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zerc
to pea
or one quarter of the
peak
to
value
of
the sinusoidal signal.
MAGN
AND PHASE COMPONENTS
The result
of
the FFT when the magnitude and
phase
optiori
is
selected is N/2 input locations
containing
tlp
magnitude
components
(Mi)
followed by
llli2 input locations containing the
phase
comppnents (P1). Magnitude is half of the
zero
to peak amplitude or one quarter of the
peak
to pea( value
of
the sinusoidal signal.
There
is
a magnitude and a phase component
for
each
bin.
The value
of
ivaries from
1
to
N/2.
The magnitupe and phase components are
related to
thQ
real
(R;)
and imaginary
(11)
components
1as
shown below:
Mi=
SQRTI(Ri.Ri) +
(l;"1)l
arctan
(l;/R;)
To
calculatelthe magnitude and phase the 21X's
FFT
a
must
first compute the real and
Conversion
from
real
to
the magnitude and phase
requires
quitb a bit more datalogger execution
time and
no
new information is
gained. lf
datalogger dxecution
time is limiting, program
the dataloggpr to store
the real and imaginary
results
and have
a
computer do the conversion
to magnitud4 and phase during the data
reduction pl'iase.The FFT assumes
the signal
was
samplefl at the beginning of each of
N
intervals. Sifice the
FFT assumes
the signal
is
periodic
witfria period
equalto
the
total sampling
period,
the rpsult of its phase calculation
at
each
frequency
component
is
the average
of
the
phase atthd, beginning
of
the
first interval with
the phase a!
the
endof the
last
interval.
The
phase
is
the angle
(0
360 degrees) of
the
cosine wave that describes the signal at a
pafticular point
in
time.
POWER
SPECTRA
The result of the FFT when the power spectra
option is
selpcted is N/2 bins of spectralenergy
(PSi) represBnting frequencies from
0
Hzto
1/2
the samplin$
frequency.
The value
of
i
varies
from
1
to
N/2. The
result in
each bin i, is related
to
the magnltude
(M;)
of the
wave
in
the
following m4nner:
14)
15I
SECTION
10.
PROCESSING INSTRUCTIONS'
where
the magnitude is half
of
the zero to peak
amplitude or one quarter
of
the peak to peak
value
of
the sinusoidal signal.
The power spectra can also be expressed as
either
of
the following:
PSi=
N*(Ui*Ui)
l7l
PS;=
F*1*19'*g';
t8l
U1
is defined as
the root mean square (RMS)
value
of
the sine
component
of
frequency
i
(f,)
(Ui
= magnitude
(M;) of
the sine wave multiplied
by
the square root of 2) in units
of
the input
signal multiplied by the scaling
multiplier.
In
equation
8, F
is
the sampling frequency (Hz) and
T
is
the duration
of
the original time series data
(seconds).
When
the FFT results are expressed
in
terms of
the power spectra, a multiplier of
1
will cause
the average of
allthe
bins
to
be
very nearly
equal to twice the variance of the original data.
FFT RESULTS
WITH BIN AVERAGING
When bin averaging is specified, the FFT results
can only
be
calculated
in
terms
of
the power
spectra. The
rest
of
this section deals with the
DC
component, bin
frequency, and the power
spectra results. An example showing
bin
averaging FFT results
is
given in Section 8.8.2.
DC
COMPONENT
Before
the FFT is applied, the average of the
originaltime
series
data is subtracted from each
value. This
is done
to maintain the resolution
of
the math
in
the rest of the FFT calculations.
When bin averaging
is
specified then
the
DC
component is not output.
BIN FREQUENCY
The band width or the frequency covered by
each averaged bin
is
equalto
FA/N
where
F
is
the sample
frequency in Hz (1/scan interval
in
seconds) and
A
is
the number of bins being
averaged.
The
frequency
(f,)
of any given averaged bin
i
where i ranges from
1
to (N/2A)-1 is given by the
following equation:
i-1
*F*A/N<fi<i*F*A/N
tgl
PSi-
2.N.(Mi.Mi)
t6l
10-9
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