Question

»i)(3pts)x(n) (i) u(n) and hn) -(G)u(n) »ii) (4pts)x(n) -(G) u(n -3) and h(n) - ()u(n-5) xiv) (Spts)x(n) -u(n) and h(n) -(u(n- 1)-() u(n-3)(Note that u(n), is unit step function, this is step response). xv)(Spts)x(n)- () u(n) and hn) -(G)un-1)-(j) u(n-3) xvi) (Spts)xn) G)u(n - 4) and hn) -(u(n-4)

Convolve the following functions

Answer 1

Z transform property: x(n) * h(n) Z.T_rightarrow X(z) H(z) using these two properties below. Because Z^T property a^r V(n) ZT_rightarrow 1/1 - az^-1 x(n) = (1/4)^n U(n) & h(n) = (1/3)^n u(n) ZT implies X(z) = 1/1 - 1/4 z^-1 & H(z) = 1/1 - 1/3 z^-1 implies Let y(n) = x(n) * h(n) implies Y(z) = x(z) H(z) = 1/1 - 1/4Z^-1 1/(1 - 1/3 z^-1) Using partial Fractions = A/1 - 1/4 z^-1 + B/1 - 1/3z^-1 A Vertical-bar _z^-1 = 4 = 1/1 - (1/3)4 = -3: B Vertical-bar _z^-1 = 3 = 1/1 - 1/4(3) = 4 therefore Y(z) = -3/1 - 1/4 z^-1 + 4/1 - 1/3 z^-1 Taking inverse Z.T. implies y(n) = -3(1/4)^n U(n) + 4(1/3)^n U(n) x(n) = (1/4)^n U(n - 3) and h(n) = (1/3)^n U(n - 5) = (1/4)^3 (1/4)^n - 3 U(n - 3) & h(n) = (1/3)^5 (1/3)^n - 5 u(n - 5) Z.T X(z) = (1/4)^3 1/1 - 1/4z^-1 (z^-3) [because x(n - n_0) Z > T_rightarrow Z^-n_0 X(z)]

Sir in the same way we can do all the remaining questions also ...so if you want to ans of remaining plz comment me then I will do then send you again...

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