Page 80 - Introduction to Information Optics

Exercises 63

1.24 With reference to the problem of observation under a microscope, if a
square (sides = r 0) instead of a circular wavelength is used, calculate the
minimum amount of entropy increase in order to overcome the thermal
background fluctuations. Show that the cost of entropy under a micro-
scope observation is even greater than kl In 2.
1.25 Show that the spatial information capacity of an optical channel under
coherent illumination is generally higher than under incoherent illumina-
tion,
1.26 Let us consider a band-limited periodic signal of bandwidth Av m , where
v m is the maximum frequency content of the signal. If the signal is passed
through an ideal low-pass filter of bandwidth Av f where the cutoff
frequency v f is lower than that of v m, estimate the minimum cost of
entropy required to restore the signal.
1.27 Refer to the photon channel previously evaluated. What would be the
minimum amount of energy required to transmit a bit of information
through the channel?
1.28 High and low signal-to-noise ratio is directly related to the frequency
transmission through the channel. By referring to the photon channels
that we have obtained, under what condition would the classical limit and
quantum statistic meet? For a low transmission rate, what would be the
minimum energy required to transmit a bit of information?
1.29 It is well known that under certain conditions a band-limited photon
channel capacity can be written as

hv
C % Av log

The capacity of the channel increases as the mean occupation number
in — S/hv increases. Remember, however, that this is valid only under the
condition Av/v « 1, for a narrow-band channel. It is incorrect to assume
that the capacity becomes infinitely large as v -» 0. Strictly speaking, the
capacity will never exceed the quantum limit as given in Eq. (1.202). Show
that the narrow-band photon channel is in fact the Shannon's continuous
channel under high signal-to-noise ratio transmission.