How is the Message Signal Recovered?
The goal of FM demodulation is to extract the original message signal $m(t)$ from the frequency-modulated carrier. The process follows these mathematical steps:
1. The FM Signal Equation
The received FM signal is represented as:
$$s_{FM}(t) = A_c \cos \left( 2\pi f_c t + 2\pi k_f \int_{0}^{t} m(\tau) d\tau \right)$$
2. Differentiation (Slope Detection)
By differentiating the FM signal with respect to time, we convert the frequency variations into amplitude variations (AM):
$$\frac{d}{dt}s_{FM}(t) = -A_c \left[ 2\pi f_c + 2\pi k_f m(t) \right] \sin \left( 2\pi f_c t + \theta(t) \right)$$
The term in the brackets, $[2\pi f_c + 2\pi k_f m(t)]$, is the envelope of the signal. It contains the message $m(t)$ riding on top of a constant carrier frequency component.
3. Envelope Detection and DC Removal
An envelope detector is used to extract the term in the brackets. After removing the constant DC component ($2\pi f_c$), the output is directly proportional to the message signal:
$$y_d(t) \approx 2\pi k_f m(t)$$
Note on Simulation: In this digital simulator, we use the Analytic Signal method. We compute the instantaneous phase using the Hilbert Transform, unwrap the phase, and then calculate the derivative (rate of change) of the phase to find the instantaneous frequency.
FM vs. PM: Spectrum Comparison
Both Frequency Modulation (FM) and Phase Modulation (PM) are forms of Angle Modulation. While their frequency components appear at the same locations, their spectral density responds differently to changes in the message frequency ($f_m$).
Frequency Modulation (FM)
Modulation Index
$$\beta = \frac{k_f A_m}{f_m}$$
"As the message frequency ($f_m$) increases, $\beta$ decreases. The sidebands spread out, but their number and total bandwidth shrink."
Phase Modulation (PM)
Modulation Index
$$\beta = k_p A_m$$
"The modulation index $\beta$ is independent of $f_m$. Increasing $f_m$ spreads sidebands apart, but the number of sidebands stays constant."
| Feature | FM Spectrum | PM Spectrum |
|---|---|---|
| Frequency Components | $$f_c \pm n f_m$$ | $$f_c \pm n f_m$$ |
| Sideband Amplitude | $$J_n(\beta)$$ | $$J_n(\beta)$$ |
| Effect of doubling $f_m$ | Sidebands spread, $\beta$ halves. | Sidebands spread, $\beta$ stays same. |
| Practical Usage | High-fidelity Analog Broadcasting. | Digital Data Transmission (PSK). |