Question

The figure of merit ratio of FM to PM for single tone modulating signal is

• A. 1
• B. 3
• C. 4
• D. \(\infty\)

Hint:

Figure of Merit (FOM) compares output SNR to some reference SNR.
For FM (single tone): \( (SNR)_o / (P_c/(N_0 B_m)) = \frac{3}{2} \beta^2 \), where \(\beta = \Delta f / f_m\).
For PM (single tone): \( (SNR)_o / (P_c/(N_0 B_m)) = \frac{1}{2} (\Delta\phi)^2 \), where \(\Delta\phi\) is peak phase deviation.
If modulation is set such that peak frequency deviation in FM is equal to peak phase deviation times \(f_m\) in PM (i.e., \(\Delta f = \Delta\phi \cdot f_m\)), then \(\beta_{FM} = \Delta\phi\). The ratio of FOMs becomes 3.

Answer 1

The Correct Option is B

The figure of merit (FOM) ratio of FM to PM for a single-tone modulating signal is derived by comparing the SNR improvement factors for both FM and PM systems. The figure of merit for FM and PM is based on their respective modulation indices and the corresponding output SNRs.

1. Figure of Merit for FM 

For Frequency Modulation (FM), the figure of merit is proportional to the square of the modulation index (\(\beta_{FM}\)). The modulation index \(\beta_{FM}\) for FM is defined as:

\(\beta_{FM} = \frac{\Delta f}{f_m}\)

Where:

• \(\Delta f\) is the peak frequency deviation.
• \(f_m\) is the frequency of the modulating signal.

 

The figure of merit for FM in terms of SNR improvement is given by:

\( \text{FOM}_{FM} \propto \beta_{FM}^2 \)

More precisely, the SNR improvement ratio for FM is:

\( \frac{\text{SNR}_{o,FM}}{\text{SNR}_{o,AM}} \approx \frac{3}{2} \beta_{FM}^2 \)

2. Figure of Merit for PM

For Phase Modulation (PM), the figure of merit is proportional to the square of the phase deviation \(\Delta\phi\). The modulation index for PM is defined as:

\(\beta_{PM} = \Delta \phi\)

Where \(\Delta \phi\) is the peak phase deviation.

The figure of merit for PM in terms of SNR improvement is given by:

\( \text{FOM}_{PM} \propto (\Delta \phi)^2 \)

More precisely, the SNR improvement ratio for PM is:

\( \frac{\text{SNR}_{o,PM}}{\text{SNR}_{o,AM}} \approx \frac{1}{2} (\Delta \phi)^2 \)

3. Comparing the Figures of Merit

To compare the figure of merit ratios of FM to PM, we look at the SNR improvement factors for both modulations. Under equivalent conditions, we compare the modulation indices in both systems:

\(\beta_{FM} = \Delta f / f_m = \Delta \phi\) for comparable peak frequency deviation and peak phase deviation times the modulating frequency.

Thus, the figure of merit ratio between FM and PM is:

\( \frac{\text{FOM}_{FM}}{\text{FOM}_{PM}} = \frac{\frac{3}{2} \beta_{FM}^2}{\frac{1}{2} (\Delta \phi)^2} = 3 \)

Conclusion

Therefore, the figure of merit ratio of FM to PM for a single-tone modulating signal is:

3