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An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. Though an ideal switch's open resistance is infinite, the formula still applies. Though an ideal switch's open resistance is infinite, the formula still applies.
In information theory, the noisy-channel coding theorem (sometimes Shannon's theorem or Shannon's limit), establishes that for any given degree of noise contamination of a communication channel, it is possible (in theory) to communicate discrete data (digital information) nearly error-free up to a computable maximum rate through the channel.
Taking into account both noise and bandwidth limitations, however, there is a limit to the amount of information that can be transferred by a signal of a bounded power, even when sophisticated multi-level encoding techniques are used. In the channel considered by the Shannon–Hartley theorem, noise and signal are combined by addition.
The noise equivalent bandwidth (or equivalent noise bandwidth (enbw)) of a system of frequency response is the bandwidth of an ideal filter with rectangular frequency response centered on the system's central frequency that produces the same average power outgoing () when both systems are excited with a white noise source. The value of the ...
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 .
An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem: C = B log 2 ( 1 + S N ) {\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)\ }
To derive the criterion, we first express the received signal in terms of the transmitted symbol and the channel response. Let the function h(t) be the channel impulse response, x[n] the symbols to be sent, with a symbol period of T s; the received signal y(t) will be in the form (where noise has been ignored for simplicity):
is the noise spectral density, the noise power in a 1 Hz bandwidth, measured in watts per hertz or joules. These are the same units as E b {\displaystyle E_{b}} so the ratio E b / N 0 {\displaystyle E_{b}/N_{0}} is dimensionless ; it is frequently expressed in decibels .