Digital Signal Processing Reference
In-Depth Information
Figure 3.10 shows the general form of the FIR lter, for K + 1 lter coecients.
(There are K + 1 of them because we start at 0 and count to K.) The number of
lter coecients is also called the number of taps. By convention, the number of
taps equals the number of lter coecients. So a lter with coecientsfb
0
, b
1
, ...
, b
K
ghas K + 1 taps, since there are K + 1 total lter coecients. However, it is
said to be of order K. In other words, the order of the lter and the taps express
the same idea, but with a dierence of 1.
With the structure of Figure 3.10 [11], it is possible to determine the output. It
is also possible to determine an equation for the output, which is
y[n] = b[0]x[n0] + b[1]x[n1] + b[2]x[n2] + ::: + b[K]x[nK]:
Notice that whatever index is used for b[] is also used in x[n]. This means we
can represent everything on the righthand side of the equation as a summation.
K
X
y[n] =
b[k]x[nk]
k=0
Example:
Given b = [0:1; 0:2; 0:3] and x = [1; 0; 0; 2; 0; 1; 4; 3], what is y?
Answer:
y[n] = x[n2]b[2] + x[n1]b[1] + x[n]b[0]
This can be found by noting the above equation, then making a chart.
y[0] = 0 b[2] + 0 b[1] + x[0]b[0]
y[1] = 0 b[2] + x[0]b[1] + x[1]b[0]
y[2] = x[0]b[2] + x[1]b[1] + x[2]b[0]
y[3] = x[1]b[2] + x[2]b[1] + x[3]b[0]
etc.
Plugging in the values of x[n] to form columns multiplied by a scalar:
