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Allan variance

The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after David W. Allan and expressed mathematically as $$\sigma_y^2(\tau)$$. The Allan deviation (ADEV), also known as sigma-tau, is the square root of the Allan variance, $$\sigma_y(\tau)$$.

The M-sample variance is a measure of frequency stability using M samples, time T between measurements and observation time $$\tau$$. M-sample variance is expressed as

 * $$\sigma_y^2(M, T, \tau).$$

The Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects. The Allan variance and Allan deviation describe frequency stability. See also the section Interpretation of value below.

There are also different adaptations or alterations of Allan variance, notably the modified Allan variance MAVAR or MVAR, the total variance, and the Hadamard variance. There also exist time-stability variants such as time deviation (TDEV) or time variance (TVAR). Allan variance and its variants have proven useful outside the scope of timekeeping and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, thus a derivative exists.

The general M-sample variance remains important, since it allows dead time in measurements, and bias functions allow conversion into Allan variance values. Nevertheless, for most applications the special case of 2-sample, or "Allan variance" with $$T = \tau$$ is of greatest interest.

[[File:5 regimes of Allan variance as a function of averaging time.png|thumb|upright=1.25|Diagram of Allan variance as a function of averaging time, showing the 5 typical regimes.

1. white/flicker phase-modulation noise (PM): At the highest frequency, phase noise dominates. This corresponds to $$\sigma(\tau) \propto \tau^{-1}$$. However, White PM has $$S[f] = f^3$$ but Flicker PM has $$S[f] = f^2$$. The Allan variance plot does not distinguish them. It requires modified Allan variance plot to distinguish them. 2. White frequency-modulation noise (FM): at a lower frequency, white noise in frequency dominates. This corresponds to $$\sigma(\tau) \propto \tau^{-1/2}, S[f] = f^0$$ 3. Flicker FM: $$\sigma(\tau) \propto \tau^0, S[f] \propto f^{-1}$$. This is also called "pink noise". 4. Random Walk FM: $$\sigma(\tau) \propto \tau^{+1/2}, S[f] \propto f^{-2}$$. This is also called "brown noise" or "brownian noise". In this regime, the frequency of the system executes a random walk. In other words, $$df/dt$$ becomes a white noise. 5. Frequency drift: $$\sigma(\tau) \propto \tau^{+1}, S[f] \propto f^{-3}$$. In this regime, the frequency of the system executes a pink noise walk. In other words, $$df/dt$$ becomes a pink noise.]]

Background
When investigating the stability of crystal oscillators and atomic clocks, it was found that they did not have a phase noise consisting only of white noise, but also of flicker frequency noise. These noise forms become a challenge for traditional statistical tools such as standard deviation, as the estimator will not converge. The noise is thus said to be divergent. Early efforts in analysing the stability included both theoretical analysis and practical measurements.

An important side consequence of having these types of noise was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources and make meaningful specifications to require from suppliers. Essentially all forms of scientific and commercial uses were then limited to dedicated measurements, which hopefully would capture the need for that application.

To address these problems, David Allan introduced the M-sample variance and (indirectly) the two-sample variance. While the two-sample variance did not completely allow all types of noise to be distinguished, it provided a means to meaningfully separate many noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any M-sample variance to any N-sample variance via the common 2-sample variance, thus making all M-sample variances comparable. The conversion mechanism also proved that M-sample variance does not converge for large M, thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure.

An early concern was related to time- and frequency-measurement instruments that had a dead time between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a systematic bias into the measurement. Great care was spent in estimating these biases. The introduction of zero-dead-time counters removed the need, but the bias-analysis tools have proved useful.

Another early aspect of concern was related to how the bandwidth of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation $$\tau$$, only low $$\tau$$ values would be affected, while higher values would be unaffected. The change of $$\tau$$ is done by letting it be an integer multiple $$n$$ of the measurement timebase $$\tau_0$$:

 * $$\tau = n \tau_0.$$

The physics of crystal oscillators were analyzed by D. B. Leeson, and the result is now referred to as Leeson's equation. The feedback in the oscillator will make the white noise and flicker noise of the feedback amplifier and crystal become the power-law noises of $$f^{-2}$$ white frequency noise and $$f^{-3}$$ flicker frequency noise respectively. These noise forms have the effect that the standard variance estimator does not converge when processing time-error samples. This mechanics of the feedback oscillators was unknown when the work on oscillator stability started, but was presented by Leeson at the same time as the set of statistical tools was made available by David W. Allan. For a more thorough presentation on the Leeson effect, see modern phase-noise literature.

Interpretation of value
Allan variance is defined as one half of the time average of the squares of the differences between successive readings of the frequency deviation sampled over the sampling period. The Allan variance depends on the time period used between samples, therefore, it is a function of the sample period, commonly denoted as τ, likewise the distribution being measured, and is displayed as a graph rather than a single number. A low Allan variance is a characteristic of a clock with good stability over the measured period.

Allan deviation is widely used for plots (conventionally in log–log format) and presentation of numbers. It is preferred, as it gives the relative amplitude stability, allowing ease of comparison with other sources of errors.

An Allan deviation of 1.3 at observation time 1 s (i.e. τ = 1 s) should be interpreted as there being an instability in frequency between two observations 1 second apart with a relative root mean square (RMS) value of 1.3. For a 10 MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used.

One may convert the Allan variance and other time-domain variances into frequency-domain measures of time (phase) and frequency stability.

M-sample variance
Given a time-series $$x(t)$$, for any positive real numbers $$T, \tau$$, define the real number sequence$$\bar y_i = \frac{x(iT + \tau) - x(iT)}{\tau} \quad i = 0, 1, 2, ...$$Then the $$M$$-sample variance is defined (here in a modernized notation form) as the Bessel-corrected variance of the sequence $$\bar y_0, ..., \bar y_{M-1}$$:$$\sigma_y^2(M, T, \tau) = \frac{M}{M - 1} \left(\frac 1M \sum_{i=0}^{M-1}\bar{y}_i^2 - \left[\frac{1}{M} \sum_{i=0}^{M-1} \bar{y}_i\right]^2 \right),$$The interpretation of the symbols is as follows:

 * $$t$$ is the reading on a reference clock (in arbitrary units).
 * $$x(t)$$ is the reading of a clock we are testing (in arbitrary units), as a function of the reference clock's reading. It can also be interpreted as the average fractional frequency time series.
 * $$\bar{y}_n$$ is the nth fractional frequency average over the observation time $$\tau$$.
 * $$M$$ is the number of clock reading intervals used in computing the $$M$$-sample variance,
 * $$T$$ is the time between each frequency sample,
 * $$\tau$$ is the time length of each frequency estimate, or the observation period.

Dead-time can be accounted for by letting the time $$T$$ be different from that of $$\tau$$.

Allan variance
The Allan variance is defined as

 * $$\sigma_y^2(\tau) = \left\langle\sigma_y^2(2, \tau, \tau)\right\rangle = \frac{1}{2} \left\langle\left(\bar{y}_{n+1} - \bar{y}_n\right)^2\right\rangle = \frac{1}{2\tau^2} \left\langle\left(x_{n+2} - 2x_{n+1} + x_n\right)^2\right\rangle$$

where $$\langle\dotsm\rangle$$ denotes the expectation operator.

The condition $T = \tau$ means the samples are taken with no dead-time between them.

Allan deviation
Just as with standard deviation and variance, the Allan deviation is defined as the square root of the Allan variance:

 * $$\sigma_y(\tau) = \sqrt{\sigma_y^2(\tau)}.$$

Oscillator model
The oscillator being analysed is assumed to follow the basic model of

 * $$V(t) = V_0 \sin (\Phi(t)).$$

The oscillator is assumed to have a nominal frequency of $$\nu_\text{n}$$, given in cycles per second (SI unit: hertz). The nominal angular frequency $$\omega_\text{n}$$ (in radians per second) is given by

 * $$\omega_\text{n} = 2\pi \nu_\text{n}.$$

The total phase can be separated into a perfectly cyclic component $$\omega_\text{n} t$$, along with a fluctuating component $$\varphi(t)$$:

 * $$\Phi(t) = \omega_\text{n}t + \varphi(t) = 2\pi \nu_\text{n}t + \varphi(t).$$

Time error
The time-error function x(t) is the difference between expected nominal time and actual normal time:

 * $$x(t) = \frac{\varphi(t)}{2\pi \nu_\text{n}} = \frac{\Phi(t)}{2\pi \nu_\text{n}} - t = T(t) - t.$$

For measured values a time-error series TE(t) is defined from the reference time function T$ref$(t) as

 * $$TE(t) = T(t) - T_\text{ref}(t).$$

Frequency function
The frequency function $$\nu(t)$$ is the frequency over time, defined as

 * $$\nu(t) = \frac{1}{2\pi} \frac{d\Phi(t)}{dt}.$$

Fractional frequency
The fractional frequency y(t) is the normalized difference between the frequency $$\nu(t)$$ and the nominal frequency $$\nu_\text{n}$$:

 * $$y(t) = \frac{\nu(t) - \nu_\text{n}}{\nu_\text{n}} = \frac{\nu(t)}{\nu_\text{n}} - 1.$$

Average fractional frequency
The average fractional frequency is defined as

 * $$\bar{y}(t, \tau) = \frac{1}{\tau} \int_0^\tau y(t + t_v) \, dt_v,$$

where the average is taken over observation time τ, the y(t) is the fractional-frequency error at time t, and τ is the observation time.

Since y(t) is the derivative of x(t), we can without loss of generality rewrite it as

 * $$\bar{y}(t, \tau) = \frac{x(t + \tau) - x(t)}{\tau}.$$

Estimators
This definition is based on the statistical expected value, integrating over infinite time. The real-world situation does not allow for such time-series, in which case a statistical estimator needs to be used in its place. A number of different estimators will be presented and discussed.

Conventions
• The number of frequency samples in a fractional-frequency series is denoted by M.

• The number of time error samples in a time-error series is denoted by N.

The relation between the number of fractional-frequency samples and time-error series is fixed in the relationship
 * $N = M + 1.$

• For time-error sample series, xi denotes the i-th sample of the continuous time function x(t) as given by

 * $x_i = x(iT),$

where T is the time between measurements. For Allan variance, the time being used has T set to the observation time τ.

The time-error sample series let N denote the number of samples (x0...xN−1) in the series. The traditional convention uses index 1 through N.

• For average fractional-frequency sample series, $\bar{y}_i$ denotes the ith sample of the average continuous fractional-frequency function y(t) as given by

 * $\bar{y}_i = \bar{y}(Ti, \tau),$

which gives

 * $\bar{y}_i = \frac{1}{\tau} \int_0^\tau y(iT + t_v) \, dt_v = \frac{x(iT + \tau) - x(iT)}{\tau}.$

For the Allan variance assumption of T being τ it becomes

 * $\bar{y}_i = \frac{x_{i+1} - x_i}{\tau}.$

The average fractional-frequency sample series lets M denote the number of samples ($\bar{y}_0 \ldots \bar{y}_{M-1}$) in the series. The traditional convention uses index 1 through M.

As a shorthand, average fractional frequency is often written without the average bar over it. However, this is formally incorrect, as the fractional frequency and average fractional frequency are two different functions. A measurement instrument able to produce frequency estimates with no dead-time will actually deliver a frequency-average time series, which only needs to be converted into average fractional frequency and may then be used directly.

• It is further a convention to let τ denote the nominal time difference between adjacent phase or frequency samples. A time series taken for one time difference τ0 can be used to generate Allan variance for any τ being an integer multiple of τ0, in which case τ = nτ0 are being used, and n becomes a variable for the estimator.

• The time between measurements is denoted by T, which is the sum of observation time τ and dead-time.

Fixed τ estimators
A first simple estimator would be to directly translate the definition into

 * $$\sigma_y^2(\tau, M) = \operatorname{AVAR}(\tau, M) = \frac{1}{2(M - 1)} \sum_{i=0}^{M-2}(\bar{y}_{i+1} - \bar{y}_i)^2,$$

or for the time series:

 * $$\sigma_y^2(\tau, N) = \operatorname{AVAR}(\tau, N) = \frac{1}{2\tau^2(N - 2)} \sum_{i=0}^{N-3}(x_{i+2} - 2x_{i+1} + x_i)^2.$$

These formulas, however, only provide the calculation for the τ = τ0 case. To calculate for a different value of τ, a new time-series needs to be provided.

Non-overlapped variable τ estimators
Taking the time-series and skipping past n − 1 samples, a new (shorter) time-series would occur with τ0 as the time between the adjacent samples, for which the Allan variance could be calculated with the simple estimators. These could be modified to introduce the new variable n such that no new time-series would have to be generated, but rather the original time series could be reused for various values of n. The estimators become

 * $$\sigma_y^2(n\tau_0, M) = \operatorname{AVAR}(n\tau_0, M) = \frac{1}{2\frac{M-1}{n}} \sum_{i=0}^{\frac{M-1}{n} - 1}\left(\bar{y}_{ni+n} - \bar{y}_{ni}\right)^2$$

with $$n \le M - 1$$,

and for the time series:

 * $$\sigma_y^2(n\tau_0, N) = \operatorname{AVAR}(n\tau_0, N) = \frac{1}{2n^2\tau_0^2\left(\frac{N - 1}{n} - 1\right)} \sum_{i=0}^{\frac{N-1}{n} - 2}\left(x_{ni+2n} - 2x_{ni+n} + x_{ni}\right)^2$$

with $$n \le \frac{N - 1}{2}$$.

These estimators have a significant drawback in that they will drop a significant amount of sample data, as only 1/n of the available samples is being used.

Overlapped variable τ estimators
A technique presented by J. J. Snyder provided an improved tool, as measurements were overlapped in n overlapped series out of the original series. The overlapping Allan variance estimator was introduced by Howe, Allan and Barnes. This can be shown to be equivalent to averaging the time or normalized frequency samples in blocks of n samples prior to processing. The resulting predictor becomes

\begin{align} \sigma_y^2(n\tau_0, M) & = \operatorname{AVAR}(n\tau_0, M) = \frac{1}{2n^2(M - 2n + 1)} \sum_{j=0}^{M-2n} \left( \sum_{i=j}^{j+n-1} y_{i+n} - y_i \right)^2 \\[5pt] & = \frac{1}{2(M - 2n + 1)} \sum_{j=0}^{M-2n} \left(\bar{y}_{j+n} - \bar{y}_j \right)^2, \end{align} $$

or for the time series:

 * $$\sigma_y^2(n\tau_0, N) = \operatorname{AVAR}(n\tau_0, N) = \frac{1}{2n^2\tau_0^2(N - 2n)} \sum_{i=0}^{N-2n-1} (x_{i+2n} - 2x_{i+n} + x_i)^2.$$

The overlapping estimators have far superior performance over the non-overlapping estimators, as n rises and the time-series is of moderate length. The overlapped estimators have been accepted as the preferred Allan variance estimators in IEEE, ITU-T and ETSI standards for comparable measurements such as needed for telecommunication qualification.

Modified Allan variance
In order to address the inability to separate white phase modulation from flicker phase modulation using traditional Allan variance estimators, an algorithmic filtering reduces the bandwidth by n. This filtering provides a modification to the definition and estimators and it now identifies as a separate class of variance called modified Allan variance. The modified Allan variance measure is a frequency stability measure, just as is the Allan variance.

Time stability estimators
A time stability (σx) statistical measure, which is often called the time deviation (TDEV), can be calculated from the modified Allan deviation (MDEV). The TDEV is based on the MDEV instead of the original Allan deviation, because the MDEV can discriminate between white and flicker phase modulation (PM). The following is the time variance estimation based on the modified Allan variance:

 * $$\sigma_x^2(\tau) = \frac{\tau^2}{3}\bmod\sigma_y^2(\tau),$$

and similarly for modified Allan deviation to time deviation:

 * $$\sigma_x(\tau) = \frac{\tau}{\sqrt{3}}\bmod\sigma_y(\tau).$$

The TDEV is normalized so that it is equal to the classical deviation for white PM for time constant τ = τ0. To understand the normalization scale factor between the statistical measures, the following is the relevant statistical rule: For independent random variables X and Y, the variance (σz2) of a sum or difference (z = x − y) is the sum square of their variances (σz2 = σx2 + σy2). The variance of the sum or difference (y = x2τ − xτ) of two independent samples of a random variable is twice the variance of the random variable (σy2 = 2σx2). The MDEV is the second difference of independent phase measurements (x) that have a variance (σx2). Since the calculation is the double difference, which requires three independent phase measurements (x2τ − 2xτ + x), the modified Allan variance (MVAR) is three times the variances of the phase measurements.

Other estimators
Further developments have produced improved estimation methods for the same stability measure, the variance/deviation of frequency, but these are known by separate names such as the Hadamard variance, modified Hadamard variance, the total variance, modified total variance and the Theo variance. These distinguish themselves in better use of statistics for improved confidence bounds or ability to handle linear frequency drift.

Confidence intervals and equivalent degrees of freedom
Statistical estimators will calculate an estimated value on the sample series used. The estimates may deviate from the true value and the range of values which for some probability will contain the true value is referred to as the confidence interval. The confidence interval depends on the number of observations in the sample series, the dominant noise type, and the estimator being used. The width is also dependent on the statistical certainty for which the confidence interval values forms a bounded range, thus the statistical certainty that the true value is within that range of values. For variable-τ estimators, the τ0 multiple n is also a variable.

Confidence interval
The confidence interval can be established using chi-squared distribution by using the distribution of the sample variance:

 * $$\chi^2 = \frac{\text{df}\,s^2}{\sigma^2},$$

where s2 is the sample variance of our estimate, σ2 is the true variance value, df is the degrees of freedom for the estimator, and χ2 is the degrees of freedom for a certain probability. For a 90% probability, covering the range from the 5% to the 95% range on the probability curve, the upper and lower limits can be found using the inequality

 * $$\chi^2(0.05) \le \frac{\text{df}\,s^2}{\sigma^2} \le \chi^2(0.95),$$

which after rearrangement for the true variance becomes

 * $$\frac{\text{df}\,s^2}{\chi^2(0.95)} \le \sigma^2 \le \frac{\text{df}\,s^2}{\chi^2(0.05)}.$$

Effective degrees of freedom
The degrees of freedom represents the number of free variables capable of contributing to the estimate. Depending on the estimator and noise type, the effective degrees of freedom varies. Estimator formulas depending on N and n has been found empirically:
 * {| class="wikitable"

!Noise type !degrees of freedom
 * + Allan variance degrees of freedom
 * white phase modulation (WPM)
 * $$\text{df} \cong \frac{(N + 1)(N - 2n)}{2(N - n)}$$
 * flicker phase modulation (FPM)
 * $$\text{df} \cong \exp\left[\left(\ln \frac{N - 1}{2n} \ln \frac{(2n + 1)(N - 1)}{4}\right)^{-1/2}\right]$$
 * white frequency modulation (WFM)
 * $$\text{df} \cong \left[ \frac{3(N - 1)}{2n} - \frac{2(N - 2)}{N}\right] \frac{4n^2}{4n^2 + 5}$$
 * flicker frequency modulation (FFM)
 * $$\text{df} \cong \begin{cases}\frac{2(N - 2)}{2.3N - 4.9} & n = 1 \\ \frac{5N^2}{4n(N + 3n)} & n \ge 2\end{cases}$$
 * random-walk frequency modulation (RWFM)
 * $$\text{df} \cong \frac{N - 2}{n}\frac{(N - 1)^2 - 3n(N - 1) + 4n^2}{(N - 3)^2}$$
 * }
 * $$\text{df} \cong \begin{cases}\frac{2(N - 2)}{2.3N - 4.9} & n = 1 \\ \frac{5N^2}{4n(N + 3n)} & n \ge 2\end{cases}$$
 * random-walk frequency modulation (RWFM)
 * $$\text{df} \cong \frac{N - 2}{n}\frac{(N - 1)^2 - 3n(N - 1) + 4n^2}{(N - 3)^2}$$
 * }
 * }

Power-law noise
The Allan variance will treat various power-law noise types differently, conveniently allowing them to be identified and their strength estimated. As a convention, the measurement system width (high corner frequency) is denoted fH.

As found in and in modern forms.

The Allan variance is unable to distinguish between WPM and FPM, but is able to resolve the other power-law noise types. In order to distinguish WPM and FPM, the modified Allan variance needs to be employed.

The above formulas assume that

 * $$\tau \gg \frac{1}{2\pi f_H},$$

and thus that the bandwidth of the observation time is much lower than the instruments bandwidth. When this condition is not met, all noise forms depend on the instrument's bandwidth.

α–μ mapping
The detailed mapping of a phase modulation of the form

 * $$S_x(f) = \frac{1}{4\pi^2} h_\alpha f^{\alpha - 2} = \frac{1}{4\pi^2} h_\alpha f^\beta,$$

where

 * $$\beta \equiv \alpha - 2,$$

or frequency modulation of the form

 * $$S_y(f) = h_\alpha f^\alpha$$

into the Allan variance of the form

 * $$\sigma_y^2(\tau) = K_\alpha h_\alpha \tau^\mu$$

can be significantly simplified by providing a mapping between α and μ. A mapping between α and Kα is also presented for convenience:

 * {| class="wikitable"

!α !β !μ !Kα
 * + Allan variance α–μ mapping
 * −2
 * −4
 * 1
 * $$\frac{2\pi^2}{3}$$
 * −1
 * −3
 * 0
 * $$2\ln 2$$
 * 0
 * −2
 * −1
 * $$\frac{1}{2}$$
 * 1
 * −1
 * −2
 * $$\frac{3[\gamma+\ln(2\pi f_H\tau)]-\ln 2}{4\pi^2}$$
 * 2
 * 0
 * −2
 * $$\frac{3f_H}{4\pi^2}$$
 * }
 * 2
 * 0
 * −2
 * $$\frac{3f_H}{4\pi^2}$$
 * }

General conversion from phase noise
A signal with spectral phase noise $$S_\varphi$$ with units rad2/Hz can be converted to Allan Variance by

 * $$\sigma^2_y(\tau) = \frac{2}{\nu_0^2} \int^{f_b}_0 S_\varphi(f) \frac{\sin^4(\pi \tau f)}{(\pi \tau)^2} \, df.$$

Linear response
While Allan variance is intended to be used to distinguish noise forms, it will depend on some but not all linear responses to time. They are given in the table:
 * {| class="wikitable"

! Linear effect ! time response ! frequency response ! Allan variance ! Allan deviation
 * + Allan variance linear response
 * phase offset
 * $$x_0$$
 * $$0$$
 * $$0$$
 * $$0$$
 * frequency offset
 * $$y_0t$$
 * $$y_0$$
 * $$0$$
 * $$0$$
 * linear drift
 * $$\frac{Dt^2}{2}$$
 * $$Dt$$
 * $$\frac{D^2\tau^2}{2}$$
 * $$\frac{D\tau}{\sqrt{2}}$$
 * }
 * $$\frac{D^2\tau^2}{2}$$
 * $$\frac{D\tau}{\sqrt{2}}$$
 * }

Thus, linear drift will contribute to output result. When measuring a real system, the linear drift or other drift mechanism may need to be estimated and removed from the time-series prior to calculating the Allan variance.

Time and frequency filter properties
In analysing the properties of Allan variance and friends, it has proven useful to consider the filter properties on the normalize frequency. Starting with the definition for Allan variance for

 * $$\sigma_y^2(\tau) = \frac{1}{2}\left\langle\left(\bar{y}_{i+1} - \bar{y}_i\right)^2\right\rangle,$$

where

 * $$\bar{y}_i = \frac{1}{\tau} \int_0^\tau y(i\tau + t) \, dt.$$

Replacing the time series of $$y_i$$ with the Fourier-transformed variant $$S_y(f)$$ the Allan variance can be expressed in the frequency domain as

 * $$\sigma_y^2(\tau) = \int_0^\infty S_y(f) \frac{2\sin^4\pi\tau f}{(\pi \tau f)^2} \, df.$$

Thus the transfer function for Allan variance is

 * $$\left\vert H_A(f)\right\vert^2 = \frac{2\sin^4\pi \tau f}{(\pi \tau f)^2}.$$

Bias functions
The M-sample variance, and the defined special case Allan variance, will experience systematic bias depending on different number of samples M and different relationship between T and τ. In order to address these biases the bias-functions B1 and B2 has been defined and allows conversion between different M and T values.

These bias functions are not sufficient for handling the bias resulting from concatenating M samples to the Mτ0 observation time over the MT0 with the dead-time distributed among the M measurement blocks rather than at the end of the measurement. This rendered the need for the B3 bias.

The bias functions are evaluated for a particular μ value, so the α–μ mapping needs to be done for the dominant noise form as found using noise identification. Alternatively, the μ value of the dominant noise form may be inferred from the measurements using the bias functions.

B1 bias function
The B1 bias function relates the M-sample variance with the 2-sample variance (Allan variance), keeping the time between measurements T and time for each measurements τ constant. It is defined as

 * $$B_1(N, r, \mu) = \frac{\left\langle\sigma_y^2(N, T, \tau)\right\rangle}{\left\langle\sigma_y^2(2, T, \tau)\right\rangle},$$

where

 * $$r = \frac{T}{\tau}.$$

The bias function becomes after analysis

 * $$B_1(N, r, \mu) = \frac{1 + \sum_{n=1}^{N-1} \frac{N - n}{N(N - 1)} \left[ 2(rn)^{\mu+2} - (rn + 1)^{\mu+2} - |rn - 1|^{\mu+2} \right]}{1 + \frac{1}{2} \left[ 2r^{\mu+2} - (r + 1)^{\mu+2} - |r - 1|^{\mu+2} \right]}.$$

B2 bias function
The B2 bias function relates the 2-sample variance for sample time T with the 2-sample variance (Allan variance), keeping the number of samples N = 2 and the observation time τ constant. It is defined as

 * $$B_2(r, \mu) = \frac{\left\langle\sigma_y^2(2, T, \tau)\right\rangle}{\left\langle\sigma_y^2(2, \tau, \tau)\right\rangle},$$

where

 * $$r = \frac{T}{\tau}.$$

The bias function becomes after analysis

 * $$B_2(r, \mu) = \frac{1 + \frac{1}{2} \left[ 2r^{\mu+2} - (r + 1)^{\mu+2} - |r - 1|^{\mu+2} \right]}{2\left(1 - 2^\mu\right)}.$$

B3 bias function
The B3 bias function relates the 2-sample variance for sample time MT0 and observation time Mτ0 with the 2-sample variance (Allan variance) and is defined as

 * $$B_3(N, M, r, \mu) = \frac{\left\langle\sigma_y^2(N, M, T, \tau)\right\rangle}{\left\langle\sigma_y^2(N, T, \tau)\right\rangle},$$

where
 * $$T = M T_0,$$
 * $$\tau = M \tau_0.$$

The B3 bias function is useful to adjust non-overlapping and overlapping variable τ estimator values based on dead-time measurements of observation time τ0 and time between observations T0 to normal dead-time estimates.

The bias function becomes after analysis (for the N = 2 case)

 * $$B_3(2, M, r, \mu) = \frac{2M + MF(Mr) - \sum_{n=1}^{M-1} (M - n) \left[ 2F(nr) - F\big((M + n)r\big) + F\big((M - n)r\big) \right]}{M^{\mu+2} [F(r) + 2]},$$

where

 * $$F(A) = 2A^{\mu+2} - (A + 1)^{\mu+2} - |A - 1|^{\mu+2}.$$

τ bias function
While formally not formulated, it has been indirectly inferred as a consequence of the α–μ mapping. When comparing two Allan variance measure for different τ, assuming same dominant noise in the form of same μ coefficient, a bias can be defined as

 * $$B_\tau(\tau_1, \tau_2, \mu) = \frac{\left\langle\sigma_y^2(2, \tau_2, \tau_2)\right\rangle}{\left\langle\sigma_y^2(2, \tau_1, \tau_1) \right\rangle}.$$

The bias function becomes after analysis

 * $$B_\tau(\tau_1, \tau_2, \mu) = \left( \frac{\tau_2}{\tau_1} \right)^\mu.$$

Conversion between values
In order to convert from one set of measurements to another the B1, B2 and τ bias functions can be assembled. First the B1 function converts the (N1, T1, τ1) value into (2, T1, τ1), from which the B2 function converts into a (2, τ1, τ1) value, thus the Allan variance at τ1. The Allan variance measure can be converted using the τ bias function from τ1 to τ2, from which then the (2, T2, τ2) using B2 and then finally using B1 into the (N2, T2, τ2) variance. The complete conversion becomes

 * $$\left\langle \sigma_y^2(N_2, T_2, \tau_2) \right\rangle = \left( \frac{\tau_2}{\tau_1} \right)^\mu \left[ \frac{B_1(N_2, r_2, \mu) B_2(r_2, \mu)}{B_1(N_1, r_1, \mu) B_2(r_1, \mu)} \right] \left\langle \sigma_y^2(N_1, T_1, \tau_1) \right\rangle,$$

where
 * $$r_1 = \frac{T_1}{r_1},$$
 * $$r_2 = \frac{T_2}{r_2}.$$

Similarly, for concatenated measurements using M sections, the logical extension becomes

 * $$\left\langle \sigma_y^2(N_2, M_2, T_2, \tau_2) \right\rangle = \left( \frac{\tau_2}{\tau_1} \right)^\mu \left[ \frac{B_3(N_2, M_2, r_2, \mu) B_1(N_2, r_2, \mu) B_2(r_2, \mu)}{B_3(N_1, M_1, r_1, \mu) B_1(N_1, r_1, \mu) B_2(r_1, \mu)} \right] \left\langle \sigma_y^2(N_1, M_1, T_1, \tau_1) \right\rangle.$$

Measurement issues
When making measurements to calculate Allan variance or Allan deviation, a number of issues may cause the measurements to degenerate. Covered here are the effects specific to Allan variance, where results would be biased.

Measurement bandwidth limits
A measurement system is expected to have a bandwidth at or below that of the Nyquist rate, as described within the Shannon–Hartley theorem. As can be seen in the power-law noise formulas, the white and flicker noise modulations both depends on the upper corner frequency $$f_H$$ (these systems is assumed to be low-pass filtered only). Considering the frequency filter property, it can be clearly seen that low-frequency noise has greater impact on the result. For relatively flat phase-modulation noise types (e.g. WPM and FPM), the filtering has relevance, whereas for noise types with greater slope the upper frequency limit becomes of less importance, assuming that the measurement system bandwidth is wide relative the $$\tau$$ as given by

 * $$\tau \gg \frac{1}{2\pi f_H}.$$

When this assumption is not met, the effective bandwidth $$f_H$$ needs to be notated alongside the measurement. The interested should consult NBS TN394.

If, however, one adjust the bandwidth of the estimator by using integer multiples of the sample time $$n\tau_0$$, then the system bandwidth impact can be reduced to insignificant levels. For telecommunication needs, such methods have been required in order to ensure comparability of measurements and allow some freedom for vendors to do different implementations. The ITU-T Rec. G.813 for the TDEV measurement.

It can be recommended that the first $$\tau_0$$ multiples be ignored, such that the majority of the detected noise is well within the passband of the measurement systems bandwidth.

Further developments on the Allan variance was performed to let the hardware bandwidth be reduced by software means. This development of a software bandwidth allowed addressing the remaining noise, and the method is now referred to modified Allan variance. This bandwidth reduction technique should not be confused with the enhanced variant of modified Allan variance, which also changes a smoothing filter bandwidth.

Dead time in measurements
Many measurement instruments of time and frequency have the stages of arming time, time-base time, processing time and may then re-trigger the arming. The arming time is from the time the arming is triggered to when the start event occurs on the start channel. The time-base then ensures that minimal amount of time goes prior to accepting an event on the stop channel as the stop event. The number of events and time elapsed between the start event and stop event is recorded and presented during the processing time. When the processing occurs (also known as the dwell time), the instrument is usually unable to do another measurement. After the processing has occurred, an instrument in continuous mode triggers the arm circuit again. The time between the stop event and the following start event becomes dead time, during which the signal is not being observed. Such dead time introduces systematic measurement biases, which needs to be compensated for in order to get proper results. For such measurement systems will the time T denote the time between the adjacent start events (and thus measurements), while $$\tau$$ denote the time-base length, i.e. the nominal length between the start and stop event of any measurement.

Dead-time effects on measurements have such an impact on the produced result that much study of the field have been done in order to quantify its properties properly. The introduction of zero-dead-time counters removed the need for this analysis. A zero-dead-time counter has the property that the stop event of one measurement is also being used as the start event of the following event. Such counters create a series of event and time timestamp pairs, one for each channel spaced by the time-base. Such measurements have also proved useful in order forms of time-series analysis.

Measurements being performed with dead time can be corrected using the bias function B1, B2 and B3. Thus, dead time as such is not prohibiting the access to the Allan variance, but it makes it more problematic. The dead time must be known, such that the time between samples T can be established.

Measurement length and effective use of samples
Studying the effect on the confidence intervals that the length N of the sample series have and the effect of the variable τ parameter n, the confidence intervals may become very large since the effective degree of freedom may become small for some combination of N and n for the dominant noise form (for that τ).

The effect may be that the estimated value may be much smaller or much greater than the real value, which may lead to false conclusions of the result.

It is recommended that:

 * The confidence interval be plotted along with the data, such that the reader of the plot knows of the statistical uncertainty of the values.
 * The length of the sample sequence (i.e. the number of samples N) must be kept as high as possible to ensure that confidence interval is small over the τ range of interest.
 * Estimators providing better degrees of freedom values be used in replacement of the Allan variance estimators or as complementing them where they outperform the Allan variance estimators. Among those the total variance and Theo variance estimators should be considered.
 * The τ range as swept by the τ0 multiplier n is limited in the upper end relative N, such that the reader of the plot may not be confused by highly unstable estimator values.

Dominant noise type
A large number of conversion constants, bias corrections and confidence intervals depends on the dominant noise type. For proper interpretation shall the dominant noise type for the particular τ of interest be identified through noise identification. Failing to identify the dominant noise type will produce biased values. Some of these biases may be of several order of magnitude, so it may be of large significance.

Linear drift
Systematic effects on the signal is only partly cancelled. Phase and frequency offset is cancelled, but linear drift or other high-degree forms of polynomial phase curves will not be cancelled and thus form a measurement limitation. Curve fitting and removal of systematic offset could be employed. Often removal of linear drift can be sufficient. Use of linear-drift estimators such as the Hadamard variance could also be employed. A linear drift removal could be employed using a moment-based estimator.

Measurement instrument estimator bias
Traditional instruments provided only the measurement of single events or event pairs. The introduction of the improved statistical tool of overlapping measurements by J. J. Snyder allowed much improved resolution in frequency readouts, breaking the traditional digits/time-base balance. While such methods is useful for their intended purpose, using such smoothed measurements for Allan variance calculations would give a false impression of high resolution,  but for longer τ the effect is gradually removed, and the lower-τ region of the measurement has biased values. This bias is providing lower values than it should, so it is an overoptimistic (assuming that low numbers is what one wishes) bias, reducing the usability of the measurement rather than improving it. Such smart algorithms can usually be disabled or otherwise circumvented by using time-stamp mode, which is much preferred if available.

Practical measurements
While several approaches to measurement of Allan variance can be devised, a simple example may illustrate how measurements can be performed.

Measurement
All measurements of Allan variance will in effect be the comparison of two different clocks. Consider a reference clock and a device under test (DUT), and both having a common nominal frequency of 10 MHz. A time-interval counter is being used to measure the time between the rising edge of the reference (channel A) and the rising edge of the device under test.

In order to provide evenly spaced measurements, the reference clock will be divided down to form the measurement rate, triggering the time-interval counter (ARM input). This rate can be 1 Hz (using the 1 PPS output of a reference clock), but other rates like 10 Hz and 100 Hz can also be used. The speed of which the time-interval counter can complete the measurement, output the result and prepare itself for the next arm will limit the trigger frequency.

A computer is then useful to record the series of time differences being observed.

Post-processing
The recorded time-series require post-processing to unwrap the wrapped phase, such that a continuous phase error is being provided. If necessary, logging and measurement mistakes should also be fixed. Drift estimation and drift removal should be performed, the drift mechanism needs to be identified and understood for the sources. Drift limitations in measurements can be severe, so letting the oscillators become stabilized, by long enough time being powered on, is necessary.

The Allan variance can then be calculated using the estimators given, and for practical purposes the overlapping estimator should be used due to its superior use of data over the non-overlapping estimator. Other estimators such as total or Theo variance estimators could also be used if bias corrections is applied such that they provide Allan variance-compatible results.

To form the classical plots, the Allan deviation (square root of Allan variance) is plotted in log–log format against the observation interval τ.

Equipment and software
The time-interval counter is typically an off-the-shelf counter commercially available. Limiting factors involve single-shot resolution, trigger jitter, speed of measurements and stability of reference clock. The computer collection and post-processing can be done using existing commercial or public-domain software. Highly advanced solutions exists, which will provide measurement and computation in one box.

Research history
The field of frequency stability has been studied for a long time. However, during the 1960s it was found that coherent definitions were lacking. A NASA-IEEE Symposium on Short-Term Stability in November 1964 resulted in the special February 1966 issue of the IEEE Proceedings on Frequency Stability.

The NASA-IEEE Symposium brought together many fields and uses of short- and long-term stability, with papers from many different contributors. The articles and panel discussions concur on the existence of the frequency flicker noise and the wish to achieve a common definition for both short-term and long-term stability.

Important papers, including those of David Allan, James A. Barnes, L. S. Cutler and C. L. Searle and D. B. Leeson, appeared in the IEEE Proceedings on Frequency Stability and helped shape the field.

David Allan's article analyses the classical M-sample variance of frequency, tackling the issue of dead-time between measurements along with an initial bias function. Although Allan's initial bias function assumes no dead-time, his formulas do include dead-time calculations. His article analyses the case of M frequency samples (called N in the article) and variance estimators. It provides the now standard α–μ mapping, clearly building on James Barnes' work in the same issue.

The 2-sample variance case is a special case of the M-sample variance, which produces an average of the frequency derivative. Allan implicitly uses the 2-sample variance as a base case, since for arbitrary chosen M, values may be transferred via the 2-sample variance to the M-sample variance. No preference was clearly stated for the 2-sample variance, even if the tools were provided. However, this article laid the foundation for using the 2-sample variance as a way of comparing other M-sample variances.

James Barnes significantly extended the work on bias functions, introducing the modern B1 and B2 bias functions. Curiously enough, it refers to the M-sample variance as "Allan variance", while referring to Allan's article "Statistics of Atomic Frequency Standards". With these modern bias functions, full conversion among M-sample variance measures of various M, T and τ values could be performed, by conversion through the 2-sample variance.

James Barnes and David Allan further extended the bias functions with the B3 function to handle the concatenated samples estimator bias. This was necessary to handle the new use of concatenated sample observations with dead-time in between.

In 1970, the IEEE Technical Committee on Frequency and Time, within the IEEE Group on Instrumentation & Measurements, provided a summary of the field, published as NBS Technical Notice 394. This paper was first in a line of more educational and practical papers helping fellow engineers grasp the field. This paper recommended the 2-sample variance with T = τ, referring to it as Allan variance (now without the quotes). The choice of such parametrisation allows good handling of some noise forms and getting comparable measurements; it is essentially the least common denominator with the aid of the bias functions B1 and B2.

J. J. Snyder proposed an improved method for frequency or variance estimation, using sample statistics for frequency counters. To get more effective degrees of freedom out of the available dataset, the trick is to use overlapping observation periods. This provides a √n improvement, and was incorporated in the overlapping Allan variance estimator. Variable-τ software processing was also incorporated. This development improved the classical Allan variance estimators, likewise providing a direct inspiration for the work on modified Allan variance.

Howe, Allan and Barnes presented the analysis of confidence intervals, degrees of freedom, and the established estimators.

Educational and practical resources
The field of time and frequency and its use of Allan variance, Allan deviation and friends is a field involving many aspects, for which both understanding of concepts and practical measurements and post-processing requires care and understanding. Thus, there is a realm of educational material stretching about 40 years available. Since these reflect the developments in the research of their time, they focus on teaching different aspect over time, in which case a survey of available resources may be a suitable way of finding the right resource.

The first meaningful summary is the NBS Technical Note 394 "Characterization of Frequency Stability". This is the product of the Technical Committee on Frequency and Time of the IEEE Group on Instrumentation & Measurement. It gives the first overview of the field, stating the problems, defining the basic supporting definitions and getting into Allan variance, the bias functions B1 and B2, the conversion of time-domain measures. This is useful, as it is among the first references to tabulate the Allan variance for the five basic noise types.

A classical reference is the NBS Monograph 140 from 1974, which in chapter 8 has "Statistics of Time and Frequency Data Analysis". This is the extended variant of NBS Technical Note 394 and adds essentially in measurement techniques and practical processing of values.

An important addition will be the Properties of signal sources and measurement methods. It covers the effective use of data, confidence intervals, effective degree of freedom, likewise introducing the overlapping Allan variance estimator. It is a highly recommended reading for those topics.

The IEEE standard 1139 Standard definitions of Physical Quantities for Fundamental Frequency and Time Metrology is beyond that of a standard a comprehensive reference and educational resource.

A modern book aimed towards telecommunication is Stefano Bregni "Synchronisation of Digital Telecommunication Networks". This summarises not only the field, but also much of his research in the field up to that point. It aims to include both classical measures and telecommunication-specific measures such as MTIE. It is a handy companion when looking at measurements related to telecommunication standards.

The NIST Special Publication 1065 "Handbook of Frequency Stability Analysis" of W. J. Riley is a recommended reading for anyone wanting to pursue the field. It is rich of references and also covers a wide range of measures, biases and related functions that a modern analyst should have available. Further it describes the overall processing needed for a modern tool.

Uses
Allan variance is used as a measure of frequency stability in a variety of precision oscillators, such as crystal oscillators, atomic clocks and frequency-stabilized lasers over a period of a second or more. Short-term stability (under a second) is typically expressed as phase noise. The Allan variance is also used to characterize the bias stability of gyroscopes, including fiber optic gyroscopes, hemispherical resonator gyroscopes and MEMS gyroscopes and accelerometers.

50th Anniversary
In 2016, IEEE-UFFC is going to be publishing a "Special Issue to celebrate the 50th anniversary of the Allan Variance (1966–2016)". A guest editor for that issue will be David's former colleague at NIST, Judah Levine, who is the most recent recipient of the I. I. Rabi Award.

title: Allan variance


text:
Ambient noise level

In atmospheric sounding and noise pollution, ambient noise level (sometimes called background noise level, reference sound level, or room noise level) is the background sound pressure level at a given location, normally specified as a reference level to study a new intrusive sound source.

Ambient sound levels are often measured in order to map sound conditions over a spatial regime to understand their variation with locale. In this case the product of the investigation is a sound level contour map. Alternatively ambient noise levels may be measured to provide a reference point for analyzing an intrusive sound to a given environment. For example, sometimes aircraft noise is studied by measuring ambient sound without presence of any overflights, and then studying the noise addition by measurement or computer simulation of overflight events. Or roadway noise is measured as ambient sound, prior to introducing a hypothetical noise barrier intended to reduce that ambient noise level.

Ambient noise level is measured with a sound level meter. It is usually measured in dB relative to a reference pressure of 0.00002 Pa, i.e., 20 μPa (micropascals) in SI units. This is because 20 μPa is the faintest sound the human ear can detect. A pascal is a newton per square meter. The centimeter-gram-second system of units, the reference sound pressure for measuring ambient noise level is 0.0002 dyn/cm2, or 0.00002 N/m2. Most frequently ambient noise levels are measured using a frequency weighting filter, the most common being the A-weighting scale, such that resulting measurements are denoted dB(A), or decibels on the A-weighting scale.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Ambient noise level


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Amplitude distortion

Amplitude distortion is distortion occurring in a system, subsystem, or device when the output amplitude is not a linear function of the input amplitude under specified conditions.

Generally, output is a linear function of input only for a fixed portion of the transfer characteristics. In this region, Ic=βIb where Ic is collector current and Ib is base current, following linear relation y=mx.

When output is not in this portion, two forms of amplitude distortion might arise
 * 1) Harmonic distortion : The creation of harmonics of the fundamental frequency of a sine wave input to a system.
 * 2) Intermodulation distortion : This form of distortion occurs when two sine waves of frequencies X and Y are present at the input, resulting in the creation of several other frequency components, whose frequencies include (X+Y), (X-Y), (2X-Y), (2Y-X), and generally (mX ± nY) for integer m and n. Generally the size of the unwanted output falls rapidly as m and n increase.

Due to the additional outputs, this form of distortion is definitely unwanted in audio, radio and telecommunication amplifiers, and it occurs for more than two waves as well.

In a narrowband system such as a radio communication system, unwanted outputs such as X-Y and 2X+Y will be remote from the wanted band and so be ignored by the system. In contrast, 2X-Y and 2Y-X will be close to the wanted signals. These so-called third order distortion products (third order as m+n = 3) tend to dominate the non-linear distortion of narrowband systems.

Amplitude distortion is measured with the system operating under steady-state conditions with a sinusoidal input signal. When other frequencies are present, the term "amplitude" refers to that of the fundamental only.

title: Amplitude distortion


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Angular misalignment loss

In waveguide design and construction, angular misalignment loss is power loss caused by the deviation from optimum angular alignment of the axes of source-to-waveguide, waveguide-to-waveguide, or waveguide-to-detector. The waveguide may be dielectric (an optical fiber) or metallic. Angular misalignment loss does not include lateral offset loss and longitudinal offset loss.

Source: from Federal Standard 1037C

title: Angular misalignment loss


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Antenna blind cone

In telecommunications, antenna blind cone (sometimes called a cone of silence or antenna blind spot) is the volume of space, usually approximately conical with its vertex at the antenna, that cannot be scanned by an antenna because of limitations of the antenna radiation pattern and mount.

An Air Route Surveillance Radar (ARSR) is an example of an antenna blind cone. The horizontal radiation pattern of an ARSR antenna is very narrow, and the vertical radiation pattern is fan-shaped, reaching approximately 70° of elevation above the horizontal plane. As the fan antenna is rotated about a vertical axis, it can illuminate targets only if they are 70° or less from the horizontal plane. Above that elevation, they are in the antenna blind cone.

The antenna blind cone is also referred to as the "cone of silence", especially in America. This term is also used for weather radars. NEXRAD radars make two-dimensional scans at varying angles ranging from 0.5° above level to 19.5° above level (during a significant weather event). These levels become much closer to the ground, and closer to each other, as they get closer to the radar site, rendering them of little use for the three-dimensional profiling such multi-level scanning is meant to provide. Thus, a weather event located very close to and/or directly overhead of the radar site will be mostly situated in the "cone of silence." This is part of the reason why most U.S. weather radars partially overlap each other's territories.

title: Antenna blind cone


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Noise temperature (antenna)

In radio frequency (RF) applications such as radio, radar and telecommunications, noise temperature of an antenna is a measure of the noise power density contributed by the antenna to the overall RF receiver system. It is defined as "the temperature of a resistor having an available thermal noise power per unit bandwidth equal to that at the antenna's output at a specified frequency". In other words, antenna noise temperature is a parameter that describes how much noise an antenna produces in a given environment. This temperature is not the physical temperature of the antenna. Moreover, an antenna does not have an intrinsic "antenna temperature" associated with it; rather the temperature depends on its gain pattern, pointing direction, and the thermal environment that it is placed in.

Mathematics
In RF applications, noise power is defined using the relationship Pnoise = kTB, where k is the Boltzmann constant, T is the noise temperature, and B is the noise bandwidth. Typically the noise bandwidth is determined by the bandwidth of the intermediate frequency (IF) filter of the radio receiver. Thus, we can define the noise temperature as:
 * $$T = \frac{P_\text{noise}}{kB}=\frac{1}{k}\frac{P_\text{noise}}{B}$$

Because k is a constant, we can effectively think of T as noise power spectral density (with unit W/Hz) normalized by k.

Antenna noise is only one of the contributors to the overall noise temperature of an RF receiver system, so it is typically subscripted, such as TA. It is added directly to the effective noise temperature of the receiver to obtain the overall system noise temperature:
 * $$T_S=T_\text{A}+T_\text{E}$$

Sources of antenna noise
Antenna noise temperature has contributions from many sources, including:
 * Cosmic microwave background radiation
 * Galactic radiation
 * Earth heating
 * The Sun
 * The Moon
 * Electrical devices
 * The antenna itself

Galactic noise is high below 1000 MHz. At around 150 MHz, it is approximately 1000 K. At 2500 MHz, it has leveled off to around 10 K.

Earth has an accepted standard temperature of 288 K.

The level of the Sun's contribution depends on the solar flux. It is given by
 * $$T_\text{A}=3.468\,F{{\lambda}^2}10^{G/10}$$

where $$F$$ is the solar flux,
 * $$\lambda$$ is the wavelength,

and $$G$$ is the logarithmic gain of the antenna in decibels.

The antenna noise temperature depends on antenna coupling to all noise sources in its environment as well as on noise generated within the antenna. That is, in a directional antenna, the portion of the noise source that the antenna's main and side lobes intersect contribute proportionally.

For example, a satellite antenna may not receive noise contribution from the Earth in its main lobe, but sidelobes will contribute a portion of the 288 K Earth noise to its overall noise temperature.

title: Noise temperature (antenna)


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Aperture-to-medium coupling loss

In telecommunication, aperture-to-medium coupling loss is the difference between the theoretical antenna gain of a very large antenna, such as the antennas in beyond-the-horizon microwave links, and the gain that can be realized in practice.

Note 1: Aperture-to-medium coupling loss is related to the ratio of the scatter angle to the antenna beamwidth.

Note 2: The "very large antennas" are referred to in wavelengths; thus, this loss can apply to line-of-sight systems also.

title: Aperture-to-medium coupling loss


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Arithmetic shift

In computer programming, an arithmetic shift is a shift operator, sometimes termed a signed shift (though it is not restricted to signed operands). The two basic types are the arithmetic left shift and the arithmetic right shift. For binary numbers it is a bitwise operation that shifts all of the bits of its operand; every bit in the operand is simply moved a given number of bit positions, and the vacant bit-positions are filled in. Instead of being filled with all 0s, as in logical shift, when shifting to the right, the leftmost bit (usually the sign bit in signed integer representations) is replicated to fill in all the vacant positions (this is a kind of sign extension).

Some authors prefer the terms sticky right-shift and zero-fill right-shift for arithmetic and logical shifts respectively.

Arithmetic shifts can be useful as efficient ways to perform multiplication or division of signed integers by powers of two. Shifting left by n bits on a signed or unsigned binary number has the effect of multiplying it by 2n. Shifting right by n bits on a two's complement signed binary number has the effect of dividing it by 2n, but it always rounds down (towards negative infinity). This is different from the way rounding is usually done in signed integer division (which rounds towards 0). This discrepancy has led to bugs in a number of compilers.

For example, in the x86 instruction set, the SAR instruction (arithmetic right shift) divides a signed number by a power of two, rounding towards negative infinity. However, the IDIV instruction (signed divide) divides a signed number, rounding towards zero. So a SAR instruction cannot be substituted for an IDIV by power of two instruction nor vice versa.

Formal definition
The formal definition of an arithmetic shift, from Federal Standard 1037C is that it is:
 * A shift, applied to the representation of a number in a fixed radix numeration system and in a fixed-point representation system, and in which only the characters representing the fixed-point part of the number are moved. An arithmetic shift is usually equivalent to multiplying the number by a positive or a negative integral power of the radix, except for the effect of any rounding; compare the logical shift with the arithmetic shift, especially in the case of floating-point representation.

An important word in the FS 1073C definition is "usually".

Equivalence of arithmetic and logical left shifts and multiplication
Arithmetic left shifts are equivalent to multiplication by a (positive, integral) power of the radix (e.g., a multiplication by a power of 2 for binary numbers). Logical left shifts are also equivalent, except multiplication and arithmetic shifts may trigger arithmetic overflow whereas logical shifts do not.

Non-equivalence of arithmetic right shift and division
However, arithmetic right shifts are major traps for the unwary, specifically in treating rounding of negative integers. For example, in the usual two's complement representation of negative integers, −1 is represented as all 1's. For an 8-bit signed integer this is 1111 1111. An arithmetic right-shift by 1 (or 2, 3, ..., 7) yields 1111 1111 again, which is still −1. This corresponds to rounding down (towards negative infinity), but is not the usual convention for division.

It is frequently stated that arithmetic right shifts are equivalent to division by a (positive, integral) power of the radix (e.g., a division by a power of 2 for binary numbers), and hence that division by a power of the radix can be optimized by implementing it as an arithmetic right shift. (A shifter is much simpler than a divider. On most processors, shift instructions will execute faster than division instructions.) Large number of 1960s and 1970s programming handbooks, manuals, and other specifications from companies and institutions such as DEC, IBM, Data General, and ANSI make such incorrect statements.

Logical right shifts are equivalent to division by a power of the radix (usually 2) only for positive or unsigned numbers. Arithmetic right shifts are equivalent to logical right shifts for positive signed numbers. Arithmetic right shifts for negative numbers in N's complement (usually two's complement) is roughly equivalent to division by a power of the radix (usually 2), where for odd numbers rounding downwards is applied (not towards 0 as usually expected).

Arithmetic right shifts for negative numbers are equivalent to division using rounding towards 0 in ones' complement representation of signed numbers as was used by some historic computers, but this is no longer in general use.

Handling the issue in programming languages
The (1999) ISO standard for the programming language C defines the right shift operator in terms of divisions by powers of 2. Because of the above-stated non-equivalence, the standard explicitly excludes from that definition the right shifts of signed numbers that have negative values. It does not specify the behaviour of the right shift operator in such circumstances, but instead requires each individual C compiler to define the behaviour of shifting negative values right.

Like C, C++ had an implementation-defined right shift for signed integers until C++20. Starting in the C++20 standard, right shift of a signed integer is defined to be an arithmetic shift.

Applications
In applications where consistent rounding down is desired, arithmetic right shifts for signed values are useful. An example is in downscaling raster coordinates by a power of two, which maintains even spacing. For example, right shift by 1 sends 0, 1, 2, 3, 4, 5, ... to 0, 0, 1, 1, 2, 2, ..., and −1, −2, −3, −4, ... to −1, −1, −2, −2, ..., maintaining even spacing as −2, −2, −1, −1, 0, 0, 1, 1, 2, 2, ... In contrast, integer division with rounding towards zero sends −1, 0, and 1 all to 0 (3 points instead of 2), yielding −2, −1, −1, 0, 0, 0, 1, 1, 2, 2, ... instead, which is irregular at 0.

title: Arithmetic shift


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Automatic repeat request

Automatic repeat request (ARQ), also known as automatic repeat query, is an error-control method for data transmission that uses acknowledgements (messages sent by the receiver indicating that it has correctly received a message) and timeouts (specified periods of time allowed to elapse before an acknowledgment is to be received) to achieve reliable data transmission over an unreliable communication channel. ARQ is appropriate if the communication channel has varying or unknown capacity. If the sender does not receive an acknowledgment before the timeout, it re-transmits the message until it receives an acknowledgment or exceeds a predefined number of retransmissions.

Variations of ARQ protocols include Stop-and-wait ARQ, Go-Back-N ARQ, and Selective Repeat ARQ. All three protocols usually use some form of sliding window protocol to help the sender determine which (if any) packets need to be retransmitted. These protocols reside in the data link or transport layers (layers 2 and 4) of the OSI model.

Examples
The Transmission Control Protocol uses a variant of Go-Back-N ARQ to ensure reliable transmission of data over the Internet Protocol, which does not provide guaranteed delivery of packets; with Selective Acknowledgement (SACK), it uses Selective Repeat ARQ.

IEEE 802.11 wireless networking uses ARQ retransmissions at the data-link layer.

ITU-T G.hn uses hybrid ARQ, a mixture of high-rate forward error correction (FEC) and ARQ. It is a high-speed local area network standard that can operate at data rates up to 1 Gbit/s over existing home wiring (power lines, phone lines and coaxial cables). G.hn uses CRC-32C for Error Detection, LDPC for FEC and selective repeat for ARQ.

ARQ systems were widely used on shortwave radio to ensure reliable delivery of data such as for telegrams. These systems came in forms called ARQ-E and ARQ-M, which also included the ability to multiplex two or four channels.

A number of patents exist for the use of ARQ in live video contribution environments. In these high throughput environments negative acknowledgements are used to drive down overheads.

title: Automatic repeat request


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Articulation score

In telecommunication, an articulation score (AS) is a subjective measure of the intelligibility of a voice system in terms of the percentage of words correctly understood over a channel perturbed by interference.

Articulation scores have been experimentally obtained as functions of varying word content, bandwidth, audio signal-to-noise ratio and the experience of the talkers and listeners involved.

title: Articulation score


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Artificial transmission line

In telecommunication, an artificial transmission line is a two-port electrical network that has the characteristic impedance, transmission time delay, phase shift, or other parameter(s) of a real transmission line. It can be used to simulate a real transmission line in one or more of these respects.

Early artificial lines were used in telephony research and took the form of a cascade of lattice phase equalisers to provide the necessary delay. The lattice phase circuit was invented by Otto Zobel in the 1920s.

title: Artificial transmission line


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Asynchronous operation

In telecommunications, asynchronous operation or asynchronous working is where a sequence of operations is executed such that the operations are executed out of time coincidence with any event. It can also be an operation that occurs without a regular or predictable time relationship to a specified event; e.g., the calling of an error diagnostic routine that may receive control at any time during the execution of a computer program.

title: Asynchronous operation


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Atmospheric duct

In telecommunications, an atmospheric duct is a horizontal layer in the lower atmosphere in which the vertical refractive index gradients are such that radio signals (and light rays) are guided or ducted, tend to follow the curvature of the Earth, and experience less attenuation in the ducts than they would if the ducts were not present. The duct acts as an atmospheric dielectric waveguide and limits the spread of the wavefront to only the horizontal dimension.

Atmospheric ducting is a mode of propagation of electromagnetic radiation, usually in the lower layers of Earth’s atmosphere, where the waves are bent by atmospheric refraction. In over-the-horizon radar, ducting causes part of the radiated and target-reflection energy of a radar system to be guided over distances far greater than the normal radar range. It also causes long-distance propagation of radio signals in bands that would normally be limited to line of sight.

Normally radio "ground waves" propagate along the surface as creeping waves. That is, they are only diffracted around the curvature of the earth. This is one reason that early long-distance radio communication used long wavelengths. The best known exception is that HF (3–30 MHz.) waves are reflected by the ionosphere.

The reduced refractive index due to lower densities at the higher altitudes in the Earth's atmosphere bends the signals back toward the Earth. Signals in a higher refractive index layer, i.e., duct, tend to remain in that layer because of the reflection and refraction encountered at the boundary with a lower refractive index material. In some weather conditions, such as inversion layers, density changes so rapidly that waves are guided around the curvature of the earth at constant altitude.

Phenomena of atmospheric optics related to atmospheric ducting include the green flash, Fata Morgana, superior mirage, mock mirage of astronomical objects and the Novaya Zemlya effect.

title: Atmospheric duct


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Attack time

In telecommunication, attack time is the time between the instant that a signal at the input of a device or circuit exceeds the activation threshold of the device or circuit and the instant that the device or circuit reacts in a specified manner, or to a specified degree, to the input. Attack time occurs in devices such as clippers, peak limiters, compressors, and voxes.

title: Attack time


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ARJ

ARJ (Archived by Robert Jung) is a software tool designed in 1991 by Robert K. Jung for creating high-efficiency compressed file archives. ARJ is currently on version 2.86 for MS-DOS and 3.20 for Microsoft Windows and supports 16-bit, 32-bit and 64-bit Intel architectures.

ARJ was one of many file compression utilities for MS-DOS and Microsoft Windows during the early and mid-1990s. Parts of ARJ were covered by (expired). ARJ is well-documented and includes over 150 command line switches.

File format support in other software
ARJ archives can be unpacked with various tools other than the ARJ software. There exists a free software re-implementation of the tool. A number of software utilities, including 7-Zip, Zipeg, and WinRAR can also unpack .arj files. For macOS, standalone utilities, such as DeArj and UnArjMac, are available.

title: ARJ


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Attenuation

In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable attenuation rates.

Hearing protectors help reduce acoustic flux from flowing into the ears. This phenomenon is called acoustic attenuation and is measured in decibels (dBs).

In electrical engineering and telecommunications, attenuation affects the propagation of waves and signals in electrical circuits, in optical fibers, and in air. Electrical attenuators and optical attenuators are commonly manufactured components in this field.

Background
In many cases, attenuation is an exponential function of the path length through the medium. In optics and in chemical spectroscopy, this is known as the Beer–Lambert law. In engineering, attenuation is usually measured in units of decibels per unit length of medium (dB/cm, dB/km, etc.) and is represented by the attenuation coefficient of the medium in question. Attenuation also occurs in earthquakes; when the seismic waves move farther away from the hypocenter, they grow smaller as they are attenuated by the ground.

Ultrasound
One area of research in which attenuation plays a prominent role, is in ultrasound physics. Attenuation in ultrasound is the reduction in amplitude of the ultrasound beam as a function of distance through the imaging medium. Accounting for attenuation effects in ultrasound is important because a reduced signal amplitude can affect the quality of the image produced. By knowing the attenuation that an ultrasound beam experiences traveling through a medium, one can adjust the input signal amplitude to compensate for any loss of energy at the desired imaging depth.

 * Ultrasound attenuation measurement in heterogeneous systems, like emulsions or colloids, yields information on particle size distribution. There is an ISO standard on this technique.
 * Ultrasound attenuation can be used for extensional rheology measurement. There are acoustic rheometers that employ Stokes' law for measuring extensional viscosity and volume viscosity.

Wave equations which take acoustic attenuation into account can be written on a fractional derivative form.

In homogeneous media, the main physical properties contributing to sound attenuation are viscosity and thermal conductivity.

Attenuation coefficient
Attenuation coefficients are used to quantify different media according to how strongly the transmitted ultrasound amplitude decreases as a function of frequency. The attenuation coefficient ($$\alpha$$) can be used to determine total attenuation in dB in the medium using the following formula:
 * $$ \text{Attenuation} = \alpha \left[\frac{\text{dB}}{\text{MHz}{\cdot}\text{cm}}\right] \cdot \ell [\text{cm}] \cdot \text{f}[\text{MHz}]$$

Attenuation is linearly dependent on the medium length and attenuation coefficient, as well as – approximately – the frequency of the incident ultrasound beam for biological tissue (while for simpler media, such as air, the relationship is quadratic). Attenuation coefficients vary widely for different media. In biomedical ultrasound imaging however, biological materials and water are the most commonly used media. The attenuation coefficients of common biological materials at a frequency of 1 MHz are listed below:

There are two general ways of acoustic energy losses: absorption and scattering. Ultrasound propagation through homogeneous media is associated only with absorption and can be characterized with absorption coefficient only. Propagation through heterogeneous media requires taking into account scattering.

Light attenuation in water
Shortwave radiation emitted from the Sun have wavelengths in the visible spectrum of light that range from 360 nm (violet) to 750 nm (red). When the Sun's radiation reaches the sea surface, the shortwave radiation is attenuated by the water, and the intensity of light decreases exponentially with water depth. The intensity of light at depth can be calculated using the Beer-Lambert Law.

In clear mid-ocean waters, visible light is absorbed most strongly at the longest wavelengths. Thus, red, orange, and yellow wavelengths are totally absorbed at shallower depths, while blue and violet wavelengths reach deeper in the water column. Because the blue and violet wavelengths are absorbed least compared to the other wavelengths, open-ocean waters appear deep blue to the eye.

Near the shore, coastal water contains more phytoplankton than the very clear mid-ocean waters. Chlorophyll-a pigments in the phytoplankton absorb light, and the plants themselves scatter light, making coastal waters less clear than mid-ocean waters. Chlorophyll-a absorbs light most strongly in the shortest wavelengths (blue and violet) of the visible spectrum. In coastal waters where high concentrations of phytoplankton occur, the green wavelength reaches the deepest in the water column and the color of water appears blue-green or green.

Seismic
The energy with which an earthquake affects a location depends on the running distance. The attenuation in the signal of ground motion intensity plays an important role in the assessment of possible strong groundshaking. A seismic wave loses energy as it propagates through the earth (seismic attenuation). This phenomenon is tied into the dispersion of the seismic energy with the distance. There are two types of dissipated energy:

 * geometric dispersion caused by distribution of the seismic energy to greater volumes
 * dispersion as heat, also called intrinsic attenuation or anelastic attenuation.

In porous fluid—saturated sedimentary rocks such as sandstones, intrinsic attenuation of seismic waves is primarily caused by the wave-induced flow of the pore fluid relative to the solid frame.

Electromagnetic
Attenuation decreases the intensity of electromagnetic radiation due to absorption or scattering of photons. Attenuation does not include the decrease in intensity due to inverse-square law geometric spreading. Therefore, calculation of the total change in intensity involves both the inverse-square law and an estimation of attenuation over the path.

The primary causes of attenuation in matter are the photoelectric effect, compton scattering, and, for photon energies of above 1.022 MeV, pair production.

Coaxial and general RF cables
The attenuation of RF cables is defined by:
 * $$\text{Attenuation (dB/100m)} = 10\times\log_{10}\left(\frac{P_1\ (W)}{P_2\ (W)}\right),$$

where $$P_1$$ is the input power into a 100 m long cable terminated with the nominal value of its characteristic impedance, and $$P_2$$ is the output power at the far end of this cable.

Attenuation in a coaxial cable is a function of the materials and the construction.

Radiography
The beam of X-ray is attenuated when photons are absorbed when the x-ray beam passes through the tissue. Interaction with matter varies between high energy photons and low energy photons. Photons travelling at higher energy are more capable of travelling through a tissue specimen as they have less chances of interacting with matter. This is mainly due to the photoelectric effect which states that "the probability of photoelectric absorption is approximately proportional to (Z/E)3, where Z is the atomic number of the tissue atom and E is the photon energy. In context of this, an increase in photon energy (E) will result in a rapid decrease in the interaction with matter.

Optics
Attenuation in fiber optics, also known as transmission loss, is the reduction in intensity of the light beam (or signal) with respect to distance travelled through a transmission medium. Attenuation coefficients in fiber optics usually use units of dB/km through the medium due to the relatively high quality of transparency of modern optical transmission. The medium is typically a fiber of silica glass that confines the incident light beam to the inside. Attenuation is an important factor limiting the transmission of a digital signal across large distances. Thus, much research has gone into both limiting the attenuation and maximizing the amplification of the optical signal. Empirical research has shown that attenuation in optical fiber is caused primarily by both scattering and absorption.

Attenuation in fiber optics can be quantified using the following equation:

 * $$\text{Attenuation (dB)} = 10\times\log_{10}\left(\frac{\text{Input intensity (W)}}{\text{Output intensity (W)}}\right)$$

Light scattering
The propagation of light through the core of an optical fiber is based on total internal reflection of the lightwave. Rough and irregular surfaces, even at the molecular level of the glass, can cause light rays to be reflected in many random directions. This type of reflection is referred to as "diffuse reflection", and it is typically characterized by wide variety of reflection angles. Most objects that can be seen with the naked eye are visible due to diffuse reflection. Another term commonly used for this type of reflection is "light scattering". Light scattering from the surfaces of objects is our primary mechanism of physical observation. Light scattering from many common surfaces can be modelled by reflectance.

Light scattering depends on the wavelength of the light being scattered. Thus, limits to spatial scales of visibility arise, depending on the frequency of the incident lightwave and the physical dimension (or spatial scale) of the scattering center, which is typically in the form of some specific microstructural feature. For example, since visible light has a wavelength scale on the order of one micrometer, scattering centers will have dimensions on a similar spatial scale.

Thus, attenuation results from the incoherent scattering of light at internal surfaces and interfaces. In (poly)crystalline materials such as metals and ceramics, in addition to pores, most of the internal surfaces or interfaces are in the form of grain boundaries that separate tiny regions of crystalline order. It has recently been shown that, when the size of the scattering center (or grain boundary) is reduced below the size of the wavelength of the light being scattered, the scattering no longer occurs to any significant extent. This phenomenon has given rise to the production of transparent ceramic materials.

Likewise, the scattering of light in optical quality glass fiber is caused by molecular-level irregularities (compositional fluctuations) in the glass structure. Indeed, one emerging school of thought is that a glass is simply the limiting case of a polycrystalline solid. Within this framework, "domains" exhibiting various degrees of short-range order become the building-blocks of both metals and alloys, as well as glasses and ceramics. Distributed both between and within these domains are microstructural defects that will provide the most ideal locations for the occurrence of light scattering. This same phenomenon is seen as one of the limiting factors in the transparency of IR missile domes.

UV-Vis-IR absorption
In addition to light scattering, attenuation or signal loss can also occur due to selective absorption of specific wavelengths, in a manner similar to that responsible for the appearance of color. Primary material considerations include both electrons and molecules as follows:

 * At the electronic level, it depends on whether the electron orbitals are spaced (or "quantized") such that they can absorb a quantum of light (or photon) of a specific wavelength or frequency in the ultraviolet (UV) or visible ranges. This is what gives rise to color.
 * At the atomic or molecular level, it depends on the frequencies of atomic or molecular vibrations or chemical bonds, how close-packed its atoms or molecules are, and whether or not the atoms or molecules exhibit long-range order. These factors will determine the capacity of the material transmitting longer wavelengths in the infrared (IR), far IR, radio and microwave ranges.

The selective absorption of infrared (IR) light by a particular material occurs because the selected frequency of the light wave matches the frequency (or an integral multiple of the frequency) at which the particles of that material vibrate. Since different atoms and molecules have different natural frequencies of vibration, they will selectively absorb different frequencies (or portions of the spectrum) of infrared (IR) light.

Applications
In optical fibers, attenuation is the rate at which the signal light decreases in intensity. For this reason, glass fiber (which has a low attenuation) is used for long-distance fiber optic cables; plastic fiber has a higher attenuation and, hence, shorter range. There also exist optical attenuators that decrease the signal in a fiber optic cable intentionally.

Attenuation of light is also important in physical oceanography. This same effect is an important consideration in weather radar, as raindrops absorb a part of the emitted beam that is more or less significant, depending on the wavelength used.

Due to the damaging effects of high-energy photons, it is necessary to know how much energy is deposited in tissue during diagnostic treatments involving such radiation. In addition, gamma radiation is used in cancer treatments where it is important to know how much energy will be deposited in healthy and in tumorous tissue.

In computer graphics attenuation defines the local or global influence of light sources and force fields.

In CT imaging, attenuation describes the density or darkness of the image.

Radio
Attenuation is an important consideration in the modern world of wireless telecommunication. Attenuation limits the range of radio signals and is affected by the materials a signal must travel through (e.g., air, wood, concrete, rain). See the article on path loss for more information on signal loss in wireless communication.

title: Attenuation


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Attenuator

Attenuator could mean:
 * Attenuator (electronics), an electronic device that reduces the amplitude of an electronic signal.
 * Optical attenuator, an electronic device that reduces the amplitude of an optical signal.
 * Attenuator (genetics), a specific regulatory sequence transcribed into RNA.
 * Impact attenuator, used on highways as a crumple zone in case of a car crash.

title: Attenuator


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Attribute

Attribute may refer to:

 * Attribute (philosophy), an extrinsic property of an object
 * Attribute (research), a characteristic of an object
 * Grammatical modifier, in natural languages
 * Attribute (computing), a specification that defines a property of an object, element, or file
 * Attribute (role-playing games), a type of statistic for a fictional character

title: Attribute


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Audit (telecommunication)

In telecommunications, an audit is one of:

 * The act of conducting a review, examination and reconciliation of Telecom, Wireless and Network customer service records, invoicing and contract agreements in order to ensure the accuracy of budgetary forecasting.
 * Independent review and examination of records and activities to assess the adequacy of system controls, to ensure compliance with established policies and operational procedures, and to recommend necessary changes in controls, policies, or procedures.
 * Analysis of invoices, lines, rates, tariffs, taxes, plans, usage, call volume, systems, and contracts resulting in cost reduction, proper invoicing and optimization of telecommunication systems often conducted by an independent telecommunications consultant or firm.

The simplest audits consist of comparing current telecommunications billing and usage to the underlying rate structure whether that is dictated by contract, tariff, or price list. Complex audits utilize software applications, direct bargaining with service providers and activity reports that include detail down to an individual employee's usage.

Auditing methods and consultants
In business, companies with significant telecommunications costs or a telecommunications focus normally either conduct audits internally or hire a consultant. No matter the method, typical audits encompass one or more of the following:

 * Telecom Expense Management (TEM): An ongoing analysis and adjustment of internal telecommunications procedures and billing designed to maximize savings. See also Wireless Expense Management
 * Telecom RFP (Request For Proposals): A proposed management plan designed to maximize efficiency, security and reliability in the business' communications. Consultants typically present an RFP for approval by their clients.
 * Management and Reporting: Ongoing telecommunications cost and activity analytics. Internal auditors and consultants both use software as part of the process: either a generalized database application or specialized applications designed for auditing.

Telecom audits can encompass every communications service that a business expends its budget on. Audits may focus on mobile phones and devices, Internet service or land line telephony, or they may encompass all three.

Current methods
Internal Audit: A business accounting department will generally only conduct a telecom bill review in the months that trigger a red flag due to the sudden spike in the expense of a communications service. This is usually contained to the one telecom service provider who triggered the red flag and the rest of the invoices are left unexamined because they fall within the considered norm of a small monthly cost increase.

Contingency Fee Based Audit:
 * In exchange for telecom consulting and bill auditing services a business can pay a percentage of all savings realized.
 * Business Process Outsourcing BPO, telecom consultant companies can provide a contingency fee based independent review and examination of records and activities to assess the compliance of telecom service providers with telecommunication contracts and State and Federal Tariffs.

title: Audit (telecommunication)


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Audit trail

An audit trail (also called audit log) is a security-relevant chronological record, set of records, and/or destination and source of records that provide documentary evidence of the sequence of activities that have affected at any time a specific operation, procedure, event, or device. Audit records typically result from activities such as financial transactions, scientific research and health care data transactions, or communications by individual people, systems, accounts, or other entities.

The process that creates an audit trail is typically required to always run in a privileged mode, so it can access and supervise all actions from all users; a normal user should not be allowed to stop/change it. Furthermore, for the same reason, the trail file or database table with a trail should not be accessible to normal users. Another way of handling this issue is through the use of a role-based security model in the software. The software can operate with the closed-looped controls, or as a 'closed system', as required by many companies when using audit trail functionality.

Industry uses
In telecommunication, the term means a record of both completed and attempted accesses and service, or data forming a logical path linking a sequence of events, used to trace the transactions that have affected the contents of a record.

In information or communications security, information audit means a chronological record of system activities to enable the reconstruction and examination of the sequence of events and/or changes in an event. Information put away or transmitted in paired structure that might be depended upon in court. An audit trail is a progression of records of computer data about a working framework, an application, or client exercises. Computer frameworks may have a few audit trails each gave to a specific sort of action. Related to proper apparatuses and systems, audit trails can help with distinguishing security infringement, execution issues and application issues. Routine log audits and investigation are valuable for distinguishing security episodes, approach infringement, fake movement, and operational issues soon after they have happened, and for giving information valuable to settling such issues. Audit logs can likewise be valuable for performing forensic investigation, supporting the associations inside examinations, setting up baselines, and distinguishing operational patterns and long run issues.

In nursing research, it refers to the act of maintaining a running log or journal of decisions relating to a research project, thus making clear the steps taken and changes made to the original protocol.

In accounting, it refers to documentation of detailed transactions supporting summary ledger entries. This documentation may be on paper or on electronic records.

In finance, it refers to an order (any firm indication of a willingness to buy or sell a security) tracking system, or consolidated audit trail, with respect to the trading of securities, that would capture order event information for orders in securities from the time of the receipt of an order, and further documenting the life of the order through the process of routing, modification, cancellation, and execution (in whole or in part) of the order.

In online proofing, it pertains to the version history of a piece of artwork, design, photograph, video, or web design proof in a project.

In clinical research, server based systems such as clinical trial management systems (CTMS) require audit trails. Anything regulatory or QA/QC related also requires audit trails.

In pharmaceutical manufacturing, it is a Good Manufacturing Practice regulatory requirement software generate audit trails, but not all software have audit trail functionality built-in. The first 'generic' audit trail generating software came out late 2021. The software is called Audit Trail Control, capable of fulfilling regulatory requirements for any software used in pharmaceutical manufacturing.

In voting, a voter-verified paper audit trail is a method of providing feedback to voters using a ballotless voting system.

title: Audit trail