GO LOOK IMPORTANTBOOK

Example plot of the Allan deviation of a clock. At very short observation time τ, the Allan deviation is high due to noise. At longer τ, it decreases because the noise averages out. At still longer τ, the Allan deviation starts increasing again, suggesting that the clock frequency is gradually drifting due to temperature changes, aging of components, or other such factors. The error bars increase with τ simply because it is time-consuming to get a lot of data points for large τ.

A clock is most easily tested by comparing it with a far more accurate reference clock. During an interval of time τ, as measured by the reference clock, the clock under test advances by τy, where y is the average (relative) clock frequency over that interval. If we measure two consecutive intervals as shown, we can get a value of (y − y′)²—a smaller value indicates a more stable and precise clock. If we repeat this procedure many times, the average value of (y − y′)² is equal to twice the Allan variance (or Allan deviation squared) for observation time τ.

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 white frequency noise and 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.^[1]^[2]
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.^[3] 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.^[4]
An early concern was related to time and frequency measurement instruments which 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^[2] 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.^[5]

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 (conveniently 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 1.3×10⁻⁹ at observation time 1 s (i.e. τ = 1 s) should be interpreted as there being an instability in frequency between two observations a second apart with a relative root mean square (RMS) value of 1.3×10⁻⁹. 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.

Definitions

M-sample variance

The

M

-sample variance is defined^[3] (here in a modernized notation form) as

\sigma _{y}^{2}(M,T,\tau )={\frac {1}{M-1}}\left\{\sum _{i=0}^{M-1}\left[{\frac {x(iT+\tau )-x(iT)}{\tau }}\right]^{2}-{\frac {1}{M}}\left[\sum _{i=0}^{M-1}{\frac {x(iT+\tau )-x(iT)}{\tau }}\right]^{2}\right\}

where

x(t)

is the phase angle (in radians) measured at time

t

, or with average fractional frequency time series

\sigma _{y}^{2}(M,T,\tau )={\frac {1}{M-1}}\left\{\sum _{i=0}^{M-1}{\bar {y}}_{i}^{2}-{\frac {1}{M}}\left[\sum _{i=0}^{M-1}{\bar {y}}_{i}\right]^{2}\right\}

where

M

is the number of frequency samples used in variance,

T

is the time between each frequency sample and

\tau

is the time-length of each frequency estimate.
An important aspect is that

M

-sample variance model can include dead-time by letting the time

T

be different from that of

\tau

.

Allan variance

The Allan variance is defined as

\sigma _{y}^{2}(\tau )=\langle \sigma _{y}^{2}(2,\tau ,\tau )\rangle

where

\langle \rangle

denotes the expectation operator. This can be conveniently expressed as

\sigma _{y}^{2}(\tau )={\frac {1}{2}}\langle ({\bar {y}}_{n+1}-{\bar {y}}_{n})^{2}\rangle ={\frac {1}{2\tau ^{2}}}\langle (x_{n+2}-2x_{n+1}+x_{n})^{2}\rangle

where

\tau

is the observation period,

{\bar {y}}_{n}

is the nth fractional frequency average over the observation time

\tau

.
The samples are taken with no dead-time between them, which is achieved by letting

T=\tau \,

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 )}}\,

Supporting definitions

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 _{\mathrm {n} }

, given in cycles per second (SI unit: hertz). The nominal angular frequency

\omega _{\mathrm {n} }

(in radians per second) is given by

\omega _{\mathrm {n} }=2\pi \nu _{\mathrm {n} }.

The total phase can be separated into a perfectly cyclic component

\omega _{n}t

, along with a fluctuating component

\phi (t)

:

\Phi (t)=\omega _{\mathrm {n} }t+\phi (t)=2\pi \nu _{\mathrm {n} }t+\phi (t).

Time error

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

x(t)={\frac {\phi (t)}{2\pi \nu _{n}}}={\frac {\Phi (t)}{2\pi \nu _{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 _{\mathrm {n} }

:

y(t)={\frac {\nu (t)-\nu _{\mathrm {n} }}{\nu _{\mathrm {n} }}}={\frac {\nu (t)}{\nu _{\mathrm {n} }}}-1.

Average fractional frequency

The average fractional frequency is defined as

{\bar {y}}(t,\tau )={\frac {1}{\tau }}\int \limits _{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 with M.
The number of time error samples in a time error series is denoted with 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, x_i 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...x_N-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 \limits _{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. This is however 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 with 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)={\text{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)={\text{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)={\text{AVAR}}(n\tau _{0},M)={\frac {1}{2n(M-1)}}\sum _{i=0}^{{\frac {M-1}{n}}-1}({\bar {y}}_{ni+n}-{\bar {y}}_{ni})^{2}

with

n\leq M-1

,
and for the time series

\sigma _{y}^{2}(n\tau _{0},N)={\text{AVAR}}(n\tau _{0},N)={\frac {1}{2n^{2}\tau _{0}^{2}({\frac {N-1}{n}}-1)}}\sum _{i=0}^{{\frac {N-1}{n}}-2}(x_{ni+2n}-2x_{ni+n}+x_{ni})^{2}

with

n\leq {\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^[6] provided an improved tool, as measurements were overlapped in n overlapped series out of the original series. The overlapping Allan variance estimator was introduced in.^[7] 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

\sigma _{y}^{2}(n\tau _{0},M)={\text{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}={\frac {1}{2(M-2n+1)}}\sum _{j=0}^{M-2n}\left({\bar {y}}_{j+n}-{\bar {y}}_{j}\right)^{2}

or for the time series

\sigma _{y}^{2}(n\tau _{0},N)={\text{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,^[4] ITU-T^[8] and ETSI^[9] 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 (σ_x) 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}}Mod\ \sigma _{y}^{2}(\tau )

and similarly for modified Allan deviation to time deviation

\sigma _{x}(\tau )={\frac {\tau }{\sqrt {3}}}Mod\ \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 of MDEV and TDEV, the following is the relevant statistical rule. For independent random variables X and Y, the variance (σ_z²) of a sum or difference (z=x-y) is the sum square of their variances (σ_z² = σ_x²+ σ_y²). The variance of the sum or difference (y = x_2τ-x_τ) of two independent samples of a random variable is twice the variance of the random variable (σ_y²=2σ_x²). The MDEV is the second difference of independent phase measurements (x) that have a variance (σ_x²). Since the calculation is the double difference which requires three independent phase measurements (x_2τ-2x_τ+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:^[4]^[7]

\chi ^{2}={\frac {(d.f.)s^{2}}{\sigma ^{2}}}

where s² is the sample variance of our estimate, σ² is the true variance value, d.f. 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)\leq {\frac {(d.f.)s^{2}}{\sigma ^{2}}}\leq \chi ^{2}(0.95)

which after rearrangement for the true variance becomes:

{\frac {(d.f.)s^{2}}{\chi ^{2}(0.95)}}\leq \sigma ^{2}\leq {\frac {(d.f.)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 empirically found^[7] to be:

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

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 f_H.

**Allan variance power-law response**
Power-law noise type	Phase noise slope	Frequency noise slope	Power coefficient	Phase noise	Allan variance	Allan deviation
white phase modulation (WPM)	$f^{0}=1$	$f^{2}$	$h_{2}$	$S_{x}(f)={\frac {1}{(2\pi )^{2}}}h_{2}$	$\sigma _{y}^{2}(\tau )={\frac {3f_{H}}{4\pi ^{2}\tau ^{2}}}h_{2}$	$\sigma _{y}(\tau )={\frac {\sqrt {3f_{H}}}{2\pi \tau }}{\sqrt {h_{2}}}$
flicker phase modulation (FPM)	$f^{-1}$	$f^{1}=f$	$h_{1}$	$S_{x}(f)={\frac {1}{(2\pi )^{2}f}}h_{1}$	$\sigma _{y}^{2}(\tau )={\frac {3[\gamma +\ln(2\pi f_{H}\tau )]-\ln 2}{4\pi ^{2}\tau ^{2}}}h_{1}$	$\sigma _{y}(\tau )={\frac {\sqrt {3[\gamma +\ln(2\pi f_{H}\tau )]-\ln 2}}{2\pi \tau }}{\sqrt {h_{1}}}$
white frequency modulation (WFM)	$f^{-2}$	$f^{0}=1$	$h_{0}$	$S_{x}(f)={\frac {1}{(2\pi )^{2}f^{2}}}h_{0}$	$\sigma _{y}^{2}(\tau )={\frac {1}{2\tau }}h_{0}$	$\sigma _{y}(\tau )={\frac {1}{\sqrt {2\tau }}}{\sqrt {h_{0}}}$
flicker frequency modulation (FFM)	$f^{-3}$	$f^{-1}$	$h_{-1}$	$S_{x}(f)={\frac {1}{(2\pi )^{2}f^{3}}}h_{-1}$	$\sigma _{y}^{2}(\tau )=2\ln(2)h_{-1}$	$\sigma _{y}(\tau )={\sqrt {2\ln(2)}}{\sqrt {h_{-1}}}$
random walk frequency modulation (RWFM)	$f^{-4}$	$f^{-2}$	$h_{-2}$	$S_{x}(f)={\frac {1}{(2\pi )^{2}f^{4}}}h_{-2}$	$\sigma _{y}^{2}(\tau )={\frac {2\pi ^{2}\tau }{3}}h_{-2}$	$\sigma _{y}(\tau )={\frac {\pi {\sqrt {2\tau }}}{\sqrt {3}}}{\sqrt {h_{-2}}}$

As found in^[10]^[11] 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 K_α is also presented for convenience:

**Allan variance α-μ mapping**
α	β	μ	K_α
-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	${\frac {3f_{H}}{4\pi ^{2}}}$

The mapping is taken from.^[4]

General Conversion from Phase Noise

A signal with spectral phase noise

S_{\phi }

with units rad²/Hz can be converted to Allan Variance by:^[13]

\sigma _{y}^{2}(\tau )={\frac {2}{\nu _{0}^{2}}}\int _{0}^{f_{b}}S_{\phi }(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:

**Allan variance linear response**
Linear effect	time response	frequency response	Allan variance	Allan deviation
phase offset	$x_{0}$	$0$	$0$	$0$
frequency offset	$y_{0}t$	$y_{0}$	$0$	$0$
linear drift	${\frac {Dt^{2}}{2}}$	$Dt$	${\frac {D^{2}\tau ^{2}}{2}}$	${\frac {D\tau }{\sqrt {2}}}$

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.^[12]

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}}\langle ({\bar {y}}_{i+1}-{\bar {y}}_{i})^{2}\rangle

where

{\bar {y}}_{i}={\frac {1}{\tau }}\int \limits _{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 address these biases the bias-functions B₁ and B₂ has been defined^[14] and allows for 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 has 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.^[15]
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 as proposed in^[3] and elaborated in^[14] 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, and is defined^[14] 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\left(rn\right)^{\mu +2}-\left(rn+1\right)^{\mu +2}-\left|rn-1\right|^{\mu +2}\right]}{1+{\frac {1}{2}}\left[2r^{\mu +2}-\left(r+1\right)^{\mu +2}-\left|r-1\right|^{\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, and is defined^[14]

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}-\left(r+1\right)^{\mu +2}-\left|r-1\right|^{\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^[15] 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=MT_{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((M+n)r)+F((M-n)r)\right]}{M^{\mu +2}\left[F(r)+2\right]}}

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 atτ₁. 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.^[10]
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^[16] 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 for 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 minimum 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 creates 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 is plotted along with the data, such that the reader of the plot is able to be aware of the statistical uncertainty of the values.
It is recommended that the length of the sample sequence, i.e. the number of samples N is kept high to ensure that confidence interval is small over the τ-range of interest.
It is recommended that the τ-range as swept by the τ₀ multiplier n is limited in the upper end relative N such that the read of the plot is not being confused by highly unstable estimator values.
It is recommended that 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.

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^[6] allowed for 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,^[17]^[18]^[19] 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. Lets 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 will the reference clock 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 should also logging and measurement mistakes 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 tau.

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

The field of frequency stability has been studied for a long time, however it was found during the 1960s that there was a lack of coherent definitions. The NASA-IEEE Symposium on Short-Term Stability in 1964 was followed with the IEEE Proceedings publishing a special issue on Frequency Stability in its February 1966 issue.
The NASA-IEEE Symposium on Short-Term Stability in November 1964^[20] brought together many fields and uses of short and long term stability with papers from many different contributors. The articles and panel discussions is interesting in that they concur on the existence of the frequency flicker noise and the wish to achieve a common definition for short and long term stability (even if the conference name only reflect the short-term stability intention).
The IEEE proceedings on Frequency Stability 1966 included a number of important papers including those of David Allan,^[3] James A. Barnes,^[21] L. S. Cutler and C. L. Searle^[1] and D. B. Leeson.^[2] These papers helped shape the field.
The classical M-sample variance of frequency was analysed by David Allan in^[3] along with an initial bias function. This paper tackles the issues of dead-time between measurements and analyses the case of M frequency samples (called N in the paper) and variance estimators. It provides the now standard α to µ mapping. It clearly builds on James Barnes work as detailed in his article^[21] in the same issue. The initial bias functions introduced assumes no dead-time, but the formulas presented includes dead-time calculations. The bias function assumes the use of the 2-sample variance as a base-case, since any other variants of M may be chosen and values may be transferred via the 2-sample variance to any other variance for of arbitrary M. Thus, the 2-sample variance was only implicitly used and not clearly stated as the preference even if the tools where provided. It however laid the foundation for using the 2-sample variance as the base case of comparison among other variants of the M-sample variance. The 2-sample variance case is a special case of the M-sample variance which produces an average of the frequency derivative.
The work on bias functions was significantly extended by James Barnes in^[14] in which the modern B₁ and B₂ bias functions was introduced. Curiously enough, it refers to the M-sample variance as "Allan variance" while referencing to Allan's paper "Statistics of Atomic Frequency Standards".^[3] With these modern bias functions, full conversion among M-sample variance measures of variating M, T and τ values could used, by conversion through the 2-sample variance.
James Barnes and David Allan further extended the bias functions with the B₃ function in^[15] to handle the concatenated samples estimator bias. This was necessary to handle the new use of concatenated sample observations with dead time in between.
The IEEE Technical Committee on Frequency and Time within the IEEE Group on Instrumentation & Measurements provided a summary of the field in 1970 published as NBS Technical Notice 394.^[10] This paper could be considered first in a line of more educational and practical papers aiding the fellow engineers in grasping the field. In this paper the 2-sample variance with T=τ is being the recommended measurement and it is referred to as Allan variance (now without the quotes). The choice of such parametrisation allows good handling of some noise forms and to get comparable measurements, it is essentially the least common denominator with the aid of the bias functions B₁ and B₂.
An improved method for using sample statistics for frequency counters in frequency estimation or variance estimation was proposed by J.J. Snyder.^[6] The trick to get more effective degrees of freedom out of the available dataset was to use overlapping observation periods. This provides a square-root n improvement. It was included into the overlapping Allan variance estimator introduced in.^[7] The variable τ software processing was also included in.^[7] This development improved the classical Allan variance estimators likewise providing a direct inspiration going into the work on modified Allan variance.
The confidence interval and degrees of freedom analysis, along with the established estimators was presented in.^[7]

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 some 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".^[10] 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^[22] from 1974, which in chapter 8 has "Statistics of Time and Frequency Data Analysis".^[23] 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.^[7] 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^[4] 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".^[12] 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 likewise telecommunication specific measures such as MTIE. It is a handy companion when looking at telecommunication standard related measurements.
The NIST Special Publication 1065 "Handbook of Frequency Stability Analysis" of W.J. Riley^[13] 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 and MEMS gyroscopes and accelerometers .

Photomultiplier

Photomultiplier tubes (photomultipliers or PMTs for short), members of the class of vacuum tubes, and more specifically vacuum phototubes, are extremely sensitive detectors of light in the ultraviolet, visible, and near-infrared ranges of the electromagnetic spectrum. These detectors multiply the current produced by incident light by as much as 100 million times (i.e., 160 dB), in multiple dynode stages, enabling (for example) individual photons to be detected when the incident flux of light is low.

Dynodes inside a photomultiplier tube

The combination of high gain, low noise, high frequency response or, equivalently, ultra-fast response, and large area of collection has maintained photomultipliers an essential place in nuclear and particle physics, astronomy, medical diagnostics including blood tests, medical imaging, motion picture film scanning (telecine), radar jamming, and high-end image scanners known as drum scanners. Elements of photomultiplier technology, when integrated differently, are the basis of night vision devices. Research that analyzes light scattering, such as the study of polymers in solution often use a laser and a PMT to collect the scattered light data.
Semiconductor devices, particularly avalanche photodiodes, are alternatives to photomultipliers; however, photomultipliers are uniquely well-suited for applications requiring low-noise, high-sensitivity detection of light that is imperfectly collimated.

Structure and operating principles

Fig.1: Schematic of a photomultiplier tube coupled to a scintillator. This arrangement is for detection of gamma rays.

Fig. 2: Typical photomultiplier voltage divider circuit using negative high voltage.

Photomultipliers are typically constructed with an evacuated glass housing (using an extremely tight and durable glass-to-metal seal like other vacuum tubes), containing a photocathode, several dynodes, and an anode. Incident photons strike the photocathode material, which is usually a thin vapor-deposited conducting layer on the inside of the entry window of the device. Electrons are ejected from the surface as a consequence of the photoelectric effect. These electrons are directed by the focusing electrode toward the electron multiplier, where electrons are multiplied by the process of secondary emission.
The electron multiplier consists of a number of electrodes called dynodes. Each dynode is held at a more positive potential, by ≈100 Volts, than the preceding one. A primary electron leaves the photocathode with the energy of the incoming photon, or about 3 eV for "blue" photons, minus the work function of the photocathode. A small group of primary electrons is created by the arrival of a group of initial photons. (In Fig. 1, the number of primary electrons in the initial group is proportional to the energy of the incident high energy gamma ray.) The primary electrons move toward the first dynode because they are accelerated by the electric field. They each arrive with ≈100 eV kinetic energy imparted by the potential difference. Upon striking the first dynode, more low energy electrons are emitted, and these electrons are in turn accelerated toward the second dynode. The geometry of the dynode chain is such that a cascade occurs with an exponentially-increasing number of electrons being produced at each stage. For example, if at each stage an average of 5 new electrons are produced for each incoming electron, and if there are 12 dynode stages, then at the last stage one expects for each primary electron about 5¹² ≈ 10⁸ electrons. This last stage is called the anode. This large number of electrons reaching the anode results in a sharp current pulse that is easily detectable, for example on an oscilloscope, signaling the arrival of the photon(s) at the photocathode ≈50 nanoseconds earlier.
The necessary distribution of voltage along the series of dynodes is created by a voltage divider chain, as illustrated in Fig. 2. In the example, the photocathode is held at a negative high voltage of order 1000V, while the anode is very close to ground potential. The capacitors across the final few dynodes act as local reservoirs of charge to help maintain the voltage on the dynodes while electron avalanches propagate through the tube. Many variations of design are used in practice; the design shown is merely illustrative.
There are two common photomultiplier orientations, the head-on or end-on (transmission mode) design, as shown above, where light enters the flat, circular top of the tube and passes the photocathode, and the side-on design (reflection mode), where light enters at a particular spot on the side of the tube, and impacts on an opaque photocathode. The side-on design is used, for instance, in the type 931, the first mass-produced PMT. Besides the different photocathode materials, performance is also affected by the transmission of the window material that the light passes through, and by the arrangement of the dynodes. Many photomultiplier models are available having various combinations of these, and other, design variables. The manufacturers manuals provide the information needed to choose an appropriate design for a particular application.

The invention of the photomultiplier is predicated upon two prior achievements, the separate discoveries of the photoelectric effect and of secondary emission.

Photoelectric effect

The first demonstration of the photoelectric effect was carried out in 1887 by Heinrich Hertz using ultraviolet light.^[1] Significant for practical applications, Elster and Geitel two years later demonstrated the same effect using visible light striking alkali metals (potassium and sodium).^[2] The addition of caesium, another alkali metal, has permitted the range of sensitive wavelengths to be extended towards longer wavelengths in the red portion of the visible spectrum.
Historically, the photoelectric effect is associated with Albert Einstein, who relied upon the phenomenon to establish the fundamental principle of quantum mechanics in 1905,^[3] an accomplishment for which Einstein received the 1921 Nobel Prize. It is worthwhile to note that Heinrich Hertz, working 18 years earlier, had not recognized that the kinetic energy of the emitted electrons is proportional to the frequency but independent of the optical intensity. This fact implied a discrete nature of light, i.e. the existence of quanta, for the first time.

Secondary emission

The phenomenon of secondary emission (the ability of electrons in a vacuum tube to cause the emission of additional electrons by striking an electrode) was, at first, limited to purely electronic phenomena and devices (which lacked photosensitivity). In 1899 the effect was first reported by Villard.^[4] In 1902, Austin and Starke reported that the metal surfaces impacted by electron beams emitted a larger number of electrons than were incident.^[5] The application of the newly discovered secondary emission to the amplification of signals was only proposed after World War I by Westinghouse scientist Joseph Slepian in a 1919 patent.^[6]

The race towards a practical electronic television camera

The ingredients for inventing the photomultiplier were coming together during the 1920s as the pace of vacuum tube technology accelerated. The primary goal for many, if not most, workers was the need for a practical television camera technology. Television had been pursued with primitive prototypes for decades prior to the 1934 introduction of the first practical camera (the iconoscope). Early prototype television cameras lacked sensitivity. Photomultiplier technology was pursued to enable television camera tubes, such as the iconoscope and (later) the orthicon, to be sensitive enough to be practical. So the stage was set to combine the dual phenomena of photoemission (i.e., the photoelectric effect) with secondary emission, both of which had already been studied and adequately understood, to create a practical photomultiplier.

First photomultiplier, single-stage (early 1934)

The first documented photomultiplier demonstration dates to the early 1934 accomplishments of an RCA group based in Harrison, NJ. Harley Iams and Bernard Salzberg were the first to integrate a photoelectric-effect cathode and single secondary emission amplification stage in a single vacuum envelope and the first to characterize its performance as a photomultiplier with electron amplification gain. These accomplishments were finalized prior to June 1934 as detailed in the manuscript submitted to Proceedings of the Institute of Radio Engineers (Proc. IRE).^[7] The device consisted of a semi-cylindrical photocathode, a secondary emitter mounted on the axis, and a collector grid surrounding the secondary emitter. The tube had a gain of about eight and operated at frequencies well above 10 kHz.

Magnetic photomultipliers (mid 1934–1937)

Higher gains were sought than those available from the early single-stage photomultipliers. However, it is an empirical fact that the yield of secondary electrons is limited in any given secondary emission process, regardless of acceleration voltage. Thus, any single-stage photomultiplier is limited in gain. At the time the maximum first-stage gain that could be achieved was approximately 10 (very significant developments in the 1960s permitted gains above 25 to be reached using negative electron affinity dynodes). For this reason, multiple-stage photomultipliers, in which the photoelectron yield could be multiplied successively in several stages, were an important goal. The challenge was to cause the photoelectrons to impinge on successively higher-voltage electrodes rather than to travel directly to the highest voltage electrode. Initially this challenge was overcome by using strong magnetic fields to bend the electrons' trajectories. Such a scheme had earlier been conceived by inventor J. Slepian by 1919 (see above). Accordingly, leading international research organizations turned their attention towards improving photomultiplers to achieve higher gain with multiple stages.
In the USSR, RCA-manufactured radio equipment was introduced on a large scale by Joseph Stalin to construct broadcast networks, and the newly formed All-Union Scientific Research Institute for Television was gearing up a research program in vacuum tubes that was advanced for its time and place. Numerous visits were made by RCA scientific personnel to the USSR in the 1930s, prior to the Cold War, to instruct the Soviet customers on the capabilities of RCA equipment and to investigate customer needs.^[8] During one of these visits, in September 1934, RCA's Vladimir Zworykin was shown the first multiple-dynode photomultiplier, or photoelectron multiplier. This pioneering device was proposed by Leonid A. Kubetsky in 1930^[9] which he subsequently built in 1934. The device achieved gains of 1000x or more when demonstrated in June 1934. The work was submitted for print publication only two years later, in July 1936^[10] as emphasized in a recent 2006 publication of the Russian Academy of Sciences (RAS),^[11] which terms it "Kubetsky's Tube." The Soviet device used a magnetic field to confine the secondary electrons and relied on the Ag-O-Cs photocathode which had been demonstrated by General Electric in the 1920s.
By October 1935, Vladimir Zworykin, George Ashmun Morton, and Louis Malter of RCA in Camden, NJ submitted their manuscript describing the first comprehensive experimental and theoretical analysis of a multiple dynode tube — the device later called a photomultiplier^[12] — to Proc. IRE. The RCA prototype photomultipliers also used an Ag-O-Cs (silver oxide-caesium) photocathode. They exhibited a peak quantum efficiency of 0.4% at 800 nm.

Electrostatic photomultipliers (1937–present)

Whereas these early photomultipliers used the magnetic field principle, electrostatic photomultipliers (with no magnetic field) were demonstrated by Jan Rajchman of RCA Laboratories in Princeton, NJ in the late 1930s and became the standard for all future commercial photomultipliers. The first mass-produced photomultiplier, the Type 931, was of this design and is still commercially produced today.^[13]

Improved photocathodes

Also in 1936, a much improved photocathode, Cs₃Sb (caesium-antimony), was reported by P. Görlich.^[14] The caesium-antimony photocathode had a dramatically improved quantum efficiency of 12% at 400 nm, and was used in the first commercially successful photomultipliers manufactured by RCA (i.e., the 931-type) both as a photocathode and as a secondary-emitting material for the dynodes. Different photocathodes provided differing spectral responses.

Spectral response of photocathodes

In the early 1940s, the JEDEC (Joint Electron Device Engineering Council), an industry committee on standardization, developed a system of designating spectral responses.^[15] The philosophy included the idea that the product's user need only be concerned about the response of the device rather than how the device may be fabricated. Various combinations of photocathode and window materials were assigned "S-numbers" (spectral numbers) ranging from S-1 through S-40, which are still in use today. For example, S-11 uses the caesium-antimony photocathode with a lime glass window, S-13 uses the same photocathode with a fused silica window, and S-25 uses a so-called "multialkali" photocathode (Na-K-Sb-Cs, or sodium-potassium-antimony-caesium) that provides extended response in the red portion of the visible light spectrum. No suitable photoemissive surfaces have yet been reported to detect wavelengths longer than approximately 1700 nanometers, which can be approached by a special (InP/InGaAs(Cs)) photocathode.^[16]

RCA Corporation

For decades, RCA was responsible for performing the most important work in developing and refining photomultipliers. RCA was also largely responsible for the commercialization of photomultiplers. The company compiled and published an authoritative and widely used Photomultiplier Handbook.^[17] RCA provided printed copies free upon request. The handbook, which continues to be made available online at no cost by the successors to RCA, is considered to be an essential reference.
Following a corporate break-up in the late 1980s involving the acquisition of RCA by General Electric and disposition of the divisions of RCA to numerous third parties, RCA's photomultiplier business became an independent company.

Lancaster, Pennsylvania facility

The Lancaster, Pennsylvania facility was opened by the U.S. Navy in 1942 and operated by RCA for the manufacture of radio and microwave tubes. Following World War II, the naval facility was acquired by RCA. RCA Lancaster, as it became known, was the base for development and production of commercial television products. In subsequent years other products were added, such as cathode ray tubes, photomultiplier tubes, motion-sensing light control switches, and closed-circuit television systems.

Burle Industries

Burle Industries, as a successor to the RCA Corporation, carried the RCA photomultiplier business forward after 1986, based in the Lancaster, Pennsylvania facility. The 1986 acquisition of RCA by General Electric resulted in the divestiture of the RCA Lancaster New Products Division. Hence, 45 years after being founded by the U.S. Navy, its management team, led by Erich Burlefinger, purchased the division and in 1987 founded Burle Industries.
In 2005, after eighteen years as an independent enterprise, Burle Industries and a key subsidiary were acquired by Photonis, a European holding company Photonis Group. Following the acquisition, Photonis was composed of Photonis Netherlands, Photonis France, Photonis USA, and Burle Industries. Photonis USA operates the former Galileo Corporation Scientific Detector Products Group (Sturbridge, Massachusetts), which had been purchased by Burle Industries in 1999. The group is known for microchannel plate detector (MCP) electron multipliers—an integrated micro-vacuum tube version of photomultipliers. MCPs are used for imaging and scientific applications, including night vision devices.
On 9 March 2009, Photonis announced that it would cease all production of photomultipliers at both the Lancaster, Pennsylvania and the Brive, France plants.^[18]

Hamamatsu

The Japan-based company Hamamatsu Photonics (also known as Hamamatsu) has emerged since the 1950s as a leader in the photomultiplier industry. Hamamatsu, in the tradition of RCA, has published its own handbook, which is available without cost on the company's website. Hamamatsu uses different designations for particular photocathode formulations and introduces modifications to these designations based on Hamamatsu's proprietary research and development.

Photocathode materials

The photocathodes can be made of a variety of materials, with different properties. Typically the materials have low work function and are therefore prone to thermionic emission, causing noise and dark current, especially the materials sensitive in infrared; cooling the photocathode lowers this thermal noise. The most common photocathode materials are^[20] Ag-O-Cs (also called S1) transmission-mode, sensitive from 300–1200 nm. High dark current; used mainly in near-infrared, with the photocathode cooled; GaAs:Cs, caesium-activated gallium arsenide, flat response from 300 to 850 nm, fading towards ultraviolet and to 930 nm; InGaAs:Cs, caesium-activated indium gallium arsenide, higher infrared sensitivity than GaAs:Cs, between 900–1000 nm much higher signal-to-noise ratio than Ag-O-Cs; Sb-Cs, (also called S11) caesium-activated antimony, used for reflective mode photocathodes; response range from ultraviolet to visible, widely used; bialkali (Sb-K-Cs, Sb-Rb-Cs), caesium-activated antimony-rubidium or antimony-potassium alloy, similar to Sb:Cs, with higher sensitivity and lower noise. can be used for transmission-mode; favorable response to a NaI:Tl scintillator flashes makes them widely used in gamma spectroscopy and radiation detection; high-temperature bialkali (Na-K-Sb), can operate up to 175 °C, used in well logging, low dark current at room temperature; multialkali (Na-K-Sb-Cs), (also called S20), wide spectral response from ultraviolet to near-infrared, special cathode processing can extend range to 930 nm, used in broadband spectrophotometers; solar-blind (Cs-Te, Cs-I), sensitive to vacuum-UV and ultraviolet, insensitive to visible light and infrared (Cs-Te has cutoff at 320 nm, Cs-I at 200 nm).

Window materials

The windows of the photomultipliers act as wavelength filters; this may be irrelevant if the cutoff wavelengths are outside of the application range or outside of the photocathode sensitivity range, but special care has to be taken for uncommon wavelengths. Borosilicate glass is commonly used for near-infrared to about 300 nm. High borate borosilicate glasses exist also in high UV transmission versions with high transmission also at 254 nm.^[21] Glass with very low content of potassium can be used with bialkali photocathodes to lower the background radiation from the potassium-40 isotope. Ultraviolet glass transmits visible and ultraviolet down to 185 nm. Used in spectroscopy. Synthetic silica transmits down to 160 nm, absorbs less UV than fused silica. Different thermal expansion than kovar (and than borosilicate glass that's expansion-matched to kovar), a graded seal needed between the window and the rest of the tube. The seal is vulnerable to mechanical shocks. Magnesium fluoride transmits ultraviolet down to 115 nm. Hygroscopic, though less than other alkali halides usable for UV windows.

Usage considerations

Photomultiplier tubes typically utilize 1000 to 2000 volts to accelerate electrons within the chain of dynodes. (See Figure near top of article.) The most negative voltage is connected to the cathode, and the most positive voltage is connected to the anode. Negative high-voltage supplies (with the positive terminal grounded) are often preferred, because this configuration enables the photocurrent to be measured at the low voltage side of the circuit for amplification by subsequent electronic circuits operating at low voltage. However, with the photocathode at high voltage, leakage currents sometimes result in unwanted "dark current" pulses that may affect the operation. Voltages are distributed to the dynodes by a resistive voltage divider, although variations such as active designs (with transistors or diodes) are possible. The divider design, which influences frequency response or rise time, can be selected to suit varying applications. Some instruments that use photomultipliers have provisions to vary the anode voltage to control the gain of the system.
While powered (energized), photomultipliers must be shielded from ambient light to prevent their destruction through overexcitation. In some applications this protection is accomplished mechanically by electrical interlocks or shutters that protect the tube when the photomultiplier compartment is opened. Another option is to add overcurrent protection in the external circuit, so that when the measured anode current exceeds a safe limit, the high voltage is reduced.
If used in a location with strong magnetic fields, which can curve electron paths, steer the electrons away from the dynodes and cause loss of gain, photomultipliers are usually magnetically shielded by a layer of soft iron or mu-metal. This magnetic shield is often maintained at cathode potential. When this is the case, the external shield must also be electrically insulated because of the high voltage on it. Photomultipliers with large distances between the photocathode and the first dynode are especially sensitive to magnetic fields.

Applications

Photomultipliers were the first electric eye devices, being used to measure interruptions in beams of light. Photomultipliers are used in conjunction with scintillators to detect Ionizing radiation by means of hand held and fixed radiation protection instruments, and particle radiation in physics experiments.^[22] Photomultipliers are used in research laboratories to measure the intensity and spectrum of light-emitting materials such as compound semiconductors and quantum dots. Photomultipliers are used as the detector in many spectrophotometers. This allows an instrument design that escapes the thermal noise limit on sensitivity, and which can therefore substantially increase the dynamic range of the instrument.
Photomultipliers are used in numerous medical equipment designs. For example, blood analysis devices used by clinical medical laboratories, such as flow cytometers, utilize photomultipliers to determine the relative concentration of various components in blood samples, in combination with optical filters and incandescent lamps. An array of photomultipliers is used in a gamma camera. Photomultipliers are typically used as the detectors in flying-spot scanners.^{[citation needed]}

High-sensitivity applications

After 50 years, during which solid-state electronic components have largely displaced the vacuum tube, the photomultiplier remains a unique and important optoelectronic component. Perhaps its most useful quality is that it acts, electronically, as a nearly perfect current source, owing to the high voltage utilized in extracting the tiny currents associated with weak light signals. There is no Johnson noise associated with photomultiplier signal currents, even though they are greatly amplified, e.g., by 100 thousand times (i.e., 100 dB) or more. The photocurrent still contains shot noise.
Photomultiplier-amplified photocurrents can be electronically amplified by a high-input-impedance electronic amplifier (in the signal path subsequent to the photomultiplier), thus producing appreciable voltages even for nearly infinitesimally small photon fluxes. Photomultipliers offer the best possible opportunity to exceed the Johnson noise for many configurations. The aforementioned refers to measurement of light fluxes that, while small, nonetheless amount to a continuous stream of multiple photons.
For smaller photon fluxes, the photomultiplier can be operated in photon-counting, or Geiger, mode (see also Single-photon avalanche diode). In Geiger mode the photomultiplier gain is set so high (using high voltage) that a single photo-electron resulting from a single photon incident on the primary surface generates a very large current at the output circuit. However, owing to the avalanche of current, a reset of the photomultiplier is required. In either case, the photomultiplier can detect individual photons. The drawback, however, is that not every photon incident on the primary surface is counted either because of less-than-perfect efficiency of the photomultiplier, or because a second photon can arrive at the photomultiplier during the "dead time" associated with a first photon and never be noticed.
A photomultiplier will produce a small current even without incident photons; this is called the dark current. Photon-counting applications generally demand photomultipliers designed to minimise dark current.
Nonetheless, the ability to detect single photons striking the primary photosensitive surface itself reveals the quantization principle that Einstein put forth. Photon counting (as it is called) reveals that light, not only being a wave, consists of discrete particles

Precision cardiac mapping system and Advisor FL circular mapping catheter, Sensor Enabled

Positron emission tomography

Positron-emission tomography (PET) is a nuclear medicine functional imaging technique that is used to observe metabolic processes in the body. The system detects pairs of gamma rays emitted indirectly by a positron-emitting radionuclide (tracer), which is introduced into the body on a biologically active molecule. Three-dimensional images of tracer concentration within the body are then constructed by computer analysis. In modern PET-CT scanners, three-dimensional imaging is often accomplished with the aid of a CT X-ray scan performed on the patient during the same session, in the same machine.
If the biologically active molecule chosen for PET is fludeoxyglucose (FDG), an analogue of glucose, the concentrations of tracer imaged will indicate tissue metabolic activity as it corresponds to the regional glucose uptake. Use of this tracer to explore the possibility of cancer metastasis (i.e., spreading to other sites) is the most common type of PET scan in standard medical care (90% of current scans). However, although on a minority basis, many other radioactive tracers are used in PET to image the tissue concentration of other types of molecules of interest. One of the disadvantages of PET scanners is their operating cost .

Uses

PET/CT-System with 16-slice CT; the ceiling mounted device is an injection pump for CT contrast agent

Whole-body PET scan using ¹⁸F-FDG

PET is both a medical and research tool. It is used heavily in clinical oncology (medical imaging of tumors and the search for metastases), and for clinical diagnosis of certain diffuse brain diseases such as those causing various types of dementias. PET is also an important research tool to map normal human brain and heart function, and support drug development.
PET is also used in pre-clinical studies using animals, where it allows repeated investigations into the same subjects. This is particularly valuable in cancer research, as it results in an increase in the statistical quality of the data (subjects can act as their own control) and substantially reduces the numbers of animals required for a given study.
Alternative methods of scanning include x-ray computed tomography (CT), magnetic resonance imaging (MRI) and functional magnetic resonance imaging (fMRI), ultrasound and single-photon emission computed tomography (SPECT).
While some imaging scans such as CT and MRI isolate organic anatomic changes in the body, PET and SPECT are capable of detecting areas of molecular biology detail (even prior to anatomic change). PET scanning does this using radiolabelled molecular probes that have different rates of uptake depending on the type and function of tissue involved. Changing of regional blood flow in various anatomic structures (as a measure of the injected positron emitter) can be visualized and relatively quantified with a PET scan.
PET imaging is best performed using a dedicated PET scanner. However, it is possible to acquire PET images using a conventional dual-head gamma camera fitted with a coincidence detector. The quality of gamma-camera PET is considerably lower, and acquisition is slower. However, for institutions with low demand for PET, this may allow on-site imaging, instead of referring patients to another center, or relying on a visit by a mobile scanner.
PET is a valuable technique for some diseases and disorders, because it is possible to target the radio-chemicals used for particular bodily functions.

Oncology

PET scanning with the tracer fluorine-18 (F-18) fluorodeoxyglucose (FDG), called FDG-PET, is widely used in clinical oncology. This tracer is a glucose analog that is taken up by glucose-using cells and phosphorylated by hexokinase (whose mitochondrial form is greatly elevated in rapidly growing malignant tumors). A typical dose of FDG used in an oncological scan has an effective radiation dose of 14 mSv.^[3] Because the oxygen atom that is replaced by F-18 to generate FDG is required for the next step in glucose metabolism in all cells, no further reactions occur in FDG. Furthermore, most tissues (with the notable exception of liver and kidneys) cannot remove the phosphate added by hexokinase. This means that FDG is trapped in any cell that takes it up, until it decays, since phosphorylated sugars, due to their ionic charge, cannot exit from the cell. This results in intense radiolabeling of tissues with high glucose uptake, such as the brain, the liver, and most cancers. As a result, FDG-PET can be used for diagnosis, staging, and monitoring treatment of cancers, particularly in Hodgkin's lymphoma, non-Hodgkin lymphoma, and lung cancer.^{[citation needed]}
A few other isotopes and radiotracers are slowly being introduced into oncology for specific purposes. For example, ¹¹C-labelled metomidate (11C-metomidate), has been used to detect tumors of adrenocortical origin.^[4]^[5] Also, FDOPA PET-CT, in centers which offer it, has proven to be a more sensitive alternative to finding, and also localizing, pheochromocytoma than the MIBG scan.^[6]^[7]^[8]

Neuroimaging

PET scan of the human brain.

Neurology: PET neuroimaging is based on an assumption that areas of high radioactivity are associated with brain activity. What is actually measured indirectly is the flow of blood to different parts of the brain, which is, in general, believed to be correlated, and has been measured using the tracer oxygen-15. However, because of its 2-minute half-life, O-15 must be piped directly from a medical cyclotron for such uses, which is difficult. In practice, since the brain is normally a rapid user of glucose, and since brain pathologies such as Alzheimer's disease greatly decrease brain metabolism of both glucose and oxygen in tandem, standard FDG-PET of the brain, which measures regional glucose use, may also be successfully used to differentiate Alzheimer's disease from other dementing processes, and also to make early diagnosis of Alzheimer's disease. The advantage of FDG-PET for these uses is its much wider availability. PET imaging with FDG can also be used for localization of seizure focus: A seizure focus will appear as hypometabolic during an interictal scan. Several radiotracers (i.e. radioligands) have been developed for PET that are ligands for specific neuroreceptor subtypes such as [¹¹C] raclopride, [¹⁸F] fallypride and [¹⁸F] desmethoxyfallypride for dopamine D2/D3 receptors, [¹¹C] McN 5652 and [¹¹C] DASB for serotonin transporters, [¹⁸F] Mefway for serotonin 5HT1A receptors, [¹⁸F] Nifene for nicotinic acetylcholine receptors or enzyme substrates (e.g. 6-FDOPA for the AADC enzyme). These agents permit the visualization of neuroreceptor pools in the context of a plurality of neuropsychiatric and neurologic illnesses.

The development of a number of novel probes for noninvasive, in vivo PET imaging of neuroaggregate in human brain has brought amyloid imaging to the doorstep of clinical use. The earliest amyloid imaging probes included 2-(1-{6-[(2-[¹⁸F]fluoroethyl)(methyl)amino]-2-naphthyl}ethylidene)malononitrile ([¹⁸F]FDDNP)^[9] developed at the University of California, Los Angeles and N-methyl-[¹¹C]2-(4'-methylaminophenyl)-6-hydroxybenzothiazole^[10] (termed Pittsburgh compound B) developed at the University of Pittsburgh. These amyloid imaging probes permit the visualization of amyloid plaques in the brains of Alzheimer's patients and could assist clinicians in making a positive clinical diagnosis of AD pre-mortem and aid in the development of novel anti-amyloid therapies. [¹¹C]PMP (N-[¹¹C]methylpiperidin-4-yl propionate) is a novel radiopharmaceutical used in PET imaging to determine the activity of the acetylcholinergic neurotransmitter system by acting as a substrate for acetylcholinesterase. Post-mortem examination of AD patients have shown decreased levels of acetylcholinesterase. [¹¹C]PMP is used to map the acetylcholinesterase activity in the brain, which could allow for pre-mortem diagnosis of AD and help to monitor AD treatments.^[11] Avid Radiopharmaceuticals of Philadelphia has developed a compound called 18F-AV-45 that uses the longer-lasting radionuclide fluorine-18 to detect amyloid plaques using PET scans.^[12]

Neuropsychology / Cognitive neuroscience: To examine links between specific psychological processes or disorders and brain activity.
Psychiatry: Numerous compounds that bind selectively to neuroreceptors of interest in biological psychiatry have been radiolabeled with C-11 or F-18. Radioligands that bind to dopamine receptors (D1,^[13] D2 receptor,^[14]^[15] reuptake transporter), serotonin receptors (5HT1A, 5HT2A, reuptake transporter) opioid receptors (mu) and other sites have been used successfully in studies with human subjects. Studies have been performed examining the state of these receptors in patients compared to healthy controls in schizophrenia, substance abuse, mood disorders and other psychiatric conditions.
Stereotactic surgery and radiosurgery: PET-image guided surgery facilitates treatment of intracranial tumors, arteriovenous malformations and other surgically treatable conditions.^[16]

Cardiology

Cardiology, atherosclerosis and vascular disease study: In clinical cardiology, FDG-PET can identify so-called "hibernating myocardium", but its cost-effectiveness in this role versus SPECT is unclear. FDG-PET imaging of atherosclerosis to detect patients at risk of stroke is also feasible and can help test the efficacy of novel anti-atherosclerosis therapies.^[17]

Infectious diseases

Imaging infections with molecular imaging technologies can improve diagnosis and treatment follow-up. PET has been widely used to image bacterial infections clinically by using fluorodeoxyglucose (FDG) to identify the infection-associated inflammatory response.
Recently, three different PET contrast agents have been developed to image bacterial infections in vivo: [¹⁸F]maltose,^[18] [¹⁸F]maltohexaose and [¹⁸F]2-fluorodeoxysorbitol (FDS).^[19] FDS has also the added benefit of being able to target only Enterobacteriaceae.

Pharmacokinetics

Pharmacokinetics: In pre-clinical trials, it is possible to radiolabel a new drug and inject it into animals. Such scans are referred to as biodistribution studies. The uptake of the drug, the tissues in which it concentrates, and its eventual elimination, can be monitored far more quickly and cost effectively than the older technique of killing and dissecting the animals to discover the same information. Much more commonly, however, drug occupancy at a purported site of action can be inferred indirectly by competition studies between unlabeled drug and radiolabeled compounds known apriori to bind with specificity to the site. A single radioligand can be used this way to test many potential drug candidates for the same target. A related technique involves scanning with radioligands that compete with an endogenous (naturally occurring) substance at a given receptor to demonstrate that a drug causes the release of the natural substance.^{[citation needed]}
The following is an excerpt from an article by Harvard University staff writer Peter Reuell, featured in HarvardScience, part of the online version of the Harvard Gazette newspaper, which discusses research by the team of Harvard Associate Professor of Organic Chemistry and Chemical Biology Tobias Ritter: "A new chemical process ... may increase the utility of positron emission tomography (PET) in creating real-time 3-D images of chemical activity occurring inside the body. This new work ... holds out the tantalizing possibility of using PET scans to peer into a number of functions inside animals and humans by simplifying the process of using “tracer” molecules to create the 3-D images." (by creating a novel electrophilic fluorination reagent as an intermediate molecule; the research could be used in drug development).^[20]

Small animal imaging

PET technology for small animal imaging: A miniature PE tomograph has been constructed that is small enough for a fully conscious and mobile rat to wear on its head while walking around.^[21] This RatCAP (Rat Conscious Animal PET) allows animals to be scanned without the confounding effects of anesthesia. PET scanners designed specifically for imaging rodents, often referred to as microPET, as well as scanners for small primates are marketed for academic and pharmaceutical research. The scanners are apparently based on microminiature scintillators and amplified avalanche photodiodes (APDs) though a new system recently invented uses single chip silicon photomultipliers.

Musculo-skeletal imaging

Musculoskeletal imaging: PET has been shown to be a feasible technique for studying skeletal muscles during exercises like walking.^[22] One of the main advantages of using PET is that it can also provide muscle activation data about deeper lying muscles such as the vastus intermedialis and the gluteus minimus, as compared to other muscle studying techniques like electromyography, which can be used only on superficial muscles (i.e., directly under the skin). A clear disadvantage, however, is that PET provides no timing information about muscle activation, because it has to be measured after the exercise is completed. This is due to the time it takes for FDG to accumulate in the activated muscles.

Safety

PET scanning is non-invasive, but it does involve exposure to ionizing radiation.^[2]
18F-FDG, which is now the standard radiotracer used for PET neuroimaging and cancer patient management,^[23] has an effective radiation dose of 14 mSv.^[3]
The amount of radiation in 18F-FDG is similar to the effective dose of spending one year in the American city of Denver, Colorado (12.4 mSv/year).^[24] For comparison, radiation dosage for other medical procedures range from 0.02 mSv for a chest x-ray and 6.5–8 mSv for a CT scan of the chest.^[25]^[26] Average civil aircrews are exposed to 3 mSv/year,^[27] and the whole body occupational dose limit for nuclear energy workers in the USA is 50mSv/year.^[28] For scale, see Orders of magnitude (radiation).
For PET-CT scanning, the radiation exposure may be substantial—around 23–26 mSv (for a 70 kg person—dose is likely to be higher for higher body weights).^[29]

Operation

Schematic view of a detector block and ring of a PET scanner

Radionuclides and radiotracers

Radionuclides used in PET scanning are typically isotopes with short half-lives ^[2] such as carbon-11 (~20 min), nitrogen-13 (~10 min), oxygen-15 (~2 min), fluorine-18 (~110 min), gallium-68 (~67 min), zirconium-89 (~78.41 hours), or rubidium-82(~1.27 min). These radionuclides are incorporated either into compounds normally used by the body such as glucose (or glucose analogues), water, or ammonia, or into molecules that bind to receptors or other sites of drug action. Such labelled compounds are known as radiotracers. PET technology can be used to trace the biologic pathway of any compound in living humans (and many other species as well), provided it can be radiolabeled with a PET isotope. Thus, the specific processes that can be probed with PET are virtually limitless, and radiotracers for new target molecules and processes are continuing to be synthesized; as of this writing there are already dozens in clinical use and hundreds applied in research. At present, however, by far the most commonly used radiotracer in clinical PET scanning is fluorodeoxyglucose (also called FDG or fludeoxyglucose), an analogue of glucose that is labeled with fluorine-18. This radiotracer is used in essentially all scans for oncology and most scans in neurology, and thus makes up the large majority of all of the radiotracer (> 95%) used in PET and PET-CT scanning.
Due to the short half-lives of most positron-emitting radioisotopes, the radiotracers have traditionally been produced using a cyclotron in close proximity to the PET imaging facility. The half-life of fluorine-18 is long enough that radiotracers labeled with fluorine-18 can be manufactured commercially at offsite locations and shipped to imaging centers. Recently rubidium-82 generators have become commercially available.^[30] These contain strontium-82, which decays by electron capture to produce positron-emitting rubidium-82.

Emission

To conduct the scan, a short-lived radioactive tracer isotope is injected into the living subject (usually into blood circulation). Each tracer atom has been chemically incorporated into a biologically active molecule. There is a waiting period while the active molecule becomes concentrated in tissues of interest; then the subject is placed in the imaging scanner. The molecule most commonly used for this purpose is F-18 labeled fluorodeoxyglucose (FDG), a sugar, for which the waiting period is typically an hour. During the scan, a record of tissue concentration is made as the tracer decays.

Schema of a PET acquisition process

As the radioisotope undergoes positron emission decay (also known as positive beta decay), it emits a positron, an antiparticle of the electron with opposite charge. The emitted positron travels in tissue for a short distance (typically less than 1 mm, but dependent on the isotope^[31]), during which time it loses kinetic energy, until it decelerates to a point where it can interact with an electron.^[32] The encounter annihilates both electron and positron, producing a pair of annihilation (gamma) photons moving in approximately opposite directions. These are detected when they reach a scintillator in the scanning device, creating a burst of light which is detected by photomultiplier tubes or silicon avalanche photodiodes (Si APD). The technique depends on simultaneous or coincident detection of the pair of photons moving in approximately opposite directions (they would be exactly opposite in their center of mass frame, but the scanner has no way to know this, and so has a built-in slight direction-error tolerance). Photons that do not arrive in temporal "pairs" (i.e. within a timing-window of a few nanoseconds) are ignored.

Localization of the positron annihilation event

The most significant fraction of electron–positron annihilations results in two 511 keV gamma photons being emitted at almost 180 degrees to each other; hence, it is possible to localize their source along a straight line of coincidence (also called the line of response, or LOR). In practice, the LOR has a non-zero width as the emitted photons are not exactly 180 degrees apart. If the resolving time of the detectors is less than 500 picoseconds rather than about 10 nanoseconds, it is possible to localize the event to a segment of a chord, whose length is determined by the detector timing resolution. As the timing resolution improves, the signal-to-noise ratio (SNR) of the image will improve, requiring fewer events to achieve the same image quality. This technology is not yet common, but it is available on some new systems.^[33]

Image reconstruction

The raw data collected by a PET scanner are a list of 'coincidence events' representing near-simultaneous detection (typically, within a window of 6 to 12 nanoseconds of each other) of annihilation photons by a pair of detectors. Each coincidence event represents a line in space connecting the two detectors along which the positron emission occurred (i.e., the line of response (LOR)).
Analytical techniques, much like the reconstruction of computed tomography (CT) and single-photon emission computed tomography (SPECT) data, are commonly used, although the data set collected in PET is much poorer than CT, so reconstruction techniques are more difficult. Coincidence events can be grouped into projection images, called sinograms. The sinograms are sorted by the angle of each view and tilt (for 3D images). The sinogram images are analogous to the projections captured by computed tomography (CT) scanners, and can be reconstructed in a similar way. However, the statistics of the data are much worse than those obtained through transmission tomography. A normal PET data set has millions of counts for the whole acquisition, while the CT can reach a few billion counts. This contributes to PET images appearing "noisier" than CT. Two major sources of noise in PET are scatter (a detected pair of photons, at least one of which was deflected from its original path by interaction with matter in the field of view, leading to the pair being assigned to an incorrect LOR) and random events (photons originating from two different annihilation events but incorrectly recorded as a coincidence pair because their arrival at their respective detectors occurred within a coincidence timing window).
In practice, considerable pre-processing of the data is required—correction for random coincidences, estimation and subtraction of scattered photons, detector dead-time correction (after the detection of a photon, the detector must "cool down" again) and detector-sensitivity correction (for both inherent detector sensitivity and changes in sensitivity due to angle of incidence).
Filtered back projection (FBP) has been frequently used to reconstruct images from the projections. This algorithm has the advantage of being simple while having a low requirement for computing resources. However, shot noise in the raw data is prominent in the reconstructed images and areas of high tracer uptake tend to form streaks across the image. Also, FBP treats the data deterministically—it does not account for the inherent randomness associated with PET data, thus requiring all the pre-reconstruction corrections described above.
Statistical, likelihood-based approaches: Statistical, likelihood-based ^[34] ^[35] iterative expectation-maximization algorithms such as the Shepp-Vardi algorithm^[36] are now the preferred method of reconstruction. These algorithms compute an estimate of the likely distribution of annihilation events that led to the measured data, based on statistical principles. The advantage is a better noise profile and resistance to the streak artifacts common with FBP, but the disadvantage is higher computer resource requirements. A further advantage of statistical image reconstruction techniques is that the physical effects that would need to be pre-corrected for when using an analytical reconstruction algorithm, such as scattered photons, random coincidences, attenuation and detector dead-time, can be incorporated into the likelihood model being used in the reconstruction, allowing for additional noise reduction. Iterative reconstruction has also been shown to result in improvements in the resolution of the reconstructed images, since more sophisticated models of the scanner Physics can be incorporated into the likelihood model than those used by analytical reconstruction methods, allowing for improved quantification of the radioactivity distribution.^[37]
Research has shown that Bayesian methods that involve a Poisson likelihood function and an appropriate prior probability (e.g., a smoothing prior leading to total variation regularization or a Laplacian distribution leading to

\ell _{1}

-based regularization in a wavelet or other domain), such as via Ulf Grenander's Sieve estimator^[38] ^[39] or via Bayes penalty methods ^[40] ^[41] or via I.J. Good's roughness method ^[42] ,^[43] may yield superior performance to expectation-maximization-based methods which involve a Poisson likelihood function but do not involve such a prior.^[44]^[45]^[46]
Attenuation correction: Attenuation occurs when photons emitted by the radiotracer inside the body are absorbed by intervening tissue between the detector and the emission of the photon. As different LORs must traverse different thicknesses of tissue, the photons are attenuated differentially. The result is that structures deep in the body are reconstructed as having falsely low tracer uptake. Contemporary scanners can estimate attenuation using integrated x-ray CT equipment, however earlier equipment offered a crude form of CT using a gamma ray (positron emitting) source and the PET detectors.
While attenuation-corrected images are generally more faithful representations, the correction process is itself susceptible to significant artifacts. As a result, both corrected and uncorrected images are always reconstructed and read together.
2D/3D reconstruction: Early PET scanners had only a single ring of detectors, hence the acquisition of data and subsequent reconstruction was restricted to a single transverse plane. More modern scanners now include multiple rings, essentially forming a cylinder of detectors.
There are two approaches to reconstructing data from such a scanner: 1) treat each ring as a separate entity, so that only coincidences within a ring are detected, the image from each ring can then be reconstructed individually (2D reconstruction), or 2) allow coincidences to be detected between rings as well as within rings, then reconstruct the entire volume together (3D).
3D techniques have better sensitivity (because more coincidences are detected and used) and therefore less noise, but are more sensitive to the effects of scatter and random coincidences, as well as requiring correspondingly greater computer resources. The advent of sub-nanosecond timing resolution detectors affords better random coincidence rejection, thus favoring 3D image reconstruction.
Time-of-flight (TOF) PET: For modern systems with a higher time resolution (roughly 3 nanoseconds) a technique called "Time-of-flight" is used to improve the overall performance. Time-of-flight PET makes use of very fast gamma-ray detectors and data processing system which can more precisely decide the difference in time between the detection of the two photons. Although it is technically impossible to localize the point of origin of the annihilation event exactly (currently within 10 cm) thus image reconstruction is still needed, TOF technique gives a remarkable improvement in image quality, especially signal-to-noise ratio.

Complete body PET-CT fusion image

Brain PET-MRI fusion image

Combination of PET with CT or MRI[

PET scans are increasingly read alongside CT or magnetic resonance imaging (MRI) scans, with the combination (called "co-registration") giving both anatomic and metabolic information (i.e., what the structure is, and what it is doing biochemically). Because PET imaging is most useful in combination with anatomical imaging, such as CT, modern PET scanners are now available with integrated high-end multi-detector-row CT scanners (so-called "PET-CT"). Because the two scans can be performed in immediate sequence during the same session, with the patient not changing position between the two types of scans, the two sets of images are more precisely registered, so that areas of abnormality on the PET imaging can be more perfectly correlated with anatomy on the CT images. This is very useful in showing detailed views of moving organs or structures with higher anatomical variation, which is more common outside the brain.
At the Jülich Institute of Neurosciences and Biophysics, the world's largest PET-MRI device began operation in April 2009: a 9.4-tesla magnetic resonance tomograph (MRT) combined with a positron emission tomograph (PET). Presently, only the head and brain can be imaged at these high magnetic field strengths.^[47]
For brain imaging, registration of CT, MRI and PET scans may be accomplished without the need for an integrated PET-CT or PET-MRI scanner by using a device known as the N-localizer.

Limitations

The minimization of radiation dose to the subject is an attractive feature of the use of short-lived radionuclides. Besides its established role as a diagnostic technique, PET has an expanding role as a method to assess the response to therapy, in particular, cancer therapy,^[64] where the risk to the patient from lack of knowledge about disease progress is much greater than the risk from the test radiation.
Limitations to the widespread use of PET arise from the high costs of cyclotrons needed to produce the short-lived radionuclides for PET scanning and the need for specially adapted on-site chemical synthesis apparatus to produce the radiopharmaceuticals after radioisotope preparation. Organic radiotracer molecules that will contain a positron-emitting radioisotope cannot be synthesized first and then the radioisotope prepared within them, because bombardment with a cyclotron to prepare the radioisotope destroys any organic carrier for it. Instead, the isotope must be prepared first, then afterward, the chemistry to prepare any organic radiotracer (such as FDG) accomplished very quickly, in the short time before the isotope decays. Few hospitals and universities are capable of maintaining such systems, and most clinical PET is supported by third-party suppliers of radiotracers that can supply many sites simultaneously. This limitation restricts clinical PET primarily to the use of tracers labelled with fluorine-18, which has a half-life of 110 minutes and can be transported a reasonable distance before use, or to rubidium-82 (used as rubidium-82 chloride) with a half-life of 1.27 minutes, which is created in a portable generator and is used for myocardial perfusion studies. Nevertheless, in recent years a few on-site cyclotrons with integrated shielding and "hot labs" (automated chemistry labs that are able to work with radioisotopes) have begun to accompany PET units to remote hospitals. The presence of the small on-site cyclotron promises to expand in the future as the cyclotrons shrink in response to the high cost of isotope transportation to remote PET machines.^[65] In recent years the shortage of PET scans has been alleviated in the US, as rollout of radiopharmacies to supply radioisotopes has grown 30%/year.^[66]
Because the half-life of fluorine-18 is about two hours, the prepared dose of a radiopharmaceutical bearing this radionuclide will undergo multiple half-lives of decay during the working day. This necessitates frequent recalibration of the remaining dose (determination of activity per unit volume) and careful planning with respect to patient scheduling.

The concept of emission and transmission tomography was introduced by David E. Kuhl, Luke Chapman and Roy Edwards in the late 1950s. Their work later led to the design and construction of several tomographic instruments at the University of Pennsylvania. In 1975 tomographic imaging techniques were further developed by Michel Ter-Pogossian, Michael E. Phelps, Edward J. Hoffman and others at Washington University School of Medicine.^[67]^[68]
Work by Gordon Brownell, Charles Burnham and their associates at the Massachusetts General Hospital beginning in the 1950s contributed significantly to the development of PET technology and included the first demonstration of annihilation radiation for medical imaging.^[69] Their innovations, including the use of light pipes and volumetric analysis, have been important in the deployment of PET imaging. In 1961, James Robertson and his associates at Brookhaven National Laboratory built the first single-plane PET scan, nicknamed the "head-shrinker."^[70]
One of the factors most responsible for the acceptance of positron imaging was the development of radiopharmaceuticals. In particular, the development of labeled 2-fluorodeoxy-D-glucose (2FDG) by the Brookhaven group under the direction of Al Wolf and Joanna Fowler was a major factor in expanding the scope of PET imaging.^[71] The compound was first administered to two normal human volunteers by Abass Alavi in August 1976 at the University of Pennsylvania. Brain images obtained with an ordinary (non-PET) nuclear scanner demonstrated the concentration of FDG in that organ. Later, the substance was used in dedicated positron tomographic scanners, to yield the modern procedure.
The logical extension of positron instrumentation was a design using two 2-dimensional arrays. PC-I was the first instrument using this concept and was designed in 1968, completed in 1969 and reported in 1972. The first applications of PC-I in tomographic mode as distinguished from the computed tomographic mode were reported in 1970.^[72] It soon became clear to many of those involved in PET development that a circular or cylindrical array of detectors was the logical next step in PET instrumentation. Although many investigators took this approach, James Robertson^[73] and Zang-Hee Cho^[74] were the first to propose a ring system that has become the prototype of the current shape of PET.
The PET-CT scanner, attributed to Dr. David Townsend and Dr. Ronald Nutt, was named by TIME Magazine as the medical invention of the year in 2000.^[75]

Cost

As of August 2008, Cancer Care Ontario reports that the current average incremental cost to perform a PET scan in the province is Can$1,000–1,200 per scan. This includes the cost of the radiopharmaceutical and a stipend for the physician reading the scan.^[76]
In England, the NHS reference cost (2015-2016) for an adult outpatient PET scan is £798, and £242 for direct access services.^[77]

Quality Control

The overall performance of PET systems can be evaluated by quality control tools such as the Jaszczak phantom.

compare :

Diffuse optical imaging
Hot cell (Equipment used to produce the radiopharmaceuticals used in PET)
Molecular Imaging
PET radiotracer
CT Scan

Class-D amplifier

A class-D amplifier or switching amplifier is an electronic amplifier in which the amplifying devices (transistors, usually MOSFETs) operate as electronic switches, and not as linear gain devices as in other amplifiers. They are rapidly switching back and forth between the supply rails, being fed by a modulator using pulse width, pulse density, or related techniques to encode the audio input into a pulse train. The audio escapes through a simple low-pass filter into the loudspeaker. The high-frequency pulses, which can be as high as 6 MHz, are blocked. Since the pairs of output transistors are never conducting at the same time, there is no other path for current flow apart from the low-pass filter/loudspeaker. For this reason, efficiency can exceed 90%.

Block diagram of a basic switching or PWM (class-D) amplifier.
Note: For clarity, signal periods are not shown to scale.

Basic operation

Class-D amplifiers work by generating a train of square pulses of fixed amplitude but varying width and separation, or varying number per unit time, representing the amplitude variations of the analog audio input signal. It is also possible to synchronize the modulator clock with an incoming digital audio signal, thus removing the necessity to convert it to analog, The output of the modulator is then used to gate the output transistors on and off alternately. Great care is taken to ensure that the pair of transistors are never allowed to conduct together. This would cause a short circuit between the supply rails through the transistors. Since the transistors are either fully "on" or fully "off", they spend very little time in the linear region, and dissipate very little power. This is the main reason for their high efficiency. A simple low-pass filter consisting of an inductor and a capacitor are used to provide a path for the low-frequencies of the audio signal, leaving the high-frequency pulses behind. In cost sensitive applications the output filter is sometimes omitted. The circuit then relies on the inductance of the loudspeaker to keep the HF component from heating up the voice coil.
The structure of a class-D power stage is somewhat comparable to that of a synchronously rectified buck converter (a type of non-isolated switched-mode power supply (SMPS)), but works backwards. Whereas buck converters usually function as voltage regulators, delivering a constant DC voltage into a variable load and can only source current (one-quadrant operation), a class-D amplifier delivers a constantly changing voltage into a fixed load, where current and voltage can independently change sign (four-quadrant operation). A switching amplifier must not be confused with linear amplifiers that use an SMPS as their source of DC power. A switching amplifier may use any type of power supply (e.g., a car battery or an internal SMPS), but the defining characteristic is that the amplification process itself operates by switching. Unlike a SMPS, the amplifier has a much more critical job to do, to keep unwanted artifacts out of the output. Feedback is almost always used, for the same reasons as in traditional analog amplifiers, to reduce noise and distortion.
Theoretical power efficiency of class-D amplifiers is 100%. That is to say, all of the power supplied to it is delivered to the load, none is turned to heat. This is because an ideal switch in its on state would conduct all the current but have no voltage loss across it, hence no heat would be dissipated. And when it is off, it would have the full supply voltage across it but no leak current flowing through it, and again no heat would be dissipated. Real-world power MOSFETs are not ideal switches, but practical efficiencies well over 90% are common. By contrast, linear AB-class amplifiers are always operated with both current flowing through and voltage standing across the power devices. An ideal class-B amplifier has a theoretical maximum efficiency of 78%. Class A amplifiers (purely linear, with the devices always "on") have a theoretical maximum efficiency of 50% and some versions have efficiencies below 20%.

Terminology

The term "class D" is sometimes misunderstood as meaning a "digital" amplifier. While some class-D amps may indeed be controlled by digital circuits or include digital signal processing devices, the power stage deals with voltage and current as a function of non-quantized time. The smallest amount of noise, timing uncertainty, voltage ripple or any other non-ideality immediately results in an irreversible change of the output signal. The same errors in a digital system will only lead to incorrect results when they become so large that a signal representing a digit is distorted beyond recognition. Up to that point, non-idealities have no impact on the transmitted signal. Generally, digital signals are quantized in both amplitude and wavelength, while analog signals are quantized in one (e.g. PWM) or (usually) neither quantity.

Signal modulation

The 2-level waveform is derived using pulse-width modulation (PWM), pulse density modulation (sometimes referred to as pulse frequency modulation), sliding mode control (more commonly called "self-oscillating modulation" in the trade.^[1]) or discrete-time forms of modulation such as delta-sigma modulation.^[2]
The most basic way of creating the PWM signal is to use a high speed comparator ("C" in the block-diagram above) that compares a high frequency triangular wave with the audio input. This generates a series of pulses of which the duty cycle is directly proportional with the instantaneous value of the audio signal. The comparator then drives a MOS gate driver which in turn drives a pair of high-power switches (usually MOSFETs). This produces an amplified replica of the comparator's PWM signal. The output filter removes the high-frequency switching components of the PWM signal and recovers the audio information that the speaker can use.
DSP-based amplifiers which generate a PWM signal directly from a digital audio signal (e. g. SPDIF) either use a counter to time the pulse length^[3] or implement a digital equivalent of a triangle-based modulator. In either case, the time resolution afforded by practical clock frequencies is only a few hundredths of a switching period, which is not enough to ensure low noise. In effect, the pulse length gets quantized, resulting in quantization distortion. In both cases, negative feedback is applied inside the digital domain, forming a noise shaper which has lower noise in the audible frequency range.

Design challenges

Switching speed

Two significant design challenges for MOSFET driver circuits in class-D amplifiers are keeping dead times and linear mode operation as short as possible. "Dead time" is the period during a switching transition when both output MOSFETs are driven into Cut-Off Mode and both are "off". Dead times need to be as short as possible to maintain an accurate low-distortion output signal, but dead times that are too short cause the MOSFET that is switching on to start conducting before the MOSFET that is switching off has stopped conducting. The MOSFETs effectively short the output power supply through themselves in a condition known as "shoot-through". Meanwhile, the MOSFET drivers also need to drive the MOSFETs between switching states as fast as possible to minimize the amount of time a MOSFET is in Linear Mode—the state between Cut-Off Mode and Saturation Mode where the MOSFET is neither fully on nor fully off and conducts current with a significant resistance, creating significant heat. Driver failures that allow shoot-through and/or too much linear mode operation result in excessive losses and sometimes catastrophic failure of the MOSFETs.^[4] There are also problems with using PWM for the modulator; as the audio level approaches 100%, the pulse width can get so narrow as to challenge the ability of the driver circuit and the MOSFET to respond. These pulses can get down to just a few nanoseconds and can result in the above undesired conditions of shoot-through and/or Linear mode. This is why other modulation techniques such as Pulse Density can get closer to the theoretical 100% efficiency than PWM.

Electromagnetic interference

The switching power stage generates both high dV/dt and dI/dt, which give rise to radiated emission whenever any part of the circuit is large enough to act as an antenna. In practice, this means the connecting wires and cables will be the most efficient radiators so most effort should go into preventing high-frequency signals reaching those:

Avoid capacitive coupling from switching signals into the wiring.
Avoid inductive coupling from various current loops in the power stage into the wiring.
Use one unbroken ground plane and group all connectors together, in order to have a common RF reference for decoupling capacitors
Include the equivalent series inductance of filter capacitors and the parasitic capacitance of filter inductors in the circuit model before selecting components.
Wherever ringing is encountered, locate the inductive and capacitive parts of the resonant circuit that causes it, and use parallel RC or series RL snubbers to reduce the Q of the resonance.
Do not make the MOSFETs switch any faster than needed to fulfil efficiency or distortion requirements. Distortion is more easily reduced using negative feedback than by speeding up switching.

Power supply design

Class-D amplifiers place an additional requirement on their power supply, namely that it be able to sink energy returning from the load. Reactive (capacitive or inductive) loads store energy during part of a cycle and release some of this energy back later. Linear amplifiers will dissipate this energy away, class-D amplifiers return it to the power supply which should somehow be able to store it. In addition, half-bridge class D amps transfer energy from one supply rail (e.g. the positive rail) to the other (e.g. the negative) depending on the sign of the output current. This happens regardless of whether the load is resistive or not. The supply should either have enough capacitive storage on both rails, or be able to transfer this energy back.^[5]

Error control

The actual output of the amplifier is not just dependent on the content of the modulated PWM signal. The power supply voltage directly amplitude-modulates the output voltage, dead time errors make the output impedance non-linear and the output filter has a strongly load-dependent frequency response. An effective way to combat errors, regardless of their source, is negative feedback. A feedback loop including the output stage can be made using a simple integrator. To include the output filter, a PID controller is used, sometimes with additional integrating terms. The need to feed the actual output signal back into the modulator makes the direct generation of PWM from a SPDIF source unattractive.^[6] Mitigating the same issues in an amplifier without feedback requires addressing each separately at the source. Power supply modulation can be partially canceled by measuring the supply voltage to adjust signal gain before calculating the PWM^[7] and distortion can be reduced by switching faster. The output impedance cannot be controlled other than through feedback.

Advantages

The major advantage of a class-D amplifier is that it can be more efficient than an analog amplifier, with less power dissipated as heat in the active devices. Given that large heat sinks are not required, Class-D amplifiers are much lighter weight than analog amplifiers, an important consideration with portable sound reinforcement system equipment and bass amplifiers. Output stages such as those used in pulse generators are examples of class-D amplifiers. However, the term mostly applies to power amplifiers intended to reproduce audio signals with a bandwidth well below the switching frequency.
Despite the complexity involved, a properly designed class-D amplifier offers the following benefits:

Reduced power waste as heat dissipation and hence:
Reduction in cost, size and weight of the amplifier due to smaller (or no) heat sinks, and compact circuitry,
Very high power conversion efficiency, usually better than 90% above one quarter of the amplifier's maximum power, and around 50% at low power levels.

Boss Audio mono amp. The output stage is top left, the output chokes are the two yellow toroids underneath.

Uses

Home theatre in a box systems. These economical home cinema systems are almost universally equipped with class-D amplifiers. On account of modest performance requirements and straightforward design, direct conversion from digital audio to PWM without feedback is most common.
Mobile phones. The internal loudspeaker is driven by up to 1 W. Class D is used to preserve battery lifetime.
Hearing aids. The miniature loudspeaker (known as the receiver) is directly driven by a class-D amplifier to maximise battery life and can provide saturation levels of 130 dB SPL or more.
Powered speakers
High-end audio is generally conservative with regards to adopting new technologies but class-D amplifiers have made an appearance^[8]
Active subwoofers
Sound Reinforcement and Live Sound. For very high power amplification the powerloss of AB amplifiers are unacceptable. Amps with several kilowatts of output power are available as class-D. The Crest Audio CD3000, for example, is a class-D power amplifier that is rated at 1500 W per channel, yet it weighs only 21 kg (46 lb).^[9] Similarly, the Powersoft K20 is a class-D power amplifier that is rated at 9000 W per 2-Ohm channel, yet it weighs only 12 kg (26.5 lb).^[10]
Bass amplifiers. Again, an area where portability is important. Example: Yamaha BBT500H bass amplifier which is rated at 500 W, and yet it weighs less than 5 kg (11 lb).^[11] The Promethean P500H by Ibanez is also capable of delivering 500 W into a 4 Ohm load, and weighs only 2.9 kg (6.4 lb). Gallien Krueger MB500 and Eden WTX500, also rated at 500 W weighs no more than 2 kg (4.4 lb).
Vacuum Tube-based Class-D Amplifier. By means of a suitable grid polarization technique and connections through controlled impedance and length equalized transmission lines, vacuum tubes can be used as power switching devices in Class-D power audio amplifiers in full-bridge and half bridge configurations

Hasil gambar untuk analysis of electronic devices spectrum of death

=============== CORTEX EYE MA THE MATIC BODY ELECTRON ===============

God JESUS says I am the door of life

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Minggu, 24 September 2017

Electronic devices spectrum of death on so intake human body AMNIMARJESLOW GOVERNMENT 91220017 LOR LCLAR ELCORTEX TO EYE ELCONTACT ELSENSOR LJBUSAF

Paralyzable and non-paralyzable behaviour

Analysis

Non-paralyzable analysis

Time-To-Count

Sources and systems

DAQ hardware

DAQ device drivers

Input devices

Hardware

DAQ software

Interpretation of value

Definitions

M-sample variance

Allan variance

Allan deviation

Supporting definitions

Oscillator model

Time error

Frequency function

Fractional frequency

Average fractional frequency

Estimators

Conventions

Fixed τ estimators

Non-overlapped variable τ estimators

Overlapped variable τ estimators

Modified Allan variance

Time stability estimators

Other estimators

Confidence intervals and equivalent degrees of freedom

Confidence interval

Effective degrees of freedom

Power-law noise

α-μ mapping

General Conversion from Phase Noise

Linear response

Time and frequency filter properties

Bias functions

B1 bias function

B2 bias function

B3 bias function

τ bias function

Conversion between values

Measurement issues

Measurement bandwidth limits

Dead time in measurements

Measurement length and effective use of samples

Dominant noise type

Linear drift

Measurement instrument estimator bias

Practical measurements

Measurement

Post-processing

Equipment and software

Research

Educational and practical resources

Uses

Structure and operating principles

Photoelectric effect

Secondary emission

The race towards a practical electronic television camera

First photomultiplier, single-stage (early 1934)

Magnetic photomultipliers (mid 1934–1937)

Electrostatic photomultipliers (1937–present)

Improved photocathodes

Spectral response of photocathodes

RCA Corporation

Lancaster, Pennsylvania facility

Burle Industries

Hamamatsu

Photocathode materials

Window materials

Usage considerations

Applications

High-sensitivity applications

Uses

Oncology

Neuroimaging

B₁ bias function

B₂ bias function

B₃ bias function