Page 49 - Introduction to Information Optics

34 1. Entropy Information and Optics

If the desired output could be obtained without the physical realizable
constraint, the input-output cross-correlation of the optimum filter is, in fact,
the input and the desired-output cross-correlation itself, which holds not only
for T ^ 0 but also T < 0. Now the problem becomes a trivial one. By taking the
Fourier transform of Eq. 1.1 17, we have

where H opt(v) is the transform function of the optimum filter, written by
H w aii9)
= -

which is, in fact, the solution of our problem.
We do not exactly know the desired output response, but we have been
demanding the filter more than our expectation; namely, by minimizing the
mean-square error, which gives rise to the best approximation. In other words,
this method attempts to get the best result we can, similar to bargaining for an
automobile with a used-car salesperson. As we demanded more than a filter
can be synthesized, the input-output crosscorrelation cannot be equal to the
crosscorrelation of the input and the desired output. According to the Wiener-
Hopf equation, the filter design should be based on the minimum error
criterion and the impulse response should be compelled to the inequality of
T ^ 0, but allows it to vary somewhat for T < 0, such as

q(t] = R id(x) - h opt(ff)R u(r - ff)d<r, (1.120)
J - on

where g{r) ^ 0 for T < 0 but vanishes for T ^ 0.

1.5.4. SIGNAL AMBIGUITY

The choice of a transmitted waveform for radar seems more trivial than it
is for communication. However, it is actually not so fundamental. For example,
it is the reflected (or echo) wave from the target that the temporal and doppler
shift (i.e., frequency) provide the range and radial velocity information of the
target. Whenever time and frequency are mentioned, we anticipate relating the
quantities to the uncertainty relationship. In other words, no wave form can
occupy a very short time while having a very narrow bandwidth. Thus, there
is a limitation imposed on the combined resolution in time and frequency.