Page 219 - Phase Space Optics Fundamentals and Applications

200 Chapter Six

6.3.1 Code Division Multiplexing
The expression of Eq. (6.18) can be reformulated as follows:

x

∗

U R ( x ) = U( ) rect G ( − x )G( − ) d d
x x x x x x
x
(6.20)
In the case of code division multiplexing there are no time, polariza-
tion, or wavelength variations. For a random encoding mask having
small spatial features, the internal integral may be approximated as
follows:

x

∗

rect G ( − x ) G( − ) d ≈ ( x − ) (6.21)

x
x
x
x
x
x
which leads to

U R ( x ) = U( ) ( x − ) d = U( x ) (6.22)
x
x
x
Note that the assumption of Eq. (6.21) is an approximation and in
practice when x is getting narrower, the right wing can better be
approximated with a spectral function which is wider than a delta. In
addition the right wing can contain another additive term that may be
approximated by a constant. The widening of the delta will blur the
spectral expression of U R which means that the super resolved image
will become field of view limited. The addition of the constant level
will reduce the contrast or the signal to noise ratio of the obtained
reconstruction.
In Fig. 6.3 we simulate numerically the proposed approach for code
division multiplexing SR. A resolution target was lowpass filtered

(a) (b)

FIGURE 6.3 (a) Low-resolution USAF target; (b) super resolved
reconstruction using code multiplexing.