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

Choices:  
A: 2 dB  
B: 3 dB  
C: 4 dB  
D: 6 dB    

Correct Answer is  D: 6 dB  

To understand how dish gain changes with frequency, use the aperture antenna relationship:

- For a parabolic reflector of fixed physical size (area A), directivity/gain scales as:
  $$G \propto \frac{A}{\lambda^2} \propto f^2$$
  since $\lambda = \frac{c}{f}$.

If you double the frequency ($f_2 = 2f_1$), then:
- $\lambda_2 = \lambda_1/2$
- $$\frac{G_2}{G_1} = \left(\frac{\lambda_1}{\lambda_2}\right)^2 = \left(\frac{\lambda_1}{\lambda_1/2}\right)^2 = 4$$
- In dB: $$\Delta G = 10\log_{10}(4) \approx 6.02\ \text{dB}$$

Therefore, the correct answer is 6 dB (D).

Why the other choices are incorrect:
- 2 dB: Would imply only a 1.58× gain increase; inconsistent with the $f^2$ scaling.
- 3 dB: Corresponds to a factor of 2 in linear gain; still short of the required factor of 4.
- 4 dB: Corresponds to about a 2.51× gain increase; also not consistent with $f^2$ scaling.

Useful formulas and rules of thumb:
- Parabolic dish gain (linear): $$G = \eta \left(\frac{\pi D}{\lambda}\right)^2$$
- Or in dBi: $$G_{\text{dBi}} \approx 10\log_{10}(\eta) + 20\log_{10}\!\left(\frac{\pi D}{\lambda}\right)$$
- For a fixed-diameter dish: gain increases by about +6 dB each time frequency doubles (and +20 dB per decade).