*Question E9D01:
How much does the gain of an ideal parabolic reflector antenna increase when the operating frequency is doubled?*  

*Correct Answer is  D: 6 dB*  

### AI Response: (may contain errors)  

To understand how the gain of an ideal parabolic reflector antenna changes when the operating frequency is doubled, we can use the formula for the gain of a parabolic antenna:

$$
G = 10 \log_{10} \left( \frac{4 \pi A_e}{\lambda^2} \right)
$$

where:
- $G$ = gain in dBi (decibels over isotropic)
- $A_e$ = effective aperture area of the antenna
- $\lambda$ = wavelength of the signal

### Explanation of the Correct Answer (6 dB)

1. **Doubling the Frequency**: When the frequency is doubled, the wavelength $\lambda$ is halved, since wavelength is inversely proportional to frequency:
   $$
   \lambda = \frac{c}{f}
   $$
   where $c$ is the speed of light and $f$ is the frequency. If $f$ is doubled, then $\lambda$ becomes $\frac{\lambda}{2}$.

2. **Effect on Gain**: Substituting $\lambda/2$ into the gain formula, we see that the gain increases because the denominator ($\lambda^2$) becomes $(\lambda/2)^2 = \frac{\lambda^2}{4}$. Thus, the gain becomes:
   $$
   G' = 10 \log_{10} \left( \frac{4 \pi A_e}{\frac{\lambda^2}{4}} \right) = 10 \log_{10} \left( 16 \frac{4 \pi A_e}{\lambda^2} \right) = G + 10 \log_{10}(4) \approx G + 6 \text{ dB}
   $$
   Therefore, the gain increases by approximately 6 dB when the frequency is doubled.

### Why the Other Choices are Incorrect

- **A: 2 dB**: This is too small of an increase. The gain does not increase linearly with frequency; it is logarithmic.
  
- **B: 3 dB**: This represents a doubling of power, which is not the case here. The gain increase due to halving the wavelength is greater than this.

- **C: 4 dB**: This is also an underestimation. The correct increase, as shown, is 6 dB due to the logarithmic relationship with the effective aperture area and the square of the wavelength.

### Summary

- Doubling the frequency results in a gain increase of 6 dB for an ideal parabolic reflector antenna.
- The gain formula shows that the gain is logarithmically related to the wavelength, and halving the wavelength leads to a significant increase in gain.