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CHAP. 11
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Slide Rule Method
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SIGNALS AND SYSTEMS
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Another method for performing convolutions, which we call the slide rule method, is particularly convenient when both x ( n ) and h ( n ) are finite in length and short in duration. The steps involved in the slide rule method are as follows: Write the values of x ( k ) along the top of a piece of paper, and the values of h ( - k ) along the top of another piece of paper as illustrated in Fig. 1-7. Line up the two sequence values x ( 0 ) and h(O), multiply each pair of numbers, and add the products to form the value of y(0). Slide the paper with the time-reversed sequence h ( k ) to the right by one, multiply each pair of numbers, sum the products to find the value y ( l ) , and repeat for all shifts to the right by n > 0. Do the same, shifting the time-reversed sequence to the left, to find the values of y ( n ) for n i0 .
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Fig. 1-7. The slide rule approach to convolution.
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In Chap. 2 we will see that another way to perform convolutions is to use the Fourier transform.
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1.5 DIFFERENCE EQUATIONS
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The convolution sum expresses the output of a linear shift-invariant system in terms of a linear combination of the input values x ( n ) . For example, a system that has a unit sample response h ( n ) = a n u ( n )is described by the equation
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Although this equation allows one to compute the output y ( n ) for an arbitrary input x ( n ) , from a computational point of view this representation is not very efficient. In some cases it may be possible to more efficiently express the output in terms of past values of the output in addition to the current and past values of the input. The previous system, for example, may be described more concisely as follows:
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Equation (I. l o ) is a special case of what is known as a linear constant coeficient difference equation, or LCCDE. The general form of a LCCDE is
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where the coefficients a ( k ) and h ( k ) are constants that define the system. If the difference equation has one or more terms a ( k ) that are nonzero, the difference equation is said to be recursive. On the other hand, if all of the coefficients a ( k ) are equal to zero, the difference equation is said to be nonrecursive. Thus, Eq. ( 1 . l o ) is an example of a first-order recursive difference equation, whereas Eq. ( 1 . 9 ) is an infinite-order nonrecursive difference equation. Difference equations provide a method for computing the response of a system, y ( n ) , to an arbitrary input x ( n ) . Before these equations may be solved, however, it is necessary to specify a set of initial conditions. For example, with an input x ( n ) that begins at time n = 0 , the solution to Eq. ( 1 . 1 1 )at time n = 0 depends on the
SIGNALS AND SYSTEMS
[CHAP. 1
values of y ( - l ) , . . . , y ( - p ) . Therefore, these initial conditions must be specified before the solution for n 2 0 may be found. When these initial conditions are zero, the system is said to be in initial rest. For an LSI system that is described by a difference equation, the unit sample response, h(n), is found by solving the difference equation for x ( n ) = 6(n) assuming initial rest. For a nonrecursive system, a ( k ) = 0, the difference equation becomes
and the output is simply a weighted sum of the current and past input values. As a result, the unit sample response is simply
Thus, h(n) is finite in length and the system is referred to as a fmite-length impulse response (FIR) system. However, if a ( k ) # 0, the unit sample response is, in general, infinite in length and the system is referred to as an infinite-length impulse response (IIR) system. For example, if
the unit sample response is h(n) = anu(n). There are several different methods that one may use to solve LCCDEs for a general input x(n). The first is to simply set up a table of input and output values and evaluate the difference equation for each value of n. This approach would be appropriate if only a few output values needed to be determined. Another approach is to use z-transforms. This approach will be discussed in Chap. 4. The third is the classical approach of finding the homogeneous and particular solutions, which we now describe. Given an LCCDE, the general solution is a sum of two parts,
where yh(n) is known as the homogeneous solution and y p ( n ) is the particular solution. The homogeneous solution is the response of the system to the initial conditions, assuming that the input x ( n ) = 0. The particular solution is the response of the system to the input x(n), assuming zero initial conditions. The homogeneous solution is found by solving the homogeneous difference equation