Communications

Syllabus

Random processes (autocorrelation, power spectral density, white noise, filtering through LTI systems); analog communications (AM and angle modulation, their spectra, demodulation, superheterodyne receivers); information theory (entropy, mutual information, channel capacity theorem); digital communications (PCM, DPCM, ASK, PSK, FSK, QAM, bandwidth, ISI, MAP and ML detection, matched filter receiver, SNR and BER); error correction, Hamming codes, CRC.

Key concepts and formulas

Random processes. The PSD is the Fourier transform of the autocorrelation (Wiener-Khinchin). White noise has a flat PSD N₀/2 and an impulse autocorrelation. Through an LTI system: S_y(f) = |H(f)|²S_x(f). Noise power in bandwidth B (white noise, ideal filter): N₀B.

Amplitude modulation. s(t) = A_c[1 + μ cos ω_mt] cos ω_ct, modulation index μ = A_m/A_c.

• Total power P_t = P_c(1 + μ²/2); power efficiency η = μ²/(2 + μ²) (33.3% at μ = 1).
• Bandwidth: DSB 2f_m, SSB f_m.
• An envelope detector needs μ ≤ 1 (no overmodulation).

Angle modulation. FM: frequency deviation Δf = k_fA_m, modulation index β = Δf/f_m. Carson's rule: B ≈ 2(Δf + f_m) = 2(β + 1)f_m. Narrowband FM (β ≪ 1) has bandwidth about 2f_m, like AM.

Superheterodyne receiver. Image frequency f_image = f_s + 2f_IF (when the local oscillator is above the signal); AM broadcast IF = 455 kHz, FM IF = 10.7 MHz.

Information theory. Entropy H = −Σ pᵢ log₂ pᵢ bits/symbol, largest (log₂ M) for M equally likely symbols. Shannon-Hartley: C = B log₂(1 + S/N) bits/s.

PCM. Bit rate = n f_s (n bits per sample); for M levels n = log₂M. SQNR ≈ 6.02n + 1.76 dB for a full-scale sine; each extra bit adds about 6 dB. Minimum (Nyquist) transmission bandwidth nf_s/2. DPCM transmits the difference between a sample and its prediction, needing fewer bits.

Digital modulation.

• BPSK and QPSK: BER = Q(√(2E_b/N₀)); QPSK has the same BER as BPSK with half the bandwidth.
• Coherent BFSK: Q(√(E_b/N₀)), 3 dB worse than BPSK.
• M-ary schemes send log₂M bits per symbol.
• The matched filter h(t) = s(T − t) maximises the output SNR, which is 2E/N₀.
• Nyquist criterion for zero ISI: with raised-cosine pulses of roll-off α, the bandwidth is R_s(1 + α)/2.

Error control. A code with minimum Hamming distance d_min detects d_min − 1 errors and corrects ⌊(d_min − 1)/2⌋. A Hamming (n, k) code with r parity bits satisfies 2ʳ ≥ n + 1 (the (7, 4) code corrects one error). CRC detects errors using polynomial division.

Worked example

An AM signal has a carrier power of 1000 W and μ = 0.8: find the total power and efficiency. An FM signal has Δf = 75 kHz and f_m = 15 kHz: find β and the Carson bandwidth. A source emits four symbols with probabilities 1/2, 1/4, 1/8 and 1/8: find its entropy. Finally, find the capacity of a 3 kHz channel with an SNR of 30 dB.

import math

Pc, mu = 1000, 0.8
print(f"AM: Pt = {Pc * (1 + mu ** 2 / 2):.0f} W, efficiency = {mu ** 2 / (2 + mu ** 2) * 100:.2f}%")
df, fm = 75, 15
print(f"FM: beta = {df / fm:.0f}, Carson bandwidth = {2 * (df + fm)} kHz")
p = [1 / 2, 1 / 4, 1 / 8, 1 / 8]
print(f"entropy = {-sum(x * math.log2(x) for x in p):.2f} bits/symbol")
B, snr = 3000, 10 ** (30 / 10)
print(f"capacity = {B * math.log2(1 + snr):.1f} bit/s")
print(f"PCM: 8 bits at 8 kHz = {8 * 8} kbit/s, SQNR = {6.02 * 8 + 1.76:.2f} dB")

AM: Pt = 1320 W, efficiency = 24.24%
FM: beta = 5, Carson bandwidth = 180 kHz
entropy = 1.75 bits/symbol
capacity = 29901.7 bit/s
PCM: 8 bits at 8 kHz = 64 kbit/s, SQNR = 49.92 dB

Practice questions