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:
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
The norm (more specifically, the norm, or Euclidean norm) of a signal is defined as the square root of its total energy:
Other Lp Norms
Since our main norm is the square root of a sum of squares,
We could equally well have chosen a normalized norm:
More generally, the (unnormalized) norm of is defined as
- : The , ``absolute value,'' or ``city block'' norm.
- : The , ``Euclidean,'' ``root energy,'' or ``least squares'' norm.
- : The , ``Chebyshev,'' ``supremum,'' ``minimax,'' or ``uniform'' norm.
There are many other possible choices of norm. To qualify as a norm on , a real-valued signal-function must satisfy the following three properties:
- , with
Mathematically, what we are working with so far is called a Banach space, which is a normed linear vector space. To summarize, we defined our vectors as any list of real or complex numbers which we interpret as coordinates in the -dimensional vector space. We also defined vector addition (§5.3) and scalar multiplication (§5.5) in the obvious way. To have a linear vector space (§5.7), it must be closed under vector addition and scalar multiplication (linear combinations). I.e., given any two vectors and from the vector space, and given any two scalars and from the field of scalars , the linear combination must also be in the space. Since we have used the field of complex numbers (or real numbers ) to define both our scalars and our vector components, we have the necessary closure properties so that any linear combination of vectors from lies in . Finally, the definition of a norm (any norm) elevates a vector space to a Banach space.
The Inner Product
Linear Vector Space