HW 08: Difference between revisions
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|<math>v(t)\,\!</math> |
|<math>v(t)\,\!</math> |
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|<math>=x(t)\,\cos(2\pi f_0 t)</math> |
|<math>=x(t)\,\cos(2\pi f_0 t)</math> |
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|x(t) is the original signal |
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|<math>v(f)\,\!</math> |
|<math>v(f)\,\!</math> |
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|<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> |
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|v(f) is x(t) after a low pass filter (cutoff frequency = f_max), multiplied by cos(2\pi f_0 t) |
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|<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> |
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|<math>w(t)\,\!</math> |
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|<math>=v(t)\cdot \cos(2\pi f_0 t)</math> |
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|w(t) is v(t) multiplied by cos(2\pi f_0 t) |
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|<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> |
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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) | ||