This set of Signals & Systems Quiz focuses on “Discrete-Time Systems in the Time-Domain – 2”.

1. The difference equation for an N^{th} order discrete-time system is ___________

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2. The response of any discrete time system can be decomposed as _____________

a) Total Response=Impulse+step

b) Total Response=Impulse+Ramp

c) Total Response=zero-output response

d) Total Response=zero-state response+zero-input response

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Explanation: There are two approaches to analyzing response of a system:

Direct solution of difference solution

Decomposing in terms of impulse signals

In the first method, the response of the system can be decomposed as:

Total Response = zero-state response + zero-input response.

3. Zero-state response of the system is _____________

a) Response of the system when initial state of the system is zero

b) Response of the system due to input alone

c) Response of the system due to input alone when initial state of the system is zero

d) Response of the system due to input alone when initial state is neglected

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Explanation: Zero-state response of the system is the response of the system due to input alone when the initial state of the system is zero. That is the system is relaxed at time n = 0.

4. Zero-input response is also known as ____________

a) zero-state response

b) Natural response

c) state-input response

d) Forced response

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Explanation: Natural response of the system is when the input x(n) = 0.

5. The general solution of natural response is of the form of _________

a) y_{h} (n)= c_{1} λ_{1}^{n}+c^{2}λ^{2}^{n}+⋯+c_{N}λ_{N}^{n}

b) y_{h} (n)= c_{1} λ_{1}^{n}+c^{2}λ^{2}^{n}+⋯+c_{N}λ_{N}^{n}

c) y_{h} (n)= c_{1} λ_{1}^{2}+c_{2}λ_{2}^{2}+⋯+c_{N}λ_{N}^{2}

d) y_{h}(n)= c_{1} λ_{1}^{n}-c^{2}λ^{2}^{n}+⋯+c_{N}λ_{N}^{n}

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Explanation: The general solution of natural response is of the form:

y

_{h}(n)= c

_{1}λ

_{1}

^{n}+c

^{2}λ

^{2}

^{n}+⋯+c

_{N}λ

_{N}

^{n}

The form will vary if the roots are repeating or complex.

6. Determine the natural response of the system: Difference equation is

y(n)-y(n-1)-2y(n-2)=x(n) and y(-1) = 1; y(-2) = 0

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Explanation: Natural Response of the system:

Homogenous equation ⇒ y(n)-y(n-1)-2y(n-2)=0

The homogenous solution: y

_{h}(n)= λ

^{n}

⇒ λ

^{n}– λ

^{(n-1)}-2λ

^{(n-2)}=0

⇒ λ

^{(n-2)}[λ

^{2}– λ

^{1}-2]=0

⇒ λ

^{2}– λ-2=0

⇒ λ

^{2}-2λ+λ-2=0

⇒ λ(λ-2)+1(λ-2)=0

⇒ (λ-2)(λ+1)=0

⇒ λ

^{1}=2,λ

^{2}=-1

General form of homogenous solution is

y

_{h}(n)= c

_{1}(2)

^{n}+c

_{2}(-1)

^{n}(1)

⇒ y(0)= c

_{1}+c

_{2}(2)

⇒ y(1)=2c

_{1}– c

_{2}(3)

⇒ y(0)-y(-1)-2y(-2)=0

Given, y(-1) = 1 and y(-2) = 0

⇒ y(0)-1=0⇒y(0)=1

Similarly, y(1)-y(0)-2y(-1)=0⇒y(1)=1+2=3

∴ y(0) = 1 and y(1) = 3

Comparing the above values with equations (2) and (3)

⇒ c_{1}+c_{2}=1 and 2c_{1}– c_{2}=3

Solving the two equations we get, c1 = 4/3 and c2 = -1/3

7. Forced Response is solution of difference equation when ____________

a) Input is zero

b) Input is given and initial conditions are zero

c) Natural Response

d) Input is given and initial conditions are non-zero

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Explanation: Forced response is solution of difference equation when input is given and initial conditions are zero. Also known as zero-state response.

8. Forced response consists of _________

a) Homogenous solution and general solution

b) General solution alone

c) Homogenous solution and particular solution

d) Particular solution alone

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Explanation: Forced response consists of homogenous solution and particular solution.

**Sanfoundry Global Education & Learning Series – Signals & Systems.**

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