Page 51 - Introduction to Information Optics

36 1. Entropy Information and Optics

known as the ambiguity function, as given by

f i2 vt f i2nv r
X(T, v)= u(t)u*(t +r)e- * dt= U*(v')U(v' + v)e ' dv'. (1.125)

The significance of /(T, v) is that the range and (radial) velocity of a target
cannot be resolved at (T O + T, v 0 + v), where T O and v 0 are the mean range and
velocity of the target. By virtue of normalization, we let

2
r, v)\ dxdv= 1, (1.126)

which is called the area of ambiguity in the time-frequency domain.
One of the simplest examples illustrating the ambiguity of a signal resulting
from a single Gaussian pulse is given by

4
u(t) = ^2e-»<\
By substituting into Eq. (1.125), we have

2
2
X(T, v) = exp(— %TiT )exp(— jnv )exp(invc).
From the preceding equation we see that the magnitude of ambiguity distribu-
tion, as given by

describes a circular pattern in the (T, v) domain, as can be seen in Fig. 1.11.
If the pulse is shortened, the circular ambiguity pattern would become
elliptical, as illustrated in Fig. 1.12.
A less trivial example is that u(t) represents a train of Gaussian pulses,
modulated by a broad Gaussian envelope, as shown in Fig. 1.13(a). The
2
corresponding ambiguity |#(T, v)| is sketched in Fig. 1.13(b), in which the
ambiguity in frequency resolution is split up into narrow bands, whereas in
time it repeats itself. Notice that the shaded areas of the redistributed
ambiguity are in fact equal to the area of the single pulse ambiguity; i.e.,

2
\x(t, v)\ di dv = 1.