Communications formula sheet

99 formulas in 5 topics, each with when to use it. Each is explained in the topic notes.

MEAN AND VARIANCE OF A RANDOM VARIABLE

μ_{X} = E [X], σ_{X}^{2} = E [(X - μ_{X})^{2}] = E [X^{2}] - μ_{X}^{2}

Holds for any random variable. $E [X^{2}]$ is the mean-square value (average power for a signal).

GAUSSIAN DENSITY

f_{X} (x) = \frac{1}{2 π σ} exp [- \frac{( x - μ ) ^{2}}{2 σ ^{2}}]

Fully described by mean $μ$ and variance $σ^{2}$ .

Amplitude and angle modulation, receivers and multiplexing

All 19 formulas

SINGLE-TONE AM WAVE

s (t) = A_{c} [1 + μ cos (ω_{m} t)] cos (ω_{c} t) = A_{c} cos ω_{c} t + \frac{μ A _{c}}{2} cos (ω_{c} + ω_{m}) t + \frac{μ A _{c}}{2} cos (ω_{c} - ω_{m}) t

Carrier plus two sidebands, each of amplitude $μ A_{c} /2$ . Bandwidth $B = 2 f_{m}$ (for a general message, $B = 2 W$ where $W$ is the highest message frequency).

POWER AND EFFICIENCY (SINGLE TONE)

P_{t} = P_{c} (1 + \frac{μ ^{2}}{2}), P_{c} = \frac{A _{c}^{2}}{2 R}, P_{S B} = \frac{μ ^{2}}{2} P_{c}, η = \frac{P _{S B}}{P _{t}} = \frac{μ ^{2}}{2 + μ ^{2}}

These power relations hold for any $μ$ . Without overmodulation ( $μ \leq 1$ ) the efficiency is at most $1/3$ , reached at $μ = 1$ . Each sideband carries $μ^{2} P_{c} /4$ . In terms of antenna (load) current, $I_{t} = I_{c} 1 + μ^{2} /2$ .