PCM Uniform Quantization: 25-Comp-B5 Computer Communications, May 2015, Q3

Question 3: PCM Uniform Quantization (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Check: assumes the "10 V full-scale" is the full input range (FSR) covered by the quantizer's $2^{12}$ codes, i.e. the quantizer maps $[0,\text{FSR}]$ (or an equivalent $10\text{ V}$ span) onto codes $0$ to $2^{12}-1$ — the paper does not state whether the range is unipolar ($0$–$10\text{ V}$) or bipolar ($-5$ to $+5\text{ V}$); the step-size arithmetic is identical either way.

Given.

Quantity	Value
Full-scale range, FSR	10 V
Code length	12 bits
Quantization	Uniform

Find. (a) normalized step size, (b) actual step size (V), (c) maximum quantized level (V), (d) actual voltage resolution.

Approach. With $n=12$ bits there are $L=2^{12}=4096$ quantization levels spanning the full-scale range; the normalized step is $1/L$ of full scale, and every other quantity follows by scaling that fraction by the $10\text{ V}$ range.

Number of quantization levels. $$L = 2^{n} = 2^{12} = 4096$$
(a) Normalized step size (fraction of full scale per code): $$\Delta_{\text{norm}} = \frac{1}{L} = \frac{1}{4096} = \boxed{2.4414\times10^{-4}}$$
(b) Actual step size in volts: $$\Delta = \frac{\text{FSR}}{L} = \frac{10\text{ V}}{4096} = \boxed{2.4414\text{ mV}}$$
(c) Maximum quantized level in volts (the top code, $L-1$, one step below full scale): $$V_{\max} = (L-1)\,\Delta = 4095 \times 2.4414\text{ mV} = \boxed{9.9976\text{ V}}$$
(d) Actual voltage resolution: the resolution — the smallest voltage change the 12-bit code can distinguish — is by definition the same quantity as the actual step size found in (b): $$\text{resolution} = \Delta = \boxed{2.4414\text{ mV}}$$

Quantity	Value
(a) Normalized step size	$1/4096 = 2.4414\times10^{-4}$
(b) Actual step size	2.4414 mV
(c) Maximum quantized level	9.9976 V
(d) Actual voltage resolution	2.4414 mV

Topic: Pulse Code Modulation (PCM) and uniform quantization.

Key relations: $L=2^n$ levels; step size $\Delta = \text{FSR}/L$; normalized step $= \Delta/\text{FSR}=1/L$; quantization error bound $\pm\Delta/2$.

Why this works: uniform quantization divides the full-scale analog range into $2^n$ equal-width intervals and assigns each an $n$-bit code; the finer the interval (more bits), the smaller the step and the closer the reconstructed voltage is to the true analog value, at the cost of a longer code word per sample.

Common pitfall: confusing the "maximum quantized level" (the highest of the $L$ discrete reconstruction levels, one step below full scale) with the full-scale value itself, or reporting the normalized step size as a voltage (it is dimensionless, a fraction of FSR).

Source: Stallings, Data and Computer Communications, 10th ed., Ch.5 (Signal Encoding Techniques) §5.4, Pulse Code Modulation.

Question 3 of 7: PCM Uniform Quantization

Question 3: PCM Uniform Quantization (20 marks)