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A robot is required to find specific brands of pop cans using a black-and-white camera and to return the cans to a recycling location. Such a statement can be the user-oriented requirements and consists of a phenomenon from the environment. However, the pop cans are hidden phenomenon in the environment. That is, the implementation will not know about pop cans; it will know about black-and-white images of pop cans. This is the visible phenomenon. When the specification that will be used as the starting point for design is written, it needs to talk in terms of these images. It will be assumed (and may need to be verified) that only real pop cans will give those images. For example, the problem will be much more difficult if the walls of the environment are covered with ads that contain images of pop cans.

Stirling s Approximation to n!

is called the half-power frequency. In this case, the name is justi ed by Problem 12.5, which shows that the power input into the circuit of Fig. 12-11 will be half-maximum when 1 j!C R1 2 that is, when ! !x . Quite generally, any nonconstant network function H ! will attain its greatest absolute value at some unique frequency !x . We shall call a frequency at which jH ! j 0:707jH !x j a half-power frequency (or half-power point), whether or not this frequency actually corresponds to 50 percent power. In most cases, 0 < !x < 1, so that there are two half-power frequencies, one above and one below the peak frequency. These are called the upper and lower half-power frequencies (points), and their separation, the bandwidth, serves as a measure of the sharpness of the peak.

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When n is large, a direct evaluation of n! may be impractical. In such cases use can be made of the approximate formula n! , 22pn n ne

(33)

The basic RL or RC network of the type examined in Section 12.2 can be generalized with Z1 and Z2 , as shown in Fig. 12-12; the load impedance ZL is connected at the output port. By voltage division, V2 Z0 V Z1 Z 0 1 or Hv V2 Z0 V1 Z1 Z 0 The other transfer

where e 2.71828 . . . , which is the base of natural logarithms. The symbol , in (33) means that the ratio of the left side to the right side approaches 1 as n S ` . Computing technology has largely eclipsed the value of Stirling s formula for numerical computations, but the approximation remains valuable for theoretical estimates (see Appendix A).

where Z 0 Z2 ZL = Z2 ZL , the equivalent impedance of Z2 and ZL in parallel. functions are calculated similarly, and are displayed in Table 12-1.

where the last step uses the fact that the standard normal density function is even. Making the change of !t in the final integral, we obtain variable x P(Y y)

Identify which phenomenon is in the environment and which is in the implementation in the library system. The physical book is an environment-hidden phenomenon. The system never knows about the book. When the librarian scans the book, he or she is really scanning a bar code. This bar code is not the ISBN but has to reflect possible multiple copies of a single book. This bar code is environment-visible. The implementation probably uses a different identifier or pointer for the book data. This internal identifier is implementation-hidden. The specification for development needs to be written in terms of the bar code on the book. Neither the physical book nor the internal identifier should be mentioned in the development specification.

But this is a chi-square distribution with 1 degree of freedom, as is seen by putting v using the fact that A 1 B !p. 2

CHAP. 12]

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