Sampling, PCM and Digital Modulation
Sampling and quantisation, PCM bit rate and SQNR, delta modulation, line coding, ISI and Nyquist pulse shaping, ASK, FSK, PSK and QAM, matched filter and bit error probability.
Digital communication sends bits. An analog message is first sampled and quantised; the bits are then shaped into pulses and modulated onto a carrier.
Sampling and PCM
- Sample at f_s ≥ 2W (Nyquist).
- Quantisation with L = 2ⁿ levels; step size Δ = (peak-to-peak range)/L; quantisation noise power Δ²/12.
- SQNR for a full-scale sinusoid: 6.02n + 1.76 dB (each extra bit adds about 6 dB).
- PCM bit rate R_b = n f_s. Minimum transmission bandwidth (Nyquist) = R_b/2.
- Companding (μ-law, A-law) improves SQNR for weak signals.
- Delta modulation: 1 bit per sample; slope overload if the step δ is too small (avoid by δ f_s ≥ max slope of the message); granular noise if δ is too large. DPCM sends quantised prediction errors.
Baseband transmission
- Line codes: NRZ, RZ, Manchester (no DC, self-clocking, double bandwidth), AMI.
- ISI (inter-symbol interference) is avoided by Nyquist pulses: a raised-cosine spectrum with roll-off α needs bandwidth B = R_s(1 + α)/2 at baseband (R_s(1 + α) for passband).
- Eye diagrams show ISI and timing margins.
Matched filter
The filter that maximises output SNR at the sampling instant has impulse response h(t) = s(T − t). Its maximum SNR = 2E/N₀ (E = symbol energy), independent of the pulse shape.
Digital modulation
| Scheme | Bits/symbol | Bandwidth (min, null-to-null approx.) | Bit error probability |
|---|---|---|---|
| BPSK | 1 | 2R_b | Q(√(2E_b/N₀)) |
| QPSK | 2 | R_b | Same as BPSK per bit |
| Coherent BFSK | 1 | Wider | Q(√(E_b/N₀)) |
| ASK (OOK) | 1 | 2R_b | Q(√(E_b/N₀)) (average energy) |
| DPSK | 1 | 2R_b | ½ e^(−E_b/N₀) |
| M-QAM | log₂ M | R_b/log₂ M | Better bandwidth efficiency, needs more SNR |
- BPSK is 3 dB better than coherent BFSK and OOK.
- Symbol rate R_s = R_b/log₂ M.
- Constellation: M-PSK points on a circle; QAM points on a grid.
Worked example
Speech band-limited to 4 kHz is sampled at the Nyquist rate with 8-bit PCM.
- f_s = 8 kHz; R_b = 64 kbit/s; SQNR ≈ 6.02 × 8 + 1.76 = 49.9 dB.
Practice
A 16-QAM system sends 40 kbit/s. What is the symbol rate in kbaud?
Show answer and explanation
Answer: 1016-QAM carries log₂16 = 4 bits per symbol: 40/4 = 10 kbaud.
With raised-cosine pulses of roll-off 0.5 at 10 kbaud, what baseband bandwidth (kHz) is needed?
Show answer and explanation
Answer: 7.5B = R_s(1 + α)/2 = 10 × 1.5/2 = 7.5 kHz.
What SQNR (dB) does a 10-bit PCM give for a full-scale sinusoid? (Give 2 decimal places.)
Show answer and explanation
Answer: 61.96SQNR = 6.02 × 10 + 1.76 = 61.96 dB.
Which of these need coherent detection (a phase-locked carrier) at the receiver? (Select all that apply.)
Show answer and explanation
Answer: A, BBPSK and QPSK need carrier phase; DPSK compares consecutive symbols and non-coherent FSK uses envelope detectors.