Digital Communication & Information Capacity – Comprehensive Study Notes

DIGITAL COMMUNICATION

• Digital communications: high-frequency analog carriers are modulated by low-frequency digital information (digital pulses).
• Information = knowledge/intelligence exchanged between two or more points.
• Digital modulation (a.k.a. digital radio): conveying digitally modulated analog carriers.
• Generic digital radio system blocks:
  • Transmitter: Input data → Precoder/Buffer → Modulator → BPF & PA → Transmission medium.
  • Synchronisation: Clock buffer, Analog carrier.
  • Receiver: BPF/Amp → Demodulator & Decoder → Output data, plus Carrier & Clock recovery.
  • Noise enters the medium; front-end BPF suppresses out-of-band noise.
• Data definitions & conversions:
  • “Data” = information to transmit.
  • Native digital if produced by computers; analog (voice) can be A/D-converted before transmission.
• Historical context: Initially for inter-computer links; later expanded to LANs/WANs & public networks.

Drivers for Growth

1. Rapid computer adoption → need for inter-computer communication.
2. Technical superiority of digital over analog transmission.
3. PSTN (the world’s largest comms system) migrated from analog to digital switching & transport.

Typical Digital Applications

• Computer-centric: File transfer, e-mail, peripheral links, Internet, LANs.
• Consumer/industrial control: TV remotes, garage-door openers, carrier-current power-line controls, radio-controlled models, remote keyless entry.

Benefits

• Noise Immunity: Binary thresholds enable regeneration instead of amplitude-limited amplification, resisting additive noise.
• Error Detection/Correction: Redundancy (parity, CRC, FEC) feasible in digital domain.
• Time-Division Multiplexing compatibility: Easier combination and switching of digitised channels.
• Digital IC economy: Smaller, cheaper, more complex & reliable than linear ICs.
• Digital Signal Processing (DSP): Enables sophisticated filtering, equalisation, mixing, phase-shift etc. in software/hardware.

Disadvantages

• Bandwidth hunger: Digitally encoded analog requires larger BW than original analog.
• Circuit complexity: Extra A/D, D/A, encoding/decoding, synchronisation blocks.

MODULATION FORMATS

• ASK, FSK, PSK, QAM (amplitude + phase) are canonical digital modulation families.

Application Examples

• Low-speed voice-band modems, DSL, digital microwave links, satellites, cellular/PCS.

INFORMATION THEORY & CAPACITY

• Information theory studies efficient bandwidth utilisation.
• Information Capacity: maximum error-free info that can cross a channel; indicator of channel “goodness”.
• Dependent on physical bandwidth $B$, signalling interval $t$, code/level count $M$, and noise.

Bit & Bit Rate

• ‘Bit’ ≡ basic unit of digital information.
• Bit rate $R_b$ (bps) = bits per second conveyed.

Hartley’s Law (baseband, noiseless)

$I \propto B \times t$
Implies capacity increases linearly with bandwidth and transmission time.

Nyquist Criterion (noiseless, binary)

“Maximum binary signalling rate in an ideal channel equals twice its bandwidth.”
$R_{b,\text{max}} = 2B$

M-ary Encoding

• $M$ = number of discrete signal conditions/levels.
• Relationship with bits per symbol $N$:
  $M = 2^{N} \quad \text{or} \quad N = \log_2 M$
  (e.g., 4-level signaling ⇒ $M=4$ ⇒ $N=2$ bits/symbol).

Symbol Rate (Baud)

• Definition: number of distinct symbols transmitted per second.
  $\text{baud} = \frac{1}{t<em>s}$ where $t</em>s$ = symbol duration.
• For binary (2-level), $\text{baud}=\text{bps}$; for M>2, \text{bps}=N\times\text{baud} > \text{baud}.

Generalised Nyquist Capacity (M-level, noiseless)

$R<em>{b,\text{max}} = 2B \log</em>2 M$
Used in examples as “Shannon-Hartley theorem” in slides, though it is actually the Nyquist formulation.

Shannon–Hartley Theorem (noisy channel, continuous levels)

$C = B \log_2 !\bigl(1 + \tfrac{S}{N}\bigr)$

• Places upper bound dictated by signal-to-noise power ratio (SNR).

Shannon Limit vs. Shannon–Hartley

• Hartley/Nyquist bound ignores noise, focuses on levels/bandwidth.
• Shannon limit bound ignores levels, focuses on $\frac{S}{N}$.
• Practical capacity = smaller of the two.
• Both are theoretical guidelines; real systems include coding overhead, implementation loss, channel distortion.

EXAMPLE PROBLEMS (from slides)

Problem 1: Noiseless 4 kHz Channel Using M-ary Signalling

Formula used: $f<em>b = 2B \log</em>2 M$

• (a) $M=2$ ⇒ $f<em>b=2(4000)\log</em>2(2)=8000\,\text{bps}=8\,\text{kbps}$.
• (b) $M=4$ ⇒ $f_b=16000\,\text{bps}=16\,\text{kbps}$.
• (c) $M=128$ ⇒ $f_b=56000\,\text{bps}=56\,\text{kbps}$.
  Observations: capacity grows logarithmically with $M$; large increases in levels yield diminishing returns.

Problem 2: 4 kHz Channel, Shannon Limit with SNR

Formula: $I = B \log_2 (1 + S/N)$
Convert SNR (dB) → absolute: $S/N = 10^{(\text{dB}/10)}$

• (a) 20 dB ⇒ $S/N=100$ ⇒ $I = 4000\log_2(101)=26.632\,\text{kbps}$.
• (b) 30 dB ⇒ $S/N=1000$ ⇒ $I = 4000\log_2(1001)=39.868\,\text{kbps}$.
• (c) 40 dB ⇒ $S/N=10000$ ⇒ $I = 4000\log_2(10001)=53.151\,\text{kbps}$.
  Trend: Capacity increases slowly beyond ~30 dB; doubling BW is more effective than boosting SNR at high values.

Problem 3: Combined Constraints

Channel: 4 kHz, $M=4$, $S/N=1000$ (30 dB)

• Nyquist (2B log2M): $R_b=16000\,\text{bps}$.
• Shannon limit: $C = 39.868\,\text{kbps}$.
  Result: Practical max rate is $16\,\text{kbps}$ (Nyquist smaller ⇒ bottleneck is finite $M$, not noise).

PRACTICAL IMPLICATIONS & INSIGHTS

• Bandwidth is a precious, regulated resource; efficient coding & modulation aim to approach theoretical limits.
• Error-control coding (block, convolutional, LDPC, Turbo) allows systems to work "closer" (within a few dB) of Shannon.
• Higher-order modulations (e.g., 64-QAM) increase $M$ to boost bit rate without extra bandwidth but require higher SNR and more linear RF chains.
• DSP and Moore’s Law make complex codecs affordable (e.g., OFDM, adaptive coding in LTE/5G).
• Ethical/ societal: More capacity enables widespread internet access, but also raises spectrum-allocation conflicts and digital divide issues.

KEY EQUATIONS SUMMARY

• Hartley: $I \propto B t$ (noiseless baseband).
• Nyquist binary: $R<em>{b\text{ max}} = 2B$. • Generalised Nyquist: $R</em>b = 2B \log<em>2 M$. • Bits/symbol: $N = \log</em>2 M$.
• Baud rate: $\text{baud}=\dfrac{1}{t<em>s}$. • Shannon Capacity: $C = B \log</em>2!\bigl(1 + \tfrac{S}{N}\bigr)$.

STUDY TIPS & CONNECTIONS

• Relate bandwidth limitations to earlier courses on filters & transmission lines.
• Connect SNR concept to fundamentals of noise (thermal, quantisation) covered in electronics.
• Practise converting between bps, baud, $M$, and SNR (linear ↔ dB).
• Apply limits to modern systems: e.g., why Wi-Fi uses 20 MHz channels and 256-QAM; evaluate required SNR.
• Recognise trade-offs: BW vs. power vs. complexity → design decision space in real products.