Digital Signal Processing Reference
In-Depth Information
e
±
j
ω
0
. This shows that the
ANC behaves as a second order IIR filter with central frequency
So the poles are approximately located at
z
=
(
1
−
α)
ω
0
. Therefore, if
ω
0
is time variant (to include in the model the potential slow drift of the PLI), the
adaptive filter has the potential to track these changes. Moreover, the step size
μ
controls the location of the poles. Since the sharpness of the notch is determined
by the closeness between the zeros and the poles of
H
leads to a
sharper notch filter. Particularly, the 3dB bandwidth can be approximated by
(
z
)
, decreasing
μ
or
μ
Hz
LA
2
2
LA
2
f
s
4
α
=
μ
Bw
LMS
≈
2
radians
.
(4.34)
π
To analyze the differences that might arise when using an NLMS instead of the LMS
algorithm, remember that the NLMS can be seen as a variable step size LMS, i.e.,
μ
NLMS
μ
LMS
(
n
)
=
2
.
(4.35)
x
(
n
)
2
2
2
However, for large adaptive filters
x
is the power of the
input process. Since in this case the input to the filter is a sinusoid with an uniformly
distributed random phase,
x
(
n
)
≈
L
σ
x
, where
σ
2
x
A
2
2. Replacing all this in (
4.34
), we see that the
bandwidth with the NLMS algorithm depends only on the step size
σ
=
/
μ
,so
or
μ
NLMS
f
s
2
Hz
Bw
NLMS
≈
μ
NLMS
radians
.
(4.36)
π
To test the ANC numerically, we use a recording of extracellular neural activity
(sampled at 28kHz) acquired froma depth electrode implanted in themedial temporal
lobe of an epileptic patient [
7
]. The high impedance electrodes used for this type of
recordings (
) would act like antennas, picking up PLI fromnearby electrical
equipments (notice that in a hospital environment there are many pieces of electrical
equipment, and switching them off might not be an option). In this application, when
possible, the reference signal can be taken from a wall outlet with proper attenuation.
We focus on the so called gamma band, which has a frequency span from 30
to 100Hz. The smaller amplitude of this signal compared to the ones associated to
slower oscillations makes it even more prone to end up masked by the PLI. As we
will see later, the power of the PLI in this recording was at least 15dB above the
power of the signal. However, it has been found that gamma oscillations are related
to several neuropsychiatric disorders (schizophrenia, Alzheimer's Disease, epilepsy,
ADHD, etc.) and memory processing [
8
,
9
]. Therefore, this application requires a
notch filter with small rejection bandwidth but with the ability to track the possibly
time varying PLI.
For comparison purposes, we also filtered the recorded signal with second order
IIR notch filters with central frequencies
f
0
of 60 and 61Hz. These filters were
designed to have a 3dB bandwidth close to 1Hz. To achieve the same bandwidth
with the NLMS algorithm we used (
4.36
), leading to
≈
500 k
μ
=
0
.
000225. We also use a
