3.13.2.1ElectronicsCore

Sampling, Quantisation and PCM

Digitising a signal makes it easier to store and regenerate, but loses information through finite sampling intervals and finite level spacing. Keep those two limits separate. This lesson is part of the Electronics option; study it if this is your centre’s chosen Paper 3 Section B topic.

40 min 3.13 Electronics

What you'll be able to do

Explain analogue and digital representations
Calculate levels, sample size and data rate
Explain pulse-code modulation and noise regeneration

Sample in time and quantise in amplitude

An analogue signal varies continuously with the measured quantity. An analogue-to-digital converter measures it at regular sample times, then assigns each sample to one of a finite set of amplitude levels. The interval between samples controls time detail; the level spacing controls amplitude detail.

An n-bit sample represents $2^n$ possible codes. Increasing bit depth provides finer quantisation over the same voltage range but produces more data. Increasing sampling rate provides more time samples but does not itself create more amplitude levels. A bit has two possible states; a byte contains eight bits.

N_{\mathrm{levels}}=2^n,\qquad \Delta V\approx\frac{V_{\mathrm{range}}}{2^n}

The precise endpoint convention depends on how the ADC range is defined.

Pulse-code modulation and data rate

In pulse-code modulation, each sample is encoded as a sequence of binary pulses representing its quantised value. The receiver identifies the bit levels, reconstructs the sample values and uses digital-to-analogue conversion and smoothing to recover an approximate original waveform.

For uncompressed data the bit rate is sampling rate times bits per sample times number of channels. Higher sampling rate and bit depth can improve fidelity but require more storage and channel capacity. Binary arithmetic beyond the limited specification is unnecessary; recognise simple codes such as 0010=2 and 1010=10.

R=f_s nN_{\mathrm{channels}}

R is in bits s⁻¹ when fs is in samples s⁻¹.

Noise and the limits of regeneration

A digital receiver can classify a noisy voltage as a 0 or 1 when noise stays within the decision margin, then regenerate clean pulses. This prevents small amplitude errors accumulating as they do in a repeatedly amplified analogue signal. Large noise can still cause bit errors; “digital” does not mean immune to interference.

Quantisation introduces an error even with a noiseless channel. Sampling must also be fast enough to represent the highest signal frequencies without aliasing; a useful band-limited design condition is a rate greater than twice the highest frequency. Filtering before sampling avoids admitting unwanted higher-frequency components.

Work through the reasoning

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.

Example 1An 8-bit ADC covers a 0–5.12 V range divided into 256 equal bins. Find bin width.

2^8=256

levels.

\Delta V=5.12/256=0.020

Answer20 mV per bin under the stated bin convention.

Example 2Stereo audio is sampled at 44000 samples s⁻¹ with 16 bits per sample per channel. Find raw bit rate and one-minute data size.

R=44000(16)(2)=1.408\times10^6

bits s⁻¹.

2One minute contains

84.48\times10^6

bits.

3Divide by 8 for bytes:

10.56\times10^6

bytes.

Answer1.408 Mbit s⁻¹; 10.56 MB for one minute, excluding overhead and compression.

Equation recap

N_{\mathrm{levels}}=2^n,\qquad \Delta V\approx\frac{V_{\mathrm{range}}}{2^n}

The precise endpoint convention depends on how the ADC range is defined.

R=f_s nN_{\mathrm{channels}}

R is in bits s⁻¹ when fs is in samples s⁻¹.

Common mistakes to avoid

Using bytes and bits interchangeably.

Eight bits make one byte.

Claiming more bits per sample increases the sampling frequency.

Bit depth and sampling rate are separate settings.

Key takeaways

Sampling limits time detail; quantisation limits amplitude detail.
PCM encodes samples as binary pulses.
Digital signals can be regenerated when noise remains within decision margins.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 10 lessons

Sampling, Quantisation and PCM

40 min 3.13 Electronics