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z 1 1:90z 2 2:61z 3 3:06z 4 3:20z 5 3:02z 6 2:54z 7 z2 1:9z 1 j z z 1:90 z 1 1:90 z 1 1:90 3:61z 1 1:90z 2 2:61z 1 1:90z 2 2:61z 1 4:96z 2 2:61z 3 3:06z 2 2:61z 3 3:06z 2 5:81z 3 3:06z 4 3:20z 3 3:06z 4 3:20z 3 6:08z 4 3:20z 5 3:02z 4 3:20z 5 3:02z 4 5:74z 5 3:02z 6 2:54z 5 3:02z 6 from which we have that the rst 7 terms of Y z in series form are Y z z 1 1:90z 2 2:61z 3 3:06z 4 3:20z 5 3:02z 6 2:54z 7 Now, comparing the above answer with the basic eq. (583), noting that in this case there is no y 0 term, we see that the sample values are y 0 0:00 y T 1:00 y 2T 1:90 y 3T 2:61 y 4T 3:06 y 5T 3:20 y 6T 3:02 y 7T 2:54

The search( ) method wraps up with the following lines of code:

Figure 338 is the result of plotting these values versus nT. Fig. 338 thus expresses the z-function z= z2 1:9z 1 in terms of time-domain sampled values of y t . (It so happens that Fig. 338 is a portion of a sine wave in sampled form.)

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As you might expect, long division is not the only way to convert a given z-transform into the equivalent time-domain sampled form. Another method, for example, makes use of partial fractions, which is a procedure for writing a given fraction as the sum of a number of fractions in which the individual fractions are each simpler than the original given fraction. The inverse of each such simpler fraction can then be found by direct inspection of a table such as Table 1. The advantage of the partial fraction method is that the answer comes out in exact or closed form, while the long-division procedure comes out in series or open form. Problem 302 Given that Y z 1 z 0:5

} }); thread.start();

nd, by means of long division, the time-domain values of y 0 , y T , y 2T , y 3T , y 4T , and y 5T . Problem 303 Given that Zvs t F z z z2 0:45

The circuitry designed to manipulate DT signals is called a DT (discrete-time) processor, or digital processor if you wish. In dealing with such processors it s common practice to associate the letter symbol x with the input DT sequence and the letter y with the resulting output DT sequence. The symbol (h) will be associated with the processor itself; that is, h will describe the digital circuitry needed to convert a given x input signal into a required y output signal. Thus, if x nT denotes an input sequence and y nT the resulting output sequence, the situation can be represented in block diagram form as shown in Fig. 339, where the box with the h notation contains the required digital circuitry.

After the Runnable implementation s run( ) method has been defined, the search thread is started with a call to thread.start( ). Upon the thread s execution, the Runnable instance s run( ) method will be invoked.

(As a memory aid, note that input to output corresponds to the natural order of x to y in the alphabet.) In the above, the quantity h is called the TRANSFER FUNCTION of the system, and is de ned as being equal to the RATIO of the OUTPUT sequence y nT to the INPUT sequence x nT ; thus y nT h 585 x nT

The showError( ) method, shown here, displays an error dialog box on the screen with the given message. This method is invoked if any required search options are missing or if there are any problems opening, writing to, or closing the log file.

that is,

y nT h x nT 586 *

// Show dialog box with error message. private void showError(String message) { JOptionPane.showMessageDialog(this, message, "Error", JOptionPane.ERROR_MESSAGE); }

Internally, a processor basically consists of the interconnections of THREE DIFFERENT TYPES or blocks of circuitry, these being ADDITION, MULTIPLICATION, and TIME-DELAY blocks. (Note that we ve not mentioned subtraction separately because in binary operations subtraction can be performed by addition, as shown in section 12.1.) Our purpose now is to investigate the manner in which these di erent boxes or blocks can be interconnected to form a processor capable of producing a desired output. In doing this, each of the three basic operations is represented by a di erent schematic symbol as follows, beginning with the addition or summer symbol (Fig. 340).

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