Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
Document source : notes.ump.edu.my
|z| = a
|z| = 1
Re[z]
Im[z]
Figure 4.2 Pole,zero,and ROC (shaded area) on the z-plane
ZTa
1
x
1
n a
2
x
2
n a
1
ZTx
1
n a
2
ZTx
2
n
a
1
X
1
z a
2
X
2
z,
4:2:4
2.
where a
1
and a
2
are constants,and X
1
z and X
2
z are the z-transforms of the
signals x
1
n and x
2
n,respectively. This linearity property can be generalized for
an arbitrary number of signals.
2. Time shifting. The z-transform of the shifted (delayed) signal yn xn À k is
Yz ZTxn À k z
Àk
Xz,
4:2:5
2.
where the minus sign corresponds to a delay of k samples. This delay property states
that the effect of delaying a signal by k samples is equivalent to multiplying its
z-transform by a factor of z
Àk
. For example,ZTxn À 1 z
À1
Xz. Thus the unit
delay z
À1
in the z-domain corresponds to a time shift of one sampling period in the
time domain.
3. Convolution. Consider the signal
xn x
1
n à x
2
n,
4:2:6
2.
where à denotes the linear convolution introduced in Chapter 3,we have
Xz X
1
zX
2
z:
4:2:7
2.
Therefore the z-transform converts the convolution of two time-domain signals to
the multiplication of their corresponding z-transforms.
Some of the commonly used signals and their z-transforms are summarized in
Table 4.3.
THE Z-TRANSFORM
135
Summary :
= 1 Re[z] Im[z] Figure 4.2 Pole,zero,and ROC (shaded area) on the z-plane ZTa 1 x 1 n a 2 x 2 n a 1 ZTx 1 n a 2 ZTx 2 n a 1 X 1 z a 2 X 2 z, 4:2:4 2.
Tags :
signals,ztransform,time,conolution,ztransforms,signal,delay,property,corresponds,ztx,samples,zdomain,exampleztxn