CM · Noise and information

Noise and SNR

A receiver adds random noise to every signal. Whether the signal survives is decided not by its raw strength but by the ratio of signal power to noise power — the SNR. Learn why the same signal, at low SNR, sinks into the noise and flips its bits.

Is the signal beating the noise?

Sweep the SNR. With high SNR the received levels hug the two values and split cleanly at the threshold (gold); as it drops they scatter, and symbols that cross the threshold turn red as errors.

Signal-to-noise ratio SNRSNR = 4.0 dB

Sweep the SNR. The same noise pattern grows and shrinks with it.

SNR and errors right now

SNR = 2.5 = 4.0 dB

Buried in noise

Noise adds on

Electrons in a wire jiggle endlessly because of temperature, and that jiggle makes a small random voltage in every receiver. The signal you receive is the one you sent with this noise added on: r(t) = s(t) + n(t). Noise cannot be erased. The only thing you can do is make the signal large enough to stand over it.

What matters is the ratio — SNR

On a channel where boosting the signal also boosts the noise, raw strength tells you nothing. What actually decides whether the receiver can split the bits is the ratio of signal power S to noise power N, SNR = S/N. When S is far above N the two levels sit well apart and clear; when N creeps up to S the two levels overlap and the distinction collapses. The ratio is usually written in decibels: SNR(dB) = 10 log₁₀(S/N).

ObserveSNR = SN

SNR is signal power over noise power.

ChooseSNR(dB) = ?

A power ratio becomes decibels via 10 log₁₀.

Thermal noise and bandwidth

Thermal noise power is N = k T B, proportional to Boltzmann’s constant k, the absolute temperature T, and the bandwidth B you admit. So the wider you open the band, the more noise comes in. There are two ways to raise SNR: increase the signal power, or narrow the band to just what you need and let in less noise. And this SNR is exactly what sets the Shannon capacity — the absolute limit on the information a channel can carry.

Fill inN = k T ?

Thermal noise power is proportional to bandwidth.

On your ownB → B/2 : N → ?

Narrow the band and the noise power falls with it.

Back to the first screen

As you lowered the SNR the received values, once snug against the two levels, scattered ever wider, and any symbol that crossed the threshold turned red as an error. Raise the SNR again and the values are pulled back into the two levels, clean and gold. Through all of it, the signal’s raw strength never once appeared on screen. The one thing to hear in noise is this: what decides whether it is read is not how large the signal is, but how high it stands above the noise — the ratio S/N.

Noise is the random fluctuation a receiver adds to every signal, so r(t) = s(t) + n(t). What decides whether the signal is read is not raw strength but the signal-to-noise ratio SNR = S/N, in decibels 10 log₁₀(S/N). Thermal noise power is N = k T B, proportional to bandwidth. At low SNR the received levels cross the decision threshold and cause errors, and SNR ultimately sets the channel’s information limit, the Shannon capacity.

In the next unit

Noise is random, so you cannot know its value at any instant in advance. To handle it you need the language of probability. Next we look at how to describe a random signal by its distribution, mean and variance — why the bell-shaped normal distribution is noise’s default face, and what average power means.