Communications

Multiplying a signal by a carrier slides its spectrum up the frequency axis with its shape intact. See for yourself why antennas shrink and many signals stack side by side without overlap.

Ride the message on the carrier’s amplitude and the waveform’s envelope becomes the message. Raise the modulation index m yourself and watch a faithful envelope at m ≤ 1 fold and distort into overmodulation past m > 1.

Ride the message on the carrier’s instantaneous frequency and the carrier crowds and thins while its amplitude stays flat. Raise the deviation Δf yourself to cross from narrowband to wideband FM, and see why information in the frequency makes FM resist noise.

Sample a continuous signal into dots and, if fs > 2 fmax, the original comes back perfectly. Sweep the rate across the Nyquist bound to watch aliasing, then follow PCM’s quantize-and-encode path that turns the signal into bits.

A sinusoidal carrier has only three knobs — amplitude, frequency, phase — and switching one with the bits gives ASK, FSK or PSK. Toggle the same bit stream through all three and see why PSK, using the phase, resists noise best.

Bandwidth B caps the symbol rate — Rs ≤ 2B (Nyquist). Raise the symbol rate yourself to the limit where pulses overlap into ISI, and see why adding bits in the same band means making each symbol heavier (larger M).

A receiver adds random noise to every signal, and whether it is read depends not on raw strength but on the signal-to-noise ratio SNR. Lower the SNR yourself to watch received levels cross the decision threshold and flip bits, and meet thermal noise N = k T B.

A single noise sample cannot be known in advance, but collect many and the histogram always converges to the same bell-shaped normal distribution. Raise the sample count yourself to watch it fit the curve, and see that the distribution is pinned by just two numbers, mean and variance.

There is a ceiling on the error-free bit rate, pinned by bandwidth and SNR alone at C = B log₂(1 + SNR). Raise the SNR yourself to watch the capacity curve cross the target rate, and see how bandwidth and power are traded.

Add redundant check bits computed from the data and you can detect and correct the bits noise flips. Toggle no-code, parity and Hamming yourself to see how minimum distance strengthens from detection to correction.