Companding

Companding is the technique of compressing the dynamic range of a signal before quantization and expanding it back after decoding, in order to achieve nearly uniform signal-to-quantization-noise ratio across the full amplitude range of the input. The word companding is formed from the combination of compressing and expanding. Companding is the practical solution to the fundamental weakness of uniform quantization, which provides poor SQNR for small-amplitude signals.

Why Companding is Needed

In telephone and speech communication systems, signal amplitudes are not uniformly distributed. Most of the time, speech has low amplitude (pauses, weak phonemes), with occasional high-amplitude bursts (vowels, stressed syllables). A uniform quantizer assigns the same step size delta to all levels, so quantization noise power is constant at delta squared divided by 12. For high-amplitude samples, this fixed noise is a small fraction of the signal, giving good SNR. For low-amplitude samples, the signal power drops but noise power stays the same, causing the SNR to fall sharply.

Companding addresses this by compressing the signal non-linearly before quantization. The compressor amplifies low-amplitude portions of the signal more than high-amplitude portions, so after compression the amplitude distribution is more uniform. When this compressed signal is uniformly quantized, the effective step size relative to the original signal is finer at low amplitudes and coarser at high amplitudes. The net effect is that SQNR is nearly constant across the dynamic range of the original signal.

mu-law Companding

The mu-law companding characteristic, standardized for North American and Japanese telephone networks, defines the compressor output C(x) for a normalized input x in the range from 0 to 1. The mu-law compressor equation is C(x) equals the natural logarithm of 1 plus mu times the absolute value of x, divided by the natural logarithm of 1 plus mu, all multiplied by the sign of x. The parameter mu equals 255 is the standard value used in practice.

For small values of x (meaning x much less than 1 divided by mu), the mu-law characteristic is approximately linear with a slope of mu divided by the natural log of 1 plus mu. This means low-amplitude signals are amplified significantly. For large values of x approaching 1, the characteristic compresses heavily, reducing the effective gain. The logarithmic shape ensures that the ratio of signal to noise remains approximately constant across the entire usable amplitude range, which is typically about 40 dB for mu equals 255.

A-law Companding

The A-law companding characteristic is the standard for European telephone networks and the international ITU-T G.711 standard. The A-law compressor is defined in two segments. For small inputs where the absolute value of x is less than or equal to 1 divided by A, the output is A times x divided by 1 plus the natural log of A, all times the sign of x. This segment is linear. For larger inputs where 1 divided by A is less than the absolute value of x and x is less than or equal to 1, the output is 1 plus the natural log of A times the absolute value of x, all divided by 1 plus the natural log of A, times the sign of x. This segment is logarithmic.

The standard value A equals 87.6 is used in practice. The linear segment at low amplitudes prevents the infinite slope that a pure logarithm would have at x equals zero, making A-law slightly simpler to implement in hardware compared to mu-law. Both A-law and mu-law provide roughly the same subjective voice quality improvement over uniform quantization, and the choice between them is primarily a regional standardization matter.

Comparison of mu-law and A-law

Both standards operate at 8 bits per sample and 8000 samples per second, producing a 64 kbps PCM bitstream for a single voice channel. Both provide an effective dynamic range of approximately 40 dB with near-constant SQNR, compared to only about 12 dB dynamic range with acceptable SNR using uniform quantization with 8 bits. The key differences are: mu-law uses a continuous logarithmic curve throughout while A-law has a piecewise linear-logarithmic curve; mu-law performs slightly better at very low amplitudes; A-law is the international standard while mu-law is the North American and Japanese standard.

Given:
Normalized input signal x = 0.1 (small amplitude)
mu-law parameter mu = 255

Why this formula applies:
mu-law compressor maps small amplitudes
to larger output values for finer quantization.

Formula:
C(x) = [ln(1 + mu * |x|) / ln(1 + mu)] * sign(x)

Substitution:
C(0.1) = ln(1 + 255 * 0.1) / ln(1 + 255)
       = ln(1 + 25.5) / ln(256)
       = ln(26.5) / ln(256)

Calculation:
ln(26.5) = 3.277
ln(256)  = 5.545
C(0.1) = 3.277 / 5.545

Final Answer:
C(0.1) = 0.591
Input 0.1 is amplified to 0.591 before
uniform quantization, giving ~6x finer
effective resolution at this low amplitude.

Exam Tip: For GATE, remember that mu-law is used in North America and Japan (mu = 255), while A-law is the international standard used in Europe and India (A = 87.6), as per ITU-T G.711. The A-law has a piecewise linear-log characteristic, while mu-law is purely logarithmic. Both use 8 bits at 8 kHz for 64 kbps voice. Companding does NOT reduce the bit rate; it improves SQNR without adding more bits.

• Companding compresses signal before quantization and expands after decoding, achieving uniform SQNR across amplitude range.
• mu-law compressor: C(x) = [ln(1 + mu times |x|) / ln(1 + mu)] times sign(x). Standard mu = 255 for North America and Japan.
• A-law compressor: piecewise linear (for |x| less than 1/A) and logarithmic (for larger |x|). Standard A = 87.6 for Europe and India.
• Both standards: 8 bits per sample, 8 kHz sampling rate, 64 kbps per voice channel.
• Companding provides approximately 40 dB of dynamic range with constant SQNR using only 8 bits.

Quick Revision

• Companding = compress before quantization + expand after decoding.
• mu-law formula: C(x) = ln(1 + mu|x|) / ln(1 + mu), mu = 255, used in North America and Japan.
• A-law formula: piecewise linear-log, A = 87.6, used in Europe, India, ITU-T G.711 international standard.
• Both operate at 8 bits, 8 kHz, 64 kbps. Companding does not change bit rate.
• Companding achieves flat SQNR over approximately 40 dB of dynamic range with 8-bit quantization.
• GATE trap: Companding improves SQNR for small signals but does not reduce quantization noise overall; it redistributes noise so SNR is equalized. Do not confuse A-law with mu-law regions in exam questions.
• A-law has a linear portion near x = 0 to avoid infinite slope; mu-law is purely logarithmic with no linear segment.