Encoding (digital)

Encoding is the final step of analog-to-digital conversion: representing each quantized sample as a string of bits. With $L$ allowed quantization levels you need $\lceil {\log }_{2}L\rceil$ bits per sample.

Take 256-level audio. Each sample is one of integers $0,1,\dots ,255$, requiring 8 bits. The encoder writes those 8 bits out in some agreed order — most-significant first or least-significant first — and the result is a serial bit stream representing the original signal.

Two reasonable conventions

There is no single “correct” encoding; the only requirement is that the transmitter and receiver agree. Common choices are unsigned binary (level $k\to$ binary representation of $k$), two’s complement (allowing negative numbers), and Gray code (consecutive levels differ in only one bit, reducing the cost of single-bit errors). See Character Representation, 2’s Complement Arithmetic.

Back to a voltage waveform

If we represent the encoded bit stream as a voltage waveform (say 0 V for a 0 and 5 V for a 1), we are back in continuous time but with only two values — the bottom-left box of the four-kinds-of-signal table from Signal. The string of pulses can be sent through any binary channel: a wire, an optical fiber, a radio link.

Where this fits

Encoding completes the conversion chain:

$\text{continuous }x(t)\stackrel{\text{[[Sampling|sample]]}}{\to }x[n]\stackrel{\text{[[Quantization|quantize]]}}{\to }q[n]\stackrel{\text{encode}}{\to }\text{bits}$

Once we have bits, the communications chapter of the story takes over: how to ship them through a noisy channel and recover them at the other end.