# Signal-to-Noise Ratio

The quantity that measures the relationship between the strength of an information-carrying signal in an electrical communications system and the random fluctuations in amplitude, phase, and frequency superimposed on that signal and collectively referred to as noise. For analog signals, the ratio, denoted S/N, is usually stated in terms of the relative amounts of electrical power contained in the signal and noise. For digital signals the ratio is defined as the amount of energy in the signal per bit of information carried by the signal, relative to the amount of noise power per hertz of signal bandwidth (the noise power spectral density), and is denoted Eb/N0. Since both signal and noise fluctuate randomly with time, S/N and Eb/N0 are specified in terms of statistical or time averages of these quantities.

The magnitude of the signal-to-noise ratio in a communications systems is an important factor in how well a receiver can recover the information-carrying signal from its corrupted version and hence how reliably information can be communicated. Generally speaking, for a given value of S/N the performance depends on how the information quantities are encoded into the signal parameters and on the method of recovering them from the received signal. The more complex encoding methods such as phase-shift keying or quadrature amplitude-shift keying usually result in better performance than simpler schemes such as amplitude- or frequency-shift keying. As an example, a digital communication system operating at a bit error rate of 10-5 requires as much as 7 dB less for Eb/N0 when employing binary phase-shift keying as when using binary amplitude-shift keying.

## Signal-to-Noise Ratio and its Application to Data Acquisition and Signal Conditioning

Signal-to-noise ratio (aka S/N or SNR) is an engineering term for the power ratio between a signal (meaningful information) and the background noise. it's a figure of importance because it describes the quality of a communication system.

Let's use NASA's robotic space explorers as an example. The signal transmitter power on these explorers are surprisingly small compared to the transmitting power of local radio and TV stations here on Earth. Yet NASA successfully receives information sent from billion of miles -- and has done so since the 1960s.

Noise sources are everywhere: noise from sensors, noise in the signal-conditioning electronics, noise in the transmitter, noise from the space, noise in the receiving antenna, noise in the receiver circuits, noise (error) in the software programs used to extract data from signals, etc. Hence, signal-to-noise ratio an important design consideration associated with communication transmitting and receiving systems.

The mathematics related to calculating total communication system S/N can is quite technical. But, the basic definition of S/N is as follows:

S/N = 10*Log(PowerSignal / PowerNoise) , or

S/N = 20*Log(VoltageSignal-RMS / VoltageNoise-RMS)

Determining the actual signal and all the noise voltage sources can gets quite technical. e.g., let's consider an analog to digital converter (ADC) which are ubiquitously found in data-acquisition equipment. These units can add "quantization noise" since there is quantization conversion uncertainty of ± LSB/2. An n-bit ADC with a sinusoidal signal input has a signal-to-noise ratio. If this ADC is operated in the over sampling mode, then the signal-to-noise ratio is given by:

S/N = 6.02*N+1.76 dB + Log(OSR), where OSR (over-sampling ratio) is defined as the ratio of sampling frequency (fs) to twice the bandwidth-limited signal frequency (fo), OSR=fs / 2*fo.

These are simple examples which demonstrate that as communication systems become more and more complicated, accurately calculating S/N becomes equally complex.

Although the S/N as a parameter of importance does not generally apply to industrial process instrumentation and control modules, you can calculate a basic value from DAQ hardware specifications. e.g., in a given Analog Voltage Input Module, "Narrow Bandwidth" has a maximum output of 1 VDC (same as RMS) and a maximum noise output of 200 micro-volts, RMS. These specifications give a S/N of 20*log (1÷200E-6) = 74dB, which denotes that a 1 volt output is 5000 times bigger than the module noise.