Digital Signal Conversion: PCM Sampling, Quantization, and A-Law Encoding

Pulse Code Modulation (PCM) Fundamentals

1. Sampling Process

Samples are taken for the analog signal, measuring its amplitude (voltage). The process involves the following steps:

1. The telephone signal has a voice frequency range of 300–3400 Hz. It passes through a low-pass filter, considering the signal from 0 to 4 kHz.
2. A sample is taken instantaneously, and the amplitude value is retained for measurement.
3. According to Nyquist's theorem, we need a sampling frequency ($f_s$) twice the maximum signal frequency ($f_s \ge 2f_{max}$). For a 4 kHz signal, this results in a sample every 125 $\mu$s (microseconds), corresponding to $f_s = 8,000$ samples/second.
4. This process yields a train of Pulse Amplitude Modulated (PAM) pulses.

2. Quantization and Companding

The sampled values are assigned digital values; the converter resolution is 8 bits. This results in $2^8 = 256$ discrete levels, called quantization intervals. These are divided into 128 positive and 128 negative levels. Decision values range from zero to the maximum signal amplitude.

Quantization Error

This occurs when an exact digital value (e.g., 2V) is assigned to a range of input analog values (e.g., 2V to 4V). The difference between the actual analog value and the assigned digital level is the error. This difference produces a distortion error signal, known as quantization noise.

Nonuniform Quantization (Companding)

Increasing the number of intervals reduces signal distortion. The human voice is an analog source of information where low-volume signals dominate, and high levels are less frequent. Therefore, to improve the signal-to-noise ratio for quiet sounds, the number of intervals is increased for small signals (narrower steps) and decreased for large signals (wider steps).

A-Law Compression Standard

The A-law characteristic curve is approximated by 16 linear segments (8 positive, 8 negative). Four segments near the origin are aligned and treated as one, effectively reducing the number of unique slopes from 16 to 13 segments. Each of the 16 segments is divided into 16 equal quantization intervals, but the step size is dissimilar from one segment to another, except for the four central segments. The total operating range uses 256 intervals (128 positive, 128 negative), grouped into 16 segments (8 positive, 8 negative).

3. PCM Encoding (Consolidation)

Quantization intervals are represented in an 8-bit binary sequence, following the A-law encoding format. The resulting 8-bit code structure is: P A A A B B B B (Polarity, Segment, Interval).

• Bit 1 (P): Indicates the polarity (1 for positive, 0 for negative).
• Bits 2, 3, 4 (A): Encode the segment number (3 bits allow for 8 segments).
• Bits 5, 6, 7, 8 (B): Encode the specific interval within that segment (4 bits allow for 16 intervals, ranging from 0000 to 1111).

The encoding process involves numbering the 128 positive and 128 negative intervals and prepending the sign bit (1 or 0).

In the receiver, the reverse process (decoding and reconstruction) is performed by a Digital-to-Analog Converter (DAC). To ensure adequate clock recovery and avoid long sequences of zeros, the resulting PCM signal is typically line-coded, often using the High-Density Bipolar 3 (HDB3) code.

4. PCM Channel Bit Rate

The resulting PCM signal transmission rate is calculated based on the sampling rate and the number of bits per sample:

$$8,000 \text{ samples/second} \times 8 \text{ bits/sample} = 64,000 \text{ bits/second (64 kbps)}$$