PCM and analog-to-digital conversion

Pulse Code Modulation (PCM) converts an analog signal to a digital (binary) one. The steps are sampling, quantization and encoding.

Analog  +---------+ +----------+ +---------+  bits
 input->| Sampler |->|Quantizer |->| Encoder |-> 0101 ...
        +---------+ +----------+ +---------+
        (low-pass filter first, fs >= 2 fm)

1. Low-pass filtering

The input is limited to its highest frequency fm (4 kHz for telephone speech) to avoid aliasing.

2. Sampling

The signal is measured at regular intervals Ts = 1/fs. By the sampling theorem, fs ≥ 2fm. Telephone speech uses fs = 8000 samples/s. The result is a PAM signal.

3. Quantization

Each sample amplitude is rounded to the nearest of L discrete levels (L = 2ⁿ). The step size is Δ = (Vmax - Vmin)/L. The rounding error is the quantization noise (maximum ±Δ/2); more levels mean less noise. The SNR is about 6n + 1.8 dB for an n-bit uniform quantizer. Non-uniform quantization (companding, μ-law/A-law) improves weak signals.

4. Encoding

Each quantized level is represented by an n-bit binary code word, for example 8 bits give 256 levels.

Bit rate

Bit rate = fs × n = 8000 × 8 = 64 kbps for one telephone voice channel (DS0).

Receiver

The decoder converts code words back to levels (D/A), and a low-pass filter reconstructs the analog signal.