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Every electronic communication system carries some kind of signal — an electrical waveform that encodes information. The distinction between analogue and digital representations of that information is one of the most consequential ideas in modern electronics. An analogue signal is continuous in both time and amplitude: at every instant the voltage takes some real value, and that value can be anywhere in a continuous range. A digital signal is discrete in both time (sampled at regular intervals) and amplitude (quantised into a finite number of levels). Once a signal is digital it can be regenerated exactly, copied an indefinite number of times without loss, processed by arithmetic operations on a computer, and protected by mathematical error-correcting codes. The whole transition of recorded audio from vinyl to CD to streaming, of television from analogue broadcast to HDTV, of telephony from copper analogue lines to packet-switched VoIP, was the transition from analogue to digital representations. This lesson develops the mathematics of sampling and quantisation, the Nyquist criterion that determines the minimum sampling rate, the role of analogue-to-digital and digital-to-analogue converters, and the practical comparison of analogue and digital systems in the presence of noise.
Spec mapping (AQA 7408 §3.13.2 — Analogue and digital signals, option E): This lesson covers the definition and properties of continuous (analogue) and discrete (digital) signals; the sampling theorem and the Nyquist criterion f_s ≥ 2 f_max; the role and operation of analogue-to-digital converters (ADCs) and digital-to-analogue converters (DACs); the relationship between bit-depth and quantisation error; the practical advantages of digital signalling for noise immunity, regenerative repeaters and error correction; typical sampling rates for audio, telephony and HDTV; and the disadvantages of digital, including sampling artefacts and aliasing. (Refer to the official AQA specification document for exact wording.)
Synoptic links: The Nyquist criterion connects directly to waves (§3.3.1) — the highest frequency that a signal contains determines the minimum sampling rate, exactly the same Fourier-decomposition logic that underlies superposition. Quantisation noise links to the logarithmic decibel scale (Lesson 3) — the signal-to-noise ratio of an N-bit quantiser is approximately 6N dB, which is why CD-quality 16-bit audio achieves ~96 dB dynamic range. The whole topic is also examined synoptically with transducers (Lesson 6): a microphone produces an analogue voltage, which is digitised by an ADC, processed digitally, and finally converted back to an analogue voltage by a DAC to drive a loudspeaker.
An analogue signal is one in which the encoded quantity (typically voltage) varies continuously with time and can take any real value within its allowed range. The classic example is the output of a microphone exposed to a sound wave: the diaphragm responds to the continuous air pressure variation, and the coil-and-magnet transducer produces a continuously varying voltage that is a scaled copy of the pressure. A vinyl record stores audio analogously: the groove width and depth vary continuously, and the stylus tracking this groove produces a continuous voltage. An analogue television broadcast varies the carrier amplitude continuously to encode the picture brightness at each line. Even a thermocouple producing a few millivolts in response to a furnace temperature is an analogue source.
The defining mathematical property of an analogue signal v(t) is that v is a continuous function of t — the value at any instant is well-defined, and tiny changes in t produce tiny changes in v.
Analogue signals carry a fundamental vulnerability: noise. Any small additional voltage added to the signal — thermal noise from a resistor, interference from a nearby mains cable, hiss from a magnetic tape — alters the signal in a way that cannot be perfectly removed. Even worse, each time the signal is copied (re-amplified, re-recorded, transmitted through a repeater) the noise present is amplified along with the wanted signal, and more noise is added. After a few generations of copies, the noise overwhelms the signal. This is why analogue tape recordings degrade with each generation and why analogue telephone calls over very long distances had to be amplified at hundreds of repeater stations — each repeater amplifying signal and noise together until the speech became unintelligible.
A digital signal is one in which the encoded quantity takes one of only a finite, discrete set of values, sampled at regular time intervals. Most commonly in electronics the alphabet is binary — the signal is either "high" (logical 1, perhaps 3.3 V or 5 V) or "low" (logical 0, perhaps 0 V). The information is encoded in the sequence of bits, not in any precise voltage level. A sequence of bits 10110100 might represent a number, a character, a sample of an audio waveform, or a pixel of an image; the meaning depends on the protocol.
The defining property of digital signalling is that the receiver only needs to make a coarse decision — "above the threshold = 1, below the threshold = 0" — and any additional noise that does not push the signal across the threshold is completely rejected. As long as the signal-to-noise ratio is reasonable (say, ≥ 10 dB), the bit stream is recovered exactly. The cost is paid up-front: you need an ADC to convert the analogue source into bits, and a DAC to convert the bits back to an analogue waveform at the destination.
To represent an analogue waveform v(t) digitally:
The result is a stream of N-bit binary numbers, one every T_s seconds — a digital representation of v(t).
graph LR
A["Analogue source<br/>v(t) continuous"] --> B["Sampler<br/>at rate f_s"]
B --> C["Quantiser<br/>N bits"]
C --> D["Encoded<br/>bit stream"]
D --> E["Transmission /<br/>storage"]
E --> F["DAC<br/>reconstruct"]
F --> G["Low-pass<br/>filter"]
G --> H["Analogue<br/>output"]
style A fill:#3498db,color:#fff
style D fill:#27ae60,color:#fff
style H fill:#3498db,color:#fff
How fast must we sample? The answer, formalised by Harry Nyquist and Claude Shannon in the 1920s and 1940s, is one of the most beautiful results in electronics:
Nyquist criterion: A signal whose highest frequency component is f_max can be sampled and perfectly reconstructed if and only if the sampling rate satisfies f_s ≥ 2 f_max.
The frequency 2 f_max is called the Nyquist rate. If the signal is sampled at a lower rate, frequency components above f_s/2 will be aliased — they will reappear in the reconstructed signal at a lower frequency, indistinguishable from a legitimate low-frequency component. Aliasing is irreversible: once the data is sampled below the Nyquist rate, the original is lost.
Intuitively: to identify a sinusoid of frequency f, you need at least two samples per cycle — one near the peak and one near the trough. A single sample per cycle gives you a constant, indistinguishable from a DC signal. Mathematically, the spectrum of a sampled signal contains copies of the original spectrum centred at every integer multiple of f_s. As long as f_s > 2 f_max these copies do not overlap, and a low-pass filter can recover the original. If f_s < 2 f_max, the copies overlap (alias) and the original cannot be recovered.
In practice, the analogue input is passed through a steep low-pass filter just before the sampler. This anti-aliasing filter removes any signal components above f_s/2, ensuring the Nyquist condition is met regardless of what the source produces. For CD audio with f_s = 44.1 kHz, the anti-aliasing filter has a cutoff just below 22.05 kHz — comfortably above the upper limit of human hearing (around 20 kHz), so the user hears no loss.
Question: A high-fidelity audio system is required to reproduce frequencies up to 20 kHz with no aliasing. (a) State the minimum sampling rate required. (b) The audio CD standard uses 44.1 kHz sampling. State the highest signal frequency that can be reproduced without aliasing. (c) Explain why CD audio uses 44.1 kHz rather than the theoretical minimum.
Solution:
(a) By Nyquist, f_s ≥ 2 × 20 kHz = 40 kHz minimum.
(b) The Nyquist frequency for f_s = 44.1 kHz is 44.1 / 2 = 22.05 kHz. Any signal up to this frequency can be reproduced without aliasing.
(c) Real anti-aliasing filters cannot have an infinitely sharp cutoff. There must be a guard band between f_max (20 kHz) and f_s/2 (22.05 kHz) over which the filter rolls off. The 44.1 kHz rate provides about 2 kHz of guard band — enough for a steep but finite-order analogue filter to attenuate the alias region by 80 dB or more. Choosing exactly 40 kHz would require an impossibly steep filter to avoid aliasing on signals just below 20 kHz.
Exam Tip: Always quote minimum sampling rate with the Nyquist factor of 2 and mention that practical systems sample slightly above this to allow a finite filter roll-off. Examiners are explicit about wanting both points.
Once a sample is taken, it must be rounded to the nearest of 2^N discrete levels for an N-bit ADC. The quantisation step size Δ is:
Δ = V_FS / 2^N
where V_FS is the full-scale voltage range. The error in any one sample is uniformly distributed between −Δ/2 and +Δ/2 — this is quantisation noise.
| Bit-depth N | Number of levels (2^N) | SNR (≈ 6N dB) | Typical use |
|---|---|---|---|
| 8-bit | 256 | 48 dB | Old digital telephony (μ-law) |
| 12-bit | 4 096 | 72 dB | Industrial sensors, oscilloscopes |
| 16-bit | 65 536 | 96 dB | CD audio, broadcast video |
| 24-bit | 16.78 M | 144 dB | Studio mastering audio |
| 32-bit float | ~ 24-bit effective | > 192 dB | High-end DAW (digital audio workstation) |
Adding one bit to the ADC halves the quantisation step size, doubling the signal-to-quantisation-noise ratio in voltage terms and quadrupling it in power. In decibels:
SNR (dB) ≈ 6.02 N + 1.76 dB ≈ 6 N dB (rule of thumb)
So 16-bit CD audio achieves about 96 dB, comfortably exceeding the dynamic range of human hearing (~120 dB at the absolute limits, but ~90 dB for non-painful listening). 24-bit mastering audio gives headroom for digital processing (each filtering or mixing operation can lose a few bits of effective resolution).
Question: An 8-bit ADC has a full-scale input range of ±5.0 V. Calculate (a) the quantisation step size, (b) the SNR in dB.
Solution:
(a) V_FS = 10.0 V (from −5 to +5). Δ = 10.0 / 2⁸ = 10.0 / 256 = 39.1 mV per step.
(b) SNR ≈ 6 × 8 = 48 dB. (More precisely, 6.02 × 8 + 1.76 ≈ 49.9 dB.)
This is why 8-bit audio sounds hissy — 48 dB of dynamic range is barely better than analogue tape, and the quantisation noise is audible on quiet passages. A 16-bit upgrade adds 48 dB of dynamic range, putting the noise floor inaudibly far below normal listening level.
An analogue-to-digital converter (ADC) converts a continuous voltage into a digital code. Several architectures exist; A-Level coverage is mostly conceptual but you should know the names:
A digital-to-analogue converter (DAC) does the inverse: it converts an N-bit binary number into an analogue voltage. The simplest design is an R-2R ladder of resistors weighted in powers of two. The DAC output is a staircase approximation to the original waveform; it is followed by a reconstruction filter (typically a low-pass with cutoff at f_s/2) that smooths the staircase into a continuous waveform.
graph TD
A["Binary input<br/>bN-1 ... b1 b0"] --> B["R-2R ladder<br/>network"]
B --> C["Summing op-amp"]
C --> D["Staircase output"]
D --> E["Reconstruction<br/>low-pass filter"]
E --> F["Smooth analogue<br/>output"]
style A fill:#27ae60,color:#fff
style F fill:#3498db,color:#fff
Once the up-front cost of ADC/DAC hardware is paid, digital signalling has enormous advantages over analogue:
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