Automotive Engineering - June 2023 - 8

SPONSORED CONTENT
The Inside Story on Fast Fourier Transform Windowing
T
he Fourier Transform (aka FFT - Fast Fourier Transform) has for decades been used by mechanical engineers to
diagnose noise, vibration and resonant fatigue issues. These domains include acoustics, rotating machinery, structural
dynamics, fatigue, and machinery diagnostics. The FFT allows engineers to quickly transform time domain
data to the frequency domain, allowing analysts to understand the frequencies of the excessive sound and vibration
which can easily be traced to resonant frequencies or a forcing function. SAE's Automotive Engineering spoke with
John Hiatt, Business Development Manager, DEWESoft LLC, who discussed possible reasons why the FFTs calculated with
two different tools do not match, with the primary focus being FFT Windowing.
What are the primary factors that contribute
to the FFT results calculated with different
algorithms not matching?
Commercial codes will require the user to input
several parameters when transforming
data to the frequency domain. Factors include:
1. What windows were applied were they the
same, and if so, were the proper window correction
factors used? 2. If measuring acceleration
(or any unit), when calculating an FFT, are
you comparing the same amplitude function
types which include: linear spectrum (g), power
spectral density (PSD (g^2/Hz)), auto power
spectrum (g^2), auto power spectral density (ASD g/
sqrt(Hz)), energy spectral density (ASD (g^2*s/Hz)). 3. Is
averaging used? If so, what type and how many blocks are
being averaged? 4. Is overlap used, and if so, what percentage?
5. How is the spectrum scaled - peak, RMS, or
peak to peak? 6. What is the frequency resolution?
Depending on the input type - deterministic or random -
the amplitude may change with frequency resolution. 7. Do
you want to compute a single or double-sided spectrum?
Typically, we use a single-sided folding of the negative
frequencies of the spectrum to the positive frequencies,
resulting in a factor of 2 difference between single and double-sided
spectrums. This typically is not an issue, as many
commercial codes only show the single-sided spectrum.
Why are FFT windows used?
There are two major assumptions made when calculating
an FFT: the signal is continuous, and the signal is periodic
in the time window. A periodic signal can be described as
a signal that starts and ends at 0, or if the signal is appended
and has no discontinuities. The FFT data is processed
in blocks of data (segments), and these blocks of
data contain a finite number of samples with a specific
duration and a sample count.
Typically, commercial code has a maximum number of
samples that can be processed called the maximum block
size (Block Size*1/Sample Rate = Block Duration). If the
data extracted for processing is not periodic (random data
is never periodic) then leakage will occur from discontinuities.
This leakage spreads energy over all the frequency
bins/lines in the spectrum because the FFT of a discontinuity
is a broadband flat spectrum in the frequency domain.
Windows are applied to force periodicity.
8 June 2023
John Hiatt, Business
Development Manager,
DEWESoft LLC
What are windows?
Windows are typically bell-shaped time-weighing
curves that force the data you are processing
to start and end at 0. Types of windows
include Hanning, Hamming, Blackman,
Blackman-Harris, Flatto, and rectangular (no
window indicates that all samples will be
weighted by a factor of 1). Examples of windows
that are not bell-shaped are exponential
and force windows, typically used in impact
testing. The window functions are simply multiplied
by the data on which the FFT will be calculated.
If the window was applied in the frequency
domain, the math process would be convolution.
How do FFT windows impact the data?
Windows mitigate the leakage at the expense of altering energy/amplitude
and frequency smearing. This is created by the
shape of the window. The change in energy/amplitude is apparent
with the removal of the signal. The frequency smearing
is not so obvious, but can be seen in the frequency domain by
the width of the main lobe. The amplitude and energy loss can
be partially corrected, but the frequency smearing cannot.
Different window types offer a different tradeoff between
the frequency smearing and the energy/amplitude reduction.
Flattop windows are designed to have very precise amplitude
at the expense of larger frequency smearing; these windows
are used for calibration. Most other windows are used for general
frequency analysis and have small tradeoffs between the
amplitude/energy loss and the frequency smearing. Most
commercial codes will provide window correction factors for
both energy (RMS level) and amplitude corrections.
When calculating the RMS value of the spectrum energy,
correction is used. When looking at a spectrum magnitude,
the amplitude correction is applied. Many commercial
codes do this automatically, but when writing your own
code, this must be accounted for.
These windows significantly alter the amplitude and energy
of the spectrum. A Hanning window has a noise
bandwidth of 1.5 delta f (frequency smearing), and max
amplitude error of 1.4 dB (16%). Correction factors for the
Hanning window are 2 for the amplitude correction and
1.63 for the energy correction (Spectrum RMS).
More Information on this topic can be found at: https://
training.dewesoft.com/online/course/fft-spectral-analysis
AUTOMOTIVE ENGINEERING

http://info.hotims.com/84495-708

Automotive Engineering - June 2023

Table of Contents for the Digital Edition of Automotive Engineering - June 2023