For tracking tasks where a sum of sine waves forcing function is used it is often difficult and/or expensive to obtain the pilot's remnant in the vicinity of the sine waves. For the case where each sine wave has at least four times an integer number of cycle per run length, this paper illustrates the serial segments method for measuring remnant power spectral density in a frequency band centered on each sine wave. This method can be implemented on digital, hybrid, or analog Fourier coefficient analyzers, and is particularly advantageous on the latter since properties of Fourier coefficients are exploited to yield both a remnant measure and an improved estimate of the correlated component

Magdaleno, R. E.

English

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3 2. 
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Serial Segment Method for Measuring Remnant 
R. E. MAGDALENO 
Systems Technology, Inc. 
For tracking tasks where a sum of sine waves forcing function is used it is often difficult 
and/or expensive to obtain the pilot's remnant in the vicinity of the sine waves. For the case 
where each sine wave has at  least four times an integer number of cycle per run length, this 
paper illustrates the Serial Segments method for measuring remnnn.? p o ~ c r  spciti ai density 
in a frqilency b a d  icikiareii on each sine wave. This method can be implemented on digital, 
hybrid, or analog Fourier coefficient analyzers, and is particularly advantageous on the latter 
since properties of Fourier coefficients are exploited to yield both a remnant measure and an 
improved estimate of the correlated component. 
INTRODUCTION 
There are many techniques available (see espe- 
cially ref. 1) for measuring the pilot's describing 
function Y ,  and remnant nc in the compensatory 
task of figure 1. Generally the remnant is more 
predictable when referred to  the input t o  the 
pilot, ne (refs. 2 and 3). However, if a digital or 
hybrid computer is not available then remnant 
measurements with analog equipment (Le., ana- 
log power spectral analyzers) may be expensive 
and/or lack the accuracy needed to  separate 
signal from noise. For the case where the input is 
a sum of sine waves, reference 4 describes input- 
referenced-Fourier coefficient measurements as 
mechanized on "standard" analog computer 
equipment as well as on a describing function 
analyzer (DFA) where the latter uses synchron- 
ous motors to  drive sine-cosine pots and multi- 
plying pots. The inherent timing precision in the 
describing function analyzer approach is such 
that suitable processing of the Fourier coeffi- 
cients (as they evolve in time) can yield an 
estimate of both the remnant and the describing 
function a t  the same frequency, thus enabling 
the remnant n, to be reinterpreted as occurring 
at  nr by +nn,=+nn,/lYp12. 
To illustrate the methods to be used, assume 
that we have a signal which is sine wave plus 
noise 
1 :e-- - - - - - 
I I  "; I 7 YP + I 
FIQURE 1.-Compensatory situation. 
where 
A =peak amplitude 
cp = phase (arbitrary) 
w1= frequency in rad/sec 
n(t) =noise component. 
The relationship between variance and power 
spectral density (PSD) used in this paper is 
uz2 = /)&(f) df (2) 
where +zz(f) =PSD of 2; units of power/Hz. 
The PSD for z(t) is 
(3) 
A2 
+ z z ( f )  =,s(f-fi)++nn(f) 
For a finite run length TR the signal zt can be 
represented by a Fourier series form 
https://ntrs.nasa.gov/search.jsp?R=19730001409 2020-03-17T07:18:54+00:00Z
~ ~~~~ ~~~ 
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266 SEVENTH CONFERENCE ON MANUAL CONTROL 
n-1  7l-1 
where the coefficients are evaluated from 
Z = mean value of z 
a,=-  ITRz( t )  cos w,t dt 
T R  0 
n ~ T R  
W n = ' n w o  
2 r  
TR 
wg=- 
Where the fundament>al frequency is one cycle 
per run length, the power in the frequency 
estimate a t  w, is 
(5 )  
For cases of interest in this paper the sine wave 
in equation ( 1 )  is assumed to  be a harmonic of 
the fundamental frequency, ~ / T R .  When dealing 
with data taken over a finite run length TR, a 
more useful definition of PSD is to  assume that  
the power in the spike is distributed over the 
smallest resolvable bandwidth, I /TR(Hz) .  This 
bandwidth follows from expanding z(t) in Fourier 
coefficients which are orthogonal a t  that  fre- 
quency spacing. Thus the PSD of z(t) is 
This is sketched in figure 2 in terms of each 
fundamental frequency band with the sine wave 
power centered on f =Ji and assumed distributed 
over 1/TR (Ha). The shaded area is the total 
power in the sine wave. The remnant sketched 
in indicates that it fluctuates about on either 
side of the sine wave portion as well as having a 
component centered there. The problem is to 
estimate both the sine wave component and a 
measure of the average remnant height. 
METHOD 
The  DFA signal processing method can be 
modified so as to  obtain remnant as well as 
FIGURE 2.-Signal and noise power spectral densities. 
describing function estimates for a 100 sec run 
length. The key point is to  select forcing function 
frequencies that are orthogonal over 25 sec so 
that independent estimates of Fourier coefficients 
can be obtained at this run length as well as for 
100 sec. Noting that each Fourier coefficient can 
be thought of as containing the power in a band- 
width of (1/TR) Hz where T E  is run length in 
seconds, then Fourier coefficients obtained over 
25 sec have an effective bandwidth four times 
that  of Fourier coefficients obtained over a 
100 sec run length. Since the power correlated 
with the forcing function is common to coeffi- 
cients obtained over 25 sec as well as those from 
100 sec run lengths, then the difference in these 
powers is that due to  remnant. 
It will be convenient to deal with twice the 
power in a frequency band rather than the PSD, 
2Pn=2[@zz(f)I Af (7) 
where Af = 1/TR. 
Inserting equations ( 5 )  and (6) into equation (7) 
yields (for a measurement frequency that  falls 
on a sine wave) 
Equation (8) relates the measured Fourier coeffi- 
cients a,, b,  to  the sine wave and remnant 
descriptors A , a,. Two independent evaluations 
of equation (8) will allow a solution to  be found. 
Equation (8) evaluated over a 100 sec run 
length a t  a frequency corresponding to  a forcing 
function frequency yields 
2@nn M = A'+- 
100 
where M = an2+bn2. 
(9) 
SERIAL SEGMENTS METHOD FOR MHASURING REMNANT 267 
Evaluating equation (8) over each 25 sec segment 
and then averaging the power in these four 
segments together yields 
29nn M ,  = A’+- 
25 
where 
4 
i = l  
a ,  is the cosine component of the ith segment 
b, is the sine component of the ith segment 
This method of calculating spectra from suc- 
cessive segments was suggested by Bartlett (ref. 
5). The short segment coe5cients are related to 
the full run length coefficients by 
-- 
an, an, 
+ $/-;+;/,:“I (11) -- 
an, an, 
4 
an=aCan ,  
i = l  
and similarly for b,. 
The right hand side of equation (10) shows 
that the average power in the four segments con- 
tains A 2  since the sine wave is identical in each 
of the four segments. The amount of measured 
remnant power is larger than in equation (9) 
since the effective bandwidth for the short run 
length segments is four times larger. However, 
the amount of remnant power in the 100 sec 
estimate is not exactly one-fourth of the remnant 
power in the averaged 25 sec segments. As shown 
in figure 2 the remnant component a t  f = fi (or 
any other component) may be higher or lower 
than the average height. Thus in equation (9) an 
additional factor E should be added on to reflect 
the difference from the average remnant, ann. 
Equations (9) and (10) can now be solved for 
A 2  and ann. However, for generality we are 
interested in the case where there are Q short 
segments in the run length TR. Thus the general- 
ized versions of equations (9) and (10) are 
respectively 
2*nn M = A’+-+€ 
2ann M,  = AZ+- 
TR/Q 
TR 
where E is the deviation of the remnant power 
from the average level. Since e is a random vari- 
able and likely to either be positive or negative 
for any one run it is neglected in the solution of 
equations (12) and (13) for average values of A 2  
and ann. Thus carrying E through the solution 
so as to see its effects yields, 
QM-M, Q E  A2 = -- 
Q-1 Q- 1 
where 
Q =number of short run lengths in the basic 
run length, TR 
Q 
i = l  
i = l  i = l  
I n  addition the sine wave input frequency f n  
must have an integer number of cycles in the 
short run length, TR/Q. Note that the remnant 
estimate contains E divided by Q-1, thus the 
remnant estimate is not sensitive to the scatter. 
The estimate for A2 is contaminated by the E 
term, yet A 2  is usually much larger than the 
remnant (or its scatter as given by E )  so that it is 
not sensitive either. 
RESULTS AND INTERPRETATION 
The serial segments method yields a remnant 
estimate (as well as describing function estimate) 
a t  each of the forcing function frequencies. This 
method of computing remnant was evaluated by 
simulating the calculations in equation (14), 
using BOMM (ref. 6) .  Using actual human oper- 
ator data we computed the Fourier coeEcientg 
SEVENTH CONFERENCE ON 
;L 
~~ ~ 
MANUAL CONTROL 
ot  Forcing Function Frequencies (0: t b;) 
Correlated Componet Plus Remnant 1 gi:ients 7Tn
I Remnont (2  Degrees of Freedom) 
3ll Smoothed Remnant (8 Degrees of Freedom) 
@,en= [~-p: (~) ]@e.  
Wie12 
p : =  G K  
Tukey Melhod.Porzen Window. Max Log of 25seC 
Frequency * & (Hz )  
Remnant Eslimote Using Serial Segments 
Tn * IOOrec 
. 
- .  ................................................................... ,'=."= 0- h n  -n e. C. c ..*.?* .I,. ._e r.r *n acc 0 ..- 4 - - - -- 4- m.n n n nn n nn n- n en - 9.7 9-n - ..NE. m e L C  0 5 "W.7 - 0  c c c  0 5  "n n- 115 c c D 0 -r.n * .. - - - . * - ** .n ". n * n .n * * a a 0 
n = Number of Cyclerl lODrec 
FIQURE X-Remnant computed from serial segments, Y, = l/(s -1). 
........... 
Remnant ( 2  Degrees of Freedom) 
Smoothed Remnant (8 Degrees of Freedom) 
@ean = [ l-pg(w)]@e* 
I@ie12 
p: = G i G  
Tukey Method, Porzen Window, Max Log Of ZSreC 
Remnant Estimote Using Serial Segmenls 
FIQURE 4.-Remnant computed from serial segments, Y ,  =2/s(s -2). 
SERIAL SEGMENTS METHOD 
of the error for TR = 100 sec and for each 25 sec 
segment. These were processed according to 
equation (14) and compared with the remnant 
found by the Classical Tukey Method (ref. 7). 
For the latter a Parzen data window was used 
with the number of lag values selected so that 
the effective bandwidth was close to that which 
results for the Serial Segments method when 
done four times per run length. 
Example calculations are shown in figures 3 
and 4, where the error power spectral density 
estimates are plotted versus frequency in units of 
cycles per 100 see run length. The Fourier com- 
ponents a t  forcing function frequencies are cir- 
cled. The pertinent computations using the Serial 
Segments method aie enclosed with a box. 
Generally these are in good agreement with the 
remnant power spectral density computed via 
the Tukey Method, which is included in figures 3 
and 4 to give an indication of remnant both at 
and between the forcing function sine waves. 
For the Tukey Method (ref. 7), the maximum 
number of lags was selected to mimic the 8 de- 
grees of freedom for the Serial Segments (the 
scatter for 8 degrees of freedom is such that 
60 percent of the values should fall within a 
4 dB range approximately centered on the true 
remnant PSD). Note that both the Tukey and 
Serial Segments methods give smoothed esti- 
mates of the remnant, and they are in good 
agreement with each other. The small differences 
are partly due to the scatter term E in equation 
(14). 
The Serial Segments method was mechanized 
using the DFA by sampling the Fourier coeffi- 
cient integrators a t  the 25, 50, 75, and 100 sec 
times and computing the serial segment values 
per equation (11). Typical error spectra results 
for a pilot (taken from ref. 8) are shown in figure 
5. The measured remnant values are generally 
well above the worst case system noise (measured 
with an analog pilot). Even a t  the highest fre- 
quency, the measured remnant is above the 
actual value of the system noise (about 3 dB 
lower than the worst case system noise). The 
normalized injected remnant is consistent with 
the reference 2 and 3 findings. 
Thus the Serial Segments method for measur- 
ing remnant is shown to yield excellent remnant 
estimates over a large dynamic range. 
FOR MEASURING REMNANT 269 
F,e;oW'"E" <r.d,llrl 
FIGURE 5.-Typical serial segments remnant data. 
REFERENCES 
1. SHIRLEY, RICHARD S.: A Comparison of Techniques 
for Measuring Human Operator Frequency Response. 
Proc. of the Sixth Annual Conference on Manual 
Control, Apr. 7-9, 1970, p. 803. 
2. LEVISON, W. H.; BARON, S.; AND KLEINMAN, D. L.: 
A Model for Human Controller Remnant. IEEE 
Trans. on Man-Machine Systems, vol. MMS-IO, no. 
4, pt. I, Dee. 1969, p. 101. 
3. JEX, H. R.; AND MAGDALENO, R. E.: Corroborative 
Data on Normalization of Human Operator Remnant. 
IEEE Trans. on Man-Machine Systems, vol. MMS- 
10, no. 4, pt. I, Dee. 1969, p. 137. 
4. ALLEN, R. WADE; AND JEX, HENRY, R.: A Simple 
Fourier Analysis Technique for Measuring the Dy- 
namic Response of Manual Control Systems. Proc. of 
the Sixth Annual Conference on Manual Control, APT. 
5. BARTLETT, M. S.: Smoothing Periodograms from 
Time-series with Continuous Spectra. Nature, vol. 
161, 1948, p. 686. 
6. BULLARD, E. C.; ET AL.: A User's Guide to BOMM, A 
System of Programs for the Analysis of Time Series. 
Published by the Institute of Geophysics and Plane- 
tary Physics, Univ. of Calif. (San Diego), Jan. 1966. 
7. BLACKMAN, R. B.; AND TUKEY, J. W.: The Measure- 
ment of Power Spectra. Dover Publications, Inc., 
1958. 
8. JEX, HENRY R.; ALLEN, R. WADE; AND MAGDALENO, 
RAYMOND E.: Display Format Effects on Precision 
Tracking Performance, Describing Functions, and 
Remnant. TR-191-1, Systems Technology, Inc., 
Mar. 1971. 
7-9, 1970, p. 785. 


Serial segment method for measuring remnant

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