The z-transform - Continuous and Discrete Time Signals and Systems - page 596

Taking the inverse transform, we obtain

800.19

y [ k ]

254.38

26
15

0 . 5 k − 7
3

2 k + 26
15

78.75

3 k y [ k ] = for k > 0 .

23.5

≈ ≈ ≈

7

1.83 3

1.03

The output response is plotted in Fig. 13.5.

k

−2 −1 0 23

Fig. 13.5. Output response of
the LTID system specified in
Example 13.12.

13.6 z-transfer function of LTID systems

In Chapters 10 and 11, we used the impulse response h [ k ] and Fourier transfer
function H ( Ω ) to represent an LTID system. An alternative representation for
an LTID system is obtained by taking the z-transform of the impulse response:

←→

h [ k ] H ( z ) .

The DTFT H (z) is referred to as the z - transfer function of the LTID system.
In conjunction with the linear convolution property, Eq. (13.24), the z-transfer
function H (z) may be used to determine the output response y [ k ]ofanLTID
system when an input sequence x [ k ] is applied at its input. In the time domain,
the output response y [ k ]isgivenby

y [ k ] = x [ k ] ∗ h [ k ] . (13.32)

Taking the z-transform of both sides of Eq. (13.32), we obtain

Y ( z ) = X ( z ) H ( z ) , (13.33)

where Y ( z ) and X ( z ) are, respectively, the z-transforms of the output response
y [ k ] and the input sequence x [ k ]. Equation (13.33) provides us with an alter-
native definition for the transfer function as the ratio of the z-transform of the
output response and the z -transform of the input signal. Mathematically, the
transfer function H (z) can be expressed as follows:
H ( z ) = Y ( z )

X ( z ) .

(13.34)

The z-transfer function of an LTID system can be obtained from its difference
equation representation, as described in the following.
Consider an LTID system whose input-output relationship is given by the
following difference equation:

y [ k + n ] + a n − 1 y [ k + n − 1] ++ a 0 y [ k ]
= b m x [ k + m ] + b m − 1 x [ k + m − 1] ++ b 0 x [ k ] .

(13.35)

By taking the z-transform of both sides of the above equation, we obtain

z n + a n − 1 z n − 1 ++ a 0 z b m z m + b m − 1 z m − 1 ++ b 0 Y ( z ) = X ( z ) ,