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For t < zero; now not outlined for t (c) zero ( d ) .sr 1.56. =0 contemplate a continuous-time procedure with the input-output relation make sure even if the program is ( a ) linear, ( b ) time-invariant, ( c ) causal. 1.57. think about a continuous-time process with the input-output relation be certain even if the program is ( a ) linear, ( b ) time-invariant. Am. 1.58. ( a ) Linear; ( b ) Time-varying give some thought to a discrete-time approach with the input-output relation y [ n ] = T { x [ n ] )= x 2 [ n ] make sure.

- eleven by way of x [ n ]and y[n],we receive nine [ n ] =f y [ n ]+ $+I q [ n - 1 1 = i y [ n ]- i x [ n ] (2.147a) (2.147b) altering n to ( n - 1) in Eq. (2.147a), now we have q [ n - 1 1 = f y [ n - eleven + i x [ n - 1 1 (2.147~) therefore, equating Eq. (2.1476) and (Eq. (2.147c), we now have ; y [ n ] - f x [ n ] = f y [ n - 1 1 + + x [ n - eleven Multiplying each side of the above equation via five and rearranging phrases, we receive y [ n ] - 2 y [ n - 1 1 = x [ n ]+ three x [ n - 1 1 (2.148) that's the necessary.

(3.161, that's, x(t - t o )H e-"oX(S) R1=R by way of definition ( three . three ) by way of the switch of variables T =t - I , we receive with a similar ROC as for X ( s ) itself. as a result, the place R and R' are the ROCs ahead of and after the time-shift operation. 3.8. be certain the time-scaling estate (3.181, that's, by means of definition ( three . three ) = a t with a > zero, we've got I w ( x a )= x(r)e-('/")'dr = - X a -, a via the switch of variables 7 ( ) - RP=aR word that a result of scaling s / a within the remodel, the.

primary classes N , and N2, respectively. lower than what stipulations is the sum x[n] =x,[n] +x2[n] periodic, and what's the elemental interval of x[n] whether it is periodic? in view that x,[n] and x2[n] are periodic with primary sessions N, and N2, respectively, now we have + N,] =x,[n +mN,] + N,] =x,[n + kN,] xI[n] =xI[n x2[n] =x,[n m ok = optimistic = confident integer integer therefore, ~ [ n =x,[n ] +mN,] + x 2 [ n + kN,] to ensure that x[n] to be periodic with interval N, one wishes + N ] =x,[n + N ]+ x 2 [ n.

( t ) dt = j w four ( t ) dt (1.98) zero the place & ( t ) is a checking out functionality that is integrable over zero < t < m. utilizing this definition, exhibit that Rewriting Eq. (1.98) as 38 [CHAP. 1 indications AND platforms we receive this is precise provided that zero =0 and okay w four ( t ) [ l- u ( t ) ] dt b ( t ) u ( t )= zero , t < zero and c $ ( t ) [ l- u ( t ) ] = zero , t > zero 1- four ( t ) u ( t )dt =0 a those stipulations suggest that considering that four ( t ) is unfair, now we have u(t)=O,tO that's, 1.25.