W
AVE
R
UNNER
X
I
S
ERIES
138
WRXi-OM-E Rev B
An effective way to reduce these effects is to maximize the acquisition record length. Record length directly
conditions the effective sampling rate of the oscilloscope and therefore determines the frequency resolution and
span at which spectral analysis can be carried out.
FFT Algorithms
A summary of the algorithms used in the oscilloscope's FFT computation is given here in a few steps:
The data are multiplied by the selected window function.
FFT is computed, using a fast implementation of the DFT (Discrete Fourier Transform):
where:
x
k
is a complex array whose real part is the modified source time domain waveform, and whose imaginary
part is 0;
X
n
is the resulting complex frequency-domain waveform;
; and
N
is the number of points in
x
k
and
X
n
.
The generalized FFT algorithm, as implemented here, works on N, which need
not
be a power of 2.
The resulting complex vector
X
n
is divided by the coherent gain of the window function, in order to compensate for
the loss of the signal energy due to windowing. This compensation provides accurate amplitude values for
isolated spectrum peaks.
The real part of
X
n
is symmetric around the Nyquist frequency, that is
R
n
= R
N-n
while the imaginary part is asymmetric, that is
I
n
= –I
N-n
The energy of the signal at a frequency
n
is distributed equally between the first and the second halves of the
spectrum; the energy at frequency 0 is completely contained in the 0 term.
The first half of the spectrum (Re, Im), from 0 to the Nyquist frequency is kept for further processing and doubled
in amplitude:
R'
n
= 2 x R
n
_0 n < N/2
I'
n
= 2 x I
n
__0 n < N/2
The resultant waveform is computed for the spectrum type selected.
If "Magnitude" is selected, the magnitude of the complex vector is computed as:
Steps 1 to 5 lead to the following result:
An AC sine wave of amplitude 1.0 V with an integral number of periods N
p
in the time window, transformed with
the rectangular window, results in a fundamental peak of 1.0 V magnitude in the spectrum at frequency N
p
x Delta
f. However, a DC component of 1.0 V, transformed with the rectangular window, results in a peak of 2.0 V
magnitude at 0 Hz.
The waveforms for the other available spectrum types are computed as follows:
Phase: angle = arctan (
I
n
/R
n
)
_M
n
> M
min
angle = 0
M
n
≤
M
min
Where
M
min
is the minimum magnitude, fixed at about 0.001 of the full scale at any gain setting, below which the
angle is not well defined.
The dBm Power Spectrum:
