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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Shannon Channel Capacity | C = B * log2(1 + S/N) | C: channel capacity (bits/s); B: bandwidth (Hz); S/N: signal-to-noise ratio (linear, not dB) |
| AM Modulated Signal | s(t) = A_c [1 + μ * m(t)] * cos(2πf_c t) | s(t): modulated signal; A_c: carrier amplitude; μ: modulation index (0 to 1); m(t): normalized message signal; f_c: carrier frequency (Hz) |
| FM Instantaneous Frequency | f_i(t) = f_c + k_f * m(t) | f_i(t): instantaneous frequency (Hz); f_c: carrier frequency (Hz); k_f: frequency sensitivity (Hz/V); m(t): message signal (V) |
| Carson's Rule (FM Bandwidth) | B_FM = 2(Δf + f_m) = 2 f_m (β + 1) | B_FM: approximate FM bandwidth (Hz); Δf: peak frequency deviation = k_f * A_m; f_m: message bandwidth (Hz); β: modulation index = Δf / f_m |
| Nyquist Sampling Rate | f_s >= 2 * f_max | f_s: sampling frequency (samples/s); f_max: highest frequency component in the signal (Hz) |
Key concepts
Amplitude Modulation (AM)
The carrier amplitude is varied in proportion to the message signal. For 100% modulation (μ = 1), 1/3 of total power is in the sidebands and 2/3 is wasted in the carrier. DSB-SC eliminates the carrier, doubling efficiency.
Frequency Modulation (FM)
The instantaneous frequency of the carrier varies with the message. FM is less susceptible to amplitude noise than AM because the information is in the frequency, not the amplitude. FM offers a bandwidth-SNR tradeoff: increasing bandwidth improves SNR.
Pulse Code Modulation (PCM)
Analog signals are sampled, quantized to 2^n levels, and encoded into n-bit binary words. Quantization noise power N_q = Δ²/12, where Δ is the step size. SNR for PCM increases by 6 dB per additional bit.
Phase Shift Keying (PSK)
Digital modulation where bits are encoded as phase shifts of the carrier. BPSK: 1 bit/symbol, 0° and 180°. QPSK: 2 bits/symbol, four phases separated by 90°. Higher-order PSK (8-PSK, 16-PSK) increases spectral efficiency but requires higher SNR.
Noise Figure
Noise figure NF = SNR_in / SNR_out (in dB: NF = SNR_in(dB) - SNR_out(dB)). It quantifies how much a component degrades the SNR. For a cascade of stages (Friis formula): F_total = F_1 + (F_2 - 1)/G_1 + (F_3 - 1)/(G_1 G_2) + ...
Tables
Modulation Comparison
| Modulation | Bandwidth | Noise Immunity |
|---|---|---|
| AM (DSB-FC) | 2 f_m | Low |
| DSB-SC | 2 f_m | Moderate |
| SSB | f_m | Moderate |
| FM (wideband) | 2(Δf + f_m) | High |
| PM | 2(β + 1) f_m | High |
PCM SNR vs Bit Depth
| Bits (n) | Levels (2^n) | SNR (approx dB) |
|---|---|---|
| 4 | 16 | 26 dB |
| 8 | 256 | 50 dB |
| 12 | 4096 | 74 dB |
| 16 | 65536 | 98 dB |
Quick facts
- Shannon capacity sets the theoretical maximum data rate; practical systems achieve 50% to 80% of this.
- AM efficiency: for sinusoidal message with μ = 1, useful power fraction = μ²/2 / (1 + μ²/2) = 1/3.
- FM modulation index β = Δf / f_m; for narrowband FM, β < 0.5.
- Telephone PCM standard: 8-bit samples, 8 kHz sampling rate gives 64 kbps per voice channel.
- In QPSK, two bits are transmitted per symbol, doubling spectral efficiency over BPSK.
- Friis formula: first stage noise figure dominates if it has sufficient gain.
- SNR improvement of FM over AM: 3β²(β + 1) for sinusoidal message; large β gives large SNR gain.
Exam shortcuts
- To find AM power quickly: total AM power P_t = P_c (1 + μ²/2), where P_c = A_c²/2R. For μ = 1, P_t = 1.5 P_c. The sideband power alone is P_c * μ²/2. No need to expand the full signal.
- Shannon capacity with SNR in dB: first convert SNR_dB to linear using S/N = 10^(SNR_dB/10), then apply C = B log2(1 + S/N). For SNR = 30 dB, S/N = 1000, so C ≈ B * 10 bits/s/Hz.
- For Carson's rule problems, identify β = Δf/f_m first. If β >> 1 (wideband FM), B_FM ≈ 2Δf. If β << 1 (narrowband FM), B_FM ≈ 2 f_m. These two approximations cover most exam cases.