Need help. Consider a discrete-time signal {x(n)} defined by x(0) = 2,x(1) = 3,x

Question

Need help.

Consider a discrete-time signal {x(n)} defined by x(0) = 2,x(1) = 3,x(2) = 1 and zero everywhere else. Assume that we have a discrete-time linear time invariant (LTI) system with unit impulse response function {h(n)} given by h(0) = 0.5, h(1) = l,h(4) = 0.2 and zero everywhere else. Express the input signal {x(n)} as the summation of three scaled and shifted impulse functions. Find the output signal {yi(n)},i = 1,2,3 corresponding to the i-th scaled and shifted impulse function you find the (a). Use the results of (b) to compute the output signal {y(n)} corresponding to input {x(n)}. Directly apply the convolution sum formula to find y(2) and y(5). Verify that you get the same answer as in (c).

Explanation / Answer

d=?

(a) x(n)= 2 d(n)+3d(n-1)+d(n-2)

(b)y0(n)=x(0)h(n)

y1(n)=x(1h(n-1)

y2(n)=x(2)h(n-2)

(c)

y(n)=y0(n)+y1(n)+y2(n)=x(0)h(n)+x(1)h(n-1)+x(2)h(n-2)

(d)y(n)=sum x(m)*h(n-m)=x(0)h(n)+x(1)h(n-1)+x(2)h(n-2)=2h(n)+3h(n-1)+h(n-2)

y(2)

from d) y(2)=2h(2)+3h(1)+h(0)=3+0.5=3.5

y(5)=2h(5)+2h(4)+h(3)=0+2*0.2+0=0.4

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