S5-1.3 Pulse Analog & Pulse Digital Modulation

What this is and why it exists

This topic is the crossing from analog to digital. Everything before it sends a continuous wave. Everything after it sends numbers.

Pulse code modulation is the crossing point. Sample the signal, round each sample to a level, and write the level as bits. Almost all modern communication sits downstream of that one idea.

The vocabulary

• PAM — pulse amplitude modulation. The height of each pulse follows the message.
• PWM — pulse width modulation. The length of each pulse follows the message.
• PPM — pulse position modulation. The timing of each pulse follows the message.
• Sampling — reading the signal at regular instants.
• Quantization — rounding each sample to the nearest of a fixed set of levels.
• Quantization noise — the error introduced by that rounding.
• Companding — compressing before quantizing and expanding after, so the levels are not evenly spaced.
• DPCM — differential pulse code modulation. The difference from a prediction is encoded instead of the sample.
• Delta modulation — the extreme case of DPCM, with one bit per sample.
• Slope overload — the failure where the staircase output cannot climb as fast as the signal.
• Granular noise — the small hunting about a slowly changing signal in delta modulation.

The mental model

The three analog pulse schemes are one idea seen three ways. A train of pulses has a height, a width and a position. Any of the three can be made to follow the message. Be able to sketch all three. They form the bridge between the continuous schemes and the digital ones.

Pulse code modulation is the important one, and it has exactly three steps. Sampling turns a continuous-time signal into a list of values. Quantizing turns each value into one of a fixed set of levels. Encoding writes each level as a group of bits.

Only the second step loses anything. Rounding to the nearest level leaves an error, and that error behaves like noise added to the signal. Adding one more bit doubles the number of levels and halves the step size. Halving the error amplitude improves the signal-to-noise ratio by about six decibels. So each extra bit is worth roughly six decibels, which is the fastest way to size a converter in your head.

Even steps are wasteful for speech. Speech spends most of its time at low amplitudes, so equally spaced levels are crowded where the signal never goes. Companding fixes this by compressing the signal before quantizing and expanding it afterwards. The result is fine steps for quiet passages and coarse steps for loud ones. It is why telephone audio sounds as good as it does at the bit rate it uses.

Prediction is the next saving. Consecutive samples of a real signal are usually similar. DPCM encodes the difference between a sample and a prediction of it, and that difference needs fewer bits. This is the first appearance of prediction as a compression idea, and information theory formalises it in the next topic.

Delta modulation is that idea at its limit. One bit per sample says only whether the signal went up or down, and the output is a staircase chasing the input. Two failures follow directly. If the signal rises faster than the staircase can climb, the output falls behind, and that is slope overload. If the signal barely moves, the staircase hunts up and down about it, and that is granular noise. Adaptive delta modulation varies the step size to reduce both. Comparing linear against adaptive on the same input shows the problem and the fix in one sitting.

Keep the two noise sources apart. Quantization noise is something you designed for and can compute. Slope overload means the design does not fit the signal you gave it.

What you should now be able to explain or do

Sketch PAM, PWM and PPM and say what varies in each. List the three steps of PCM and name the one that loses information. Estimate how many bits a required signal-to-noise ratio needs, using about six decibels per bit. Explain why speech needs companding. Describe slope overload and granular noise, and say how adaptive delta modulation reduces each.

Check yourself