Digital Signal Processing Reference
In-Depth Information
1
the Same as a Delay by One Unit?
8.4
Why Is z
1
in the frequency-
domain delays a signal by one sample. To see why this is the case, let's examine
two similar FIR lters, one with coecientsf1, 0g, and the other with coecients
f0, 1g. These trivial lters are shown in Figure 8.2.
1
in a box, since multiplying by z
A delay unit is often shown as z
1
x[n]
y [n]
1
input
output
0
D
0
x[n]
y [n]
2
input
output
1
D
Figure 8.2: Two trivial FIR lters.
By inspection, we see that the rst lter eectively does nothing, and the output
is the same as the input, y
1
[n] = x[n]. The second lter really does not do anything
either, except that the output is a delayed version of the input, i.e., y
2
[n] = x[n1].
We can remove the multipliers, since multiplying by zero gives us a zero result, and
multiplying a number by one gives the same number back. Since one of the adder's
inputs is zero, we can remove the adders from the diagram as well. This leaves us
with a very reduced form of these two lters, Figure 8.3.
The rst lter has a transfer function H
1
(z) = 1z
0
+ 0z
1
, while the second one
has H
2
(z) = 0z
0
+ 1z
1
. Writing the output in terms of the transfer function and
input gives us:
1X(z)z
0
+ 0X(z)z
1
Y
1
(z)
=
= X(z)z
0
= X(z)
0X(z)z
0
+ 1X(z)z
1
Y
2
(z)
=
