the signal-to-noise ratio

A whisper is easy to hear in a quiet library and impossible at a rock concert. What matters is not how loud the whisper is in absolute terms, but how it compares to the background din. The signal-to-noise ratio, SNR, captures exactly that: how big your wanted signal is compared to the unwanted noise riding along with it. It is the single most useful number for judging whether a measurement, a recording, or a link is actually any good.

SNR is a ratio of powers, or equivalently of voltages squared, and it is almost always quoted in decibels: SNR(dB) = 20 times log10(Vsignal / Vnoise) for voltages, or 10 times log10(Psignal / Pnoise) for powers. Example: a 1 V signal sitting on 1 mV of noise has a voltage ratio of 1000, so SNR = 20 times log10(1000) = 60 dB. Every 20 dB is a factor of 10 in voltage, and 6 dB is a factor of 2. A good audio system reaches 90 to 110 dB; a marginal one struggles below 40 dB and sounds plainly hissy.

SNR matters because it sets what you can actually resolve. Each stage in a signal chain adds its own noise, and the FIRST stage usually dominates the total, which is why a low-noise front end matters most of all. You improve SNR by raising the signal (more gain early, a bigger sensor output), by lowering the noise (narrower bandwidth, lower source resistance, cooler parts), or by averaging, since averaging N independent readings improves SNR by sqrt(N). The honest limit: you can never recover a signal already buried below the noise floor simply by amplifying, because gain lifts the signal and the noise together.