Digital Signal Processing Reference
In-Depth Information
Estimates
a
j
and
b
j
of the Fourier coefficients characterizing the sinusoid at
frequency
f
j
drawn through the sample points with the least square error are
obtained by solving the system of equations corresponding to Equation (15.2):
b
ji
a
ji
α
−
α
s
cs
a
j
=
,
α
c
α
s
−
α
cs
(15.3)
b
ji
α
−
a
ji
α
b
j
c
cs
=
,
α
c
α
s
−
α
cs
where
N
2
N
sin(2
π
f
i
t
k
+
ϕ
i
) cos 2
π
f
j
t
k
,
a
ji
=
k
=
1
(15.4)
N
2
N
b
ji
sin(2
π
f
i
t
k
i
) sin 2
π
f
j
t
k
=
+
ϕ
k
=
1
and
N
2
N
cos
2
2
π
f
j
t
k
α
=
,
c
k
=
1
N
2
N
sin
2
2
π
f
j
t
k
α
=
,
(15.5)
s
k
=
1
N
2
N
α
=
sin 2
π
f
j
t
k
cos 2
π
f
j
t
k
.
cs
k
=
1
The solution to the minimization task allows the degree of aliasing
γ
ji
to be
defined as follows:
a
j
+
b
j
.
γ
ji
=
(15.6)
=
Now consider the notation in Equation (15.5). Obviously,
α
1 and interval
c
=
α
0. This suggests that aliasing can also be evaluated by the following
approximate equality:
cs
b
ji
.
a
ji
+
γ
=
(15.7)
ji
Substitution of Equation (15.7) for Equation (15.6) is very desirable, because
it considerably simplifies further analysis. The errors arising in this case were
evaluated by computer simulations and they are small enough to justify this
substitution. Therefore there is no need to use the more correct but computation-
ally significantly more complicated best-fitting approach to estimate the Fourier
