Paradoxically, signal detection is concerned with fitting models to
supposedly random series similarly to mathematical proofs by
reductio ad absurdum of the antithesis. That is, the hypothesis
(antithesis) 
is made, that the observed series 
has
properties of a pure noise series, 
. Then, a model is fitted
and series 
and 
are obtained. If the quality of the
model fits to the observations 
does not significantly differ
from the quality of a fit to pure noise 
, then 
is true
and we say that 
contains no signal but noise. In the
opposite case of model fitting 
significantly better than
, we reject 
and say that the model signal was detected
in 
. The difference is significant (at some level) if it is
not likely (at this level) to occur between two different realizations
of the noise 
.
The quality of the fit is evaluated using a function S of the series
, 
, and 
. A function of random variables,
such as 
, is a random variable itself and is called a
statistic. A random variable S is characterized by its probability
distribution function. Following 
we use the distribution of S
for pure noise signal 
, 
and 
, to be
denoted 
or simply 
. Precisely, we shall use the
cumulative probability distribution function which for a given
critical value of the statistic 
supplies the probability
for the observed S to fall on one side of the 
.
The observed value of the statistic and its probability distribution,
and 
respectively, are used to obtain the
probability 
of 
being true. That is, if p turns out
small, 
, 
is improbable and 
has no properties
of 
. Then we say that the model signal has been detected at
the confidence level 
. The smaller 
is, the more
convincing (significant) is the detection. The special realization of
a random series which consists of independent variables of common
(gaussian) distribution is called (gaussian) white noise. We assume
here that the noise 
is white noise. Note that in the signal
detection process, frequency 
and lag l are considered
independent variables and do not count as parameters.
Summarizing, the basis for the determination of the properties of a an
observed time series is a test statistic, S, with known probability
distribution for (white) noise, 
.
Let 
consist of 
random variables and let a given model
have 
parameters. Then the modeled series 
corresponds
to a combination of 
random variables and the residual series
corresponds to a combination of 
random
variables. The proof rests on the observation that orthogonal
transformations convert vectors of independent variables into vectors
of independent variables. Let us consider an approximately linear
model with matrix 
so that 
,
where P is a vector of 
parameters. Then 
spans a
vector space with no more than 
orthogonal vectors (dimensions).
The numbers 
, 
and 
are called the numbers of degrees
of freedom of the observations, the model fit, and the residuals,
respectively.