文档介绍：Chapter 5 Discrete-Time System Analysis in the z-Domain
§5-1 Transfer Function Representation
§5-2 Transform of the Input/output Convolution Sum
§5-3 Stability of Discrete-time Systems
§5-4 Frequency Response of Discrete-Time Systems
Problems
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§5-1 Transfer Function Representation
In this section the transfer function representation is generated for the class of causal linear time-invariant discrete-time systems. The development begins with discrete-time systems defined by an input/output difference equation. Systems given by a first-order input/output difference equation are considered first.
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First-order Case
Consider the linear time-invariant discrete-time system given by the first-order input/output difference equation
y[n]+ay[n–1] = bx[n] ()
where a and b are real numbers, y[n] is the output, and x[n] is the input. Taking the z-transform of both sides of () and using the right-shift property gives
where Y(z) is the z-transform of the output response y[n] and X(z) is the z-transform of the input x[n]. Solving () for Y(z) yields
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()
Equation () is the z-domain representation of the discrete-time system defined by the input/output difference equation (). The first term on the right-hand side of () is the z-transform of the part of the output response resulting from the initial condition y[–1], and the second term on the right-hand side of () is the z-transform of the part of the output response resulting from the input x[n] applied for n=0, 1, 2, ….
The system given by () has no initial energy at time n=0 if y[–1] =0, in which case () reduces to
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() es
()
Defining
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The function H(z) is called the transfer function of the system since it specifies the transfer from the input to output in the z-domain assuming no initial energy (y[–1] =0). Equation () is the transfer function representation of the system.
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Example Step response
For the system given by (), suppose that a0 and x[n] i