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Chapter 2: Problem 7

Multipath fading is maximized when the two beams arrive 180 degrees out of phase. How much of a path difference is required to maximize the fading for a 100 -km-long 1-GHz microwave link?

Short Answer

Expert verified
A path difference of 0.15 meters is required to maximize the fading.

Step by step solution

01

Understanding Multipath Fading and Phase Difference

In multiply path scenarios, fading is maximized when two signals arriving at a receiver are 180 degrees out of phase. This implies a phase difference of half a wavelength between the two signals.
02

Determine Wavelength from Frequency

We use the speed of light to find the wavelength of the microwave link: \[ c = f \lambda \] where \( c = 3 \times 10^8 \text{ m/s} \), and \( f = 1 \times 10^9 \text{ Hz} \).Substituting the values, \[ \lambda = \frac{c}{f} = \frac{3 \times 10^8}{1 \times 10^9} = 0.3 \text{ meters} \]
03

Calculate Path Difference for 180 Degrees Phase Shift

A 180-degree phase shift corresponds to a path difference of half a wavelength. Therefore, the path difference required is:\[ \text{Path difference} = \frac{\lambda}{2} = \frac{0.3}{2} = 0.15 \text{ meters} \]
04

Final Path Difference Conclusion

To maximize fading, the two beams need to have a path difference of 0.15 meters to produce a 180-degree phase shift.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phase Difference
Phase difference in the context of multipath fading refers to the discrepancy in the phase of two or more converging signals. Signals can have different phases depending on the distance they travel. When two signals that take different paths reach the receiver out of phase, they can either amplify or cancel each other.
A phase difference of 180 degrees means the waves are perfectly out of phase, leading to destructive interference. This is when one wave peaks where the other troughs, effectively canceling each other out and creating a point of significant fading in the signal strength.
  • Out of phase: Signals don't match up in their wave pattern.
  • Destructive interference: Two waves overlap to produce a lesser or zero effect.
This phase discrepancy is critical when considering multipath fading as it directly affects signal quality and strength.
Wavelength Calculation
To find the wavelength of a signal, you need to know its frequency and the speed at which it travels. In the case of a microwave link, these signals usually travel at the speed of light.
The formula to calculate wavelength is:
\[ \lambda = \frac{c}{f} \]where
  • \( c \) is the speed of light \((3 \times 10^8\,\text{m/s})\)
  • \( f \) is the frequency of the signal
In our specific case of a 1 GHz microwave link, we substitute:
\[ \lambda = \frac{3 \times 10^8}{1 \times 10^9} = 0.3\,\text{meters}\]Understanding this calculation helps us know how far apart signals should be to experience complementary (or harmful) interference effects, such as fading, due to phase differences.
Microwave Link Path Difference
For a microwave link to experience maximized fading due to multipath, the path difference must lead to a 180-degree phase difference in arriving signals.
Given a wavelength of 0.3 meters, a full 360-degree cycle refers to one complete wave. Therefore, a 180-degree phase shift means the two waves are offset by half a wavelength.
This translates directly to the required path difference:
\[ \text{Path difference} = \frac{\lambda}{2} \]Substituting our wavelength:
\[ \text{Path difference} = \frac{0.3}{2} = 0.15\,\text{meters}\]
  • Microwave links apply in long-distance communication, where phase differences arise due to multiple paths.
  • Recognizing path difference can help mitigate fading effects.
In practical terms, understanding this path difference helps in designing networks that maintain signal quality and reduce unintended signal loss.

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