_{Problem 4 (60 points): Find explicit expressions for the convolution y(t) = (x1 x2)(t) for the following choices of x1 and x2. The exact form in which you are supposed to specify y may be different in different parts of this problem. Note: You must obtain these expressions analytically and show your work in order to receive any credit. If you need to make any approximations, state them clearly. 4(a) 10 points x1(t) = I[0,2](t), x2(t) = I[0,2](t) 2I[2,4](t). Specify y(t) using a sketch, rather than writing out an analytical expression as a function of time. 4(b) 10 points x1(t) = I[0,2](t), x2(t) = sin 11t. Specify y(t) explicitly. 4(c) 10 points x1(t) = sinc2 (t) sin 10t, x2(t) = sin 11t. Specify y(t) explicitly. 4(d) 10 points x1(t) = sinc2 (t) sin 10t, x2(t) = P n= p(t n), where p(t) = I[0, 1 2 ] . Specify y(t) explicitly. 4(e) 10 points x1(t) = sinc2 (t) sin 10t, x2(t) = sinc(t) sin 11t. Specify |Yb(f)| and Yb(f). 4(f) 10 points x1(t) = sinc2 (t) sin 10t, X2(s) = 100 s+100 , Re(s + 100) > 0. Specify y(t) explicitly, stating any approximations you might be making.}

Problem 4 (60 points): Find explicit expressions for the convolution y(t) = (x1 * x2)(t) for the following choices of ci and X2. The exact form in which you are supposed to specify y may be different in different parts of this problem. Note: You must obtain these expressions analytically and show your work in order to receive any credit. If you need to make any approximations, state them clearly. 4(a) 10 points xi(t) = [[0,21(t), xz(t) = [[0,2)(t) 21(2,4)(t). Specify y(t) using a sketch, rather than writing out an analytical expression as a function of time. 4(b) 10 points xi(t) = 1(0,2)(t), xz(t) = sin 11nt. Specify y(t) explicitly. 4(c) 10 points xi(t) = sinc(t) sin 107t, x2(t) sin 11at. Specify y(t) explicitly. 4(d) 10 points x1(t) = sinc? (t) sin 107t, x2(t) = En=-- Oop(t n), where p(t) = 1[0, 1]. Specify y(t) explicitly. 4(e) 10 points xi(t) = sinc(t) sin 107t, xz(t) = sinc(t) sin 119t. Specify S (f)| and / (f). 4(f) 10 points xi(t) sinc?(t) sin 10t, X2(s) Re(s + 1007) > 0. Specify y(t) explicitly, stating any approximations you might be making. 1007 s+100%) Problem 4 (60 points): Find explicit expressions for the convolution y(t) = (x1 * x2)(t) for the following choices of ci and X2. The exact form in which you are supposed to specify y may be different in different parts of this problem. Note: You must obtain these expressions analytically and show your work in order to receive any credit. If you need to make any approximations, state them clearly. 4(a) 10 points xi(t) = [[0,21(t), xz(t) = [[0,2)(t) 21(2,4)(t). Specify y(t) using a sketch, rather than writing out an analytical expression as a function of time. 4(b) 10 points xi(t) = 1(0,2)(t), xz(t) = sin 11nt. Specify y(t) explicitly. 4(c) 10 points xi(t) = sinc(t) sin 107t, x2(t) sin 11at. Specify y(t) explicitly. 4(d) 10 points x1(t) = sinc? (t) sin 107t, x2(t) = En=-- Oop(t n), where p(t) = 1[0, 1]. Specify y(t) explicitly. 4(e) 10 points xi(t) = sinc(t) sin 107t, xz(t) = sinc(t) sin 119t. Specify S (f)| and / (f). 4(f) 10 points xi(t) sinc?(t) sin 10t, X2(s) Re(s + 1007) > 0. Specify y(t) explicitly, stating any approximations you might be making. 1007 s+100%)