Observed signals often look random
and are often modeled by filtered random numbers.
In this chapter we will see many examples of signals
built from random numbers
and discover how the nomenclature of statistics applies
to them.
Fundamentally, this chapter characterizes ``**resolution**,''
resolution of frequency and arrival time,
and the statistical resolution of signal amplitude and power
as functions of time and frequency.

We will see popping up everywhere.
This enters our discussion when we look at spectra
of signals built from random numbers.
Also, signals that are theoretically uncorrelated
generally appear to be weakly correlated
at a level of , where *n*
is the number of independent points in the signal.

Measures of resolution
(which are variously called
**variance**s,
**tolerance**s,
uncertainties,
**bandwidth**s,
**duration**s,
**spread**s,
**rise time**s,
spans, etc.)
often interact with one another, so that
experimental change to reduce one
must necessarily increase another or some combination of the others.
In this chapter we study basic cases
where such conflicting interactions occur.

To avoid confusion I introduce the unusual notation where is commonly used. Notice that the letter resembles the letter ,and connotes length without being confused with wavelength. Lengths on the time and frequency axes are defined as follows:

dt,dfmesh intervals in time and frequency , mesh intervals in time and frequency extent of time and frequency axis , time duration and spectralbandwidthof a signal

There is no mathematically tractable
and universally acceptable definition
for time span and spectral bandwidth .A variety of defining equations are easy to write,
and many are in general use.
The main idea is that the time span
or the frequency span should be able to include most of the energy
but need not contain it all.
The time duration of a damped exponential function is infinite
if by duration we mean the span of nonzero function values.
However, for practical purposes
the time span is generally defined as the time required
for the amplitude to decay to *e ^{-1}* of its original value.
For many functions the span is defined
by the span between points
on the time or frequency axis
where the curve (or its envelope) drops to half of the maximum value.
Strange as it may sound,
there are certain concepts about the behavior of
and
that seem appropriate for ``all'' mathematical choices
of their definitions,
yet these concepts can be proven only for special choices.

- TIME-FREQUENCY RESOLUTION
- FT OF RANDOM NUMBERS
- TIME-STATISTICAL RESOLUTION
- SPECTRAL FLUCTUATIONS
- CROSSCORRELATION AND COHERENCY
- SMOOTHING IN TWO DIMENSIONS
- PROBABILITY AND CONVOLUTION
- THE CENTRAL-LIMIT THEOREM

10/21/1998