In digital communication or data transmission, (energy per bit to noise power spectral density ratio) is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit". It is especially useful when comparing the bit error rate (BER) performance of different digital modulation schemes without taking bandwidth into account.

is the signal energy associated with each user data bit; it is equal to the signal power divided by the user bit rate (not is in units of joules is the noise spectral density, the noise power in a 1Hz bandwidth, measured in watts per hertz or joules.

is dimensionless; it is frequently expressed in decibels directly indicates the power efficiency of the system without regard to modulation type, error correction coding or signal bandwidth (including any use of spread spectrum). This also avoids any confusion as to which of several definitions of "bandwidth" to apply to the signal.

is also equal to the signal-to-noise ratio (SNR) in that bandwidth divided by the "gross" link spectral efficiency in (bit/s)/Hz, where the bits in this context again refer to user data bits, irrespective of error correction information and modulation type.

) is assumed, and interference is not always noise-like. In spread spectrum systems (e.g., CDMA), the interference is.

Relation to carrier-to-noise ratio

is closely related to the carrier-to-noise ratio), i.e. the signal-to-noise ratio (SNR) of the received signal, after the receiver filter but before detection:

where is the channel data rate (net bit rate) and is the channel bandwidth.

The equivalent expression in logarithmic form (dB):

when negative frequencies and complex-valued equivalent baseband signals are considered rather than passband signals, and in that case, there will be a 3dB difference.

Relation to Es/N0

can be seen as a normalized measure of the energy per symbol to noise power spectral density):

is the energy per symbol in joules and is the nominal spectral efficiency in (bits/s)/Hz. is also commonly used in the analysis of digital modulation schemes. The two quotients are related to each other according to the following:

where for 8PSK.

This is the energy per bit, not the energy per information bit.

can further be expressed as:

where is the carrier-to-noise ratio or signal-to-noise ratio, is the channel bandwidth in hertz, and is the symbol rate in baud or symbols per second.

Shannon limit

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:

where is the information rate in bits per second excluding error-correcting codes, is the bandwidth of the channel in hertz, is the total signal power (equivalent to the carrier power ), and is the total noise power in the bandwidth.

for any system that achieves reliable communication, by considering a gross bit rate equal to the net bit rate , a bandwidth utilization parameter of bits per second per half hertz, or bits per dimension (a signal of bandwidth dimensions, according to the Nyquist–Shannon sampling theorem). Making appropriate substitutions, the Shannon limit is:

:

is near zero, the bound, sometimes called the ultimate Shannon limit, is:

which corresponds to −1.59dB.

This often-quoted limit of −1.59dB applies only to the theoretical case of infinite bandwidth. The Shannon limit for finite-bandwidth signals is always higher.

Cutoff rate

For any given system of coding and decoding, there exists what is known as a cutoff rate about 2 dB above the Shannon capacity limit. The cutoff rate used to be thought of as the limit on practical error correction codes without an unbounded increase in processing complexity, but has been rendered largely obsolete by the more recent discovery of turbo codes, low-density parity-check (LDPC) and polar codes.

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