Consider the IIR recursive filter shown in figure and let hF(n), hR(n), and h(n) denote the impulse responses of the FIR section, the recursive section, and the overall filter, respectively.

(a) Find all the causal and stable recursive second-order sections with integer coefficients (a1, a2) and determine and sketch their impulse responses and frequency responses. These filters do not require complicated multiplications or quantization after multiplications.

(b) Show that three of the sections obtained in part (a) can be obtained by interconnection of other sections.

(c) Find a difference equation that describes the impulse response h(n) of the filter and determine the conditions for the overall filter to be FIR.

(d) Rederive the results in parts (a) to (c) using z-domainconsiderations.

(a) H(z) = bo(1 + bz-(1+ bzz-(1 + byz"1)) H(2) = bo(z- + (b)z-2 + (bz+ b3))) (b) 0(, +2) + no1_2 = (2)H kal