Problem 4 There are three causal systems with the following difference equations labeled (1), (2), and (3): 4y[n]+2y[n 1] = 3x[n]−13x[n−1]+4x[n-2] (1) 3y[n]-4y[n-1]+y[n−2] = 2x[n] + x[n−1] (2) 15y[n]+2y[n-1]−y[n−2] = 30x[n]+15x[n-1] (3) a. Find the expression for the transfer function of the all three systems, H₁(z), H₂(z), H₂(z) and plot their pole-zero graphs, and determine whether each of them is BIBO stable or not. b. Two of these Filters are then connected in cascade, so that the cascaded system is BIBO stable. Select the two systems that will result in a BIBO stable system when connected in cascade. Find the expression for this system's transfer function, and then plot its pole-zero graph. c. Find the difference equation of the cascaded system.

a, b eR olmak üzere, X1+ 0x2+ 0x3 = 2 = 2 %3D Ti + x2 + 0x3 = b-2 =6-2 Ox1 + 2x2 + (2a – 2)r3 = b -xi+ 0x2 + (a – 1)x3 %3D = -2 lineer denklem sistemi veriliyor. Gauss yok etme yöntemini kullanarak, a ve b değerleri için aşağıdaki ifadelerden hangisinin doğru olması gerektiğini be- lirleyiniz. I. Sistem a 1, b=8 için tutarsızdır. %3D II. Sistem a çözüme sahiptir. 1, b = 8 için sonsuz sayıda III. Sistem a #1, b=8 için tek çözüme sahiptir. IV. Sistem a = 1, b 8 için sonsuz sayıda çözüme sahiptir.

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a non- trivial linear combination of the given functions that vanishes identically. 1. f(x) = 2x, g(x) = 3x², h(x) = 5x – 8x² 2. f(x) = 5, g(x) = 2 – 3x², h(x) = 10 + 15x2 3. f(x) = 0, g(x) = sin x, h(x) = e* 4. f(x) = 17, g(x) = 2 sin² x, h(x) = 3 cos? x 5. f(x) = 17, g(x) = cos² x, h(x) = cos 2x 6. f(x) = e*, g(x) = cosh x, h(x) = sinh x %3D %3D

4. For what value(s) of k, if any, will the following linear system x+ y+kz 1 x+ky+ z 1 kx+ y+ z -2 has- (a) no solution (b) a unique solution, and (c) infinitely many solution?

Problem 2. # 7.2.7 Heart disease patients often experience spasms of the coronary arteries. Be- cause biological amines may play a role in these spasms, a research team measured amine levels in coronary arteries that were obtained postmortem from patients who had died of heart discase and also from a control group of patients who had died from other causes. The following table summarizes the concentration of the amine serotonin (NG/GM). Use a t-test to compare the means at the 5% significance level. Control 12 Heart Disease 3840 5310 SE 850 640

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7. The general solution to the system of equations given by 21 - 22 + x3 + T4 = 1, 4x – 4x2 + 3r3 + 6x, = 0, %3D - 2x2 + x3 +3x4 = 0, %3D is 3 -1 a +b 3 1' a +b -1 3002