To determine whether the system y(t) = 4x(2t-1) - 16 satisfies the properties of being memoryless, causal, time-invariant, and linear, let's define each property and evaluate the given system:
(a) Memoryless:
A system is memoryless if the output at a particular time depends only on the input at that same time and does not depend on any past or future inputs. In other words, the system does not have memory of past inputs.
The given system y(t) = 4x(2t-1) - 16 does not satisfy the memoryless property. This is because the output y(t) depends not only on the input x(t) at time t but also on the value of x(t) at time 2t-1, which is a past time.
(b) Causal:
A system is causal if the output at a particular time depends only on the input at or before that same time. In other words, the system does not anticipate future inputs.
The given system y(t) = 4x(2t-1) - 16 satisfies the causal property. The output y(t) depends only on the input x(t) at or before time t, as the expression 2t-1 is a linear function of time t.
(c) Time-invariant:
A system is time-invariant if a time shift in the input signal results in an equivalent time shift in the output signal. In other words, the system's behavior remains the same over time.
The given system y(t) = 4x(2t-1) - 16 is time-invariant. If we shift the input signal x(t) by a time delay, the output y(t) will also be shifted by the same amount of time delay. This is because the system's behavior is solely determined by the linear transformation of the input.
(d) Linear:
A system is linear if it satisfies the properties of additivity and homogeneity.
Additivity: If the input is the sum of two signals, the output is the sum of the individual outputs.
Homogeneity: Scaling the input signal scales the output signal proportionally.
The given system y(t) = 4x(2t-1) - 16 is linear. We can see that it satisfies both additivity and homogeneity. When the input signal x(t) is the sum of two signals, the output y(t) will be the sum of the individual outputs. Similarly, scaling the input signal x(t) by a factor scales the output signal y(t) by the same factor.
In summary:
(a) The given system is not memoryless.
(b) The given system is causal.
(c) The given system is time-invariant.
(d) The given system is linear.
I hope this explanation is clear and helpful! Let me know if you have any further questions.