Biomedical Engineering Reference
In-Depth Information
TABLE 13.1:
Spectral Window vs. Frequency Resolution
f
sampling rate
N
Δ
T
1.
1 sec
−
2000 samples/sec
2000
0.0005
2.
1 sec
−
1000 samples/sec
1000
0.001
3.
2 sec
−
1000 samples/sec
2000
0.05
Since
w
n
=
2
π/
N
=
2
π
T
. The frequency resolutions are shown below.
ω
0
1.
0.003141592
f
res
=
=
1.0 Hz
2
π
T
n
∗
f
samp
2
f
res
=
ω
2.
0.006283185
=
1.0 Hz
π
3.
0.00314592
=
0.5 Hz
e
−
j
2
π/
N
is called phase or
The Twiddle Factor
,
x
(
k
) is called the
N
-point
DFT, and the distance between successive samples in the frequency domain gives the
fundamental frequency of
x
(
t
).
In terms of normalized frequency units, the fundamental frequency is given by
where
W
N
=
2
N
. Note in the example given in Table 13.1 that a one-second window results in a
1-Hz frequency resolution regardless of the sampling rate.
By definition, the Frequency Resolution (
f
res
) or the fundamental frequency of the
spectral (
f
0
) is “The minimum frequency that a signal can be resolved in the frequency
domain, and is equal to the inverse of the record length or the signal window length
being analyzed (in seconds) in time domain” (13.4):
π/
1
R
1
N
T
f
res
=
L
=
(13.4)
.
Record length
: The length of the sampled analog signal with a total of
N
samples is given
as
R
.
L
NT
.
Expanding the DFT equation results in (13.5):
=
N
−
1
(Re[
x
(
n
)]Re[
W
n
N
]
Im[
x
(
n
)]Im[
W
n
N
])
x
(
k
)
=
0
N
{
−
n
=
(13.5)
(Re[
x
(
n
)]Im[
W
n
N
]
Im[
x
(
n
)]Re[
W
n
N
])
+
+
}
where
k
=
0
,
1
,...,
N
−
1
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