LINEAR TIME-INVARIANT SYSTEMS in VS .NET

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LINEAR TIME-INVARIANT SYSTEMS
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=Kx[n] (2.42) where K is a (gain) constant. Thus, the corresponding impulse response is simply h [ n ]= K 6 [ n ] (2.43)
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Therefore, if h[n,] # 0 for n,
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0, the discrete-time LTI system has memory.
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Causality:
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Similar to the continuous-time case, the causality condition for a discrete-time LTI system is Applying the causality condition ( 2 . 4 4 ) to Eq. (2.391, the output of a causal discrete-time LTI system is expressed as
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k =O
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 continuous-time 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 discrete-time LTI system is given by
C. Stability:
It can be shown (Prob. 2.37) that a discrete-time LTI system is B I B 0 stable if its impulse response is absolutely summable, that is,
EIGENFUNCTIONS OF DISCRETE-TIME LTI SYSTEMS
In Chap. 1 (Prob. 1.45) we saw that the eigenfunctions of discrete-time 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 TIME-INVARIANT 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 discrete-time 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 z-transform and discrete-time 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 continuous-time systems is played by difference equations for discrete-time systems.
A. Linear Constant-Coefficient Difference Equations:
The discrete-time counterpart of the general differential equation (2.25) is the Nthorder linear constant-coefficient 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 constant-coefficient difference equations is given in Chap. I (Prob. 1.37). Analogous to the continuous-time case, the solution of Eq. (2.53) and all properties of systems, such as linearity, causality, and time-invariance, 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 TIME-INVARIANT 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 continuous-time case, the impulse response h[n] of a discrete-time LTI system described by Eq. (2.53)or, equivalently, by Eq. (2.54)can be determined easily as
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