barcode print in asp net THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS in .NET

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THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
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[CHAP. 4
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where X,(z ), . . . , Xn(z ) are functions with known inverse transforms x,[n], . ..,xn[n]. From the linearity property (4.17) it follows that
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C. Power Series Expansion:
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The defining expression for the z-transform [Eq. (4.3)] is a power series where the sequence values x[n] are the coefficients of z-". Thus, if X( z) is given as a power series in the form
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we can determine any particular value of the sequence by finding the coefficient of the appropriate power of 2 - ' . This approach may not provide a closed-form solution but is very useful for a finite-length sequence where X(z) may have no simpler form than a polynomial in z - ' (see Prob. 4.15). For rational r-transforms, a power series expansion can be obtained by long division as illustrated in Probs. 4.16 and 4.17.
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D. Partial-Fraction Expansion:
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As in the case of the inverse Laplace transform, the partial-fraction expansion method provides the most generally useful inverse z-transform, especially when X t z ) is a rational function of z. Let
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Assuming n
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and all poles pk are simple, then
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where
Hence, we obtain
Inferring the ROC for each term in Eq. (4.35) from the overall ROC of X(z) and using Table 4-1, we can then invert each term, producing thereby the overall inverse z-transform (see Probs. 4.19 to 4.23). If rn > n in Eq. (4.321, then a polynomial of z must be added to the right-hand side of Eq. (4.351, the order of which is (m - n). Thus for rn > n, the complete partial-fraction
CHAP. 41
THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
expansion would have the form
If X(Z) has multiple-order poles, say pi is the multiple pole with multiplicity r, then the expansion of X(z)/z will consist of terms of the form
where
THE SYSTEM FUNCTION OF DISCRETE-TIME LTI SYSTEMS
A. The System Function:
In Sec. 2.6 we showed that the output y[n] of a discrete-time LTI system equals the convolution of the input x[n] with the impulse response h[n]; that is [Eq. (2.3511,
Applying the convolution property (4.26) of the z-transform, we obtain
where Y(z), X(z), and H(z) are the z-transforms of y[n], x[n], and h[n], respectively. Equation (4.40) can be expressed as
The z-transform H(z) of h[n] is referred to as the system function (or the transfer function) of the system. By Eq. (4.41) the system function H(z) can also be defined as the ratio of the z-transforms of the output y[n] and the input x[n.l. The system function H ( z ) completely characterizes the system. Figure 4-3 illustrates the relationship of Eqs. (4.39) and (4.40).
X(Z)
H(z)
Y(z)=X(z)H(z)
Fig. 4-3
Impulse response and system function.
THE Z-TRANSFORMAND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
B. Characterization of Discrete-Time LTI Systems:
Many properties of discrete-time LTI systems can be closely associated with the characteristics of H(z) in the z-plane and in particular with the pole locations and the ROC.
1. Causality:
For a causal discrete-time LTI system, we have [Eq. (2.4411
since h[n] is a right-sided signal, the corresponding requirement on H(z) is that the ROC of H ( z ) must be of the form
That is, the ROC is the exterior of a circle containing all of the poles of H ( z ) in the z-plane. Similarly, if the system is anticausal, that is,
then h[n] is left-sided and the ROC of H ( z ) must be of the form
That is, the ROC is the interior of a circle containing no poles of H ( z ) in the z-plane. 2. Stability: In Sec. 2.7 we stated that a discrete-time LTI system is BIB0 stable if and only if [Eq. (2.4911
The corresponding requirement on H(z) is that the ROC of H(z1 contains the unit circle (that is, lzl= 1). (See Prob. 4.30.)
3. Ctzusal and Stable Systems:
If the system is both causal and stable, then all of the poles of H ( z ) must lie inside the unit circle of the z-plane because the ROC is of the form lzl> r,,, and since the unit circle is included in the ROC, we must have r,, < 1. C. System Function for LTI Systems Described by Linear Constant-Coefficient Difference Equations: ! Sec. 2.9 we considered a discrete-time LTI system for which input x[n] and output n y[n] satisfy the general linear constant-coefficient difference equation of the form
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