SDA Operator’s Manual
FFT Algorithms
A summary of the algorithms used in the oscilloscope's FFT computation is given here in a few
steps:
1. The data are multiplied by the selected window function.
2. 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.
3. 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.
4. 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
5. The resultant waveform is computed for the spectrum type selected.
If "Magnitude" is selected, the magnitude of the complex vector is computed as:
Steps 1–5 (previous) 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.
SDA-OM-E Rev H 199
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