A 25-week gestаtiоnаl аge newbоrn is receiving mechanical ventilatiоn with an inspired fraction of oxygen (FIO2) of 0.70 and high inspiratory pressures. Which of the following conditions is likely to develop?
Questiоn 1 (5 pоints) A cоntinuous-time periodic signаl (x(t)) is reаl-vаlued and has a fundamental period [T=8.] The nonzero complex Fourier series coefficients are [a_{1}=a_{-1}=2,qquada_{3}=4j,qquada_{-3}=a_{3}^{*}=-4j.] Express (x(t)) in the form [x(t)=sum_{k}A_kcos!left(omega_k t+phi_kright),] where [omega_k=komega_0,qquadomega_0=frac{2pi}{T}.] Select the correct answer. (A) [x(t)=4cos!left(frac{pi}{4}tright)-8cos!left(frac{3pi}{4}t+frac{pi}{2}right)] (B) [x(t)=2cos!left(frac{pi}{4}tright)-4cos!left(frac{3pi}{4}t-frac{pi}{2}right)] (C) [x(t)=4cos!left(frac{pi}{4}tright)-8cos!left(frac{3pi}{4}t-frac{pi}{2}right)] (D) [x(t)=2cos!left(frac{pi}{4}tright)+8cos!left(frac{3pi}{4}t+frac{pi}{2}right)] (E) None of the above. Question 2 (5 points) Consider the discrete-time periodic signal [x[n]=1+sin!left(frac{2pi}{N}nright)+3cos!left(frac{2pi}{N}nright)+cos!left(frac{4pi}{N}n+frac{pi}{2}right),] where the fundamental period is (N). Which one of the following sets of DTFS coefficients is correct? (A) [a_0=1,qquada_{pm1}=frac{3}{2}mpfrac{j}{2},qquada_2=frac{j}{2},qquada_{-2}=-frac{j}{2},]with all remaining coefficients equal to zero. (B) [a_0=1,qquada_{pm1}=frac{3}{2}pmfrac{j}{2},qquada_2=-frac{j}{2},qquada_{-2}=frac{j}{2},]with all remaining coefficients equal to zero. (C) [a_0=0,qquada_{pm1}=3mp j,qquada_2=j,qquada_{-2}=-j,]with all remaining coefficients equal to zero. (D) [a_0=1,qquada_1=3,qquada_{-1}=1,qquada_2=frac12,qquada_{-2}=frac12,]with all remaining coefficients equal to zero. (E) None of the above. Question 3 (5 points) Determine the complex exponential Fourier series representation of the following continuous-time periodic signal: [x(t)=sin^2(t).] Select the correct answer. (A) [omega_0=2,qquada_0=frac12,qquada_{pm1}=-frac14,]with all remaining coefficients equal to zero. (B) [omega_0=1,qquada_0=frac12,qquada_{pm2}=-frac14,]with all remaining coefficients equal to zero. (C) [omega_0=2,qquada_0=frac12,qquada_{pm2}=-frac12,]with all remaining coefficients equal to zero. (D) [omega_0=1,qquada_0=1,qquada_{pm2}=-frac14,]with all remaining coefficients equal to zero. (E) None of the above. Question 4 (10 points) A periodic signal[x(t)=sum_{k=-3}^{3} a_k e^{jk2pi t}]with fundamental frequency (omega_0=2pi) is applied to an LTI system with impulse response[h(t)=e^{-t}u(t).] The nonzero Fourier series coefficients of (x(t)) are[a_0=1,qquada_1=a_{-1}=frac{1}{4},qquada_2=a_{-2}=frac{1}{2},qquada_3=a_{-3}=frac{1}{3}.] If[y(t)=sum_{k=-3}^{3} b_k e^{jk2pi t},]which of the following correctly gives the Fourier series coefficients of the output (y(t))? (A) [b_k=a_k(1+jk2pi), qquad -3le kle 3] (B) [b_k=frac{a_k}{1+jk2pi}, qquad -3le kle 3] (C) [b_k=frac{a_k}{1-jk2pi}, qquad -3le kle 3] (D) [b_k=a_k e^{-jk2pi}, qquad -3le kle 3] (E) None of the above Question 5 (15 points) Consider a continuous-time LTI system with the input-output relation [y(t)=int_{-infty}^{t} e^{-(t-tau)}x(tau),dtau.] Answer the following questions. (A) Find the impulse response (h(t)) of this system. (B) Show that the complex exponential function (e^{st}) is an eigenfunction of the system. (C) Using the impulse response obtained in part (a), determine the eigenvalue corresponding to the eigenfunction (e^{st}). Question 6 (15 points) Suppose we are given the following facts about a signal (x(t)): 1. (x(t)) is a real signal. 2. (x(t)) is periodic with period (T=4), and it has Fourier series coefficients (a_k). 3. (a_k=0) for (|k|>1). 4. The signal with Fourier series coefficients [ b_k=e^{-jkpi/2}a_{-k} ] is odd. 5. [ frac{1}{4}int_{4}|x(t)|^2,dt=frac{1}{2}. ] Determine the signal (x(t)). Question 7 (20 points) Consider a continuous-time signal (x(t)), whose CTFT is [X(jomega)=begin{cases}0.25(omega+4), & -4