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 [
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:
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
Since our main
norm is the square root of a sum of squares,
we are using what is called an
norm and we may write

to emphasize this fact.
We could equally well have chosen a
normalized
norm:
which is simply the ``RMS level'' of

(``Root Mean Square'').
More generally, the (unnormalized)
norm of

is defined as
(The normalized case would include

in front of the summation.)
The most interesting

norms are
: The
, ``absolute value,'' or ``city block'' norm.
: The
, ``Euclidean,'' ``root energy,'' or ``least squares'' norm.
: The
, ``Chebyshev,'' ``supremum,'' ``minimax,''
or ``uniform'' norm.
Note that the case

is a limiting case which becomes
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
-
-
,
The first property, ``positivity,'' says the norm is nonnegative, and
only the zero vector has norm zero. The second property is
``subadditivity'' and is sometimes called the ``
triangle inequality''
for reasons that can be seen by studying
Fig.
5.6. The third property says the norm is
``absolutely homogeneous'' with respect to
scalar multiplication. (The
scalar 
can be complex, in which case the angle of

has no effect).
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 ProductPrevious Section: Linear Vector Space