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Problem 3-2 Consider the following equation: $z^4 = \frac{\sqrt{2}}{2} - j\frac{\sqrt{2}}{2}$. (a) How many solutions are there to this equation? (b) Determine all the solutions and express your answers in polar form. (c) Sketch your solutions in the complex plane. Make sure to label the axes of your sketch. Problem 3-3 A simple linear time-invariant circuit has the frequency response given below: $H(\omega) = \frac{1}{1 + j\omega \frac{\sqrt{3}}{700}}$. Determine the output $y(t)$ of the circuit when the input is $x(t) = cos(700t)$. Problem 3-4 Consider the signal $y(t)$ given below: $y(t) = \sqrt{3} cos \left(2\pi (47)t + \frac{5\pi}{6}\right) + cos \left(2\pi (47)t + \pi\right) + \sqrt{3} cos \left(2\pi (47)t + \frac{\pi}{6}\right)$. Simplify $y(t)$ and express it as a single sinusoid: $y(t) = A cos(2\pi f_0t + \phi)$ where the values of A, $f_0$, and $\phi$ are specified. Problem 3-5 Consider the signal $x(t) = 2 cos(2\pi (30)t) + 6 cos(2\pi (50)t)$. Sketch the spectrum of the signal $x(t)$. Show the spectrum as a function of f in Hz.

          Problem 3-2
Consider the following equation:
$z^4 = \frac{\sqrt{2}}{2} - j\frac{\sqrt{2}}{2}$.
(a) How many solutions are there to this equation?
(b) Determine all the solutions and express your answers in polar form.
(c) Sketch your solutions in the complex plane. Make sure to label the axes of your sketch.
Problem 3-3
A simple linear time-invariant circuit has the frequency response given below:
$H(\omega) = \frac{1}{1 + j\omega \frac{\sqrt{3}}{700}}$.
Determine the output $y(t)$ of the circuit when the input is $x(t) = cos(700t)$.
Problem 3-4
Consider the signal $y(t)$ given below:
$y(t) = \sqrt{3} cos \left(2\pi (47)t + \frac{5\pi}{6}\right) + cos \left(2\pi (47)t + \pi\right) + \sqrt{3} cos \left(2\pi (47)t + \frac{\pi}{6}\right)$.
Simplify $y(t)$ and express it as a single sinusoid: $y(t) = A cos(2\pi f_0t + \phi)$ where the values of A, $f_0$,
and $\phi$ are specified.
Problem 3-5
Consider the signal
$x(t) = 2 cos(2\pi (30)t) + 6 cos(2\pi (50)t)$.
Sketch the spectrum of the signal $x(t)$. Show the spectrum as a function of f in Hz.

Problem 3-2
Consider the following equation:
z^4 = (√(2))/(2) - j(√(2))/(2).
(a) How many solutions are there to this equation?
(b) Determine all the solutions and express your answers in polar form.
(c) Sketch your solutions in the complex plane. Make sure to label the axes of your sketch.
Problem 3-3
A simple linear time-invariant circuit has the frequency response given below:
H(ω) = (1)/(1 + jω(√(3))/(700)).
Determine the output y(t) of the circuit when the input is x(t) = cos(700t).
Problem 3-4
Consider the signal y(t) given below:
y(t) = √(3) cos (2π (47)t + (5π)/(6)) + cos (2π (47)t + π) + √(3) cos (2π (47)t + (π)/(6)).
Simplify y(t) and express it as a single sinusoid: y(t) = A cos(2π f0t + ϕ) where the values of A, f0,
and ϕ are specified.
Problem 3-5
Consider the signal
x(t) = 2 cos(2π (30)t) + 6 cos(2π (50)t).
Sketch the spectrum of the signal x(t). Show the spectrum as a function of f in Hz.

Added by Ryan L.

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