Most signals that occur in practice are analog. Since the issues that arise in digital processing of analog signals are essentially the same in both the 1-D and 2-D cases. Let’s briefly summarize the 2-D results.

Consider an analog 2-D signal x_{c}(t_{1},t_{2}).
We’ll denote its analog Fourier transform by X_{c}(W_{1},W_{2}). Suppose we obtain a discrete-space
signal x(n_{3}. n_{2}) by sampling the analog signal x(n_{1},n_{2})
wth sampling period (T_{1}, T_{2}) as follows:

_{} (1.56)

Equation (1.56) represents the input-output relationship
of an ideal analog-to-digital (A/D) converter. The relationship between _{}, the
discrete-space Fourier transform of x(n_{1},n_{2}), and X_{c}(W_{1},W_{2}), the continuous-space Fourier transform
of x_{c}(t_{1},t_{2}) is given by

_{} (1.57)

Two examples of X_{c}(W_{1},W_{2}) and _{} are shown in
Figure 1.40. Figure 1.40(a) shows a case in which 1/T_{1 }> W’_{c}/ p and 1/T_{2 }>
W’’_{c}/
p where W’_{c}
and W’’_{c} are the cutoff frequencies of X_{c}(W_{1},W_{2}), as shown in the figure. Figure
1.44(b) shows a case in which 1/T_{1 }< W’_{c}/ p and 1/T_{2 }<
W’’_{c}/
p. From the
figure, when 1/T_{1 }> W’_{c}/ p and 1/T_{2 }> W’’_{c}/ p, x_{c}(t_{1},t_{2})
can be recovered from x(n_{1},n_{2}). Otherwise, x_{c}(t_{1},t_{2})
cannot be exactly recovered from x(n_{1},n_{2}) without
additional information on x_{c}(t_{1},t_{2}). This is
the 2-D sampling theorem, and is a straightforward extension of the 1-D result.

An ideal digital-to-analog (D/A)
converter recovers x_{c}(t_{1},t_{2}) from x(n_{1},n_{2})
when the sampling frequencies 1/T_{1 }and 1/T_{2} are high
enough to satisfy the requirements of the sampling theorem. The output of the
ideal D/A converter, y_{c}(t_{1},t_{2}), is given by

_{} (1.58)

The function y_{c}(t_{1},t_{2})
is identical to x_{c}(t_{1},t_{2}) when the sampling
frequencies used in the ideal A/D converter are sufficiently high. Otherwise, y_{c}(t_{1},t_{2})
is an aliased version of x_{c}(t_{1},t_{2}). Equation
(1.58) is a straightforward extension of the 1-D result.

An analog signal can often be
processed by digital processing techniques using the A/D and D/A converters
discussed above. The digital processing of analo_{1} signals can. in
general. he represented by the system in Figure 1.45. The analol lowpass filter
limits the bandwidth of the analog signal to reduce the effect o aliasing. In
digital image processing, the analog prefiltering operation is often performed
by a lens and the scanning aperture used in converting an optical image to an
electrical signal. The importance of the antialiasing filter is illustrated in
Figure 1.46. Figure 1.46(a) shows an image of 128 x 128 pixels with little
aliasing due to an effective antialiasing filter used. Figure 1.46(b) shows an
image of 128 x 128 pixels with noticeable aliasing.

The A/D converter of (1.56) is based on sampling on
the Cartesian grid. Th analog signal can also be sampled on a different type of
grid.

**(a)**