Question

6. (15) Consider the following causal linear time-invariant (LTT) discrete-time filter with input in and output yn described by y[n] = x[n] – rn - 2 for n 20 . Is this a finite impulse response (FIR) or infinite impulse response (IIR) filter? Why? • What are the initial conditions and their values for this causal and linear time-invariant system? Why? • Draw the block diagram of the filter relating input x[n) and output y[n] • Derive a formula for the transfer function in the z-domain and the region of conver- gence. • Give a formula for the discrete-time frequency response of the filter.

Answer 1

Geven causal LT I system is Yiny = 846] -[n-] In order to comment whether it is an FIR filter | ITR filter, we need to know about impulse response. Impulse response (hGJ) is the output of the system when input is unit impulse signal (&(ni) ... n(n) = s[n]-S[n- I Impulse Response has only two peo, finite impulso Hence given filter is an & FIR filter ③ Block diagram : y[n] = x[n] - [1-] Apply z transform 3 4CZ) = x(z)-7-2X(7) => 467)= x(z) [1-7?] 0 Transfer function & y[n] = x [n] = a[n-2] Apply z transform 4(7) = x(7)-7? XC7) = 1-7-2 Roc plane is the entire z

@ for enoteal conditions, YCH - H(7) = 1-7-20 By using initial value theorem, & Initial value = t ) H(E) = (-2)(1-22) = (1-0)(1-0) Ft -> gnitial value = 1 response ② for discrete, time frequency H(Z) = 1-7-2 Put z = egon - H(z)= 1-Ce sun) = iné ajun = 1 [cos awn - j ginawn] = (1-683wn) + j(snack) Hote) = (asm*on)+ j (sina.com)