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 $x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{\Omega }(\beta )\mathrm{\Phi }(\beta ,t)d\beta$ and $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\mathrm{\Phi }}^{*}(\beta ,t)\mathrm{\Phi }(\lambda ,t)dt=\delta (\beta -\lambda )$ where $x(t)$ is any real function of t. If we have a linear time invariant system where an input of $\mathrm{\Phi }(\lambda ,t)$ produces an output of $\mathrm{\Psi }(\lambda ,t)$.

• How do you find $\mathrm{\Omega }(\beta )$ if you are given $x(t)$?
• What is the output due to $cos(2\pi ft)$?

Question 3

If a signal x(t) only has frequency components near DC, $|X(f)|=0$ for $|f|>f_{max}$, then x(t) is known as a baseband signal. When x(t) is a baseband signal, $x(t)\cos (2\pi f_{0}t)$ 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, $v(t)=x(t)\cos (2\pi f_{0}t)$.
• What is the lowest $f_0$ 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).