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CHAPTER 4
The z-Transform
In Chapter 3 we studied the discrete-time Fourier transform approach for
representing discrete signals using complex exponential sequences. This
representation clearly has advantages for LTI systems because it describes
systems in the frequency domain using the frequency response function
H(e jω ). The computation of the sinusoidal steady-state response is greatly
facilitated by the use of H(e jω ). Furthermore, response to any arbitrary
absolutely summable sequence x(n) can easily be computed in the frequency
domain by multiplying the transform X(e jω ) and the frequency
response H(e jω ). However, there are two shortcomings to the Fourier
transform approach. First, there are many useful signals in practice—
such as u(n) and nu(n)—for which the discrete-time Fourier transform
does not exist. Second, the transient response of a system due to initial
conditions or due to changing inputs cannot be computed using the
discrete-time Fourier transform approach.
Therefore we now consider an extension of the discrete-time Fourier
transform to address these two problems. This extension is called the
z-transform. Its bilateral (or two-sided) version provides another domain
in which a larger class of sequences and systems can be analyzed, and its
unilateral (or one-sided) version can be used to obtain system responses
with initial conditions or changing inputs.
4.1 THE BILATERAL z-TRANSFORM
The z-transform of a sequence x(n) isgiven by
X(z) = △ Z[x(n)] =
∞∑
x(n)z −n (4.1)
n=−∞
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