CALIBRATION FACTOR (also called SCALE FACTOR)
Data acquired by an ADC is in terms of binary counts representing some fraction of a specified full-scale voltage input. The calibration factor is the multiplier that must be applied to the digitized (See DIGITIZE) value to convert it to engineering units. For example, an acceleration signal of 5-g/v is to be digitized by a 12-bit (11 bits + sign) ADC having a full input of 10 V. The Calibration Factor is:
An ADC count of 1562 would then be equivalent to
A characteristic of a bandpass or band reject filter or a constant percentage
filter of these types, the center frequency is the geometric mean of the upper
and the lower 
cutoff frequencies:
For a constant bandwidth filter, the center frequency is the arithmetic mean (See AVERAGE, ARITHMETIC) of the upper and lower frequencies:
See FILTER, CHEBYSHEV.
See GOODNESS OF FIT.
CIRCULAR CONVOLUTION (PERIODIC CONVOLUTION)
The periodic convolution 
of
two time sequences 
and 
is defined as:
where 
is a periodic extension
of the sample set 
, i.e.:
The inverse discrete Fourier transform (DFT-1) of the product of two DFTs is a periodic convolution of the original time sequences (See DFT and WRAP-AROUND ERROR.).
A correlation function computed in terms of a circular convolution.
The term applied to the generally undesirable (but sometimes intentional) circumstance when an output signal is limited in some sense by the full-scale range of an amplifier, ADC, or other device. Clipping may be hard, that is, when the signal is strictly limited at some level; or it may be soft, in which case the clipped signal continues to follow the input at some reduced gain (See GAIN).
The real part of a complex function, or the component that is in phase with the input excitation. In frequency domain analysis, the coincident terms are the cosine terms of the Fourier transform (See FREQUENCY DOMAIN, FOURIER TRANSFORM).
A frequency domain function generally computed to show the degree of a linear, noise-free relationship between a system input and output. Values of coherence satisfy the relationship:
where a value of 0.0 indicates no causal relationship between an input and the output, and 1.0 indicates the existence of linear noise-free frequency response function between input and output (See FREQUENCY DOMAIN).
For a system having multiple inputs 
and
one output 
, the multiple coherence
represents the fraction of power in the output accounted for by simultaneous
linear filter relationships with all the inputs. This coherence function obeys
the usual inequality, and will be unity under noise-free ideal conditions
when there is a true linear relationship occurring in a multiple-input/single-output
system.
For a system having a single input 
and
output , the ordinary coherence
is defined as:
where
For a linear system 
can be
interpreted as the fractional portion of the power output 
which
is contributed by the input 
at
frequency 
. The coherence function
is a measure of the statistical validity of the transfer function estimate.
A value of 
indicates the existence
of a nonlinear system, the presence of extraneous noise, or the existence
of other uncorrelated inputs. Note that the coherence function is independently
normalized at each frequency and is therefore independent of the shape of
the frequency response function between measurements points (See FREQUENCY
RESPONSE FUNCTION.).
For a system having multiple inputs and
one output , the partial coherence
is the coherence computed between any individual input and the output when
the effect of all other inputs is removed from the output by a linear least
squares prediction. This coherence obeys the usual inequality, and will reveal
the existence of a linear relationship between a particular residual input
and the output even when the relationship is not apparent from the ordinary
coherence function (See LEAST SQUARES PREDICTION).
The result of multiplying the imaginary part of a complex quantity by -1 (See QUAD and COMPLEX FUNCTION).
See ZOOM ANALYSIS.
A complex function is any mathematically defined relationship of the form:
,
where:
Complex functions are often represented in terms of their amplitude:
and phase:
See CONFIDENCE INTERVAL.
The range with a specified value of uncertainty within which the true value
of a measured quantity will lie. For example, consider a random variable 
with a true mean of 
. Note
that the true mean is never known, but can only be estimated. An estimate
of the mean, 
, is then made
with a 90% confidence interval of 
.
The confidence statement means that if we made 100 such independent estimates
of 
, about 90 would be in
the range of 
, and about 10
would lie outside this range. Note that the confidence statement does not
state how close a particular estimate
is to the true value (
is a function of the numbered samples and the variance of the samples).
The type of spectrum produced from nonperiodic data. The spectrum is continuous in the frequency domain (See LINE SPECTRUM, FREQUENCY DOMAIN).
A mathematical concept. The convolution of continuous functions x(t) and y(t) is defined as:
Let 
be the Fourier transforms
of 
, respectively. It can
be shown that if 
, then 
.
If 
is the input to a linear
system whose impulse response is 
,
then the output of the system is the convolution 
,
(See CIRCULAR CONVOLUTION).
See AUTOCORRELATION FUNCTION; CROSS CORRELATION FUNCTION.
A statistical concept. The covariance of two random variables, x and y, is defined as:
where 
is the expected value
(mean) of the quantity in brackets. The covariance is a measure of the correlation
of the two variables. 
, where
are the standard deviations
of 
, the signals are fully
correlated. If 
, the signals
are uncorrelated. A normalized quantity, 
,
where:
is called the correlation coefficient.
Critical damping is the smallest amount of damping at which a system will respond to a step function without overshoot (See DAMPING). The critical damping coefficient for a linear, viscously damped, single-degree-of-freedom mechanical system is defined as:
where:
(See DEGREES OF FREEDOM: VISCOUSLY DAMPED).
A measure of the similarity of two functions with the time displacement (lag)
between them used as an independent variable, The sample cross correlation
function 
between two sequences
and 
is usually computed as:
where
For discrete "stationary" random quantities where 
must be finite, the formula gives an estimate only with a statistical uncertainty
which increases as decreases.
CROSS SPECTRAL DENSITY FUNCTION
A measure in the frequency domain of the similarity of two functions. It
is usually computed from the Fourier transforms of two discrete functions
and 
to:
where:
is a "raw" cross
spectral estimate. The cross spectral density is then estimated by averaging
frames of 
.
It can also be computed from the "cross-correlation function" as:
where
For discrete "stationary" random quantities, the formulas give an estimate
only with a statistical uncertainty (error) which increases as decreases.
Typically, the data is also multiplied by a window and this will affect the
effective resolution of the estimate, (See FOURIER TRANSFORM WINDOW).
The process whereby coefficients of an arbitrary function (usually a polynomial) are computed such that the function approximates the values in a given data set. A mathematical function, such as the minimum mean squared error, is used to judge the goodness of fit (See MEAN SQUARED ERROR).
The frequency at which the rolloff skirt of the filter shape is down from the nominal unity gain passband level by a specified amount.