Systems: Properties

Question 1

When is a system memoryless?

Answer 1

If the output only depends only on the present input.

Question 2

When is a system said to have memory?

Answer 2

A system is said to have memory if the current output depends on past as well as the present input.

Question 3

When is a system causal?

Answer 3

A system is causal if and only if the current output is only a function of past and present inputs and not future inputs.

Question 4

When is a system invertible?

Answer 4

If the input signal can be recovered from the output signal. For this, distinct inputs have to lead to distinct outputs (one-to-one mapping). e.g. y = x^2 is non invertible as we cannot determine the sign of the input.

Question 5

What is BIBO stability?

Answer 5

For every bounded input x, the output y is also bounded. (Bounded Input-Bounded Output)

Question 6

How can we confirm whether a system is BIBO unstable?

Answer 6

To show a system is not BIBO stable, we only need to find a single input which leads to an unbounded output.

Question 7

Is the following system BIBO stable? Explain why. y(t) = tx(t)

Answer 7

No - it is not BIBO stable. For a constant input x = 1 (bounded), output y = t (unbounded).

Question 8

How do we determine whether a system is linear?

Answer 8

If the following to properties hold:

1. Homogeneity: F(au) = aF(u)
2. Additivity: F(u+v) = F(u) + F(v)

Question 9

What does linearity mean for a signal?

Answer 9

1. Scaling before or after the system is the same
2. Summing before or after the system is the same

Question 10

What is linearity also known as?

Answer 10

Superposition property

Question 11

What is the method for testing the additivity of a system?

Answer 11

1. Define 2 arbitrary inputs (called x₁ and x₂)
2. Find their associated outputs (called y₁ and y₂)
3. Check what the output y₃ is if the input is x₃ = x₁+x₂
4. The system is linear if superposition principle holds for y₃ = y₁+y₂

Question 12

When is a system time invariant?

Answer 12

A system is time invariant if a time shift in the input results in a time shift in the output.

y(t - a) = F[x(t - a)]

Question 13

What is the method for determining whether a system is time invariant?

Answer 13

1. Input x(t) into the system
2. Obtain y(t)
3. Shift y(t) in time by replacing all t with t - a, which is the output we expect to get if the system is time invariant
4. Shift x in time so that x(t-a)
5. Input x(t - a) into the system to get an output y
6. Compare the y from step 3 and step 5, if they are the same the system is time invariant.