Digital Signal Processing Reference
In-Depth Information
3.2.5 MA Filter with Complex Coefficients
There is another important breed of MA filters based on the concept of super-
heterodyning. This consists of a local oscillator, a mixer, and a filter. The signal x
ð
t
Þ
is multiplied with a local oscillator e
j2
ft
and passed through an integral or lowpass
filter. Mathematically,
3
it is given by
R
P
u
ð
t
Þ
u
k
e
j2
ft
dt
e
j2
fk
U
a
k
þ
jb
k
¼
ð
3
:
11
Þ
Equation (3.11) depicts a mapping from continuous to discrete. This means that we
beat the given signal u
k
with an oscillator and pass the result through a summation.
This results in a pair of MA filters:
a
k
¼
X
u
k
cos
ð
2
b
k
¼
X
u
k
sin
ð
2
fk
Þ
and
fk
Þ:
ð
3
:
12
Þ
Figure 3.5 is an implementation of (3.12) with non-linear devices such as multipliers.
Figure 3.5 Heterodyning
When we sweep the oscillator to generate a set of rows resulting in a pair of
matrices, one for real (in-phase or cosine) and one for imaginary (quadrature or
sine), we can also observe the output as Fourier series coefficients.
Instead of using an infinite sine wave, we can sample the sine wave to generate
the coefficients of the MA filter. The filter characteristics depend on the number of
coefficients and how many samples are taken per cycle. Consider a matrix
A
with
each row containing the coefficients of an MA filter.
0
1
1
1
1
1
1
1
1
1
@
A
10
:
7071
0
0
:
7071
1
0
:
7071
0
0
:
7071
1
0
1
0
1
0
1
0
1
0
:
7071
0
0
:
7071
10
:
7071
0
0
:
7071
A ¼
ð
3
:
13
Þ
1
1
1
1
1
1
1
1
1
0
:
7071
0
0
:
7071
10
:
7071
0
0
:
7071
1
0
1
0
1
0
1
0
:
0
:
1
0
:
:
10
7071
0
7071
7071
0
0
7071
3
Note that we have avoided using limits.
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