This section defines some useful functions of signals (vectors).

The mean of a
signal (more precisely the ``sample mean'') is defined as the
average value of its samples:^{5.5}

The total energy
of a signal is defined as the sum of squared moduli:

In physics, energy (the ``ability to do work'') and work are in units
of ``force times distance,'' ``mass times velocity squared,'' or other
equivalent combinations of units.^{5.6} In digital signal processing, physical units are routinely
discarded, and signals are renormalized whenever convenient.
Therefore,
is defined above without regard for constant
scale factors such as ``wave impedance'' or the sampling interval .

The average power of a signal is defined as the energy
per sample:

Another common description when is real is the mean square.
When is a complex sinusoid, i.e.,
, then
; in other words, for complex sinusoids,
the average power equals the instantaneous power which is the
amplitude squared. For real sinusoids,
re, we have
.

Power is always in physical units of energy per unit time. It therefore
makes sense to define the average signal power as the total signal energy
divided by its length. We normally work with signals which are functions
of time. However, if the signal happens instead to be a function of
distance (e.g., samples of displacement along a vibrating string), then the
``power'' as defined here still has the interpretation of a spatial
energy density. Power, in contrast, is a temporal energy density.

The root mean square (RMS) level of a signal is simply
. However, note that in practice (especially in audio
work) an RMS level is typically computed after subtracting out any
nonzero mean value. Here, we call that the variance.

The variance (more precisely the sample variance) of the
signal is defined as the power of the signal with its mean
removed:^{5.7}

It is quick to show that, for real signals, we have

which is the ``mean square minus the mean squared.'' We think of the
variance as the power of the non-constant signal components (i.e.,
everything but dc). The terms ``sample mean'' and ``sample variance''
come from the field of statistics, particularly the theory of
stochastic processes. The field of statistical signal
processing [25,32,63] is firmly rooted in
statistical topics such as ``probability,'' ``random variables,''
``stochastic processes,'' and ``time series analysis.'' In this book,
we will only touch lightly on a few elements of statistical signal
processing in a self-contained way.

The norm (more specifically, the norm, or
Euclidean norm) of a signal is defined as the square root
of its total energy:

We think of as the length of the vector in -space.
Furthermore, is regarded as the distance between
and . The norm can also be thought of as the ``absolute value'' or
``radius'' of a vector.^{5.8}