Problem 1

3. Given the signal flow graph below determine the transfer matrix A where Aij = Yi/Xj. Note that Aij = Yi/Xj given that all other inputs equal to zero.are \left.\left[\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right]=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] \begin{array}{l} X_{1} \\ X_{2} \end{array}\right]

4 Find and sketch the Fourier transforms for the following signals. u(t)=(1-|t|) I_{[-1,1]}(t) v(t)=\operatorname{sinc}(2 t) \operatorname{sinc}(4 t) \text {. } \text { (c) } s(t)=v(t) \cos (200 \pi t) \text {. } (d) Classify cach of the signals in (a) (c) as baseband or passband.

A signal x(t) has a Fourier Transform given by X(\omega)=\frac{5(1+j \omega)}{8-\omega^{2}+6 j \omega} Without finding x(t), find the Fourier Transform of the following: а. х(t-3) b. x(4t) C. ei1.e12x(t) d. x(-2t)

1.Consider the coupled acoustic-mechanical system. The velocity of the masses are given by u and applied force by the variable f. The variables k represent the mechanical stiffness, M the mass and b the damping coefficient. The closed open pipe is filled with a fluid having mass density Po, sound speed c,cross sectional area A , length L. а. Using mobility analogy where the velocity as the "across" variable, determine the an equivalent circuit for the system. b. Determine the equations of motion in the Laplace-domain. c. Determine the equations of motion in the time-domain. d. Find the transfer function U2(s)/Uo(s).

. Obtain the Fourier Transform of the following functions \text { a. } x(t)=\left\{\begin{array}{l} e^{a t}, t<0 \\ e^{-a t}, t>0 \end{array}\right\} b. Signal shown below:

4. Given the system equations a. Using only amplifiers and integrators draw a signal-flow graph representation of the system where U(s) is the input and X2(s) is the output. You may assume zero initial conditions. b. Find the transfer function X2(s)/U(s) using Mason's Gain formula. Check your result using an algebraic approach. \frac{d x_{1}}{d t}=x_{1}+5 x_{2} \frac{d x_{2}}{d t}=2 x_{1}+u

Problem 2 Fourier transform analysis using Library of transforms. This is another Fourier transform analysis problem. Consider a one degree-of-freedom damped spring-mass system governed by the differential equation ÿ+2y+26y = 26u, where y is the position of the mass relative to it's equilibrium position and u is a force that is applied to the mass. The force input u is the same as the previous problem, i.e. (1). Solve for y on the time interval (-∞, ∞). Graph y on the interval [-3,3] second. Hint: Once ŷ is determined, use a partial fraction expansion and the "Library" from Homework 6 to reverse-engineer the time functions associated with the terms in the partial fraction expansion.

2. Let s be a periodic signal with period To = 2 and s(t)=\left\{\begin{array}{ll} -t(t-1) & 0 \leq t<1 \\ (t-1)(t-2) & 1 \leq t<2 \end{array}\right. ) Find the first, second, and third derivatives r = Ds, u= D²s, and v = D³s. s) Find the Fourier coefficients of each of the four signals: §, î, û, and û. :) (12 pts) For each of the four signals, compute the power with a time domain calculation and compute the power in frequencies ±1/2 (the positive and negative fundamental frequencies) with a frequency domain calculation. What fraction of the power is in the positive and negative fundamental frequencies? Express all answers both symbolically and with an approximate decimal representation.

1. (30 pts) For each of the following systems, determine whether it is linear and whether it is time-invariant. Justify your answers. If it is LTI, find the impulse response function h(t). Each system is specified by the output y that is produced from an input r. \text { (a) } y(t)=x(t+7) \text { (b) } y(t)=x(3 t) \text { (c) } y(t)=|x(10)| y(t)=\int_{-\infty}^{\infty} I_{[0,+\infty)}(t-\tau) \exp (\tau-t) x(\tau) d \tau y(t)=\int_{-\infty}^{\infty} \frac{1}{1+\tau^{2}} x(\tau-t) d \tau y(t)=\int_{-1}^{0}(\tau-1) x(t+\tau) d \tau y(t)=\min (1, \max (-1, x(t-4))) n) Let (a1,. , ak) be a vector of k nonnegative reals and let (T1,.., Tk) E R*. y(t)=\underset{x \in \mathbb{R}}{\operatorname{argmin}} \sum_{i=1}^{k} a_{j}\left(z-x\left(t-\tau_{i}\right)\right)^{2} The argmin, is the value of z (the argument) that minimizes the expression.

Problem 5 A system is tested on a moving platform u -moving base The position of the mass relative to an inertial frame is y. The actual measurement of the mass motion, however, is its acceleration, ÿ, which is provided by an accelerometer attached to the mass. The mass, spring rate, and damping rate are 1, 4 and 8, respectively, so the ODE is ÿ + 4y + 8y = 4ů + 8u. 5 1. Solve the unforced (u= 0) IVP for y with y(0) = -1, g(0) = 1, by computing the characteristic roots A₁ and ₂ and letting y(t) = Aet + Bet, t≥ 0. The parameters A and B are determined by enforcing the initial conditions. Once y(t) is determined (it is a real-valued signal), compute the mass acceleration by differentiation (note: there are no discontinuities in y or any of its time derivatives in a neighborhood of t = 0). 2. Now use the unilateral Laplace transform to find the acceleration due to these initial con- ditions (u= 0 still) by following these steps: 1) apply the transform to the ODE, 2) iso- late the expression for ŷ, 3) use the Derivative Theorem again for the mass acceleration, ỹ = s²ŷ - sy(0) - (0), 4) "invert" the expression for to determine ÿ(t) for, t≥ 0. Compare to the result from Part 1. Note that when differentiating a dependent variable it is always necessary to apply the Derivative Theorem to account for possible non-zero initial conditions associated with it. 3. Now consider a forced IVP in which the ICs are the same as Part 1, however, u(t) = tµ(t), t20. Use the unilateral Laplace transform to determine ÿ, t≥ 0. Note that "external" inputs are always considered "abrupt", so u(0) = 0, ú(0) = 0, etc. Hint: develop the expression for first, then apply the Derivative Theorem to determine , then reverse engineer to time-domain signals. Make sure to distinguish which signals in ÿ are abrupt, versus those that are not.