Appendix

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Thus Fig. 331, in the main text, can be described in algebraic form by eq. (571) or in graphical form as below, starting at t 0.

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Note: Impulse strength is also called impulse weight. Thus the above sequence is also referred to as a sequence of weighted impulses. In conclusion, let us note that it is not correct to write that t 1 for t 0, or that t nT 1 for t nT. All we say is that t exists only for t 0, and that t 0 for all other values of t. Thus impulse notation, as we re using it here, is useful in the mathematical description of impulse-type sampled signals. The term unit impulse refers to the fact that t is de ned to enclose unit area and it is this conception that leads to very useful results when, in the calculus, the process of integration is applied to the study of sampled signals. Right now, however, we ll just view t and t nT as useful shorthand notations.

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Note 33.

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Basic terminology:

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Algebraic Long Division

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dividend numerator quotient: divisor denominator If the numerator and denominator are both algebraic polynomials in x, a useful procedure to nd the quotient can be summarized as follows. 1. 2. Arrange both numerator and denominator in descending powers of x.* DIVIDE the FIRST TERM OF THE NUMERATOR by the FIRST TERM OF THE DENOMINATOR. 3. Now MULTIPLY the ENTIRE DENOMINATOR by the result of step (2), then SUBTRACT the result from the numerator. 4. Now consider the result of step (3) as being a new numerator, and repeat steps (2) and (3).

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* To keep track of the work it s helpful to write in any missing powers of x with zero coe cients. For example, x3 x 1 would be written as x3 0x2 x 1.

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Appendix

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The following examples will help you to check your understanding of the above four steps. It should be noted that it s generally not necessary, or even desirable, to apply long division to a given algebraic fraction; it depends upon the particular situation, such as, here, nding an inverse z-transform. It should also be noted that algebraic fractions are classi ed as being proper or improper as follows. If the highest power of x is located in the denominator the fraction is said to be proper, but if this is not true the fraction is said to be improper. Both types will appear in the following examples, with suitable comments. Example 1

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Write the improper algebraic fraction 5x 3 13x 2 2x 2 x 2 in a form that contains only a proper fraction.

Solution This can be done by using algebraic long division, as follows. First, for this operation, let us begin by writing the indicated division in the more convenient form x 2 j 5x3 13x2 2x 2 where x 2 is the divisor. Now carefully follow the prescribed procedure until the new numerator becomes free of the variable x. The detailed results are as follows.

5x2 3x 4 5x3 13x2 2x 2 3x2 2x 3x2 6x 4x 2 4x 8 10

5x3 10x2

In the above result 10 is called the remainder, and all we can do is write 10= x 2 to indicate that the remainder, 10, is to be divided by the divisor, x 2. Thus the nal answer is 5x3 13x2 2x 2 10 5x2 3x 4 x 2 x 2 10 ; is a proper fraction, the problem requirement Since the only fraction, x 2 is met. Example 2

Apply algebraic long division to the proper algebraic fraction

1 . x2 2

Appendix

Solution We can begin by writing the indicated division in the convenient form x2 2 j 1 , then applying the four-step procedure, as follows. First, by step (2), 1 x 2 x2 then, applying step (3), the work appears as x2 2 j 1 x 2

1 2x 2 2x 2 The new numerator is thus 2x 2 , to which, by step (4), we must again apply steps (2) and (3). Doing this, the work now appears as x2 2