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LINEAR TIMEINVARIANT SYSTEMS in VS .NET
LINEAR TIMEINVARIANT SYSTEMS Scan QR In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Generate QRCode In .NET Framework Using Barcode printer for VS .NET Control to generate, create Quick Response Code image in .NET applications. [CHAP. 2
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C h [ k ] x [ n k ] (2.45) Alternatively, applying the causality condition ( 2 . 4 4 ) to Eq. (Z..V), we have
Equation (2.46) shows that the only values of the input x[n] used to evaluate the output y[n] are those for k I n. As in the continuoustime case, we say that any sequence x[n] is called causal if and is called anticausal if Then, when the input x[n] is causal, the output y[n] of a causal discretetime LTI system is given by C. Stability: It can be shown (Prob. 2.37) that a discretetime LTI system is B I B 0 stable if its impulse response is absolutely summable, that is, EIGENFUNCTIONS OF DISCRETETIME LTI SYSTEMS
In Chap. 1 (Prob. 1.45) we saw that the eigenfunctions of discretetime LTI systems represented by T are the complex exponentials z n ,with z a complex variable. That is, T(zn)= Azn (2.50) CHAP. 21
LINEAR TIMEINVARIANT SYSTEMS
where A is the eigenvalue of T associated with zn. Setting x[n] = z n in Eq. (2.391, we have
= H ( z ) z n = Azn
(2.51) where
A =H(z) = h [ k ]z P k
(2.52) Thus, the eigenvalue of a discretetime LTI system associated with the eigenfunction z n is given by H ( z ) which is a complex constant whose value is determined by the value of z via Eq. (2.52). Note from Eq. (2.51) that y[O] = H(z) (see Prob. 1.45). The above results underlie the definitions of the ztransform and discretetime Fourier transform which will be discussed in Chaps. 4 and 6. 2.9 SYSTEMS DESCRIBED BY DIFFERENCE EQUATIONS
The role of differential equations in describing continuoustime systems is played by difference equations for discretetime systems. A. Linear ConstantCoefficient Difference Equations: The discretetime counterpart of the general differential equation (2.25) is the Nthorder linear constantcoefficient difference equation given by where coefficients a, and b, are real constants. The order N refers to the largest delay of y[n] in Eq. (2.53). An example of the class of linear constantcoefficient difference equations is given in Chap. I (Prob. 1.37). Analogous to the continuoustime case, the solution of Eq. (2.53) and all properties of systems, such as linearity, causality, and timeinvariance, can be developed following an approach that directly parallels the discussion for differential equations. Again we emphasize that the system described by Eq. (2.53) will be causal and LTI if the system is initially at rest. B. Recursive Formulation: An alternate and simpler approach is available for the solution of Eq. (2.53). Rearranging Eq. (2.53) in the form we obtain a formula to compute the output at time n in terms of the present input and the previous values of the input and output. From Eq. (2.54) we see that the need for auxiliary conditions is obvious and that to calculate y[n] starting at n = no, we must be given the values of y[n,,  11, y[no  21,. . . , y[no  N ] as well as the input x[n] for n 2 n,,  M. The general form of Eq. (2.54) is called a recursiue equation since it specifies a recursive procedure for determining the output in terms of the input and previous outputs. In the LINEAR TIMEINVARIANT SYSTEMS
[CHAP. 2
special case when N = 0, from Eq. (2.53)we have
which is a nonrecursice equation since previous output values are not required to compute the present output. Thus, in this case, auxiliary conditions are not needed to determine Y ~ I .
C. Impulse Response: Unlike the continuoustime case, the impulse response h[n] of a discretetime LTI system described by Eq. (2.53)or, equivalently, by Eq. (2.54)can be determined easily as

