(a) LetS denote an incrementally linear system, and let x 1 [n] be an arbitrary input

Chapter 1, Problem 1.47

(choose chapter or problem)

(a) LetS denote an incrementally linear system, and let x 1 [n] be an arbitrary input signal to S with corresponding output y 1 [n]. Consider the system illustrated in Figure Pl.47(a). Show that this system is linear and that, in fact, the overall input-output relationship between x[n] and y[n] does not depend on the particular choice of x 1 [ n]. (b) Use the result of part (a) to show that Scan be represented in the form shown in Figure 1.48. (c) Which ofthe following systems are incrementally linear? Justify your answers, and if a system is incrementally linear, identify the linear system Land the zeroinput response y0 [n] or y0(t) for the representation of the system as shown in Figure 1.48. (i) y[n] = n + x[n] + 2x[n + 4] { n/2, (ii) [ ] (n-1 )/2 Y n = (n - 1)/2 + k~-n x[k], n odd ( . ") [ ] { x[n] - x[n - 1] + 3, if x[O] 2: 0 111 Y n = x[n] - x[n- 1] - 3, if x[O] < 0 (iv) The system depicted in Figure P1.47(b). (v) The system depicted in Figure P1.47(c). (d) Suppose that a particular incrementally linear system has a representation as in Figure 1.48, with L denoting the linear system and y0 [n] the zero-input response. Show that Sis time invariant if and only if Lis a time-invariant system and y0 [ n] is constant.