Information Technology Reference
In-Depth Information
2
N
1
ˆ
∑
=
P
(
f
)
=
x
(
n
)
exp(
−
j
2
π
fn
)
(1)
PER
N
n
1
ˆ
P
PER
is the estimation of periodogram. In the Welch method, signals are
divided into overlapping segments, each data segment is windowed, periodograms are
calculated and then average of periodograms is found.
(
f
)
Where
{
}
x
l
, l=1,…,S are data
segments and each segment's length equals M. Note that, the overlap is often chosen
to be 50%. The Welch spectrum estimate is given by:
(
n
)
1
s
ˆ
ˆ
∑
=
P
(
f
)
=
P
(
f
)
(2)
w
l
S
l
1
2
1
1
M
ˆ
∑
=
P
(
f
)
=
v
(
n
)
x
(
n
)
exp(
−
j
2
π
fn
)
(3)
l
l
M
P
n
1
ˆ
TH
P
l
(
f
)
l
where
is the periodogram estimate of
segment,
v(n)
is the data-window,
∑
=
M
n
2
ˆ
P
=
1
/
M
v
(
n
)
P
w
is the
Welch PSD estimate,
M
is the length of each signal segment and S is the number of
segments.
Then, Evaluation of
(
f
)
P
is total average of
v(n)
and given as
,
1
ˆ
P
w
at the frequency samples basically requires the
computation of the following discrete Fourier transform (DFT):
(
f
)
N
2
π
∑
=
nk
X
(
k
)
=
x
(
n
)
exp(
−
j
)
, k=0,…, N-1
(4)
N
n
1
Where
X(k)
is expressed as the discrete Fourier coefficient,
N
is the length of avail-
able data and
x(n)
is the input signal on the time domain. The procedure that com-
putes Eq. (4) is called as FFT algorithm. The Welch PSD can be efficiently computed
by the FFT algorithm. Variance of an estimator is one of the measures often used to
characterize its performance. For 50% overlap and triangular window, variance for
the Welch method is given by;
9
ˆ
ˆ
var(
P
(
f
))
=
var(
P
(
f
))
(5)
w
l
8
S
ˆ
ˆ
P
w
(
f
)
P
l
(
f
)
Where
the Welch PSD is estimate and
is the periodogram estimate of
each signal interval [4-10].
