View source for HW 08

===Question 1===
If the sound track of a movie was played into a high fidelity playback system at twice the correct speed, what happens to a sine wave's frequency, amplitude and phase, relative to what happens at the correct speed? Explain your answers.
===Answer 1===
Frequency: The frequency is doubled

Amplitude:

Phase:

===Question 2===
Suppose <math>x(t)=\int_{-\infty}^{\infty}\Omega(\beta)\,\Phi(\beta,t)\,d\beta</math> and <math>\int_{-\infty}^{\infty}\Phi^*(\beta,t)\,\Phi(\lambda,t)\,dt=\delta(\beta-\lambda)</math> where <math>x(t)\,\!</math> is any real function of t. If we have a linear time invariant system where an input of <math>\Phi(\lambda,t)\,\!</math> produces an output of <math>\Psi(\lambda,t)\,\!</math>.
*How do you find <math>\Omega(\beta)\,\!</math> if you are given <math>x(t)\,\!</math>?
*What is the output due to <math>cos(2\pi ft)\,\!</math>?
===Answer 2===
===Question 3===
If a signal x(t) only has frequency components near DC, <math>\left|X(f)\right| = 0</math> for <math>|f|>f_{max}\,\!</math>, then x(t) is known as a baseband signal. When x(t) is a baseband signal, <math>x(t)\,\cos(2\pi f_0 t)</math> is known as a double sideband (DSB) signal. Sometimes a double sideband signal is used to send information over a radio frequency communications link. The transmitter and receiver are shown below.
*Find the Fourier Transform of the DSB signal, <math>v(t) = x(t)\,\cos(2\pi f_0 t)</math>.
*What is the lowest <math>f_0\,\!</math> that can be used and still have the communications system work?
*How does the bandwidth of v(t) compare to the bandwidth of x(t)?
*What does the spectrum of w(t) look like and how does it compare to that of x(t)? A graph would be appropriate showing the spectrum of x(t) and that of w(t).
===Answer 3===
{| border="0" cellpadding="0" cellspacing="0"
|-
|<math>v(t)\,\!</math>
|<math>=x(t)\,\cos(2\pi f_0 t)</math>
|-
|<math>v(f)\,\!</math>
|<math>=\int_{-\infty}^{\infty}x(t)\,\cos(2\pi f_0 t)e^{-j2\pi ft}</math>
|}