The z-transform - Continuous and Discrete Time Signals and Systems - page 566

applications of the z-transform. Section 13.5 applies the z-transform to calculate
the output of an LTID system from the input sequence and the impulse response
of the LTID system. The relationship between the Laplace transform and the
z-transform is discussed in Section 13.6. Stability analysis of the LTID system in
the z-domain is presented in Section 13.7, while graphical techniques to derive
the frequency response from the z-transform are discussed in Section 13.8.
Section 13.9 compares the DTFT and z-transform in calculating the steady state
and transient responses of an LTID system. Section 13.10 introduces important
M ATLAB library functions useful in computing the z-transform and in the
analysis of LTID systems. Finally, the chapter is concluded in Section 13.11
with a summary of important concepts.

13.1 Analytical dev elopment

Section 11.1 defines the synthesis and analysis equations of the DTFT pair
x [ k ]

DTFT
←−−→

X ( Ω ) as follows:

1
2 π

X ( Ω )e j Ω k d Ω ; x [ k ] =

DTFT synthesis equation (13.1)

2 π

k =−∞ x [ k ]e

∞

− j Ω k .

DTFT analysis equation X ( Ω ) = (13.2)

To derive the expression for the bilateral z-transform, we calculate the DTFT
of the modified version x [ k ]e

−σ k of the DT signal. Based on Eq. (13.2), the
DTFT of the modified signal is given by

k =−∞ x [ k ]e k =−∞ x [ k ]e

∞ ∞

x [ k ]e −σ k

−σ k e − j Ω k − ( σ + j Ω ) k .

ℑ = = (13.3)

Substituting e σ + j Ω = z in Eq. (13.3) leads to the following definition for the
bilateral z-transform:

k =−∞ x [ k ] z

∞

x [ k ]e −σ k

− k .

z-analysis equation X ( z ) =ℑ = (13.4)

It may be noted that the summation in Eq. (13.4) is absolutely summable only
for selected values of z . For other values of z , the infinite sum in Eq. (13.4)
may not converge to a finite value, and hence X ( z ) becomes infinite. The region
in the complex z-plane, where summation (13.4) is finite, is referred to as the
region of convergence (ROC) of the z-transform X ( z ).
By following a similar derivation for the DTFT synthesis equation, Eq. (13.1),
the expression for the inverse z-transform is given by

1
2 π j

X ( z ) z k − 1 d z , z-synthesis equation x [ k ] = (13.5)

C