Digital Signal Processing Reference
In-Depth Information
These equations interpret the cross-interference coefficients characterizing in-
terference between the signal components introduced in Chapter 15. Therefore
expressions (18.4) can be given in the following form:
M
−
1
a
i
=
+
,
=
,
−
,
[
a
m
(
A
i
C
m
)
b
m
(
A
i
S
m
)]
i
0
M
1
m
=
0
(18.5)
M
−
1
b
i
=
[
a
m
(
B
i
C
m
)
+
b
m
(
B
i
S
m
)]
,
i
=
0
,
M
−
1
,
m
=
0
where the coefficients
N
−
1
2
N
(
A
i
C
m
)
=
cos(2
π
f
m
t
k
) cos(2
π
f
i
t
k
)
,
k
=
0
N
−
1
2
N
(
B
i
C
m
)
=
cos(2
π
f
m
t
k
) sin(2
π
f
i
t
k
)
,
=
k
0
(18.6)
N
−
1
2
N
(
A
i
S
m
)
=
sin(2
π
f
m
t
k
) cos(2
π
f
i
t
k
)
,
k
=
0
N
−
1
2
N
(
B
i
S
m
)
=
sin(2
π
f
m
t
k
) sin(2
π
f
i
t
k
)
.
k
=
0
These coefficients, reflecting the impact of the sampling imperfections, are
actually the weights of the errors that corrupt the estimation of a Fourier coefficient
a
i
(or
b
i
) at frequency
f
i
and are related to the sampling nonuniformities of the
sine (or cosine) component present in the signal at frequency
f
m
. Another set of
cross-interference coefficients, specifically the coefficients
A
m
C
i
,
B
m
C
i
A
m
S
i
and
B
m
S
i
, characterize interference acting in the inverse direction from the signal
component at frequency
f
i
to the component at frequency
f
m
. It follows from
Equations (18.6) that
,
=
,
=
,
A
i
C
m
A
m
C
i
B
i
C
m
A
m
S
i
(18.7)
A
i
S
m
=
B
m
C
i
,
B
i
S
m
=
B
m
S
i
.
Therefore it is not necessary to calculate the coefficients
A
m
C
i
A
m
S
i
and
B
m
S
i
on the basis of formulae similar to Equations (18.6), which is a great help.
,
B
m
C
i
,
18.1.2 Interpretation
To get a better idea of exactly how the sampling irregularities impact pro-
cessing of the nonuniformly sampled signals, Equations (18.6) defining the
