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.
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 [28,34,68] 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