Digital Signal Processing Reference
In-Depth Information
x[n]
y[n]
D
Figure 3.7: Signal y is a delayed version of x.
3.2
FIR Filter Structures
Now that we have seen how the parts make a lter, we will demonstrate some
FIR lters and discuss some important characteristics: describing FIR lters by
equations, and how the unit impulse function works with them. The FIR lter
structure is quite regular, and once the numbers corresponding to the multipliers
are known, the lter can be completely specied. We call these numbers the lter
coecients. One unfortunate use of the terminology is to call the lter's outputs
coecients as well.
In this topic, \coecients" will be used to denote the lter
coecients only.
In Figure 3.8, we see an example FIR lter. Notice that the arrows show direction
as left to right only, typical of FIR lters. Sure, an arrow points from the input
down to the delay unit, but signicantly no path exists from the output side back
to the input side. It earns the name feed-forward for this reason.
Suppose Figure 3.8 has an input x[n] = [1; 0]. The outputs would be [0:5; 0:5],
and zero thereafter. We nd these outputs by applying them, in order, to the lter
structure. At time step 0, x[0] appears at the input, and anything before this is zero,
which is what the delay element will output. At the next time step, everything shifts
down, meaning that x[0] will now be output from the delay element, meanwhile x[1]
appears as the current input.
0.5
x[n]
y[n]
0.5
D
Figure 3.8: FIR lter with coecientsf0.5, 0.5g.
How would you express y[n] as an equation? As we observe from calculating y[0]
and y[1], we really have the same pattern:
y[0] = 0:5x[0] + 0:5x[1]
