Digital Signal Processing Reference
In-Depth Information
X
k
X
k-1
X
k-2
X
k-3
X
k-4
reg
dly
reg
dly
reg
dly
reg
dly
C
0
C
1
C
2
C
3
C
4
X
X
X
X
X
∑
5-Tap FIR Filter
Y
k
Figure 4.2. FIR
lter diagram.
The coefficients are usually static (they do not change over
time), and determine the filter's frequency response.
In equation form, the filter could be represented as:
y
k
¼
x
k
4
This is just the sum of the multipliers. This could get rather
tedious to write as the number of taps gets larger, so the following
short-hand summation is often used:
y
k
¼
P
i
¼
0to4
C
i
x
k
i
We can also make the equation for any length of filter. To make
our filter of length“N”, we simply replace the 4 (5-1 taps)withN
C
0
x
k
þ
C
1
x
k
1
þ
C
2
x
k
2
þ
C
3
x
k
3
þ
C
4
1.
y
k
¼
P
i
¼
0toN
1
C
i
x
k
i
Another way to look at this is that the data stream x
k
þ
2
,x
k
þ
1
,
x
k
,x
k
1
,x
k
2
.
is sliding past a fixed array of coefficients. At each
clock-cycle, the data and coefficients are cross-multiplied and the
outputs of all multipliers for that clock-cycle are summed to form
a single output (this process also known as dot product). Then, on
the next clock-cycle, the data is shifted one place relative to the
coefficients (which are fixed), and the process repeated. This
process is known as convolution.
The FIR structure is very simple, yet has the ability to create
almost any frequency response, given a sufficient number of taps.
This is very powerful, but unfortunately not at all intuitive. It's
somewhat analogous to the brain
a very simple structure of
interconnected neurons, yet the combination can produce
amazing results. In the course of the rest of this chapter, we will
try to gain some understanding of how this happens.
Below is an example using actual numbers, to illustrate this
process called convolution.
We will define a filter of 5 coefficients {C
0
,C
1
,C
2
,C
3
,C
4
}
e
¼
{1,3,5,3,1}.
