#04 Phase derivative, FM/PM equivalence and conditional bandwidth tradeoffs

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Define angle modulation and frequency as phase derivative

   

   The preceding AM-family examples vary the carrier envelope; here the carrier amplitude is fixed.

   Use an ideal real carrier with positive amplitude and continuously defined phase in radians.

   The constant-envelope angle-modulated signal is:

   $\displaystyle s\left(t\right)=A_{c} \cos \left(\theta \left(t\right)\right)$

   Separate the carrier phase and the message-dependent phase deviation:

   $\displaystyle \theta \left(t\right)=2\pi f_{c} t+\varphi \left(t\right)$

   Instantaneous frequency in hertz is the phase derivative divided by two pi:

   $\displaystyle f_{i}\left(t\right)=\frac{\theta '\left(t\right)}{2\pi }$

   FM with positive frequency sensitivity follows the message value:

   $\displaystyle f_{i}\left(t\right)=f_{c}+k_{f} m\left(t\right)$

   PM with phase sensitivity changes phase directly:

   $\displaystyle \varphi \left(t\right)=k_{p} m\left(t\right)$

   Frequency and phase are related quantities, not two derivatives of phase; frequency sensitivity is in hertz per message unit, phase sensitivity in radians per message unit.

   Narration transcript

   In the previous lessons, every scheme was built on top of carrier amplitude. Now we change the rule. We keep the amplitude constant and push the message into the angle of the carrier. The angle has two natural handles, frequency and phase. That single decision splits into two related schemes. Frequency Modulation, or FM, lets the message control the instantaneous frequency. Phase Modulation, or PM, lets the message control the instantaneous phase. Both belong to the angle modulation family.

2. 2. Keep the message and instantaneous frequency phase consistent

   

   Frequency determines the local rate of phase advance:

   $\displaystyle \theta '\left(t\right)=2\pi f_{i}\left(t\right)$

   A zero message leaves the FM carrier at its unmodulated frequency:

   $\displaystyle m\left(t\right)=0, f_{i}\left(t\right)=f_{c}$

   For positive frequency sensitivity, a larger message value means a larger instantaneous frequency; the message derivative is not the FM control signal.

   Use a positive-frequency cosine message to keep phase and frequency consistent:

   $\displaystyle m\left(t\right)=A_{m} \cos \left(2\pi f_{m} t\right)$

   With zero initial phase deviation, the corresponding FM phase deviation is:

   $\displaystyle \varphi \left(t\right)=\beta \sin \left(2\pi f_{m} t\right)$

   Differentiating this phase gives the same cosine message dependence:

   $\displaystyle f_{i}\left(t\right)=f_{c}+\beta f_{m} \cos \left(2\pi f_{m} t\right)$

   The envelope stays constant while phase spacing changes:

   $\displaystyle s\left(t\right)=A_{c} \cos \left(2\pi f_{c} t+\beta \sin \left(2\pi f_{m} t\right)\right)$

   Narration transcript

   The key idea behind angle modulation is the concept of instantaneous frequency. Imagine a pure carrier oscillating at a fixed rate. When the message rises, we make the carrier oscillate slightly faster. When the message falls, we make it oscillate a bit slower. So at every instant, the carrier has its own local frequency that follows the message. The amplitude never changes, only how fast the wave advances in time. That is why an FM waveform looks compressed in some places and stretched in others, even though every cycle keeps exactly the same height.

3. 3. Compare FM and PM with units and derivative gain

   

   FM makes frequency deviation proportional to the message; PM makes phase deviation proportional to the message.

   For positive FM sensitivity and message peak amplitude, the peak frequency deviation is:

   $\displaystyle \Delta f=k_{f} A_{m}$

   Scaling the same message waveform changes the peak frequency swing in the same proportion.

   Write the PM phase deviation explicitly:

   $\displaystyle \varphi _{P}\left(t\right)=k_{p} m\left(t\right)$

   To realize PM with an FM modulator, drive it with the properly scaled derivative and match the initial phase:

   $\displaystyle x_{D}\left(t\right)=\left(\frac{k_{p}}{2\pi k_{f}}\right)m'\left(t\right)$

   Broadcast and voice FM and phase-based digital modulation are application examples; waveform similarity alone does not identify the modulation law.

   Narration transcript

   FM and PM look very similar on a screen, but they encode the message differently. In Frequency Modulation, the instantaneous frequency deviation is proportional to the message itself. So a louder message pushes the frequency further away from the carrier. In Phase Modulation, the instantaneous phase deviation is proportional to the message. That subtle change makes PM equivalent to applying FM after a derivative of the message. In practice, FM became dominant in broadcast and voice radio, while PM appears more often as a building block inside digital communication.

4. 4. State what is held fixed when changing modulation index

   

   For a single tone, hold its positive frequency fixed while comparing peak deviations:

   $\displaystyle \beta =\frac{\Delta f}{f_{m}}$

   When beta is much smaller than one, the first-order narrowband approximation retains the carrier and first sideband pair; higher pairs are small, not exactly absent.

   Narrowband FM has AM-like spectral locations for one tone, but different relative sideband phases and a different modulation law.

   At the fixed message frequency, increasing beta increases the peak deviation:

   $\displaystyle \Delta f=\beta f_{m}$

   Larger beta generally requires more significant sideband pairs; individual sideband amplitudes need not grow monotonically.

   At fixed message frequency, the Carson bandwidth estimate grows with beta; noise improvement additionally requires adequate receiver conditions:

   $\displaystyle B_{C}=2 f_{m}\left(\beta +1\right)$

   Narration transcript

   Inside FM, one number controls how aggressive the modulation is, the modulation index, written as beta. When beta is small, only a small frequency deviation is used, and the spectrum looks almost like AM with a single sideband pair. This is narrowband FM. When beta grows, the carrier swings far in frequency, and the spectrum opens up into many sideband pairs around the carrier. This is wideband FM. A larger beta improves audio quality and resistance to noise, but it always costs more bandwidth.

5. 5. Use Carson bandwidth without an unconditional noise-quality guarantee

   

   Wider-deviation FM can improve noise-limited reception above the receiver threshold with adequate channel and IF bandwidth; it is not an unconditional audio-quality guarantee.

   For peak frequency deviation and highest message frequency W, define the Carson bandwidth estimate:

   $\displaystyle B_{C}=2\left(\Delta f+W\right)$

   Carson bandwidth is a practical occupied-band estimate, not a strict finite spectral support or an exact regulatory bandwidth.

   For the single-tone comparison, the highest message frequency equals the tone frequency:

   $\displaystyle W=f_{m}$

   Increasing deviation at fixed message bandwidth increases this estimate. Changing message frequency as well can raise beta while reducing bandwidth.

   FM demodulation and the phase-locked loop are the next receiver concepts; bandwidth, threshold and tracking conditions remain part of the design.

   Narration transcript

   To wrap up, FM and PM trade simplicity for noise resistance and quality. The bandwidth is no longer just twice the message bandwidth. A practical rule of thumb is the Carson approximation. The FM bandwidth is roughly twice the sum of the peak frequency deviation and the highest message frequency. So a wider deviation buys cleaner reception, but it always costs more spectrum. In the next lesson, we look at how to actually demodulate FM, and how a receiver locks onto a carrier with a Phase Locked Loop, or PLL.

Source video: Analog Communication #04 | Frequency Modulation (FM) and Phase Modulation (PM) Intuition (3:19)