Question: Suppose we cascade a differentiator and a smoother. The equations for the differentiator is w[n] = x[n] x[n 1] where w[n] is the output and

Suppose we cascade a differentiator and a smoother. The equations for the differentiator is w[n] = x[n] ˆ’ x[n ˆ’ 1] where w[n] is the output and x[n] the input, and for the smoother the equation is

where y[n] is the output and w[n] the input.

(a) Obtain the difference equation relating the input x[n] to the output y[n] of the whole system.

(b) If x[n] = u[n], calculate y[n]. Assume zero initial conditions.

y[n] = {y[n – 1]+ w[n]

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