solving (b) and (c) using matlab So i want the code of this answer 150 2.43. One

solving (b) and (c) using matlab So i want the code of this answer

Transcribed Image Text:

150 2.43. One of the important properties of convolution, in both continuous and discrete time, is the associativity property. In this problem, we will check and illustrate this prop- erty. (a) Prove the equality 0 1 51- by showing that both sides of eq. (P2.43-1) equal (b) Consider two LTI systems with the unit sample responses h [n] and h[n] shown in Figure P2.43(a). These two systems are cascaded as shown in Figure P2.43(b). Let x[n] = u[n]. and 2 3 4 01234 (a) [x(t) * h(t)] * g(t) = x(t) * [h(t) * g(t)] h [n] = (-)^u[n] : +00 + | | x()ho)g(t + )d+da. -00 h[n] =u[n] +u[n-1] and where the input is Figure P2.43 n h[n] h [n] = sin 8n x[n] n = = x[n] h [n] (i) Compute y[n] by first computing w[n] = x[n] *h [n] and then computing y[n] = w[n] * h[n]; that is, y[n] = [x[n] * h[n]] * h[n]. (ii) Now find y[n] by first convolving h [n] and h[n] to obtain g[n] h [n] *h[n] and then convolving x[n] with_g[n] to obtain y[n] = x[n] * [h [n] * h[n]]. The answers to (i) and (ii) should be identical, illustrating the associativity prop- erty of discrete-time convolution. (c) Consider the cascade of two LTI systems as in Figure P2.43(b), where in this case w[n] Linear Time-Invariant Systems au[n], |a| <1, (b) 8[n] - ad[n 1]. (P2.43-1) h[n] Chap. 2 y[n] 150 2.43. One of the important properties of convolution, in both continuous and discrete time, is the associativity property. In this problem, we will check and illustrate this prop- erty. (a) Prove the equality 0 1 51- by showing that both sides of eq. (P2.43-1) equal (b) Consider two LTI systems with the unit sample responses h [n] and h[n] shown in Figure P2.43(a). These two systems are cascaded as shown in Figure P2.43(b). Let x[n] = u[n]. and 2 3 4 01234 (a) [x(t) * h(t)] * g(t) = x(t) * [h(t) * g(t)] h [n] = (-)^u[n] : +00 + | | x()ho)g(t + )d+da. -00 h[n] =u[n] +u[n-1] and where the input is Figure P2.43 n h[n] h [n] = sin 8n x[n] n = = x[n] h [n] (i) Compute y[n] by first computing w[n] = x[n] *h [n] and then computing y[n] = w[n] * h[n]; that is, y[n] = [x[n] * h[n]] * h[n]. (ii) Now find y[n] by first convolving h [n] and h[n] to obtain g[n] h [n] *h[n] and then convolving x[n] with_g[n] to obtain y[n] = x[n] * [h [n] * h[n]]. The answers to (i) and (ii) should be identical, illustrating the associativity prop- erty of discrete-time convolution. (c) Consider the cascade of two LTI systems as in Figure P2.43(b), where in this case w[n] Linear Time-Invariant Systems au[n], |a| <1, (b) 8[n] - ad[n 1]. (P2.43-1) h[n] Chap. 2 y[n]