Wave packets & Fourier analysis: minimum pulse duration w/ given frequency

AI Thread Summary
The discussion revolves around calculating the minimum pulse duration for transmitting signals over a telephone line and a space probe. For the telephone line with a frequency range of 2500 Hz, the shortest pulse duration is determined to be approximately 31.8 microseconds. For the space probe transmitting a picture with 2 million pulses at a bandwidth of 1000 Hz, the minimum time required to send the image is calculated to be at least 318 seconds, or about 5.3 minutes. The calculations involve applying Fourier analysis principles and ensuring proper pulse spacing to avoid overlap. Overall, the thread emphasizes the relationship between bandwidth, pulse duration, and transmission time in signal processing.
WJSwanson
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Homework Statement


I: A telephone line can transmit a range of frequencies \Delta f = 2500 Hz. Roughly what is the duration of the shortest pulse that can be sent over this line?

II: A space probe sends a picture containing 500 by 500 elements, each containing a brightness scale with 256 possible levels. This scale requires eight binary digits. Suppose that the transmitter uses a bandwidth of 1000 Hz. (For the faint signals from distant space probes the bandwidth must be kept small to reduce the effects of electronic noise that is present at all frequencies.) Roughly how long is needed to send one picture? Note that the center-to-center separation, a, of adjacent pulses must be at least the total width of anyone pulse.


Homework Equations


I:
\Delta\omega = 2\pi\Delta f
\Delta t \geq \frac{1}{2\Delta\omega}

II:
N (number of pulses) = 500 * 500 * 8 = 2.00 * 10^{6} pulses
a \geq 2\Delta t = \frac{2}{2\Delta\omega} = \frac{1}{2\pi\Delta f}

The Attempt at a Solution



I:
\Delta\omega = 2\pi\Delta f = 6.28 * 2.50 * 10^{3} Hz = 1.57 * 10^{4} Hz

\Delta t \geq \frac{1}{2\Delta\omega} = \frac{1}{3.14 * 10^{4}} s^{-1}

\Delta t \geq 3.18 * 10^{-5} s

II:
I'm not quite sure how to approach this. I know that there will have to be 2.00 * 106 pulses per picture and that the individual pulses have to be spaced by a = 2\Delta t. I also would imagine that I would want to use the relation

\Delta x\Delta k \geq 1/2

by deriving

\Delta k = \frac{2\pi}{\Delta\lambda} = \frac{2\pi\Delta f}{c} = \frac{2\pi}{c\Delta t}

or else perhaps use

\Delta\omega\Delta t = \Delta t * \Delta(\frac{1}{2\pi t}) \geq 1/2

but I really don't know where to go from there, aside from multiplying my minimum time interval by 2 million.
 
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Okay, I ended up figuring out part II.

Since we know that
\Delta\omega = 2\pi\Delta f = 6.28 * 10^{3} s^{-1}

we infer that

\Delta t \geq \frac{1}{2\Delta\omega}<br /> <br /> and since the probe sends the message at a rate of 2\Delta t = \frac{t}{2.0 * 10^{6}} we know that<br /> <br /> \Delta t = \frac{t}{4.0 * 10^{6}}<br /> <br /> and since<br /> <br /> \Delta t\Delta\omega = \Delta\omega * \frac{t}{4.0 * 10^{6}} \geq \frac{1}{2}<br /> <br /> we can solve for t<sub>min</sub>:<br /> <br /> t_{min} \geq \frac{4.0 * 10^{6}}{2\Delta\omega} \Rightarrow t_{min} \geq \frac{2.0 * 10^{6}}{6.28 * 10^{3} s^{-1}}<br /> <br /> thus,<br /> <br /> t_{min} \geq 3.18 * 10^{2} s<br /> <br /> Or alternatively,<br /> <br /> t_{min} \geq 5.3 min
 
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