Question

1-Take the following system with the input x, and output y, y(n) = x(n) + 2x(n − 1) + 2x(n − 2) + x(n − 3). Find its output, if 2(n) = cos(won), , and . Is this system a filter, and if so what kind? where wo, takes values of 0, 2-Take the following system with the input x, and output y, y(n) = cy(n − 1) + x(n). Take y(n) = 0, for n < 0. Find the output if (a) x(n) = (n), (b) x(n) = 8(n) – 8(n − 1). Note that c is a constant. Evaluate your results in part (a) when (1) c= 0.5, (2) c= 2. Do you see similar trends in the two cases? If not what is the critical value of the constant c, which results in a different qualitative behavior of the system,

solve only question 1 please !!!!

Answer 1

y yom) = a(+2am-1 +22(n2)+acm-3) Applying Z-transform on both sides 462) = x(2) + 22'x12)+22?x(2)+?x(2) Y(2) = H(2)= 1t2Z'+2=²+z3 Frequency response is obtained by substituting مانع =3 HeiW) = ltzeju rze ziwt e ziw given a (n)= Coswon. If the input acn) is of the form a coswon and the frequency response is Heuw), then the output yn) is given by ym) - Al Hiwo) | Cos (won + <Hlevwo)). At Wo=0 ) = 1+ 2 +2 +1 = 6

e y(n)= 663 (0)n = 6 At Wo = 773 Meub)= 1+z2+1% +2€ 2175 -3,17 - 1+=[+-69}=(1-19)+(-1) Hey b) = =j213 = 215 21 yen) = 253 64(5n-1) = 1+2 output At Wo = 1 jme -2;% Ho/2) = 1tze tze - 3j % te =1+26+j) +2(-1) +(-3(-)). =-1-j He: "32) - vzej

output yo) = 13 Q3 (In - 3). when woah -j21 -j 317 Hlejt) = 1to jt = 1+2 (-1)+2 (1) + (-) - Heur) =0 Output ym) = 0. Cos(10) yem) -0. 6 ; Wo=0 HCelu) | = { 215 283 : Wo = "73 vz ; wo = 1 lo ; wo at (hce vw) decreases as oo increases from o to It. so the filter is a Lowpass filter.