Such an analysis begins by definitions of time duration and spectral bandwidth.
The time duration of a damped exponential function is infinite if by
duration you mean the span of nonzero function values.
However,
for nearly all
practical purposes the time span is chosen 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) drop to half of the maximum value.
The
main idea is that the time span or the frequency span should be able to include most of the total energy but need not contain
all of it.
The precise definition of and is
somewhat arbitrary and may be chosen to simplify analysis.
The general
statement is that for any function the time duration and the
spectral bandwidth are related by

(1) |

A similar and perhaps more basic concept than the product of time and
frequency spreads is the relationship between spectral bandwidth
and *rise time* of a system response function.
The *rise time* of a system response is also defined somewhat arbitrarily,
often as the time
span between the time of excitation and the time at which the system
response is half its ultimate value.
In principle, a broad frequency
response can result from a rapid decay time as well as from a rapid
rise time.
*Tightness* in the inequality (1) may be associated
with situations in which a certain rise time is quickly
followed by an equal decay time.
*Slackness* in the inequality
(1) may be associated with increasing inequality between rise
time and decay time.
Slackness could also result from other combinations
of rises and falls such as random combinations.
Many systems respond very
rapidly compared to the rate at which they subsequently decay.
Focusing
our attention on such systems,
we can now seek to derive the inequality
(1) applied to rise time and bandwidth.
The first step is to choose
a definition for rise time.
The choice is determined not only for clarity and usefulness
but also by the need to ensure tractability of the subsequent analysis.
I have found a reasonable definition of rise time to be

(2) |

Although the *Z* transform method is a great aid in studying situations where
divergence (as 1/*t*) plays a key role,
it does have the disadvantage that it destroys the formal identity
between the time domain and the frequency domain.
Presumably this disadvantage is not fundamental since we can always
go to a limiting process in which the discretized time domain tends to a
continuum.
In order to utilize the analytic simplicity of the *Z* transform
we now consider the dual to the rise-time problem.
Instead of a time function
whose square vanishes identically at negative time we now consider a spectrum
which vanishes at negative frequencies.
We measure how fast this spectrum can rise after .We will find this to
be related to the time duration of the complex time function *b*_{t}.
More precisely, we will now define the lowest significant frequency
component in the spectrum analogously to
(2) to be

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

- Consider
*B*(*Z*) = [1 - (*Z*/*Z*)_{0}^{n}]/(1 -*Z*/*Z*) in the limit_{0}*Z*goes to the unit circle. Sketch the time function and its squared amplitude. Sketch the frequency function and its squared amplitude. Choose and ._{0} - A time series made up of two frequencies may be written as
Given , ,
*b*,_{0}*b*,_{1}*b*,_{2}*b*show how to calculate the amplitude and phase angles of the two sinusoidal components._{3} - Consider the frequency function graphed below.
**E4-1-3**Exercise 4.1.3.

Figure 1Describe the time function in rough terms indicating the times corresponding to 1/

*f*, 1/_{1}*f*, 1/_{2}*f*. Try to avoid algebraic calculation. Sketch an approximate result._{3}

**PROBLEM FOR RESEARCH**

Can you find a method of defining and of one-sided wavelets in such a way that for minimum-phase wavelets only the uncertainty principle takes on the equality sign?

10/30/1997