Companding

The µ-law algorithm is the dominant companding standard in telecommunications, defined by a logarithmic compression curve. It applies heavy compression to high-amplitude samples while preserving low-amplitude detail.

• Compression function: y = sgn(x) * ln(1 + µ|x|) / ln(1 + µ) where µ typically equals 255
• Effect: Maps a wide dynamic range into a narrower one, directly reducing PAPR
• Origin: Developed for pulse code modulation (PCM) in digital telephony to improve signal-to-noise ratio
• Trade-off: Introduces intentional in-band distortion that increases Error Vector Magnitude (EVM)

Companding is a two-stage process: compression at the transmitter and expansion at the receiver. The compressor reduces PAPR before the power amplifier, while the expander restores the original signal dynamic range after the receiver's analog-to-digital conversion.

• Transmit side: Nonlinear compression reduces peak amplitudes, allowing the PA to operate closer to saturation with higher efficiency
• Receive side: Inverse expansion function reconstructs the original signal envelope, recovering the compressed samples
• Symmetry requirement: The compression and expansion curves must be exact inverses to avoid net distortion
• Practical limitation: In wireless systems, channel noise and PA nonlinearity can corrupt the expanded signal, making perfect reconstruction difficult

Companding achieves PAPR reduction at the direct expense of Error Vector Magnitude (EVM) degradation. The compression function deliberately distorts the signal constellation, and this distortion cannot be fully recovered after expansion in the presence of channel impairments.

• PAPR gain: Typically 2-4 dB reduction at the 10⁻⁴ CCDF point, depending on companding aggressiveness
• EVM penalty: Increases proportionally with the compression ratio; aggressive µ-law values cause constellation point spreading
• Spectral efficiency: Unlike clipping, companding produces smooth nonlinearity with less severe out-of-band spectral regrowth
• Design parameter: The compression parameter (µ or A) directly controls the PAPR-EVM trade-off point

Two standardized logarithmic companding curves exist: µ-law (North America, Japan) and A-law (Europe, ITU-T G.711). Both achieve similar PAPR reduction but differ in their approximation of the logarithmic curve.

• µ-law: Continuous logarithmic function, smoother compression at low amplitudes, µ=255 standard
• A-law: Piecewise linear approximation with 13 segments, A=87.6 standard, simpler hardware implementation
• Performance: Nearly identical PAPR reduction and EVM characteristics for wireless applications
• Selection: Often dictated by regional standards compliance or hardware implementation complexity in the digital baseband processor

Companding and clipping-based Crest Factor Reduction (CFR) are competing PAPR reduction techniques with fundamentally different distortion characteristics.

• Clipping: Hard amplitude threshold creates sharp discontinuities, causing severe spectral regrowth that requires iterative filtering
• Companding: Smooth nonlinear transformation produces less out-of-band emission but more uniform in-band distortion
• Peak regrowth: Clipping suffers from peak regrowth after filtering; companding does not require post-filtering
• Combined approaches: Modern systems often cascade companding with light clipping for aggressive PAPR targets while managing both EVM and ACLR

In Orthogonal Frequency Division Multiplexing (OFDM) systems, the PAPR varies per symbol due to subcarrier phase alignment. Adaptive companding adjusts the compression parameter dynamically based on instantaneous PAPR statistics.

• Symbol-by-symbol adaptation: Compression level varies per OFDM symbol to target a consistent output PAPR
• Side information: The receiver requires knowledge of the compression parameter used, transmitted as overhead or estimated blindly
• CCDF optimization: Adaptive schemes achieve better CCDF shaping than fixed companding, targeting specific probability points
• Implementation: Requires real-time PAPR estimation and a lookup table of compression curves indexed by the estimated peak statistics