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Fig. 339
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(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
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CHAPTER 13 The Digital Processor
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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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The purpose of an adder is clear from Fig. 340, in which there could, of course, be more than just two input lines. In the gure, it s understood that the adder handles x1 independently of the presence of x2 , and x2 independently of the presence of x1 ; that is, it s understood that the principle of superposition applies to Fig. 340 and hence, as far as doing binary arithmetic is concerned, adders are linear devices. Next, the block diagram symbol for MULTIPLICATION is shown in Fig. 341, in which an input signal x nT is multiplied by a CONSTANT FACTOR a to produce an output signal a times the input signal, as shown.
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The third basic requirement is that sample TIME DELAY must be provided for in a DT processor. Time delay is generated in multiples of T, where, as usual, T is the time between samples. The block diagram symbol for time delay has already been given in Fig. 336, but is repeated here in Fig. 342, where k 1; 2; 3; . . . , depending upon the number of sample periods a signal is to be delayed. (Note that whereas Fig. 336 is in the z-domain, Fig. 342 is in the nT-domain ; but it s usual, in block diagrams, to use the same z k notation in both cases.)
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Fig. 342
* We sometimes like to say that h operates on x nT to produce y nT . y To indicate that x2 is to be subtracted from x1 a minus sign would be placed alongside the input arrow for x2 in Fig. 340, in which case y nT x1 nT x2 nT .
CHAPTER 13 The Digital Processor
The above delay operation is said to be time-invariant, because the BASIC INFORMATION carried by the signal is not altered by the time delay. That is, even though the output signal lags kT seconds behind the input signal, both signals still carry the same basic information. The basic reason why time delay is used in digital processors is because the present or now output, y nT , of a processor is generated NOT ONLY by the present or now value of the input sequence x nT but also by PAST VALUES of x nT and, in some cases, by present and past values of the output sequence y nT . We must remember that the job of a processor is to electronically carry out whatever MATHEMATICAL OPERATION is speci ed by the transfer function h in eq. (586). In general, the required mathematical operation will be too complicated to allow a processor to produce each now term of the output sequence, given only a single now value of the input sequence. Thus, to do its job, a processor requires more information than just each single now value. Fortunately, the additional information needed can be obtained by making use of PAST VALUES of the input sequence and, in some cases, also the present and past values of the output sequence. These things will be taken up in more detail in the next section.
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