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In the above formula, P is measured in units of power, such as watts (W) or milliwatts (mW), and the signal-to-noise ratio is a pure number. However, when the signal and noise are measured in volts (V) or amperes (A), which are measures of amplitude, [note 1] they must first be squared to obtain a quantity proportional to power, as shown below:
The Shannon–Hartley theorem says that the limit of reliable information rate (data rate exclusive of error-correcting codes) of a channel depends on bandwidth and signal-to-noise ratio according to:
BER comparison between BPSK and differentially encoded BPSK with gray-coding operating in white noise. In a noisy channel, the BER is often expressed as a function of the normalized carrier-to-noise ratio measure denoted Eb/N0, (energy per bit to noise power spectral density ratio), or Es/N0 (energy per modulation symbol to noise spectral density).
If the noise source is correlated with the signal, such as in the case of quantisation error, the intentional introduction of additional noise, called dither, can reduce overall noise in the bandwidth of interest. This technique allows retrieval of signals below the nominal detection threshold of an instrument.
Noise reduction, the recovery of the original signal from the noise-corrupted one, is a very common goal in the design of signal processing systems, especially filters. The mathematical limits for noise removal are set by information theory .
Peak signal-to-noise ratio is a metric used to measure the maximum power of a signal to the noise. It is commonly used in image signals because the pixel intensity in an image does not directly represent the actual signal value. Instead, the pixel intensity corresponds to color values, such as white being represented as 255 and black as 0
In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09 dB. The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle or sawtooth. For complex signals in high-resolution ADCs this is an accurate model.
A minimum detectable signal is a signal at the input of a system whose power allows it to be detected over the background electronic noise of the detector system. It can alternately be defined as a signal that produces a signal-to-noise ratio of a given value m at the output. In practice, m is usually chosen to be greater than unity.