P-5.8 The impulse response h[n] of an FIR filter is shown in Fig. P-5.8. Determine the filter coefficients {bk} of the difference equation for the FIR filter. If an input x[n] = {2, 1, -1} is applied to it, obtain the output.
y[n] = sum_{k=0}^{L-1} x[n - k]
For the input sequence x[n] = (-0.5)^n u[n], compute the values of y[n] over the index range -2 ≤ n ≤ 8, assuming that L = 2.
(c) For the input sequence x[n] = a^n u[n], derive a general formula for the output of the running sum y[n] that applies for any value a, for any length L, and for the index range n ≥ -2. In doing so, you may have use for the formula:
sum_{k=M}^{N} a^k = frac{a^M - a^{N+1}}{1 - a}