ECE 426 Digital Communication Departmetof Electrical & Communication Egineering
Extends the effective dynamic range of quantization, allowing both weak and strong signals to be encoded with adequate precision.
Maintains relatively constant signal-to-quantization-noise ratio across a wide range of input amplitudes.
Implements non-uniform quantization through uniform quantization of compressed signal.
Companding (COMpressing and exPANDING) is a technique used in digital communications to improve the dynamic range of signals during quantization. The term combines "compression" at the transmitter and "expansion" at the receiver.
In PCM (Pulse Code Modulation) systems, companding allows a larger dynamic range to be achieved with a given number of bits than would be possible with uniform quantization. This is particularly important for voice signals, which have a large dynamic range (40-50 dB).
The companding process involves two operations:
y = f(x)
where x is input signal, y is compressed signal
x̂ = f-1(ŷ)
where ŷ is quantized compressed signal, x̂ is recovered signal
x(t) with wide dynamic range
Non-linear compression y = f(x)
Linear quantization of y
Digital bitstream
Recover quantized ŷ
x̂ = f-1(ŷ)
Reconstructed x̂(t)
Standardized in ITU-T G.711, primarily used in North American and Japanese telephone systems.
Compression Characteristic:
y = sgn(x) · ln(1 + μ|x|) / ln(1 + μ)
where |x| ≤ 1, μ = 255 (standard value)
Standardized in ITU-T G.711, used in European and most international telephone systems.
Compression Characteristic:
For |x| ≤ 1/A: y = A·x / (1 + ln(A))
For 1/A < |x| ≤ 1: y = sgn(x)·(1+ln(A|x|))/(1+ln(A))
where A = 87.6 (standard value)
| Characteristic | μ-law | A-law |
|---|---|---|
| Origin | Bell Labs (1970) | European standards |
| Formula Continuity | Continuous everywhere | Piecewise linear approximation |
| Small Signal Slope | μ/ln(1+μ) ≈ 16 | A/(1+ln(A)) ≈ 16 |
| Idle Channel Noise | Better performance | Slightly higher |
| Implementation | Logarithmic | 13-segment piecewise linear |
ITU-T G.711 standard for PCM voice transmission at 64 kbps. Used in PSTN, VoIP gateways, and PBX systems.
Used in 2G (GSM) and 3G voice codecs to maximize voice quality over limited bandwidth channels.
Professional audio equipment uses companding to increase dynamic range in digital recording systems.
Maximizes link budget efficiency by maintaining signal quality across varying channel conditions.
Secure voice systems use companding to ensure intelligibility under adverse signal conditions.
Data acquisition systems measuring signals with wide dynamic ranges (sensors, seismic, etc.).