Companding

The compression curve defines the nonlinear mapping between input and output amplitudes. The two dominant international standards are µ-law, used in North America and Japan, and A-law, used in Europe. µ-law provides a slightly wider dynamic range and is defined by the continuous logarithmic formula F(x) = sgn(x) * ln(1 + µ|x|) / ln(1 + µ) with µ typically set to 255. A-law is a piecewise approximation optimized for PCM voice telephony, offering superior small-signal performance with a reduced computational load. Both curves are designed to make the quantization step size proportional to the signal amplitude, effectively allocating more bits to quiet passages.

Companding directly reduces the peak-to-average power ratio (PAPR) by compressing the dynamic range of the signal before it enters the power amplifier. By attenuating high-amplitude peaks and boosting low-amplitude valleys, the signal's envelope becomes more uniform. This allows the power amplifier to operate closer to its 1dB compression point (P1dB) without clipping, improving power efficiency. The reduction in PAPR is a direct trade-off: the signal's crest factor is lowered, but the process introduces in-band distortion that degrades the error vector magnitude (EVM).

The nonlinear compression function generates intermodulation products that cause spectral regrowth into adjacent channels. This is the primary penalty of companding. The sharp transitions in the compression curve, particularly near the saturation point, create high-frequency spectral components that degrade the adjacent channel leakage ratio (ACLR). To mitigate this, companding is often paired with pulse shaping filters or peak windowing techniques. The design challenge is to balance the PAPR reduction benefit against the resulting spectral mask violations, ensuring compliance with regulatory spectral mask requirements.

Companding fundamentally trades quantization noise for companding distortion. In a uniform quantizer, low-amplitude signals suffer from poor signal-to-quantization noise ratio. Companding improves this by effectively expanding the quantization levels for small signals. However, the nonlinear mapping introduces harmonic and intermodulation distortion that is signal-dependent. This distortion is not random like quantization noise; it is deterministic and correlated with the signal envelope. For communication systems, this means the distortion can be partially compensated for at the receiver if the companding parameters are known, a technique known as decompanding.

Orthogonal Frequency Division Multiplexing (OFDM) signals exhibit inherently high PAPR due to the summation of multiple independent subcarriers. Companding is a popular, low-complexity PAPR reduction technique for OFDM. The process is applied to the complex baseband time-domain samples after the IFFT operation. Key design parameters include the companding threshold and the compression ratio. A high compression ratio aggressively reduces PAPR but causes severe spectral regrowth. Adaptive companding schemes dynamically adjust these parameters based on the instantaneous signal statistics to maintain an acceptable balance between power efficiency and ACLR.

Companding and digital predistortion (DPD) are complementary linearization strategies. Companding addresses the signal's statistical properties before amplification, while DPD corrects the amplifier's nonlinear transfer characteristic. In a modern transmitter chain, companding is applied first to reduce PAPR, allowing the power amplifier to operate at a higher average power. The residual nonlinearity, including the distortion introduced by the compander itself, is then corrected by the DPD engine. This cascade requires the DPD model to be trained on the companded signal to effectively linearize the entire chain, including the AM-AM and AM-PM distortion introduced by both the compander and the PA.

The fundamental problem that companding addresses. PAPR is the ratio of a signal's instantaneous peak power to its average power, expressed in dB. High-PAPR signals like OFDM force power amplifiers to operate with significant power back-off to avoid nonlinear distortion. Companding directly reduces PAPR by compressing high-amplitude peaks and expanding low-amplitude valleys, enabling more efficient amplifier operation at the cost of introduced distortion.

A broader class of signal conditioning techniques that includes companding. CFR reduces PAPR before the power amplifier to prevent clipping distortion and spectral regrowth. While companding uses a smooth, non-uniform quantization curve, other CFR methods include:

• Hard clipping: Abruptly truncates peaks, causing severe spectral regrowth
• Peak windowing: Applies smooth time-domain windows to peaks
• Tone reservation: Uses reserved OFDM subcarriers for peak cancellation Companding is often cascaded with these techniques in modern transmitter chains.

The primary impairment companding seeks to avoid, yet paradoxically introduces in a controlled manner. When a power amplifier is driven beyond its 1dB compression point (P1dB), waveform peaks are abruptly truncated, generating severe intermodulation distortion (IMD) and spectral regrowth. Companding applies a smooth, logarithmic compression curve that softens this clipping effect, redistributing distortion energy in a more spectrally benign way. The trade-off is increased in-band Error Vector Magnitude (EVM).

A signal processing technique that works synergistically with companding to manage spectral regrowth. After companding introduces quantization distortion, noise shaping intentionally redistributes that error energy away from critical in-band frequencies to less sensitive out-of-band regions. This improves Adjacent Channel Leakage Ratio (ACLR) performance. Common implementations use delta-sigma modulation with feedback loops that high-pass filter the quantization noise spectrum.

The nonlinear amplifier characteristics that companding pre-compensates for. AM-AM distortion is the deviation of output amplitude from a linear relationship with input amplitude, causing gain compression. AM-PM distortion is the variation of phase shift with instantaneous signal envelope. Companding's compression curve can be designed to pre-distort the signal envelope, partially linearizing the combined compander-amplifier cascade. However, the expansion process at the receiver introduces additional complexity in managing these distortions.

A power amplifier phenomenon that complicates companding design. Memory effects cause the amplifier's current output to depend on past input states due to thermal dynamics, electrical biasing, and charge trapping in semiconductor materials. This creates frequency-dependent nonlinear behavior that a static companding curve cannot fully compensate for. Advanced systems combine companding with memory polynomial models or neural network linearization to address these time-dispersive effects and control spectral regrowth across wide bandwidths.