# Signals and Systems.

by sahil_time
Tags: signals, systems
 P: 140 1) The system is not memoryless. The definition of the derivative can be expressed in two ways. What it was before, and what it will be in the future, like the following: $\frac{dx}{dt}=\lim{Δt\rightarrow0}\frac{x(t)-x(t-Δt)}{Δt}$ or $\frac{dx}{dt}=\lim{Δt\rightarrow0}\frac{x(t+Δt)-x(t)}{Δt}$ The two definitions will give the same result. However, the first one is has memory (in that it uses a past value of x(t)) and the second one is non-causal (in that it uses a future value of x(t)). If you had a system that needed to compute the derivative, the first definition is the only one you would be able to use, and that would give you a system with memory. 2) If the system is linear it must satisfy superposition and homogeneity. Homogeneity: It's easy to see that the system is homogeneous. If an input x(t) gives y1(t) = x(t)+x(t-2) then the input αx(t) will give y2(t) = αx(t)+αx(t-2) = α(x(t)+x(t-2)) = αy1(t). This a scaled input will produce a scaled output. What about superposition? Well here we run into problems. Because if x1(t) < 0 and x2(t) < 0 for all t, but x1(t) + x2(t) > 0 for all t, then the response for each of the inputs will be 0, but if you add them together, they will provide an output. Thus, the system is not linear.