HW 08: Difference between revisions

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|<math>v(t)\,\!</math>
|<math>v(t)\,\!</math>
|<math>=x(t)\,\cos(2\pi f_0 t)</math>
|<math>=x(t)\,\cos(2\pi f_0 t)</math>
|x(t) is the original signal
|-
|-
|<math>v(f)\,\!</math>
|<math>v(f)\,\!</math>
|<math>=\int_{-\infty}^{\infty}x(t)\cos(2\pi f_0 t)e^{-j2\pi ft}\,dt</math>
|<math>=\int_{-\infty}^{\infty}x(t)\cos(2\pi f_0 t)e^{-j2\pi ft}\,dt</math>
|v(f) is x(t) after a low pass filter (cutoff frequency = f_max), multiplied by cos(2\pi f_0 t)
|-
|-
|
|
Line 36: Line 38:
|
|
|<math>=\frac{1}{2}x(t)\left[\delta(f_0-f)+\delta(f_0+f)\right]</math>
|<math>=\frac{1}{2}x(t)\left[\delta(f_0-f)+\delta(f_0+f)\right]</math>
|-
|<math>w(t)\,\!</math>
|<math>=v(t)\cdot \cos(2\pi f_0 t)</math>
|w(t) is v(t) multiplied by cos(2\pi f_0 t)
|-
|
|<math>=\int_{-\infty}^{\infty}\frac{1}{2}x(t)\left[\delta(f_0-f)+\delta(f_0+f)\right]\frac{e^{j2\pi f_0 t}+e^{-j2\pi f_0 t}}{2}e^{j2\pi ft}\,df</math>
|}
|}

Revision as of 15:32, 7 December 2008

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 and where is any real function of t. If we have a linear time invariant system where an input of produces an output of .

  • How do you find if you are given ?
  • What is the output due to ?

Answer 2

Question 3

If a signal x(t) only has frequency components near DC, for , then x(t) is known as a baseband signal. When x(t) is a baseband signal, 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, .
  • What is the lowest 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

x(t) is the original signal
v(f) is x(t) after a low pass filter (cutoff frequency = f_max), multiplied by cos(2\pi f_0 t)
w(t) is v(t) multiplied by cos(2\pi f_0 t)