Question

7. Determine the system function, magnitude response, and phase response of the fol- lowing systems and use the pole-zero pattern to explain the shape of their magnitude response (a) y[n] = 1(x(n]-x(n-1), ln -2

no need for pole-zero plot

Answer 1

y[n]) = }(x[n] - x[n-11) Apply DTFT. Y(elº) = x (ed") - e-bºx(ev) Y(e) 1 16-ja H (e%) = (1-eso) h(n) = }[7(n)-3(n-1)] The magnitude of H (ej®) is, 187 (24) = = The phase of H (el) is, ZH (el®)=0° +180º – tan ZH (e!")=180° – 0 sin cos o (b) v[r]= _(*[n]-x[n–2) Apply DTFT. Yle%) = x (010)_*(e)

H (es")= }(1-220) n(n) = }[(n)-8(n-2)] The magnitude of H (em) is. |H (el)= The phase of H (ei®) is, ZH (et“)=0° +180° – tan" sm20) ZH (el®)=180°–20 y[1] = (*[r]+x[n–1]) – (x[n–2] + x[1–3]) Apply DTFT. y (etc) = __x (eo) + *'° X (et)-20210X (est") - 122372X(elo)

H (e)) = (1+2+1° -e719-e350) h(n) = 3 [3(n) +8(n-1)-8(n-2)-8(n-3)] The magnitude of H (em") is, |H (16) = 4 + 4 4 =0 The phase of H (cm) is, Cos 0 0 0° -0+(180°- 20)+(180°-30)= 360° -60

(d) y [n] = {(x[i]+x[n-1) - ([n–3]+x[n–4]) Apply DTFT. Yle%) = x(e1")+-5° X(eº) -Le** X (e)0) -- */ X (el) H (e) = {1+** -e39 –+50) h(n) = } [7(n) +8(n-1)-8(n-3)-8(n=4)] The magnitude of H (et) is, The phase of H (el) is, ZH (65")=0° - tan" ( sag en ) +(180°-tan" ( (cos30 2H (el® )= 0° –~+(180°–30) +(180° – 40) = 360° – 80