qr code vb.net library The Form of, and Basic Equations for, a DT Processor in .NET

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The Form of, and Basic Equations for, a DT Processor
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Electronic circuitry, both CT and DT, often makes use of FEEDBACK, which involves a condition in which a portion of the system OUTPUT signal is fed back into the INPUT of the system. Such feedback, when properly used, can in some cases produce very bene cial results. In regard to DT processors, those that do use feedback are said to be recursive, while those that do not are non-recursive. Consider, now, examples of both types, beginning with Fig. 343.
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Fig. 343
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In the gure note, rst of all, that the output signal y nT is not in any way fed-back into the system; thus Fig. 343 is an example of a non-recursive system. Note also that the output sequence y nT is the sum of the present or now value 2x nT and two PAST values, 3x nT T and 7x nT 2T , which occurred T and 2T sample times ago relative to the now time of nT seconds. (In the time-delay boxes the exponent 1 means unit time delay, that is, a time delay of T seconds.)
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CHAPTER 13 The Digital Processor
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In regard to interpreting a gure such as Fig. 343, we should note that, while a notation such as x nT really denotes an entire SEQUENCE of values, n 0; 1; 2; 3; . . . , we can, for convenience, think of v nT as denoting some particular sample value existing at a time nT. One more point to note is that Fig. 343 is classi ed as a second-order processor, because it uses two delays. Next, as a second example, consider Fig. 344. Note that the output sequence y nT is fed back to the input adder after being delayed 1 sample period. Thus Fig. 344 is a simple form of recursive DT processor (of rst order, because only one delay is used).
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Fig. 344
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Figures 343 and 344 are simple examples of DT processors and we need not, at this point in our study, worry about how they are put to work. It is important, however, that you understand the meaning of the notation. In general, processors will consist of a combination of non-recursive and recursive types; for example, the combination of Figs. 343 and 344 to make a single processor gives Fig. 345, in which y nT y 0 nT 10y nT T
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Fig. 345
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and thus y nT 2x nT 3x nT T 7x nT 2T 10y nT T With the foregoing in mind, let s agree now to adopt the notation illustrated in Fig. 346 for the general form of a DT processor. (There is a non-recursive part to the LEFT of point R and a recursive part to the RIGHT of point R, where R is just a reference point.) Note, rst, that the value of the output at the point R (the output of the rst adder) is R nT
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k p X k 0
bk x nT kT
CHAPTER 13 The Digital Processor
Fig. 346
Thus we have that the nal output value of the processor is y nT
k p X k 0
bk x nT kT
k q X k 1
ak y nT kT
587
or, written out in expanded form, eq. (587) becomes y nT b0 x nT b1 x nT T b2 x nT 2T bp x nT pT a1 y nT T a2 y nT 2T aq y nT qT 588
Equations (587) and (588) are called di erence equations because of the presence of the past history forms having the di erence notation nT kT . Let us now operate on eqs. (587) and (588) in such a way as to express things in terms of the z-transform. This can be done as follows. First note that, in the above equations, y nT and x nT denote sampled values of y t and x t for any particular value of n we might be interested in. However, to bring the ztransform into the picture we must summate the values of y nT and x nT over the entire range of n that there is, for n 0 to n ! 1 (in accordance with the basic de nition of eq. (573) in section 13.3). Therefore (so that we can apply eq. (573)) let us multiply both sides of eq. (588) by z n and then summate from n 0 to n ! 1; doing this, eq. (588) becomes X X X X x nT z n b1 x nT T z n bp x nT pT z n y nT z n b0 X X X a1 y nT T z n a2 y nT 2T z n aq y nT qT z n where all summations are understood to be from n 0 to n 1. Now note that (with same summation from n 0 to n 1) by eq. (573) X y nT z n Y z and by item (4), Table 2 X y nT kT z n Y z z k
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