For the given system, determine whether they are time invariant, causal, and/or

Question

For the given system, determine whether they are time invariant, causal, and/or memoryless



For the given system,

1) If the input is..
x1(t)=u(t)-u(t-1)


then the output is...
y1(t)=t[u(t)-u(t-1)]



2) If the input is..
x2(t)=u(t-1) -u(t-2)

then the output is...
y2(t)=[u(t)-(t-1)u(t-1)+(t-2)u(t-3)]



3) If the input is..
x3(t)=u(t-1)-u(t-2)


then the output is
y3(t)=(t-1) u(t-1) - (t-2)u(t-2)-u(t-3)



Is this system

1) time invariant: yes or no, why?
2) causal: yes or no, why?
3) memoryless: yes or no, why?

Explanation / Answer

1)y(n)=S{x(t)}=tx(t) S{x(t-t0)}=t(x(t-t0))--1 y(t-t0)=(t-1)x(t-t0)--2 1 and 2 are not equal so not Time Invarient casual only depends on present input and memory less (same) 2)Time invarient because notice the output is only depended on different component of input, no other variable is no multiplying like the previous example. so if input shifts the output will shift too. Casual: example t=0 x(0)=u(-1)+u(-2) ;y(0)=u(0)-u(-1)-2u(-3) output never depends on the future input value Memory:it has memory. it depends on previous input like u(-1) u(-3) for t=0;

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