Consider a system with input x[k] and outputy[k] that satisfy thedifference equa
ID: 1830497 • Letter: C
Question
Consider a system with input x[k] and outputy[k] that satisfy thedifference equation y[k] =ky[k -1]+ x[k]The system is causal and satisfies initial-rest conditions, i.e.if x[k] = 0 fork <k0 then y[n] = 0 for k <k0 Determine if the system is a) linearb)time-invariant. justify your answersfor part a and part b Consider a system with input x[k] and outputy[k] that satisfy thedifference equation y[k] =ky[k -1]+ x[k]
The system is causal and satisfies initial-rest conditions, i.e.if x[k] = 0 fork <k0 then y[n] = 0 for k <k0 Determine if the system is a) linearb)time-invariant. justify your answersfor part a and part b justify your answersfor part a and part b
Explanation / Answer
y[k] = ky[k - 1] + x[k] for a system to be linear it must follow the principle ofsuper-position i.e additivity and homogeneity ax1[n]+bx2[n]=ay1[n]+by2[n] now the given system does not follow principle ofsuperrposition so its non-linear b)for a system to be time invariant, a time shift in the i/p signalproduces an identical shift in o/p singal here,if we shift i/p sigal by 1the i/p signal becomes [k-1]y[k-1]+x[k-1] ..it is shifted by 1 ...the o/p signal becomesy[k-1] i.e it is also shifted by 1 hence it is a time-invariant signal