Page 50 - Introduction to Information Optics

1.5. Signal Analysis 35

To simplify the analysis, we set scalar wave field u(t) be constrained by the
following equation:

f 2 f 2
\u(t)\ dt = \U(v)\ dv= 1. (1.121)

In view of the Wiener-Khinchin and Parseval's Theorems, we see that in time
domain, we have

2 l2
u(t)u*(t + T)dt = \ \U(v) e ™dv, (1.122)
%J
and

2
T= I |K(T)| <*T =

where the superasterisk denotes the complex conjugate. Similarly, in frequency
domain we have

2
K(v) = | U*(v')U(v' + v)dv' = | \u(t}\ e i2nvt dt, (1.123)
and

2
4
F = I \K(v)\ dv = I \u(t)\ dt,
where F is defined as the frequency resolution cell, which provides the
well-known uncertainty relationship as given by

(frequency resolution cell) x (time span) = 1. (1.124)

Notice that Eq. (1.122) applies frequently to the resolution of detecting
stationary targets at different ranges, while Eq. (1.123) applies to the resolution
of detecting moving targets at different radial velocities. However, when the
targets are both at different ranges, and moving with different radial velocities,
(neither quantity being known in advance), the separation of time (e.g., range)
and frequency (i.e., radial velocity) resolution do not always provide the actual
resolving power of the target. Thus, we need a more general description of time
and frequency shift to interpret the resolving power of the signal, which is