Signals and Systems (System Properties)

[rht1369]

Homework Statement

Indicate following properties for the system with the given input-output equation:
-Linearity, Causality, Time Invariance, memoryless or not

and explain why?

Homework Equations

y(t) = ∫^t_-∞ (t - λ) x(λ) dλ

The Attempt at a Solution

lets say X(t) is the antiderivative of x(t), then:

y(t) = t ∫^t_-∞ x(λ) - ∫^t_-∞ λ x(λ) dλ
= t (X(t) - X(-∞)) - (t X(t) - X(t) - (-∞ X(-∞) - X(-∞)))

now we have the relation between y(t) and X(t) but how can we obtain the relation between y(t) and x(t) to find out about the causality and memory properties?
can this be a clue or I am doing wrong?

 

[KataKoniK]

Based on the given input-output equation, the system can be described as linear, causal, and time-invariant. This is because the equation follows the principles of linearity, causality, and time invariance.

Firstly, the equation is linear because it is a linear combination of the input signal x(t). This means that the output y(t) is directly proportional to the input x(t) and any scaling or addition of the input will result in a corresponding scaling or addition of the output. This is evident in the equation where the integral of the input signal x(λ) is multiplied by t, showing that any changes in the input signal will result in corresponding changes in the output signal.

Secondly, the equation is causal because the output y(t) only depends on the input x(λ) for values of λ less than or equal to t. This means that the output is not affected by any future values of the input, which is a characteristic of causal systems.

Thirdly, the equation is time-invariant because the output y(t) is not affected by any time shifts or delays in the input signal x(t). This can be seen in the equation where the integral is taken from -∞ to t, meaning that the output is a function of the input at all time points, not just a specific time point.

As for the memoryless property, it is difficult to determine based on the given equation alone. However, based on the principles of linearity, causality, and time-invariance, it can be inferred that the system is memoryless. This is because a memoryless system is a special case of a causal and time-invariant system, where the output at any given time is only dependent on the input at that same time. Since the given system is both causal and time-invariant, it can be concluded that it is also memoryless.