CM · Noise and information

Noise and SNR

Every signal that enters a receiver has random noise added to it. Whether the signal survives is decided not by its absolute strength but by the ratio of signal power to noise power, the SNR. This unit shows why, even for the same signal, a low SNR buries it in noise and flips bits.

Is the signal beating the noise?

The two faint horizontal lines are the two transmitted levels, and the dashed line in the middle is the decision threshold. Each dot at the tip of a bar is one received value with noise added; at a high SNR the dots hug the two levels and split cleanly at the threshold. Lower the SNR and the dots scatter, and any dot that crosses the threshold to the wrong side turns into a red error.

Signal-to-noise ratio SNRSNR = 4.0 dB

At the starting 4 dB, one of the 14 received values has crossed the threshold and shows red; above 10 dB the errors disappear. The noise pattern is fixed, so changing the SNR only makes the same jitter larger or smaller.

SNR and errors right now

SNR = 2.5 = 4.0 dB

Buried in noise

Noise adds on

Electrons in a wire jiggle ceaselessly because of temperature, and that jiggling produces a small random voltage in every receiver. So the received signal is the transmitted signal plus noise, r(t) = s(t) + n(t). Noise cannot be erased, so the only thing to do is make the signal large enough to stand well above it.

What matters is the ratio — SNR

On a channel where boosting the signal boosts the noise too, absolute strength alone says nothing. What decides whether the receiver can tell the bits apart is the ratio of signal power to noise power, SNR = S/N. When signal power far exceeds noise power, the two levels sit well apart and stay distinct; as noise power creeps up toward signal power, the two levels blur together and the distinction collapses. The ratio is usually written in decibels, SNR(dB) = 10 log₁₀(S/N).

ObserveSNR = SN

SNR is the signal power divided by the noise power.

ChooseSNR(dB) = ?

Taking 10 log₁₀ of a power ratio turns it into decibels.

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 the receiver lets in. Opening a wider band therefore lets in that much more noise. That leaves two ways to raise the SNR: increase the signal power, or narrow the band to only what is needed and so cut the noise. And this SNR in turn 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 → ?

Halve the band and the noise power halves too.

Back to the first screen

As you lowered the SNR, the received values that had hugged the two levels scattered more and more, and any value that crossed the threshold became a red error. Raising the SNR again drew the values back onto the two levels, clean and gold. Through all of this, the absolute strength of the signal never appeared anywhere on the screen. One thing to remember about noise: what decides whether a signal can be read is not how large it is but how far 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 can be read is not its absolute strength but the signal-to-noise ratio SNR = S/N, or in decibels 10 log₁₀(S/N). Thermal noise power is N = k T B, proportional to bandwidth. When the SNR is low, received values cross the decision threshold and errors occur, and in the end the SNR sets the Shannon capacity, the information limit of the channel.

In the next unit

Noise is random, so its value at any single instant cannot be known in advance. Handling noise therefore needs the language of probability. The next unit shows how to describe random signals with distributions, means, and variances. The starting point is why the bell-shaped normal distribution is the basic face of noise, and what average power means.