*Question E4C06:
How much does increasing a receiver’s bandwidth from 50 Hz to 1,000 Hz increase the receiver’s noise floor?*  

*Correct Answer is  D: 13 dB*  

### AI Response: (may contain errors)  

To determine how much increasing a receiver's bandwidth affects the noise floor, we use the relationship between bandwidth and noise power. The noise power in a receiver is proportional to its bandwidth. Specifically, when bandwidth increases, the noise power increases logarithmically.

### Explanation:

1. **Formula for Noise Power Increase:**

   The increase in noise power due to an increase in bandwidth can be calculated using the formula:

   $$ \Delta \text{dB} = 10 \log_{10} \left( \frac{B_2}{B_1} \right) $$

   where:
   - $B_1$ is the initial bandwidth (50 Hz)
   - $B_2$ is the final bandwidth (1,000 Hz)

2. **Calculation:**

   Plugging in the values:

   $$ \Delta \text{dB} = 10 \log_{10} \left( \frac{1000}{50} \right) $$

   $$ \Delta \text{dB} = 10 \log_{10} (20) $$

   $$ \Delta \text{dB} \approx 10 \times 1.3010 $$

   $$ \Delta \text{dB} \approx 13 \text{ dB} $$

   Therefore, increasing the bandwidth from 50 Hz to 1,000 Hz increases the noise floor by approximately 13 dB.

3. **Why the Correct Answer is Correct:**

   - **Choice D: 13 dB** is correct because the calculation shows that the noise floor increases by approximately 13 dB when the bandwidth is increased from 50 Hz to 1,000 Hz.

4. **Why the Other Choices are Incorrect:**

   - **Choice A: 3 dB** is incorrect because a 3 dB increase would correspond to doubling the bandwidth, not increasing it by a factor of 20.
   - **Choice B: 5 dB** is incorrect because it does not match the calculated increase for a 20-fold increase in bandwidth.
   - **Choice C: 10 dB** is incorrect because a 10 dB increase would correspond to a tenfold increase in bandwidth, not a 20-fold increase.

Understanding this relationship helps in predicting how changes in bandwidth affect the noise performance of a receiver.