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Note 31.
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Shifting Theorem
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Here we wish to establish a relationship called the shifting theorem, which can be done with the aid of Fig. 37-A. In the gure, let t be time measured from the origin, as shown. Let curve A be a portion of the curve of some function y f t . Now let it be given that curves A and B are identical except that curve B is shifted horizontally T units to the right of curve A, as shown in the gure. That is, curve B lags T seconds behind curve A; notice that curve A has the value y0 at t 0, but curve B does not reach that value until T seconds later.
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Fig. 37-A
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Now let t 0 ( t prime ) be time measured from the instant t=T, as shown in the gure. Thus t 0 0 when t T. If we use t for time in curve A, and t 0 for time in curve B, then the equations for curves A and B will be identical; thus y f t y f t
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for curve A for curve B
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From the gure note that t T t , so that t 0 t T. Using this relation, we can write the equations of both curves in terms of t; thus y f t y f t T equation of curve A equation of curve B
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What we ve tried to demonstrate here is called the shifting theorem, which can be summarized as follows. For any function f t , the substitution of t T in place of t has the e ect of shifting the curve of f t horizontally T units to the right. That is, the curve of f t T is exactly the same as the curve of f t except that it is shifted T units to the right, as illustrated in Fig. 37-A. Note: If, in Fig. 37-A, curve B had been drawn to the left of curve A, then, t 0 t T, showing that substitution of t T in place of t in f t has the e ect of shifting the curve of f t T units to the left of its original position.
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Note 32.
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Unit Impulse
Here we wish to introduce the very useful concept of unit impulse. Let us begin with drawings A, B, and C in Fig. 38-A. As illustrated, the particular form of pulse we ll be interested in here will always have a time duration of a seconds and a value equal to 1=a; thus, since, a 1=a 1, such a pulse will always enclose unit area, as shown in the gure.
Appendix
Fig. 38-A
Now, in C, let us allow a to approach zero as a limiting value a ! 0 which, in turn, will make the value of the pulse, 1=a, become in nitely great 1=a ! 1), with the area enclosed by the pulse always remaining equal to 1. In words, we are hypothesizing, at t 0, the existence of a pulse of in nitely great amplitude but vanishingly short time duration, the pulse always enclosing unit area. The hypothetical pulse so described is called a UNIT IMPULSE, and is denoted by the symbol  t , which can be read as delta of t ( d is the small Greek letter delta ). Such a pulse cannot, of course, exist in the real physical world. It is, nevertheless, a very useful mathematical device, for the following reasons. First of all, very short, high-valued pulses of voltage and current do exist in the real world, and such actual pulses, when applied to a network, have the same general e ect as would the application of a theoretical impulse to the network. In other words, the theoretical analysis of a network to applied  t yields results that will closely approximate the actual results produced by the application of a very high, sharp, pulse to the network. A second reason lies in the fact that a function of time t can be expressed in terms of a particular secondary variable s, where s is a complex number of the form s a j!. This is important, because the work required in circuit analysis can often be greatly reduced when carried out in terms of s instead of t. This is especially true if we re investigating the e ect of applying an impulse type of signal to a network. This is because it turns out that  t is simply replaced by 1 when working in terms of s, a fact that can considerably reduce the algebraic complications in impulse-type problems. Lastly,  t is a convenient symbol to use in writing sampling equations, which let us discuss in more detail, as follows. First, it will be convenient to refer to  t as an impulse of unit strength. Then the notation A t will naturally be called an impulse of strength A. Hence, by note 31, the notation A t nT will denote an impulse of strength A shifted nT units to the right. GRAPHICALLY, an impulse is represented by a vertical line with arrowhead, with notation given alongside the line. Thus, graphically, the three impulse cases just mentioned above would appear as illustrated in the following.
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