S5-1.2 Angle Modulation

What this is and why it exists

A sine wave offers two things you can change. One is its amplitude, which the previous topic used. The other is its angle. Angle modulation varies the angle and leaves the amplitude alone.

That choice buys noise immunity and pays for it in bandwidth. Understanding the size of that bargain is the point of this topic.

The vocabulary

• Phase modulation — the carrier phase moves in step with the message.
• Frequency modulation — the carrier frequency moves in step with the message.
• Frequency deviation — the largest shift the carrier frequency makes away from its resting value.
• Modulation index — for a single tone, the frequency deviation divided by the message frequency.
• Bessel function of the first kind — the function whose values give the sideband amplitudes of single-tone FM.
• NBFM — narrowband FM, meaning a modulation index well below one.
• WBFM — wideband FM, meaning a large modulation index.
• Carson's rule — an estimate of transmission bandwidth. It is about twice the sum of the peak deviation and the highest message frequency.
• Armstrong method — building a wideband signal from a narrowband one by multiplying the frequency.
• Phase locked loop — a loop that tracks an input's phase and can be used to recover FM.
• Pre-emphasis and de-emphasis — lifting the high frequencies before transmission and cutting them by the same amount after.

The mental model

Start by joining the two schemes. Frequency is the rate at which phase advances. So frequency modulating a carrier with a message is the same as phase modulating it with the running total of that message. Work in either view and convert when it is convenient. That saves learning the second scheme from scratch.

Take a single tone next. The modulation index is the deviation divided by the tone frequency. Everything else in the topic follows from that one number.

The spectrum is where the mathematics gets heavier. Sidebands appear at every multiple of the tone frequency, above and below the carrier. Their amplitudes are Bessel function values of the first kind. The modulation index is the argument and the sideband number is the order. There are infinitely many of them. The important lesson is that you decide where to stop, because the signal has no natural edge. Carson's rule is the usual place to stop, and it keeps almost all of the power.

Narrowband and wideband are not two mechanisms. They are the same mechanism at different modulation indices. At a small index only the first pair of sidebands matters, and the spectrum resembles AM. Raise the index and more sideband pairs grow. Students often file the two as separate schemes and then cannot say where the boundary lies.

Power is the part that surprises people. An angle-modulated wave has a constant envelope. Its transmitted power therefore does not change with modulation at all. Raising the index does not raise the power. It redistributes the same power over more sidebands, and so over more bandwidth. That is the bargain in one sentence. Noise immunity is bought with spectrum, not with power.

Generation and detection are practical problems. Producing a large deviation directly is hard. The Armstrong method makes a narrowband signal first, then multiplies the frequency, which multiplies the deviation with it. Reaching a specification indirectly like this is a habit worth noticing. On the receiving side, a phase locked loop follows the incoming phase, and its control voltage is the recovered message. That same loop returns in the digital half of this course.

Pre-emphasis and de-emphasis close the topic. Noise at the output of an FM detector grows with frequency, so the top of the audio band suffers most. The fix is to lift the high frequencies before transmission and cut them by the same amount afterwards. The message comes out unchanged and the noise comes out reduced. Plotting the two responses together shows them mirroring each other, which is why the pair leaves the signal alone.

What you should now be able to explain or do

Convert between the phase and frequency views of one modulated wave. Compute a modulation index from a deviation and a tone frequency. Say why the FM spectrum has infinitely many sidebands and where Carson's rule chooses to stop. Explain why transmitted power stays constant while bandwidth grows. Describe the Armstrong method and the phase locked loop detector. Say why pre-emphasis helps and why the pair leaves the message unchanged.

Check yourself