--- title: "Non-uniform quantization using uniform quantizer; μ-law" subject: "Communication Systems" chapter: 4 key: "semester_6/communication_systems:nonuniform-quantization-mu-law" source: https://www.bctengineer.com/q/semester_6/communication_systems/nonuniform-quantization-mu-law site: BCT Engineering --- # Non-uniform quantization using uniform quantizer; μ-law Communication Systems · Chapter 4: Pulse Modulation - Asked in: 2081 Bhadra (CS II), 5+3 marks - Video: [μ law Companding in Non-uniform Quantization | Basics, Formula & Importance | Digital Communication](https://www.youtube.com/watch?v=ByMoziFuKVI) — 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/nonuniform-quantization-mu-law - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4 ## Question Elaborate on the technique of implementing non-uniform quantization using uniform quantization technique. With a suitable diagram, illustrate the working of μ law. ## Short answer Non-uniform quantization is implemented by companding: a compressor (nonlinear mapping) followed by a uniform quantizer, with an expander at the receiver. μ-law is y = [ln(1+μ|x|)/ln(1+μ)] sgn(x); with μ=255, small signals are amplified more, giving nearly constant SQNR across a wide dynamic range. ## Answer #### Non-uniform quantization using a uniform quantizer A non-uniform quantizer is equivalent to a **compressor**, followed by a **uniform quantizer**, with an **expander** at the receiver. This is called **companding**. ```text x(t)->[Compressor]--y-->[Uniform ]-->[Encoder]-->channel y = c(x) quantizer channel->[Decoder]-->[Expander x = c^-1(y)]-->x^(t) ``` **Technique:** 1. The input $x$ (normalised to $|x| \le 1$) is passed through a non-linear **compressor** $y = c(x)$ with large slope near zero and small slope near full scale. 2. A uniform quantizer with step $\Delta$ operates on $y$. Step $\Delta$ in $y$ corresponds to a step in $x$ of $$ \Delta_x \approx \frac{\Delta}{c'(x)} $$ so where $c'(x)$ is large (small $x$) the effective step is fine, and where $c'(x)$ is small (large $x$) the step is coarse. 3. At the receiver, the **expander** $c^{-1}(y)$ undoes the compression. Compressor + expander together have a linear overall response. 4. If $c(x)$ is logarithmic, $\Delta_x \propto |x|$, so the noise is proportional to the signal and **SQNR is nearly constant** for all levels. In practice $c(x)$ is implemented digitally as a piecewise-linear segmented characteristic, so standard uniform ADC/DACs can be used. #### μ-law $$ y = \frac{\ln(1+\mu|x|)}{\ln(1+\mu)}\,\text{sgn}(x), \qquad 0 \le |x| \le 1 $$ - $\mu = 0$: uniform (linear). Standard **$\mu = 255$** for 8-bit PCM (North America, Japan). - Small $|x|$ ($\mu|x| \ll 1$): $y \approx \frac{\mu x}{\ln(1+\mu)}$, linear with large gain (46 for $\mu = 255$). - Large $|x|$ ($\mu|x| \gg 1$): $y \approx \frac{\ln(\mu|x|)}{\ln(1+\mu)}$, logarithmic. Example ($\mu = 255$): $x = 0.01 \to 0.228$, $x = 0.1 \to 0.591$, $x = 0.5 \to 0.876$, $x = 1 \to 1$. ```text y 1.0| ____---- mu = 255 | __---- __-- mu = 5 | _-' ___--- | / __--- . mu = 0 (y = x) | / _-- . |/- . +---------------------> x 0 1.0 ``` Larger $\mu$ gives more compression and a wider dynamic range with nearly flat SQNR (about 38 dB over 40 dB input range for 8-bit $\mu = 255$). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a non-uniform quantizer consist of? A. Two uniform quantizers B. Compressor, uniform quantizer, expander C. Compressor only D. Expander, uniform quantizer, compressor Answer: B. Compressor-uniform quantizer at Tx, expander at Rx. (Key point: Compander structure) 2. If compressor slope is c'(x), the effective step in x is: A. Delta B. Delta/c'(x)^2 C. Delta c'(x) D. Delta/c'(x) Answer: D. Large slope near zero gives fine steps. (Key point: Effective step) 3. For mu|x| much less than 1, mu-law behaves like: A. A logarithm B. A constant C. A square law D. A linear characteristic with large gain Answer: D. y is approximately mu x/ln(1+mu). (Key point: mu-law small x) 4. Why does a logarithmic compressor give constant SQNR? A. Noise is zero B. Signal is fixed C. Step is constant D. Step in x is proportional to |x|, so noise scales with signal Answer: D. Delta_x proportional to |x|. 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