## Signal Metrics

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*:

*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}

*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*[27,33,65] 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:

*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}

### Other Lp Norms

Since our main norm is the square root of a sum of squares,

*norm*and we may write to emphasize this fact.

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.

### Norm Properties

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
- ,

### Banach Spaces

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*.

**Next Section:**

The Inner Product

**Previous Section:**

Linear Vector Space