Question E8B01:
What is the modulation index of an FM signal?  

Choices:  
A: The ratio of frequency deviation to modulating signal frequency  
B: The ratio of modulating signal amplitude to frequency deviation  
C: The modulating signal frequency divided by the bandwidth of the transmitted signal  
D: The bandwidth of the transmitted signal divided by the modulating signal frequency    

Correct Answer is  A: The ratio of frequency deviation to modulating signal frequency  

To find the FM modulation index, use:
$$\beta = \frac{\Delta f}{f_m}$$
where:
- $\Delta f$ = peak frequency deviation (Hz)
- $f_m$ = modulating signal frequency (Hz)

1) Why the correct answer is correct
- In FM, the carrier frequency swings by a peak amount $\Delta f$ at a rate set by the modulating frequency $f_m$. The modulation index $\beta$ quantifies how “deeply” the carrier is frequency-modulated and is defined as the ratio of these two quantities:
  $$\beta = \frac{\Delta f}{f_m}$$
- This ratio is dimensionless (Hz/Hz) and directly describes FM deviation relative to the modulating frequency.

2) Why the other choices are incorrect
- B: “The ratio of modulating signal amplitude to frequency deviation”  
  FM index does not depend on the amplitude of the modulating signal; it depends on frequency deviation and modulating frequency. Also, units would be inconsistent (volts/Hz).
- C: “The modulating signal frequency divided by the bandwidth of the transmitted signal”  
  Using Carson’s Rule, bandwidth is approximately $B \approx 2(\Delta f + f_m) = 2f_m(\beta + 1)$. Thus,
  $$\frac{f_m}{B} \approx \frac{1}{2(\beta + 1)} \neq \beta.$$
- D: “The bandwidth of the transmitted signal divided by the modulating signal frequency”  
  From Carson’s Rule, 
  $$\frac{B}{f_m} \approx 2(\beta + 1) \neq \beta.$$

3) Useful formulas and rules of thumb
- Modulation index (single-tone FM): 
  $$\beta = \frac{\Delta f}{f_m}$$
- Carson’s Rule (occupied bandwidth estimate):
  $$B \approx 2(\Delta f + f_{\max}) = 2f_{\max}(\beta + 1)$$
- Narrowband FM: $\beta \ll 1$; Wideband FM: $\beta \gg 1$.