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Linear Systems

Consider the circuit shown below. Find using circuit analysis techniques,

Vab, the voltage across the terminals a and b,

PRL, the power dissipated by the load resistor, RL,

The power delivered to the load resistor by the voltage source, V1, and

The The venin equivalent circuit presented to the load resistor, RL.

Validate all of your answers with Multisim circuit analysis. Submit both sowing they produce the same result.

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Linear Systems

5 Use Parseval's identity to compute the following integrals.

\text { a) } \int_{-\infty}^{\infty} \operatorname{sinc}^{2}(2 r) d t \text {, }

\text { (b) } \int_{0}^{\infty} \operatorname{sinc}(t) \operatorname{sinc}(2 t) d t \text {. }

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Linear Systems

4 Find and sketch the Fourier transforms for the following signals.

\text { (c) } s(t)=v(t) \cos (200 \pi t) \text {. }

u(t)=(1-|t|) I_{[-1,1]}(t)

(d) Classify cach of the signals in (a) (c) as baseband or passband.

v(t)=\operatorname{sinc}(2 t) \operatorname{sinc}(4 t) \text {. }

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Linear Systems

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.

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Linear Systems

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.

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Linear Systems

Accurately sketch the spectrum (magnitude only) of the following function

y(t)=x(t) \cos (2 \pi 10 t) \text { if } x(t)=3 \operatorname{rect}\left(\frac{t}{2}\right)

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Linear Systems

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)

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Linear Systems

Find the inverse Fourier Transform of

G(\omega)=\frac{10 j \omega}{(-j \omega+2)(j \omega+3)}

Y(\omega)=\frac{\delta(\omega)}{(j \omega+1)(j \omega+2)}

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Linear Systems

. 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:

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Linear Systems

Consider the following linear program:

\begin{array}{l} \max x_{1}+x_{2} \\ \text { s.t.: } \\ \text { Cl } 3 x_{1}+2 x_{2} \leq 12 \\ \text { C2 } 2 x_{1}+3 x_{2} \leq 12 \\ \text { C3 } 2 x_{1}+2 x_{2} \leq 9 \\ \text { C4 } 2 x_{1}+2 x_{2} \geq 3 \\ \text { CS } \quad x_{1}, x_{2} \geq 0 \end{array}

a) (5 points) Graph the feasible region of the LP. Is the feasible region unbounded?

b) (35 points) Solve this problem using Simplex algorithm. Make sure to indicate:

* Whether or not you need to perform Phase

* Show the BV and NBV for each iteration of Simplex

* Show the steps of the algorithm in the graph from point a)

* Is this a unique or multiple solution?

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