Consider a linear time-invariant system whose impulse response is 

h[n] = (1/2)ⁿ u[n] + (1/3)ⁿ u[n].

 The input x[n] is zero for n < 0, but in general, may be non zero for 0 ≤ n ≤ ∞. We would like to compute the output y[n] for 0 ≤ n ≤ 10⁹, and in particular, we want to compare the use of an FIR filter with that of an IIR filter for obtaining y[n] over this interval.

(a) Determine the linear constant-coefficient difference equation for the IIR system relating x[n] and y[n].

(b) Determine the impulse response h₁[n] of the minimum-length LTI FIR filter whose output y₁[n] is identical to the output y[n] for 0 ≤ n ≤ 10⁹.

(c) Specify the linear constant-coefficient difference equation associated with the FIR filter in Part (b).

(d) Compare the number of arithmetic operations (multiplications and additions) required to obtain y[n] for 0 ≤ n ≤ 10⁹ using the linear constant-coefficient difference equations in part (a) and in part (c).  

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Part B 1 -1