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QUESTION: 1

What is the rule h*(x+y) = (y+x)*h called?

Solution:

By definition, the commutative rule h*x=x*h.

QUESTION: 2

Does the system h(t) = exp([-1-2j]t) correspond to a stable system?

Solution:

The system corresponds to an oscillatory system, this resolving to a marginally stable system.

QUESTION: 3

What is the rule h*(x*c) = (x*h)*c called?

Solution:

By definition, the commutative rule i h*x=x*h and associativity rule = h*(x*c) = (h*x)*c.

QUESTION: 4

Is y[n] = n*cos(n*pi/4)u[n] a stable system?

Solution:

The ‘n’ term in the y[n] will dominate as it reaches to infinity, and hence could reach infinite values.

QUESTION: 5

What is the rule (h*x)*c = h*(x*c) called?

Solution:

By definition, the associativity rule = h*(x*c) = (h*x)*c.

QUESTION: 6

Is y[n] = n*sin(n*pi/4)u[-n] a stable system?

Solution:

The ‘n’ term in the y[n] will dominate as it reaches to negative infinity, and hence could reach infinite values. Eventhough + infinity would not be a problem, still the resultant system would be unstable.

QUESTION: 7

What is the following expression equal to: h*(c*(b+d(t))), d(t) is the delta function

Solution:

Apply commutative and associative rules

QUESTION: 8

Does the system h(t) = exp([1-4j]t) correspond to a stable system?

Solution:

The system corresponds to an unstable system, as the Re(exp) term is a positive quantity.

QUESTION: 9

The system transfer function and the input if exchanged will still give the same response.

Solution:

By definition, the commutative rule i h*x=x*h=y. Thus, the response will be the same.

QUESTION: 10

For an LTI discrete system to be stable, the square sum of the impulse response should be

Solution:

If the square sum is infinite, the system is an unstable system. If it is zero, it means h(t) = 0 for all t. However, this cannot be possible. Thus, it has to be finite.

QUESTION: 11

What is the rule h*x = x*h called?

Solution:

By definition, the commutative rule h*x=x*h.

QUESTION: 12

For an LTI discrete system to be stable, the square sum of the impulse response should be

Solution:

If the square sum is infinite, the system is an unstable system. If it is zero, it means h(t) = 0 for all t. However, this cannot be possible. Thus, it has to be finite.

QUESTION: 13

What is the rule (h*x)*c = h*(x*c) called?

Solution:

By definition, the associativity rule = h*(x*c) = (h*x)*c.

QUESTION: 14

Does the system h(t) = exp(-7t) correspond to a stable system?

Solution:

The system corresponds to a stable system, as the Re(exp) term is negative, and hence will die down as t tends to infinity.

QUESTION: 15

What is the following expression equal to: h*(d+bd), d(t) is the delta function

Solution:

Apply commutative and associative rules and the convolution formula for a delta function

QUESTION: 16

Does the system h(t) = exp([14j]t) correspond to a stable system?

Solution:

The system corresponds to an oscillatory system, this resolving to a marginally stable system.

QUESTION: 17

What is the rule c*(x*h) = (x*h)*c called?

Solution:

By definition, the commutative rule i h*x=x*h and associativity rule = h*(x*c) = (h*x)*c.

QUESTION: 18

Is y[n] = n*sin(n*pi/4)u[-n] a causal system?

Solution:

The anti causal u[-n] term makes the system non causal.

QUESTION: 19

The system transfer function and the input if exchanged will still give the same response.

Solution:

By definition, the commutative rule i h*x=x*h=y. Thus, the response will be the same.

QUESTION: 20

Is y[n] = nu[n] a linear system?

Solution:

The system is linear s it obeys both homogeneity and the additive rules.

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