--- title: "Companding and its types" subject: "Communication Systems" chapter: 4 key: "semester_6/communication_systems:companding-types" source: https://www.bctengineer.com/q/semester_6/communication_systems/companding-types site: BCT Engineering --- # Companding and its types Communication Systems · Chapter 4: Pulse Modulation - Asked in: 2076 Asoj (CS II), 2+5 marks; 2069 Chaitra (CS II), 3+4 marks (asked 2 times) - Video: [Companding (Basics, Block Diagram, Working, Characteristics & Importance) Explained](https://www.youtube.com/watch?v=q39NTeIIIZ4) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/companding-types - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4 ## Question What is companding and why it is necessary? Explain any two types of companding techniques. ## Short answer Companding compresses signal before uniform quantization and expands at receiver to achieve non-uniform quantization, necessary for constant SQNR over wide dynamic range with fewer bits. μ-law (North America, μ=255) and A-law (Europe, A=87.6) are standard companding laws per ITU-T G.711. ## Answer **Companding** (compressing + expanding) is a way to obtain non-uniform quantization: the signal is **compressed** by a non-linear amplifier before a uniform quantizer at the transmitter, and **expanded** by the inverse characteristic at the receiver. ```text m(t)->[Compressor]->[Uniform Q]->channel->[Expander]->m(t) y = c(x) x = c^-1(y) ``` #### Why it is necessary - With uniform quantization, noise $\Delta^2/12$ is fixed, so weak signals have poor SQNR. - Speech has a large dynamic range and small amplitudes are most frequent. - The compressor boosts small amplitudes and reduces large ones, so effectively small inputs see fine steps and large inputs coarse steps. The SQNR stays nearly constant (within a few dB) over about 40 dB of input range with only 8 bits. #### μ-law companding (North America, Japan) $$ y = \frac{\ln(1 + \mu|x|)}{\ln(1+\mu)}\,\text{sgn}(x), \qquad 0 \le |x| \le 1 $$ - $\mu = 0$ gives uniform quantization; the standard value is $\mu = 255$ (8-bit PCM). - For small $x$ it is nearly linear ($y \approx \mu x/\ln(1+\mu)$); for large $x$ it is logarithmic. - Example with $\mu = 255$: $x = 0.01 \to y = 0.228$; $x = 0.1 \to y = 0.591$; $x = 0.5 \to y = 0.876$. #### A-law companding (Europe, India, Nepal) $$ y = \begin{cases} \dfrac{A|x|}{1+\ln A}\,\text{sgn}(x), & 0 \le |x| \le \dfrac{1}{A} \\[2mm] \dfrac{1+\ln(A|x|)}{1+\ln A}\,\text{sgn}(x), & \dfrac{1}{A} \le |x| \le 1 \end{cases} $$ - Standard value $A = 87.6$. It is linear for small signals and logarithmic for large ones. Both A-law and μ-law are specified in ITU-T G.711. - In practice both laws are realised by a **13-segment (A-law) or 15-segment (μ-law) piecewise-linear** approximation. ```text y (out) 1 | . . . . <- compressed | . . | . .... <- uniform (y = x) | . .... |. .... |. .. +-------------------- x (in) 0 1 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is companding? A. Sampling at a variable rate B. Quantizing with a fixed step and no non-linearity C. Compression before a uniform quantizer and expansion at the receiver D. Expansion at the transmitter and compression at the receiver Answer: C. The compressor y=c(x) precedes the uniform quantizer; the inverse expander follows the channel. (Key point: Companding) 2. What is the benefit of companding for speech? A. Nearly constant SQNR over a wide input range with only 8 bits B. Higher SQNR for loud signals only C. Perfect zero quantization noise D. Reduced sampling rate to 2 kHz Answer: A. Small inputs see fine steps and large ones coarse steps, keeping SQNR nearly flat. (Key point: Why necessary) 3. In mu-law companding, what does mu = 0 give and what is the standard mu? A. Uniform quantization; mu = 87.6 B. Maximum compression; mu = 255 C. Uniform quantization; mu = 255 D. Logarithmic compression; mu = 87.6 Answer: C. At mu = 0 the law is linear; the standard value for 8-bit PCM is 255. (Key point: mu-law) 4. Which statement about A-law (A = 87.6) is correct? A. It is linear for |x| <= 1/A and logarithmic above that, and is used in Europe/Nepal B. It is logarithmic for small signals and linear for large ones C. It is used in North America and Japan D. It is linear for all x Answer: A. A-law has a linear segment for small signals and a log segment for large ones (ITU-T G.711). 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