INTRODUCTION - Signals, Systems, and Transforms - page 17

The integral of x(t) from t = 0 to t = nH in Figure 1.20 can be expressed as the in-

tegral from t = 0 to t = (n - 1)H plus the integral from (n - 1)H to nH . Thus,

in (1.21),

nH

y(t)| t =nH = y(nH) = L x(t)dt

0

(n- 1)H nH

= L x(t)dt + L x(t)dt (1.22)

0 (n- 1)H

L y[(n - 1)H] + Hx[(n - 1)H].

Ignoring the approximations involved, we express this equation as

y(nH) = y[(n - 1)H] + Hx[(n - 1)H]. (1.23)

However, is only an approximation to the integral of at Equa-
tion (1.23) is called a first-order linear difference equation with constant coefficients .
Usually, the factor H that multiplies the independent variable n in (1.23) is omitted,
resulting in the equation

y(nH) x(t) t = nH.

y[n] - y[n - 1] = Hx[n - 1]. (1.24)

We can consider the numerical integrator to be a system with the input x [ n ] and out-
put y [ n ] and the difference-equation model (1.24). A system described by a differ-
ence equation is called a discrete-time system .
Many algorithms are available for numerical integration [5]. Most of these al-
gorithms have difference equations of the type (1.23). Others are more complex and
cannot be expressed as a single difference equation. Euler's rule is seldom used in
practice, because faster or more accurate algorithms are available. Euler's rule is
presented here because of its simplicity.

Picture in a Picture

We now consider a television system that produces a picture in a picture [6]. This
system is used in television to show two frames simultaneously, where a smaller pic-
ture is superimposed on a larger picture. Consider Figure 1.21, where a TV picture
is depicted as having six lines. (The actual number of lines is greater than 500.) Sup-
pose that the picture is to be reduced in size by a factor of three and inserted into
the upper right corner of a second picture.
First, the lines of the picture are digitized (sampled). In Figure 1.21, each line
produces six samples (the actual number can be more than 2000), which are called
picture elements ( pixels ). Both the number of lines and the number of samples per
line must be reduced by a factor of three to reduce the size of the picture. Assume
that the samples retained for the reduced picture are the four circled in Figure 1.21.
(In practical cases, the total number of pixels retained may be greater than 100,000.)