Investigate analogue-to-digital and digital-to-analogue conversion, the Nyquist sampling theorem, quantisation and noise, pulse-code modulation and its alternatives, companding, line codes, and error detection and correction

What this dot point is asking

The communication systems fundamentals dot point introduced the analogue / digital distinction. This one goes deeper: how engineers actually convert between the two domains. You need the sampling theorem, the quantisation arithmetic that bounds signal-to-noise ratio, the standard pulse-code modulation chain, the alternatives (delta, sigma-delta), the voice-companding shortcuts (mu-law, A-law), the line codes that put bits onto wires, and the error detection / correction layer that makes digital links reliable.

The answer

Analogue-to-digital conversion (ADC) and the sampling theorem

An ADC samples a continuous-time analogue signal at regular intervals and quantises each sample to a discrete numerical value. The Nyquist sampling theorem sets the minimum sample rate:

f_sample >= 2 x f_max

where f_max is the highest frequency in the signal. Sampling below 2 x f_max produces aliasing: high-frequency components fold down and corrupt the band of interest. A real ADC therefore precedes its sample-and-hold circuit with an anti-aliasing low-pass filter that suppresses any energy above f_sample / 2.

The DAC at the receiver inverts the process: convert each numerical sample to a voltage, then low-pass filter to reconstruct the smooth analogue waveform.

Quantisation and bit depth

Each sample is rounded to one of 2^N discrete levels, where N is the bit depth. This rounding introduces quantisation noise that bounds the achievable signal-to-noise ratio (SNR). For a full-scale sinusoidal input, the approximate SNR limit is:

SNR_dB ~ 6.02 x N + 1.76

So each extra bit adds about 6 dB of dynamic range. Common bit depths and their SNR ceilings:

• 8 bit: about 50 dB. Adequate for telephony voice.
• 12 bit: about 74 dB. Common in industrial data acquisition.
• 16 bit: about 98 dB. CD audio standard.
• 24 bit: about 146 dB (in theory; thermal noise floors dominate in practice). Studio recording.

The engineering choice trades bit depth (more storage and bandwidth) against quantisation noise (more bits = less noise).

Pulse-code modulation (PCM)

PCM is the classical digital-encoding scheme: sample at f_sample, quantise to N bits, transmit the bits as a binary stream. The standard telephony PCM rate is:

8 kHz sampling x 8 bits per sample = 64 kbps per voice channel

This is the digital signal level zero (DS0) in legacy telco hierarchies, and remains the reference point for narrowband voice.

CD audio uses linear PCM at:

44.1 kHz x 16 bits x 2 channels (stereo) = 1411.2 kbps

The CD sample rate of 44.1 kHz is above 2 x 20 kHz (human hearing limit) with a guard band for the anti-aliasing filter.

Delta modulation and sigma-delta

Delta modulation transmits only one bit per sample, indicating whether the input went up or down since the last sample. The receiver integrates this bit stream to reconstruct the signal. Simple hardware, but limited dynamic range and slope overload at high signal slew rates.

Sigma-delta (or delta-sigma) modulation oversamples the input at many times the Nyquist rate, uses noise-shaping feedback to push quantisation noise out of the signal band, then digitally filters and decimates the result. Modern audio ADCs (and the codec chips in mobile phones) are overwhelmingly sigma-delta because they extract high resolution from simple one-bit modulators and digital signal processing.

Companding for voice (mu-law and A-law)

Linear 8-bit PCM gives unacceptably noisy quiet speech. Companding applies a non-linear amplitude curve before quantisation (compress at the encoder) and the inverse curve after the DAC (expand at the decoder). The result: more quantisation steps allocated to low-amplitude signals, fewer to large-amplitude signals.

• mu-law (mu = 255). North American and Japanese telephony standard.
• A-law (A = 87.6). European and Australian telephony standard.

Both standards achieve voice quality equivalent to about 12-bit linear PCM at the 8-bit transmission rate.

Line codes

A line code maps bits to electrical waveforms on a transmission medium. Choices balance DC balance, clock recovery, bandwidth efficiency and error detection:

• NRZ (Non-Return-to-Zero). Bit 1 = high voltage; bit 0 = low voltage. Simple but no transitions during long runs of identical bits, so the receiver can lose clock sync. No DC balance.
• Manchester encoding. Each bit period contains a mid-bit transition (0 = low-to-high; 1 = high-to-low, or vice versa). Self-clocking and DC balanced, but doubles the bandwidth requirement. Used in classic 10 Mbps Ethernet.
• 4B/5B and 8B/10B. Map blocks of data bits to slightly longer code words chosen to guarantee transitions and DC balance. 8B/10B is used in Gigabit Ethernet, Fibre Channel, PCI Express gen 1-2, and many SerDes links.
• 64B/66B. Used in 10 Gigabit Ethernet and beyond; lower overhead than 8B/10B.

Error detection and error correction

Error detection lets the receiver discover that a packet is corrupt and request retransmission.

• Parity bit. One extra bit set so the total number of 1s is even (even parity) or odd. Detects any odd number of bit errors. Cheap; weak.
• Checksum. Sum the data words mod some base. Detects most random errors; weak against burst errors.
• Cyclic redundancy check (CRC). Treat the data as a polynomial and divide by a generator polynomial; transmit the remainder. CRC-32 (used in Ethernet) detects all single-bit, two-bit, odd-count and burst-up-to-32-bit errors. Strong and computable with shift-register hardware.

Forward error correction (FEC) adds redundancy so the receiver can correct errors without a retransmit:

• Hamming codes. Add log_2(n) parity bits to correct a single bit error in n bits. Classical and pedagogical.
• Reed-Solomon (RS). Block code over a finite field. Excellent against burst errors. Used in CD audio, DVDs, QR codes, satellite links, deep-space probes.
• Convolutional codes with Viterbi decoding. Continuous coding; used heavily in 2G, 3G and satellite links.
• Turbo codes. Used in 3G and 4G LTE; approach the Shannon limit.
• Low-density parity-check (LDPC) codes. Used in 5G NR data channels, Wi-Fi 6, DVB-S2 satellite, and 10G+ Ethernet. State-of-the-art for high-throughput links.

The engineering trade-off: more redundancy means more reliable reception but lower net throughput. Modern adaptive systems vary the FEC rate based on measured channel quality.

Examples in context

Example 1. The GSM voice codec chain. A 2G GSM phone samples voice at 8 kHz and 13-bit linear, then runs RPE-LTP (regular pulse excitation long-term prediction) speech compression to 13 kbps net, adds CRC and convolutional FEC for the noisy radio channel, and modulates the resulting protected bit stream onto the GMSK carrier. Every layer in this dot-point appears: ADC, source coding (companding's cousin), channel coding (CRC plus FEC), modulation. The same pattern (sample, source-code, channel-code, modulate) recurs in 4G LTE and 5G NR with different specific choices.

Example 2. Deep-space probe telemetry. NASA's Voyager 1, transmitting from beyond the solar system, has extraordinarily low received signal power. The probe uses concatenated Reed-Solomon and convolutional FEC so that received bits decode reliably at signal-to-noise ratios where uncoded transmission would be unusable. The engineering choice trades raw throughput (the FEC overhead reduces information bits per second) for reliable communication over the most demanding channel imaginable. The same Reed-Solomon idea inside everyday CD-ROMs and DVB-T broadcast TV is a direct descendant of the deep-space coding research.

Try this

Q1. State the Nyquist sampling theorem and apply it to a voice signal with components up to 3.4 kHz. [3 marks]

• Cue. The sample rate must be at least twice the highest frequency in the signal. For voice with f_max = 3.4 kHz, f_sample >= 6.8 kHz. Telephony standards specify 8 kHz, with the extra margin allowing a realisable anti-aliasing filter.

Q2. Calculate the maximum SNR of a 12-bit linear PCM system, and state one application where this bit depth is appropriate. [3 marks]

• Cue. SNR_dB ~ 6.02 x 12 + 1.76 = 74 dB. Appropriate for industrial data acquisition, professional voice recording, or transcoding intermediate stages where 8-bit voice is insufficient but 16-bit audio is over-specified.

Q3. Compare error detection (CRC) and forward error correction (Reed-Solomon) for a noisy radio link, identifying when each is preferable. [6 marks]

• Cue. CRC detects errors and triggers retransmission via the higher protocol layer. Effective when retransmission is cheap (low-latency link, both ends available) and channel errors are intermittent. FEC corrects errors at the receiver with no retransmit. Essential when retransmission is impossible (broadcast TV, deep-space probes, one-way links) or expensive (high-latency satellite). Real systems often combine both: FEC at the physical layer corrects common errors; CRC at the link layer detects the rare cases the FEC cannot correct and triggers a retransmit. The engineering trade-off is redundancy overhead versus achievable throughput and latency.