Solution for both the question is provided below:
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Problem 3.12 Find the DTFT of the following time-domain signals: (b) x[n] = alu. lal <...
2. Given x[n]— 1-ae-ja' find the DTFT of: (a) y[n] = nx[n],(b) z[n] = (n − 1)x[n] dX(92) Hint: nx[n]< > ; dΩ
Problem 3.) Find and plot X(w) and X(w), the magnitude and DTFT for the signal x[n] given by a) b) x[n]= cos(-n) x[n]-(-1)" (a)"u[n] for 0< a〈 1
(a) Find the z-transform of (i) x[n] = a"u[n] + B^u[n] + cºul-n – 1], lal< 161 < le| (ii) x[n] = n-au[n (iii) x[n] = {** [cos (Tan)]u[n] -em* [cos (fin)]u[n – 1]
Let x[n] and y[n] be periodic signals with common period N, and let z[n] = { x[r]y[n – r) r=<N> be their period convolution. Let z[n] = sin(7") and y[n] = { . 0 <n<3 4 <n <7 Asns? be two signals that are periodic with period 8. Find the Fourier series representation for the periodic convolution of these signals.
(a) Find the z-transform of (i) x[n] = a"u[n] +b"u[n] + cºul-n – 1], lal <151 < le|| (ii) x[n] = n*a"u[n] (iii) x[n] = en* [cos (în)]u[n] – en" (cos (ien)] u[n – 1] (b) 1. Find the inverse z-transform of 1-jz-1 X(2) = (1+{z-1)(1 – {z-1) 2. Determine the inverse z-transform of x[n] is causal X(x) = log(1 – 2z), by (a) using the power series log(1 – x) = - 95 121 <1; (b) first differentiating X(2)...
Problem 10.7. Show that azn-e* has n roots with Izl < 1 if lal > e. Problem 10.8. Suppose that f is holomorphic inside and on a toy contour γ. Suppose f has no zeros on the contour γ and that zi, ,«n are the zeros of f inside γ, the order of being kj. Show that j-1 for any function g which is holomorphic inside and on γ. Problem 10.9. Let u(z,y)-zy(x2-уг). Find the maximum and minimum values of...
Problem 2 Determine the signals having the following Fourier transforms. So, 0 < 1W < Wo (a) X(w) = { (a) A 10) | 1, wo < lw <a (b) X(w) = cos?w (hint: expand first X (w) in terms of ejw) (c) X(W) = { 1, wo – dw/25\w Swo + dw/2 10, elsewhere
24. The joint cdf of (X,Y) is Find a) Joint pdf of (X, Y) b) Marginal pdf of X and Y c) PI(X s 1) n (Y s 1) d) PI(1 < X <3) n (1 <Y <2)] Page 4 of5
Problem 5 Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) e(,0x< o0, zero elsewhere X- n (Y1 0), find the cumulative distribution function (cdf) for Zn = n (Y1 - 0), and Let Zn find the limiting cdf of Zn as n >oo.
For each of the following, find the pdf of Y (e) Y =In X and (п+ m+1)! -"(1 - x)" fx (x) = n!m! <1 and m and n are positive integers where 0 (f) Y e and 1 fx(x) e)2/2 where 0 < oo and o2 is a positive constant (i) Y 1 X2 and fx (x)=3(x 1)2, -1 < x <1