FLEXIBLE DATA SIZE FAST FOURIER TRANSFORM PROCESS FOR SPARSE FREQUENCY SINUSOIDAL AND HARMONIC SIGNAL ANALYSIS

TRANSFORMATION DE FOURIER RAPIDE A VOLUME DE DONNEES VARIABLE POUR L'ANALYSE DE SIGNAUX SINUSOIDAUX ET HARMONIQUES

English Abstract

Conventional digital measurement instruments and software use radix-2 fast Fourier
transform (FFT) to analyze steady state sinusoidal and harmonic signals. Steady state sinusoidal
and harmonic signals occurs commonly in rotating machine vibration, mechanical and electrical
modal oscillation, and electric power system. Analogue anti-aliasing filters and digital time domain
windows, such as Hanning and Gaussian windows, are applied to the signal to reduce the error
resulted from the radix-2 FFT. The use of these windows can be eliminated if the data size contains
exact number of integral multiples of the sinusoidal and harmonic signals. Radix-2 FFT forces the
user to limit data size (number of data points) to power of 2 (for example ...., 256, 512, 1024, 2048,
...). It is impossible to match a limited number of data sizes to the aforementioned signals with
readily available data-sampling rates (such as 1000, 2000, 5000, 10000, ... samples per second). The
following disclosure describe that through multiple application of flexible data size mixed radix FFT
can eliminate the use of the windows and produce more accurate results.

Claims

CLAIMS
The embodiments of the invention in which an exclusive property or privilege is claimed are
as follows:
1. A process to calculate accurately the magnitude and phase of sparse frequency sinusoidal
or harmonic signals, which comprises varying the size of the data set to contain an integral number
of signal periods, selecting the radices for the mixed radix fast Fourier transform calculation to
closely match the size of the data set, and resolving its Fourier transform through mixed radix fast
Fourier transform.

Description

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DIS C L O S U R E
This invention relates to a specially tle~igned computer software procedure to obtain
maxil~lulll possible data accuracy for steady state sinusoidal and h~rmonic signals through the
technique of flexible data size fast Fourier transform (FFT).
Steady state ~iml~oicl~l and harmonic signals occurs commonly in rotating m~chine vibration,
me~h~ni~l and electric~l modal oscill~tinn, and electric power system harmonics and resonances.
All of these signals are periodic signals. If the digital data set of these signals colltains exactly
integral number of periods of the signals to be tr~ rn~ " led by the ~ 1 to frequency domain, the
resulting frequency component of the periodic signal is exact. This special condition of "integral
periods" is almost never achieved through norm~lly available products. Because their data sizes are
fixed to 256 points, 512 points, 1024 points, ... etc. and their time sampling rates are also fixed to
1000, 2000, 5000, 10000, .... points per second. It is not likely that the limited number of
combination of the fixed data size and sampling rate can produce the desired integral periods. Time
tl-)m~in windows, such as H~nning and G~ls~i~n windows, were introduced to reduce the ~parellt
magnitude error resulted from the condition of "fractional periods" in the data set. But the a~ale
phase error remains lln(h~nged by the windows.
One method to elimin~te the apparent error and the use of the windows is to provided
con~illuously "adjustable s~mrling rate" such as 999, 1000, 1001, 1002,....s~mrl~s per second to
cause the fixed number of data points to contain "integral periods". The cost of added hardware
and the unusual s~mrling rates have deterred any wide use to this method.
This invention resolves the requirement of "integral periods" by adjusting or selecting the
number of data points to get integral multiples of periods. The sampling rates are m~int~ined at
normal increments. The norm~l method of radix-2 FFT is replaced by the mixed radix FFT. The
following block diagram is the flow chart for the FLEXIBLE DATA SIZE FAST FOURIER
TRANSFORM PROCESS for SPARSE FREQUENCY SINUSOIDAL and HARMONIC SIGNAL
ANALYSIS.
The procedures of applying the FLEXIBLE DATA SIZE FAST FOURIER TRANSFORM
PROCESS for SPARSE FREQUENCY SINUSOIDAL and HARMONIC SIGNAL ANALYSIS
is also outlined in the block diagram below.
~LI~XIBLI~ DATASIZE E/AST l~OURIBR TR~NSEIORM PROOESS for SPA~S13 ~REQIJE~NCY SINllSOIDAL and HARMONIC SIGNAL ANALYSIS

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BLOCK DIAGRAM and ~;LOW CHART of the
FIEXIBLE DATA SIZE FA~T FOURIER TRANSFORM PROCESS
for SPARSE FREQUENCY SINUSOIDAL and HARMONIC SIGNAL ANALYSIS.
SPARSE 1 Z
FREQUENCY DIGITAL DIGITAL
SINUSOIDAL DIGITIZER DATA COMPUTER
or HARMONIC SAMPLING SAMPLING ~ SIMULATION
SIGNAL INTERVAL--dt INTERVAL=dt PROGRAM
FROM PHYSICAL
SYSTEM
SELECT
DATA SIZE, N
N
4\1 5 \/
CALCULATE TOTAL
FFT DATA TIME
Nxdt
6~I
DISPLAY
MAGNITUDE Nxdt
SPECTRUM
7 ~ 8 ~ V
r T~ TrT~
~ b'TMn ln
~, FREQUENCY f 1 = T .. , l~
FOR THE EXACT f f T n x T ~ N x dt
FFT
¦, n
FIND NI
WHERE NI~N
NI=(
I NI
13 12 1 1 W
FIND NID
DISPLAY OR MIXED RADIX WHERE NID=NI~5
RESULTS c FFT ~ &NID=Radix 1 x
..... RadixI
EXIT
PL13XIBL13 DATA SIZE~ IIAST ~OUR113R TRANSI~ORM PROOESS ~or SPARS13 ~REOUE~NCY SINUSOIDAL and HARMONIC SIGNAL ANALYSIS

21~6209
The functions for each of the blocks are described as follows.
Block Number Description
This is a collve~-Lion~l analogue to digital converter. It has fixed sampling
rates, ie 1,000 samples per second, 2000 s~mples per second, ...., 50,000
samples per second, .. "dt" is the time between sample and is the sampling
period. A large number of digitally sampled data is stored here.
2 The digital data output from digital computer ~im~ ti~n programs are stored
in the data storage device.
3 This is the first selection of the data size "N". "N" is an integer and usually a
single radix based number such as 1024 for radix-2 or 1000 for radix-10.
4 Simple radix fast Fourier transform (~ l ), for e~r~mple radix-2 for 1024 data
points programs or radix-10 for 1000 data point data. It will perform the FF~
within a relatively short time.
S Calculate the total data time (N x dt) for the "N" number of data points.
6 Display the m~gnit-1-ie spectrum of the FFT result. Screen cursers and
programmed intli~ting devices on screen are used to assist the function of
block number 7.
7 Through manually controlled curser movement on screen or programmed
selection routine, select the desired frequency "f" for the exact FFT
calculation in block number 12.
8 Calculate the period "T" of the selected frequency.
9 Find the largest possible integer multiple "n" of period "T" such that
nxT s Nxdt
Calculate the number of data points "NI" that will COll~ill the integral
number of periods. NI = (n x T) / dt
DA~ASlZI~ FASr ~OURII~R TRANS~ORM PROCESS for SPARSE ~REOIJ~NCY SINUSOIDAL an~ HARI~50NIC SIGNAL ANALYSIS

2116~0g
,~ck Number Description
11 Search for an integer "NID" within the range of NI + 5, such that "NID"
can be factored in to practical and workable radices for the mixed radix FFT
in the next block.
NID = NI + S = Radixl x Radix2 x Radix3 x .... x RadixI
Where Radixl, Radix2, Radix3, ..., RadixI are positive integers.
12 Perform mixed radix FFT on the NID number of data points with the radix
sequence given in block number 11.
13 Display or print out the mixed radix FFT results.
Exit, or return to block number 7 to select a di~erellt frequency,
or return to block number 3 to select a new data size.
This invention is most effectively applied to sparse frequency sinusoidal and hslrmnnic
signals. A modern desk-top digital computer can be programmed to execute the process. It is
distinguished from the other inventions in the following manner:
1. The use of time flnm~in windows, such as the E~nnin~ and the G~ n windows, on the
data is elimin~terl
2. The ma2~ullu~l possible frequency accuracy can be achieved with e~i~ting data set.
3. The maxilllum possible magnitude and phase accuracy can be achieved with e~ tin~ data
set.
4. The use of signal input analogue anti-z~ ing filter can be elimin~te~l or relaxed. Because
the high frequency harmonics do not fold-back onto the same frequency line at the low frequency
end.
XIBLE DATASIZE FAST ~OURIER TRANSFORM PROCESS Eor SPARSE FREOllENCY SINUSOIDAL and HARMONIC SIGNAL ANALYSIS
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