Digital Signal Processing Reference
In-Depth Information
Once the last non-zero input data x
k
has shifted all the way
through the filter taps, the output data y
k
will go to zero (this
starts at k
10 in our example).
Now let's consider a special case, where x
k
¼
¼
1 for k
¼
0, and
x
k
¼
0. This means that we only have one non-zero input
sample, and it is equal to 1. Now if we compute the output, which
is simpler this time, we get:
y
-1
¼
P
i
¼
0oN
0 for k
s
1
C
i
x
-1
i
¼
(1)(0)
þ
(3)(0)
þ
(5)(0)
þ
(3)(0)
0
y
0
¼
P
i
¼
0toN
þ
(1)(0)
¼
x
0-i
¼
þ
þ
þ
1
C
i
(1)(1)
(3)(0)
(5)(0)
(3)(0)
1
y
1
¼
P
i
¼
0oN
þ
(1)(0)
¼
1
C
i
x
1
i
¼
(1)(0)
þ
(3)(1)
þ
(5)(0)
þ
(3)(0)
3
y
2
¼
P
i
¼
0oN
þ
(1)(0)
¼
1
C
i
x
2
i
¼
(1)(0)
þ
(3)(0)
þ
(5)(1)
þ
(3)(0)
5
y
3
¼
P
i
¼
0oN
þ
(1)(0)
¼
1
C
i
x
3
i
¼
(1)(0)
þ
(3)(0)
þ
(5)(0)
þ
(3)(1)
3
y
4
¼
P
i
¼
0oN
þ
(1)(0)
¼
C
i
x
4
i
¼
(1)(0)
þ
(3)(0)
þ
(5)(0)
þ
(3)(0)
1
1
y
5
¼
P
i
¼
0oN
þ
(1)(1)
¼
x
5
i
¼
þ
þ
þ
1
C
i
(1)(0)
(3)(0)
(5)(0)
(3)(0)
0
y
6
¼
P
i
¼
0oN
þ
(1)(0)
¼
1
C
i
x
6
i
¼
(1)(0)
þ
(3)(0)
þ
(5)(0)
þ
(3)(0)
0
Notice that the output is the same sequence as the coeffi-
cients. This should come as no surprise once you think about it.
This output is defined as the filter's impulse response, so named
as it occurs when the filter input is an impulse, or a single non-
zero input equal to one. This gives the FIR filter its name. By
Finite Impulse Response” or FIR, this indicates that if this type of
filter is driven with an impulse, we will see a response (the output)
has a finite length, after which it becomes zero. This may seem
trivial, but it is a very good property to have, as we will see in the
chapter on infinite impulse response filters.
þ
(1)(0)
¼
4.4 Computing Frequency Response
So far we have covered the mechanics of building the filter,
and how to compute the output data, given the coefficients and
input data. But we do not have any intuitive feeling as to how this
operation can allow some frequencies to pass through, and block
other frequencies. A very basic understanding of a low pass filter
can be gained by the concept of averaging. We all know that if we
average multiplication results, we get a smoother, more
