 - b2z-1)].

Step 3: Match the given H(z)

The given H(z) is:

H(z) = 4 / (4 - z-2).

To match this with the derived expression, rewrite the denominator of the given H(z):

4 - z-2 = (2 - z-1)(2 + z-1).

So, the given H(z) can be expressed as:

H(z) = 4 / [(2 - z-1)(2 + z-1)].

Comparing this with the derived H(z), we identify:

b1 = 1/2, b2 = -1/2, and A2 = 1.

Thus, A = 1 (since A must be positive), b1 = 1/2, and b2 = -1/2.

Correct Option: Option 3 (A = 1, b1 = 1/2, b2 = -1/2)

Properties of Z Transform Question 3:

The 𝑍-transform of a discrete signal 𝑥[𝑛] is

 with ROC = R.

Which one of the following statements is true?

  1. Discrete-time Fourier transform of x[n] converges if R is |𝑧| > 3
  2. Discrete-time Fourier transform of x[n] converges if R is 
  3. Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a left-sided sequence
  4. Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a right-sided sequence

Answer (Detailed Solution Below)

Option 2 : Discrete-time Fourier transform of x[n] converges if R is 

Properties of Z Transform Question 3 Detailed Solution

Given:

 

Poles of X(z) are located at z = , z =  and z = 3.

For DTFT to converge, the ROC of Z-transform of x() should contain unit circle.

If x(n) is a right sided sequence then the ROC is |z|>3 which does not include unit circle. So, option (D) and (A) are wrong.

If R.O.C. is  

If x(n) is a left sided then R.O.C will be |z|  which does not include unit circle. So, option (C) is wrong.

Hence, the correct option is (B).

Properties of Z Transform Question 4:

Which one of the following statements not correct for convolution?

  1. The convolution of an odd and even function is an odd function
  2. The convolution of two odd functions is an even function. 
  3. The convolution. of two even functions is an even function.
  4. The convolution of two odd functions is an odd function.

Answer (Detailed Solution Below)

Option 4 : The convolution of two odd functions is an odd function.

Properties of Z Transform Question 4 Detailed Solution

 Properties of convolution of two signals:

→ Convolution of an odd and even function is an odd function.

→ Convolution of two odd functions is an even function.

→ Convolution of two even function is an even function

 Convolution of two causal functions is causal function.

→ Convolution of two Anti causal function is Anticausal function.

→ Convolution of two unequal length rectangular pulse is trapezium.

→ Convolution of two equal length rectangular pulses is triangle.

→ Convolution of signal with periodic train of impulses is periodic representation of signal.

Therefore option (4) statement is not correct.

Properties of Z Transform Question 5:

Two systems with impulse responses h1(t) and h2 (t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

  1. product of h1 (t) and h2 (t)
  2. sum of h1 (t) and h2 (t)
  3. convolution of h1 (t) and h2 (t)
  4. subtraction of h1 (t) and h2 (t)

Answer (Detailed Solution Below)

Option 3 : convolution of h1 (t) and h2 (t)

Properties of Z Transform Question 5 Detailed Solution

Concept:

  • The overall impulse response of systems connected in cascade in the time domain is given by the convolution of the two impulse responses, i.e. convolution h1 (t) and h2 (t).
  • In the frequency domain, the overall impulse response is the multiplication of the impulse responses in the frequency domain.

 

Given two impulse responses are h1(t) and h2(t).

These are cascaded, so the resulting system response will be a convolution of these two systems.

Properties:

Commutative property:

 x(t) ∗ h(t) = h(t) ∗ x(t)

Associative property: 

x(t) ∗ [h1(t) ∗ h2(t)] = [x(t) ∗ h1(t)] ∗ h2(t)

Distributive property:

x(t) ∗ [h1(t) + h2(t)] = x(t) ∗ h1(t) + x(t) ∗  h2(t)

Convolution with impulse property:

 x(t) ∗ δ (t) = x(t)

Width property: if the durations(widths) of x(t) and h(t) are finite and given by Wx and Wh,

Then the duration(width) of the x(t) ∗ h(t) is Wx + Wh

Important points:

Convolution of two equal width rectangles will give resultant as a triangular profile.

Convolution of two unequal width rectangles will give resultant as a trapezoidal profile.

Top Properties of Z Transform MCQ Objective Questions

Consider the following statements regarding a linear discrete-time system

H(z) = (z2 + 1)/[(z + 0.5) (z – 0.5)]

A. The system is stable

B. The initial value h(0) of the impulse response is -4

C. The steady-state value of the impulse response is zero.

Which of these statements is/are correct?

  1. A, B and C
  2. A and B
  3. A and C
  4. B and C

Answer (Detailed Solution Below)

Option 3 : A and C

Properties of Z Transform Question 6 Detailed Solution

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Concept:

For a causal signal x(n), the initial value theorem states that:

For a causal signal x(n), the final value theorem states that:

Calculation:

Given:

∴ Poles = 0.5 and -0.5

Zeros = ±j

Hence all the poles are lying inside the unit circle. Therefore, the system is stable.

Now by using the initial value theorem, we get

Final Value theorem:

Hence, statement C is also correct.

A discrete-time signal x[n] = δ [n - 3] + 2δ [n - 5] has a z-transform X(z). If Y(z) = X(-z) is the z-transform of another signal y[n], then

  1. y[n] = x[n]
  2. ​y[n] = x[-n]
  3. ​y[n] = - x[n]
  4. ​y[n] = - x[-n]

Answer (Detailed Solution Below)

Option 3 : ​y[n] = - x[n]

Properties of Z Transform Question 7 Detailed Solution

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Concept:

The z-transform of a unit impulse function x[n] = δ [n] is given as:

X(z) = 1

Also, the time-shifting affects the z-transform as:

x[n - n0] = z -n0 X(z)

Application:

Given:

x[n] = δ [n - 3] + 2δ [n - 5] 

Taking the z transform, we get:

X(z) = z-3 + 2 z-5

Replacing z by -z we get:

We observe that:

Taking inverse z-transfrom of the above, we get:

If Z transform of x(n) is X(z) then the Z transform of x(n - k) is _______

  1. X(z-k z)
  2. X(zk z)
  3. z-k X(z)
  4. zk X(z)

Answer (Detailed Solution Below)

Option 3 : z-k X(z)

Properties of Z Transform Question 8 Detailed Solution

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Definition:

Z transform is defined as

Time-shifting:

If X(z) is a z transform of x(n), then the z transform of x(n – n0) is,

Analysis:

The output of a linear time invariant system can be obtained from its unit impulse response function and the input function by

  1. Convolution
  2. Addition
  3. Multiplication
  4. Autocorrelation

Answer (Detailed Solution Below)

Option 1 : Convolution

Properties of Z Transform Question 9 Detailed Solution

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Discrete-Time System:

The convolution of two a signal with a system with impulse response h(n) is represented as:

y[n] = x[n] ∗ h[n]

Continuous Time System:

The convolution of two a signal with a system with impulse response h(t) is represented as:

y(t) = x(t) ∗ h(t)

Let x[n] = x[-n] Let X(z) be the Z-transform of x[n]. if 1 + j2 is a zero of X(z). Which one of the following must also be a zero of X(z)

  1. 0.2 + j 0.4
  2. 0.2 – j 0.4
  3. 1 + j 2
  4. 1 – j 0.5

Answer (Detailed Solution Below)

Option 2 : 0.2 – j 0.4

Properties of Z Transform Question 10 Detailed Solution

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Concept:

Time reversal property of Z-transform

X[n] ↔ X(z)

X[-n] ↔ X(z-1)

Calculation:

Given that x[n] = x[-n]

⇒ X(z) = X(z-1)

Zero of X(z) = 1 + j2

Then another zero will be:

Two systems with impulse responses h1(t) and h2 (t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

  1. product of h1 (t) and h2 (t)
  2. sum of h1 (t) and h2 (t)
  3. convolution of h1 (t) and h2 (t)
  4. subtraction of h1 (t) and h2 (t)

Answer (Detailed Solution Below)

Option 3 : convolution of h1 (t) and h2 (t)

Properties of Z Transform Question 11 Detailed Solution

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Concept:

  • The overall impulse response of systems connected in cascade in the time domain is given by the convolution of the two impulse responses, i.e. convolution h1 (t) and h2 (t).
  • In the frequency domain, the overall impulse response is the multiplication of the impulse responses in the frequency domain.

 

Given two impulse responses are h1(t) and h2(t).

These are cascaded, so the resulting system response will be a convolution of these two systems.

Properties:

Commutative property:

 x(t) ∗ h(t) = h(t) ∗ x(t)

Associative property: 

x(t) ∗ [h1(t) ∗ h2(t)] = [x(t) ∗ h1(t)] ∗ h2(t)

Distributive property:

x(t) ∗ [h1(t) + h2(t)] = x(t) ∗ h1(t) + x(t) ∗  h2(t)

Convolution with impulse property:

 x(t) ∗ δ (t) = x(t)

Width property: if the durations(widths) of x(t) and h(t) are finite and given by Wx and Wh,

Then the duration(width) of the x(t) ∗ h(t) is Wx + Wh

Important points:

Convolution of two equal width rectangles will give resultant as a triangular profile.

Convolution of two unequal width rectangles will give resultant as a trapezoidal profile.

The z-transform of  is 

  1.  if |z| > 1
  2.  if |z| ≠ 0
  3.  if |z| > 1
  4. z(1) - n

Answer (Detailed Solution Below)

Option 3 :  if |z| > 1

Properties of Z Transform Question 12 Detailed Solution

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The z-transform for 1/n does not exist for n=0, but it exists for n > 0 and for n

Let n > 0

X(z) = 

Differentiating both sides w.r.t z we get,

   |z-1| 1

Integrating both sides, we get

 for |z| > 1

Let n

X(z) = 

Differentiating both sides w.r.t z we get,

Let n = -p

     |z|

Integrating both sides, we get

 for |z|

∴ By looking at the options the correct answer is 3 (given for n > 0)

Which one of the following statements not correct for convolution?

  1. The convolution of an odd and even function is an odd function
  2. The convolution of two odd functions is an even function. 
  3. The convolution. of two even functions is an even function.
  4. The convolution of two odd functions is an odd function.

Answer (Detailed Solution Below)

Option 4 : The convolution of two odd functions is an odd function.

Properties of Z Transform Question 13 Detailed Solution

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 Properties of convolution of two signals:

→ Convolution of an odd and even function is an odd function.

→ Convolution of two odd functions is an even function.

→ Convolution of two even function is an even function

 Convolution of two causal functions is causal function.

→ Convolution of two Anti causal function is Anticausal function.

→ Convolution of two unequal length rectangular pulse is trapezium.

→ Convolution of two equal length rectangular pulses is triangle.

→ Convolution of signal with periodic train of impulses is periodic representation of signal.

Therefore option (4) statement is not correct.

The ROC of the Z-Transform of the signal 2nu(n) - 3nu(-n - 1):

  1. is |Z| > 1
  2. does not exist
  3. is 2 < |Z| < 3
  4. is |Z| < 1

Answer (Detailed Solution Below)

Option 3 : is 2 < |Z| < 3

Properties of Z Transform Question 14 Detailed Solution

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Concept:
 

 ;  ROC = |z| > a

 ; ROC = |z|

From property of linearity:

x1(n) → X1(z);  ROC: R1​​

x2(n) → X2(Z);  ROC: R2

If g(n) = x1(n) + x2(n) then ROC of g(n) is Rintersection R2​​​

If R1 intersection R2 is nullity then that expression's Z transform does not exist. 

Analysis:

g(n) = 2nu(n) - 3n(n) u(-n-1)

ROC: |z| > 2 intersection |z|

∴ The resultant ROC will be 2

If ROC1 is the region of convergence of x(n) and ROC2 is the region of convergence of y(n), then x(n) * y(n) is _______. (Where '*' represents the convolution opetation)

  1. ROC1 - ROC2
  2. ROC1 Intersection ROC2
  3. ROC1 Union ROC2
  4. ROC1 + ROC2

Answer (Detailed Solution Below)

Option 2 : ROC1 Intersection ROC2

Properties of Z Transform Question 15 Detailed Solution

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Concept:

Let,

yo(n) = x(n) * y(n)

Then to find Yo(z)

In terms of Z transform we can write:

x(n) * y(n) = X(z) × Y(z)

The convolution of two discrete time signals gives the ROC as the intersection of respective ROCs. 

Additional Information

Properties of the region of convergence:

The properties of the ROC depend on the nature of the signal. Assuming that the signal has a finite amplitude and that the z-transform is a rational function.

  • The ROC is a ring or disk in the z-plane, centred on the origin (0 R L ≤ ∞)
  • The Fourier transform of x[n] converges absolutely if and only if the ROC of the z-transform includes the unit circle.
  • The ROC cannot contain any poles.
  • If x[n] is finite duration, then the ROC is the entire z-plane except perhaps at z = 0 or z = ∞
  • If x[n] is a right-sided sequence then the ROC extends outward from the outermost finite pole to infinity.
  • If x[n] is left-sided then the ROC extends inward from the innermost nonzero pole to z = 0
  • A two-sided sequence (neither left nor right-sided) has a ROC consisting of a ring in the z-plane, bounded on the interior and exterior by a pole (and not containing any poles).
  • The ROC is a connected region.

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