4.4 Communications

Modulating a carrier

The Communications area is 5-8 questions and is highly formula-driven, which makes it a reliable point source if you know where the equations live in the FE Reference Handbook. Modulation impresses a low-frequency message signal onto a high-frequency carrier so it can be transmitted efficiently. The three classic analog methods change different carrier properties:

• Amplitude modulation (AM) varies the carrier amplitude in proportion to the message.
• Frequency modulation (FM) varies the carrier frequency.
• Phase modulation (PM) varies the carrier phase.

FM and PM are together called angle modulation: they keep amplitude constant and encode information in the angle, which makes them far more resistant to amplitude noise than AM — the reason FM radio sounds cleaner. The cost is bandwidth.

Bandwidth

Bandwidth questions reduce to a few rules. For standard double-sideband AM, with a message band-limited to fm, the transmission bandwidth is:

BW(AM) = 2 · fm

For FM, the bandwidth is estimated with Carson's rule, where Δf is the peak frequency deviation:

BW(FM) ≈ 2(Δf + fm)

Channel capacity and SNR

The headline result of information theory is the Shannon–Hartley theorem, giving the maximum error-free data rate (channel capacity) over a band-limited noisy channel:

C = B · log2(1 + S/N)

Here C is capacity in bits per second, B is the channel bandwidth in hertz, and S/N is the linear signal-to-noise power ratio. Capacity grows linearly with bandwidth but only logarithmically with SNR — doubling bandwidth doubles capacity, while doubling SNR barely changes it.

The most common trap is the signal-to-noise ratio (SNR) unit. SNR is usually quoted in decibels:

SNR(dB) = 10 · log10(S/N)

Before plugging into Shannon's formula, convert dB to a linear ratio: S/N = 10^(SNR_dB/10). For example, 30 dB means S/N = 10^3 = 1000, so the (1 + S/N) term is 1001.

Digital modulation and multiplexing

Digital systems map bits onto a carrier. The basic schemes mirror the analog ones:

• ASK (amplitude-shift keying) — bits set amplitude levels.
• FSK (frequency-shift keying) — bits select among frequencies.
• PSK (phase-shift keying) — bits set carrier phase (BPSK = 1 bit/symbol, QPSK = 2 bits/symbol).
• QAM (quadrature amplitude modulation) — combines amplitude and phase; 16-QAM carries 4 bits/symbol, 64-QAM carries 6.

Higher-order schemes pack more bits per symbol (bits/symbol = log2 M for M states) but demand a higher SNR to keep symbols distinguishable. Multiplexing shares one physical channel among many signals: FDM (frequency-division) assigns separate frequency bands, TDM (time-division) assigns time slots, and CDM/CDMA (code-division) assigns orthogonal codes.