Digital Signal Processing Reference
In-Depth Information
lter. With IIR lters, it is possible for the output to approach innity (or negative
innity), which can be seen with the transfer function, H(z).
The z-transform describes the frequency response of lters, simply by nding z-
transform of the lter's coecients. For an FIR lter, we nd the frequency response
with:
N
X
k
H(z) =
h[k]z
k=0
where h[k] are the lter's coecients.
For an IIR lter, the situation is a bit more complicated, since we have both
a feed-forward part (coecients b[k]) and a feed-back part (coecients a[k]). We
start by looking at the time-domain equation describing the lter's output.
K
M
X
X
y[n] =
b[k]x[nk] +
a[k]y[nk]
k=0
k=1
Remember that the feed-back part starts with a[1] instead of a[0], since there is
no multiplier at that corresponding location. Next, we nd the z-transform of this
equation,
N
M
X
X
k
+ Y (z)
k
;
Y (z) = X(z)
b[k]z
a[k]z
k=0
k=1
where z = re
j!
.
The transfer function gives us the IIR lter's response. We can specify the eect
of the lter without having to give an input:
H(z) =
Y (z)
X(z)
| or |
P
N
k
k=0
b[k]z
k=1
a[k]z
k
:
Notice that the denominator could be zero under the right circumstances (values for
z). If this happens, our frequency response, H(z), is innite. We call such a value
for z a pole, borrowing from the image of a circus tent, where the tent fabric has a
pronounced shape from the tent poles, like a three-dimensional plot of H(z)'s surface
would for certain values of z. Also, some values of z could result in a zero value for
the function as a whole, from the numerator. These spots we call zeros. When we
talk about the frequency response for IIR lters, we are primarily concerned with
H(z) =
P
M
1
