Question E7C10:
Which of the following filters is used in a 2-meter band repeater duplexer?  

Choices:  
A: A crystal filter  
B: A cavity filter  
C: A DSP filter  
D: An L-C filter    

Correct Answer is  B: A cavity filter  

Correct answer: B: A cavity filter

Why B is correct:
- 2-meter repeaters must transmit and receive simultaneously with only ±600 kHz separation. They need very high isolation (often 80–100 dB) with low insertion loss and high power handling.
- Cavity filters (quarter-wave resonators in large conductive cans) have very high Q, sharp skirts, deep notches, and can handle repeater power levels. They are the standard choice for VHF/UHF duplexers.

Why the others are incorrect:
- A: Crystal filter
  - Crystals are used at IF stages (e.g., 10.7 MHz, 455 kHz), not typically at 144–148 MHz RF for high power.
  - Poor power handling at repeater RF levels; impractical to build a VHF duplexer with crystals.
- C: DSP filter
  - DSP operates after downconversion inside the receiver; it cannot provide RF isolation between a transmitter and receiver sharing the same antenna/line. It does nothing to prevent receiver front-end overload.
- D: L-C filter
  - Discrete inductors and capacitors at VHF have limited Q and parasitics, yielding insufficient selectivity and isolation for a 600 kHz split at 146 MHz, especially at high power.

Useful formulas and rules of thumb:
- Repeater split on 2 m: ±600 kHz. Fractional offset:
  $$\text{FBW} = \frac{\Delta f}{f_0} \approx \frac{0.6\ \text{MHz}}{146\ \text{MHz}} \approx 0.0041 \ (0.41\%)$$
- Required loaded Q for a simple resonator at that bandwidth (order-of-magnitude):
  $$Q \approx \frac{f_0}{\text{BW}} \approx \frac{146\ \text{MHz}}{0.6\ \text{MHz}} \approx 240$$
  Note: Duplexers need much deeper rejection than implied by a 3 dB bandwidth; multiple high-Q cavities are cascaded to achieve 80–100 dB isolation with low loss.
- Wavelength at 146 MHz:
  $$\lambda = \frac{c}{f} \approx \frac{3.0\times 10^8\ \text{m/s}}{146\times 10^6\ \text{Hz}} \approx 2.05\ \text{m}$$
  Quarter-wave cavity length:
  $$\ell \approx \frac{\lambda}{4} \approx 0.51\ \text{m}$$
- Typical Q:
  - Cavity resonators: $Q \sim 2{,}000$–$10{,}000$ (high selectivity, high power).
  - L-C at VHF: $Q \lesssim 100$–$200$ (insufficient selectivity/isolation).
  - Crystals: very high Q but not suitable for high-power VHF duplexing and not used at RF front ends.