[3.12 APPENDIX: IMPULSE](#page-9-0) RESPONSE FOR A SPECIAL CASE

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3.12 APPENDIX: IMPULSE RESPONSE FOR A SPECIAL CASE

When aN =0, A⁰ =bN/aN becomes indeterminate, and the procedure needs to be modified slightly. When aN = 0, Q[E] can be expressed as EQˆ [E], and Eq. (3.26) can be expressed as

$$E\tilde{Q}[E]h[n] = P[E]\delta[n] = P[E]\{E\delta[n-1]\} = EP[E]\delta[n-1]$$

Hence,

$$\hat{Q}[E]h[n] = P[E]\delta[n-1]$$

In this case the input vanishes not for n ≥ 1, but for n ≥ 2. Therefore, the response consists not only of the zero-input term and an impulse A0δ[n] (at n = 0), but also of an impulse A1δ[n−1] (at n = 1). Therefore,

$$h[n] = A_0 \delta[n] + A_1 \delta[n-1] + y_c[n]u[n]$$

We can determine the unknowns A0, A1, and the N − 1 coefficients in yc[n] from the N + 1 number of initial values h[0], h[1], … , h[N], determined as usual from the iterative solution of the equation Q[E]h[n] = P[E]δ[n]. † Similarly, if aN = aN−¹ = 0, we need to use the form h[n] = A0δ[n]+A1δ[n−1]+A2δ[n−2]+yc[n]u[n]. The N +1 unknown constants are determined from the N +1 values h[0], h[1], … , h[N], determined iteratively, and so on.