Signal-to-noise ratio is a measure used in science and engineering that compares the
level of a desired signal to the level of background noise. It is defined as the ratio
of signal power to the noise power, often expressed in decibels. A ratio higher than
1:1 indicates more signal than noise. While SNR is commonly quoted for electrical signals,
it can be applied to any form of signal. The signal-to-noise ratio, the bandwidth,
and the channel capacity of a communication channel are connected by the Shannon–Hartley
theorem. Signal-to-noise ratio is sometimes used informally
to refer to the ratio of useful information to false or irrelevant data in a conversation
or exchange. For example, in online discussion forums and other online communities, off-topic
posts and spam are regarded as "noise" that interferes with the "signal" of appropriate
discussion.
Definition Signal-to-noise ratio is defined as the power
ratio between a signal and the background noise:
where P is average power. Both signal and noise power must be measured at the same or
equivalent points in a system, and within the same system bandwidth.
If the variance of the signal and noise are known, and the signal is zero-mean:
If the signal and the noise are measured across the same impedance, then the SNR can be obtained
by calculating the square of the amplitude ratio:
where A is root mean square amplitude. Because many signals have a very wide dynamic range,
SNRs are often expressed using the logarithmic decibel scale. In decibels, the SNR is defined
as
which may equivalently be written using amplitude ratios as
The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic
range measures the ratio between the strongest un-distorted signal on a channel and the minimum
discernible signal, which for most purposes is the noise level. SNR measures the ratio
between an arbitrary signal level and noise. Measuring signal-to-noise ratios requires
the selection of a representative or reference signal. In audio engineering, the reference
signal is usually a sine wave at a standardized nominal or alignment level, such as 1 kHz
at +4 dBu. SNR is usually taken to indicate an average
signal-to-noise ratio, as it is possible that instantaneous signal-to-noise ratios will
be considerably different. The concept can be understood as normalizing the noise level
to 1 and measuring how far the signal 'stands out'.
Difference from conventional power In Physics, the average power of an AC signal
is defined as the average value of voltage times current; for resistive circuits, where
voltage and current are in phase, this is equivalent to the product of the rms voltage
and current:
But in signal processing and communication, one usually assumes that so that factor is
usually not included while measuring power or energy of a signal. This may cause some
confusion among readers, but the resistance factor is not significant for typical operations
performed in signal processing, or for computing power ratios. For most cases, the power of
a signal would be considered to be simply
where 'A' is the amplitude of the AC signal. Alternative definition
An alternative definition of SNR is as the reciprocal of the coefficient of variation,
i.e., the ratio of mean to standard deviation of a signal or measurement:
where is the signal mean or expected value and is the standard deviation of the noise,
or an estimate thereof. Notice that such an alternative definition is only useful for
variables that are always non-negative. Thus it is commonly used in image processing, where
the SNR of an image is usually calculated as the ratio of the mean pixel value to the
standard deviation of the pixel values over a given neighborhood. Sometimes SNR is defined
as the square of the alternative definition above.
The Rose criterion states that an SNR of at least 5 is needed to be able to distinguish
image features at 100% certainty. An SNR less than 5 means less than 100% certainty in identifying
image details. Yet another alternative, very specific and
distinct definition of SNR is employed to characterize sensitivity of imaging systems;
see Signal-to-noise ratio. Related measures are the "contrast ratio"
and the "contrast-to-noise ratio". SNR for various modulation systems
Amplitude modulation Channel signal-to-noise ratio is given by
where W is the bandwidth and is modulation index
Output signal-to-noise ratio is given by
Frequency modulation Channel signal-to-noise ratio is given by
Output signal-to-noise ratio is given by
Improving SNR in practice
All real measurements are disturbed by noise. This includes electronic noise, but can also
include external events that affect the measured phenomenon — wind, vibrations, gravitational
attraction of the moon, variations of temperature, variations of humidity, etc., depending on
what is measured and of the sensitivity of the device. It is often possible to reduce
the noise by controlling the environment. Otherwise, when the characteristics of the
noise are known and are different from the signals, it is possible to filter it or to
process the signal. For example, it is sometimes possible to use
a lock-in amplifier to modulate and confine the signal within a very narrow bandwidth
and then filter the detected signal to the narrow band where it resides, thereby eliminating
most of the broadband noise. When the signal is constant or periodic and the noise is random,
it is possible to enhance the SNR by averaging the measurement. In this case the noise goes
down as the square root of the number of averaged samples.
Digital signals When a measurement is digitized, the number
of bits used to represent the measurement determines the maximum possible signal-to-noise
ratio. This is because the minimum possible noise level is the error caused by the quantization
of the signal, sometimes called Quantization noise. This noise level is non-linear and
signal-dependent; different calculations exist for different signal models. Quantization
noise is modeled as an analog error signal summed with the signal before quantization.
This theoretical maximum SNR assumes a perfect input signal. If the input signal is already
noisy, the signal's noise may be larger than the quantization noise. Real analog-to-digital
converters also have other sources of noise that further decrease the SNR compared to
the theoretical maximum from the idealized quantization noise, including the intentional
addition of dither. Although noise levels in a digital system
can be expressed using SNR, it is more common to use Eb/No, the energy per bit per noise
power spectral density. The modulation error ratio is a measure of
the SNR in a digitally modulated signal. Fixed point
For n-bit integers with equal distance between quantization levels the dynamic range is also
determined. Assuming a uniform distribution of input signal
values, the quantization noise is a uniformly distributed random signal with a peak-to-peak
amplitude of one quantization level, making the amplitude ratio 2n/1. The formula is then:
This relationship is the origin of statements like "16-bit audio has a dynamic range of
96 dB". Each extra quantization bit increases the dynamic range by roughly 6 dB.
Assuming a full-scale sine wave signal, the quantization noise approximates a sawtooth
wave with peak-to-peak amplitude of one quantization level and uniform distribution. In this case,
the SNR is approximately
Floating point Floating-point numbers provide a way to trade
off signal-to-noise ratio for an increase in dynamic range. For n bit floating-point
numbers, with n-m bits in the mantissa and m bits in the exponent:
Note that the dynamic range is much larger than fixed-point, but at a cost of a worse
signal-to-noise ratio. This makes floating-point preferable in situations where the dynamic
range is large or unpredictable. Fixed-point's simpler implementations can be used with no
signal quality disadvantage in systems where dynamic range is less than 6.02m. The very
large dynamic range of floating-point can be a disadvantage, since it requires more
forethought in designing algorithms.
Optical SNR Optical signals have a carrier frequency that
is much higher than the modulation frequency. This way the noise covers a bandwidth that
is much wider than the signal itself. The resulting signal influence relies mainly on
the filtering of the noise. To describe the signal quality without taking the receiver
into account, the optical SNR is used. The OSNR is the ratio between the signal power
and the noise power in a given bandwidth. Most commonly a reference bandwidth of 0.1 nm
is used. This bandwidth is independent of the modulation format, the frequency and the
receiver. For instance an OSNR of 20dB/0.1 nm could be given, even the signal of 40 GBit
DPSK would not fit in this bandwidth. OSNR is measured with an optical spectrum analyzer.
See also
Notes
References
External links Taking the Mystery out of the Infamous Formula,"SNR
= 6.02N + 1.76dB," and Why You Should Care. [1] Analog Devices
ADC and DAC Glossary – Maxim Integrated Products
Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don't get lost in the noise
floor – Analog Devices The Relationship of dynamic range to data
word size in digital audio processing Calculation of signal-to-noise ratio, noise
voltage, and noise level Learning by simulations – a simulation showing
the improvement of the SNR by time averaging Dynamic Performance Testing of Digital Audio
D/A Converters Fundamental theorem of analog circuits: a
minimum level of power must be dissipated to maintain a level of SNR