Communications Fundamentals FAQ / Chapter 3

3.8.7 What Is Companding?

1. What Does Companding Mean?
2. Why Is Companding Needed?
3. Why Are Low-Level Signals More Affected by Quantization Noise?
4. What Is Dynamic Range?
5. How Does Compression Improve Quantization?
6. What Happens After Transmission?
7. How Does Companding Affect Quantization Levels?
8. Why Is Human Hearing Relevant?
9. What Is μ-law Companding?
10. What Is A-law Companding?
11. Why Are There Two Standards?
12. How Much Improvement Does Companding Provide?
13. Does Companding Increase Bit Rate?
14. Is Companding Lossless?
15. Where Is Companding Used?
16. Is Companding Still Relevant Today?
17. How Does Companding Compare with Increasing the Number of Bits?
18. Why Is Companding Important?

Digital communications systems often involve compromises between signal quality, transmission bandwidth, and equipment complexity. One of the most successful examples of such a compromise is companding, a technique that significantly improves the perceived quality of digitized speech without increasing the transmission bit rate.

Companding has been used extensively in digital telephony for decades and remains an important concept in communications engineering. It enabled traditional pulse-code modulation (PCM) telephone systems to deliver acceptable speech quality using only 8 bits per sample and a 64 kbps channel. Without companding, digital telephony would either have required substantially higher bit rates or would have suffered from noticeably poorer performance.

The importance of companding extends beyond telephone systems. The underlying principles continue to influence modern speech coders, audio systems, and digital signal-processing techniques.

What Does Companding Mean?

The word companding is derived from two operations:

• Compressing before transmission.
• Expanding after reception.

The compressor reduces the dynamic range of the signal before quantization.

The expander restores the original dynamic range after decoding.

Together, these processes improve the signal-to-quantization-noise ratio for weak signals while maintaining the same overall bit rate.

The term is therefore a contraction of: COMPressing + exPANDING = COMPANDING

Why Is Companding Needed?

To understand why companding is useful, it is first necessary to understand a limitation of ordinary PCM systems.

In a uniform PCM system, the quantization levels are equally spaced.

For example, an 8-bit quantizer provides:

$2^8=256$quantization levels.

These levels are distributed uniformly across the allowable signal range. While this approach is simple, it does not match the characteristics of speech. Speech signals exhibit a very large dynamic range. A person may:

• Whisper quietly.
• Speak normally.
• Raise their voice.
• Shout loudly.

The amplitude difference between these signals may be tens of decibels. Unfortunately, uniform quantization treats all amplitudes equally.

As a result, weak signals experience proportionally larger quantization errors than strong signals.

Why Are Low-Level Signals More Affected by Quantization Noise?

Consider a quantization step size of Δ.

For a large-amplitude signal, the quantization error may represent only a tiny fraction of the signal amplitude. For example: Signal amplitude = 100 units. Quantization error = 1 unit. Relative error: 1/100 = 1%.

Now consider a weak signal: Signal amplitude = 5 units. Quantization error = 1 unit. Relative error: 1/5 = 20%. Although the absolute error is unchanged, the relative error is much greater. Consequently:

• Strong signals sound relatively clean.
• Weak signals suffer more noticeable distortion.

This characteristic is particularly undesirable in speech communications because speech contains many low-level sounds that contribute significantly to intelligibility and naturalness.

What Is Dynamic Range?

Dynamic range refers to the ratio between the largest and smallest signal amplitudes that a system must accommodate. Speech possesses a substantial dynamic range.

For example:

• A whisper may be barely audible.
• A shout may be hundreds of times larger in amplitude.

A quantizer designed to accommodate the largest signals must therefore spread its quantization levels over a wide range. The resulting level spacing may be unnecessarily coarse for weak signals.

Companding addresses this problem by modifying the amplitude distribution before quantization.

How Does Compression Improve Quantization?

The compressor reduces the dynamic range of the signal before it reaches the quantizer. Large amplitudes are compressed more than small amplitudes.

As a result:

• Weak signals are effectively magnified.
• Strong signals are compressed.

The compressed signal therefore occupies a smaller dynamic range and when this compressed signal is quantized, more quantization levels become available for low-level signals.

This reduces their quantization error and improves perceived quality.

What Happens After Transmission?

The compressed signal is transmitted through the PCM system in the normal manner.

At the receiver:

1. The PCM samples are decoded.
2. The quantized amplitudes are reconstructed.
3. An expander restores the original amplitude relationships.

The expander performs the inverse operation of the compressor. Weak signals are reduced to their original levels. Strong signals are restored to their original amplitudes.

To the listener, the signal appears normal. However, because the quantization process occurred in the compressed domain, the quantization noise affecting weak signals is greatly reduced.

How Does Companding Affect Quantization Levels?

Without companding, quantization levels are uniformly spaced.

For example:

With companding, the effective spacing becomes non-uniform.

Near zero amplitude: Quantization levels are closely spaced. At high amplitudes: Quantization levels are more widely spaced. This arrangement better matches the characteristics of speech and human hearing.

The result is often called non-uniform quantization.

Why Is Human Hearing Relevant?

Human hearing responds approximately logarithmically rather than linearly. A doubling of sound pressure does not necessarily sound twice as loud.

Instead, people tend to perceive loudness changes in relative rather than absolute terms.

Companding exploits this characteristic. By allocating more quantization precision to weak signals and less to strong signals, companding produces a result that is better aligned with human perception.

Consequently, substantial improvements in perceived quality can be achieved without increasing transmission bandwidth.

What Is μ-law Companding?

One of the two principal companding standards is μ-law (pronounced "mu-law").

μ-law is used primarily in North America and Japan.

The compression characteristic is approximately logarithmic and is described mathematically by:

$y=\frac{\mathrm{l}\mathrm{n}(1+\mu x)}{\mathrm{l}\mathrm{n}(1+\mu )}$where:

• $x$ is the normalized input amplitude.
• $y$ is the compressed output.
• $\mu$ is the compression parameter.

For telephone systems:

$\mu =255$is commonly used.

The resulting characteristic provides substantial compression for large signals while preserving resolution for weak signals.

What Is A-law Companding?

The second major standard is A-law companding.

A-law is used throughout:

• Europe.
• Australia.
• Much of Asia.
• Many other regions.

The mathematical relationship differs slightly from μ-law but achieves a similar objective.

For telephone systems:

$A=87.6$is typically used.

Although the two systems are not identical, their performance is broadly comparable.

The existence of two standards reflects historical developments in different parts of the world.

Why Are There Two Standards?

The development of digital telephony occurred independently in different regions.

North American systems adopted μ-law. European systems adopted A-law.

Each standard became deeply embedded in regional telecommunications infrastructure.

Although international gateways must occasionally convert between the two formats, both continue to be used successfully.

How Much Improvement Does Companding Provide?

The exact improvement depends on signal amplitude.

For weak speech signals, companding can provide a substantial increase in effective SQNR.

An 8-bit companded PCM system often performs similarly to a uniform PCM system with significantly more effective resolution at low amplitudes.

In practical terms, companding makes:

• Quiet speech clearer.
• Soft consonants easier to hear.
• Speech quality more consistent.

This improvement was a major factor in the success of digital telephony.

Does Companding Increase Bit Rate?

No.

This is one of its most attractive features. A conventional PCM channel uses 8 bits per sample before companding, the same channel still uses 8 bits per sample after companding. The transmission rate stays the same at 64 kbps. No additional bandwidth is required.

The improvement comes entirely from using the available quantization levels more effectively.

Is Companding Lossless?

No.

Companding does not eliminate quantization noise. Quantization errors still occur. However, the errors are redistributed so that they become less objectionable. The objective is not perfect reconstruction but improved perceived quality. This distinction is important.

Companding is an optimization technique rather than a lossless coding method.

Where Is Companding Used?

Companding has been used extensively in:

• Digital telephony. Traditional PCM telephone systems rely heavily on μ-law or A-law companding.
• PBX systems. Private telephone exchanges frequently employ companded PCM channels.
• Digital voice systems. Many voice-processing systems incorporate companding principles.
• Audio processing. Some audio systems employ dynamic-range compression techniques based on similar concepts.
• Communications equipment. Numerous voice communications devices incorporate companding to improve performance.

Is Companding Still Relevant Today?

Yes.

Although modern speech coders often use more sophisticated compression techniques, companding remains important because:

• Legacy telephone systems still exist.
• Many digital voice interfaces employ PCM.
• The underlying principles continue to influence modern signal processing.

Understanding companding also provides valuable insight into:

• Human perception.
• Quantization.
• Dynamic range.
• Speech processing.

For these reasons, companding remains a standard topic in communications engineering courses.

How Does Companding Compare with Increasing the Number of Bits?

One way to improve PCM quality is to increase the number of bits per sample.

For example, 8 bits produces 256 samples; 12 bits 4,096 samples, 16 bits 65,536 samples, and so on.

Increasing the number of bits improves performance across all signal amplitudes. However, it also increases:

• Data rate.
• Storage requirements.
• Transmission bandwidth.

Companding provides a more economical solution. By reallocating quantization precision, it achieves much of the perceived benefit without increasing the bit rate.

This made it particularly attractive when transmission resources were expensive.

Why Is Companding Important?

Companding represents one of the most elegant solutions in communications engineering.

Rather than increasing bandwidth or bit rate, it improves performance by exploiting:

• The characteristics of speech.
• The properties of quantization.
• The behavior of human hearing.

The result is a substantial improvement in perceived quality at essentially no cost in transmission capacity.

This clever engineering compromise helped make digital telephony practical and remains one of the classic examples of efficient communications-system design.

Summary

Companding is a technique that improves the quality of PCM speech systems by compressing signal amplitudes before quantization and expanding them after decoding. By allocating more effective quantization levels to weak signals and fewer to strong signals, companding reduces the impact of quantization noise where it is most noticeable.

The two principal standards are μ-law and A-law, which remain widely used throughout the world. Companding played a crucial role in the success of 64 kbps digital telephony and continues to provide valuable insight into source coding, quantization, and digital speech communications.

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