Question

Determine which of these properties (Memoryless, Time invariant, Linear, Causal, and Stable) hold and which do not hold for each of the continuous-time system, y[n] = x [4n + 1]. Justify your answers. y(t) denotes the system output and x(t) is the system input

Answer 1

Given that etty yen=relunto For memoryleas put nao soy co= reco put nal , y co= x (5) so herce output is depanding on input, not on present input. it is not memoryless- future That's why For time invariant - To be time invariant yen, no = yen-no) There ycn,no) a recun-noti but yenimo) yen-nos a rełucn-nost1] - yon-nos = x(un-unotu so yon, nos t yonimo So Not time invariant

for causal- To be causal present outputy should depand on present input and past input not on future input. so yonoarecumto n=oj yo future input - bresentoutput That's why it is not a causal system. pest = recib for sebable - For stable, if input is bounded then outbut shots should be bounded - It is BIBO stable. If recno = cosna -) bounded then yono = cos [acant) ] - yono = cos(unata) a-simuna - so yenna-sinunk - bounded So for every bounded input, output is bounded. That's why it is stable system. for linear- To be linear - yi(m) + y 2 (0) = yo(m) bere if recoo =266m) then yicho=24 Chamunt . If rean) = 22cm them 42(M)=racunt so yi(moty (n = arcuntitracuntu If ainstream=reem), 3(m) = rei cunty tra (unto so yin + 420 = 43(n) So it is linear system.