7. Derive the time domsin representation of the Sollowing Lapince transiorm espressicnbasi on e sivee Roc...
7. Find the inverse Laplace Transform of X(so2 with ROC-1< Rels) 1.
1. Derive the Taylor series representation for in the z -il <v2.
1. Let x[n] be a periodic sequence with period N with Fourier series representation x[n] = akek(34)n k=<N> Assume that N is even. Derive the expressions for the following signals (a) x[n] – x[n – (b) x[n] + x[n + 1 (Note that this signal is periodic with period ) (c) (-1)" x[n]
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
7. For the probability density function f(x) = for 0 <<<2 (a) Find P(x < 1) (b) Find the expected value. (c) Find the variance.
Consider the discrete-time signal given below. Ş ()", n20 X = 0 n < 0 where a=8. Find the average power Poo
2. Circle the causal BIBO stable ROC below. a) 1.1<\리<1.2 b) Izk1/201zP1/2 d) 0.5<Izl<0.9 e) none above 3. A linear time-invariant IIR system is always BIBO stable a) True b) False 4. If a fiter has z-transform H(z)05, then the fiter s ;z>0.5, then the filter is zz-0.5z a) Nonlinear b)FIR )R d) two-sided e) none above 5. The discrete-time frequency o in rad/ sample of the sinusoid hin] below is d) T2 e) none above hIn] -1
7. Suppose that the joint density of X and Y is given by f(x,y) = e-ney, if 0 < x < f(z, y) = otherwise. Find P(X > 1|Y = y)
(10 pts) The joint distribution of X and Y is given by: f(x,y) = 1/y, 0 < x < y < 1. Derive the distribution of Z= Y/X. You must use both the methods (CDF & Transforma- tion).
( xy 7. CHALLENGE: fxy(x, y) = 0< < 2, 0 <y <1 otherwise 0 Find P(X+Y < 1) HINT: consider the region of the XY plane where the inequality is true.