# Digital Signal Processing Questions and Answers – State-Space System Analysis and Structures

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This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “State Space System Analysis and Structures”.

1. The state space or the internal description of the system still involves a relationship between the input and output signals, what are the additional set of variables it also involves?
a) System variables
b) Location variables
c) State variables
d) None of the mentioned

Explanation: Although the state space or the internal description of the system still involves a relationship between the input and output signals, it also involves an additional set of variables, called State variables.

2. State variables provide information about all the internal signals in the system.
a) True
b) False

Explanation: The state variables provide information about all the internal signals in the system. As a result, the state-space description provides a more detailed description of the system than the input-output description.

3. Which of the following gives the complete definition of the state of a system at time n0?
a) Amount of information at n0 determines output signal for n≥n0
b) Input signal x(n) for n≥n0 determines output signal for n≥n0
c) Input signal x(n) for n≥0 determines output signal for n≥n0
d) Amount of information at n0+input signal x(n) for n≥n0 determines output signal for n≥n0

Explanation: We define the state of a system at time n0 as the amount of information that must be provided at time n0, which, together with the input signal x(n) for n≥n0 determines output signal for n≥n0.

4. From the definition of state of a system, the system consists of only one component called memory less component.
a) True
b) False

Explanation: According to the definition of state of a system, the system consists of two components called memory component and memory less component.

5. If we interchange the rows and columns of the matrix F, then the system is called as ______________
a) Identity system
b) Diagonal system
c) Transposed system
d) None of the mentioned

Explanation: The transpose of the matrix F is obtained by interchanging its rows and columns, and it is denoted by FT. The system thus obtained is known as Transposed system.
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6. A single input-single output system and its transpose have identical impulse responses and hence the same input-output relationship.
a) True
b) False

Explanation: If h(n) is the impulse response of the single input-single output system, and h1(n) is the impulse response of the transposed system, then we know that h(n)=h1>(n). Thus, a single input-single output system and its transpose have identical impulse responses and hence the same input-output relationship.

7. A closed form solution of the state space equations is easily obtained when the system matrix F is?
a) Transpose
b) Symmetric
c) Identity
d) Diagonal

Explanation: A closed form solution of the state space equations is easily obtained when the system matrix F is diagonal. Hence, by finding a matrix P so that F1=PFP-1 is diagonal, the solution of the state equations is simplified considerably.

8. What is the condition to call a number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector?
a) (F+λI)U=0
b) (F-λI)U=0
c) F-λI=0
d) None of the mentioned

Explanation: A number λ is an Eigen value of F and a nonzero vector U is the associated Eigen vector if
FU=λU
Thus, we obtain (F-λI)U=0.

9. The determinant |F-λI|=0 yields the characteristic polynomial of the matrix F.
a) True
b) False

Explanation: We know that (F-λI)U=0
The above equation has a nonzero solution U if the matrix F-λI is singular, which is the case if the determinant of (F-λI) is zero. That is, |F-λI|=0.
This determinant yields the characteristic polynomial of the matrix F.

10. The parallel form realization is also known as normal form representation.
a) True
b) False

Explanation: The parallel form realization is also known as normal form representation, because the matrix F is diagonal, and hence the state variables are uncoupled.

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