APPENDIX
E
TABLE OF RANDOM-INPUT
DESCRIBING FUNCTIONS (RIDFs)
In this table we employ the probability function (cf. Sec. 7.2) denoted by
and its integral, the probability integral, denoted by
1
1"
exp
PZ(X) = =
4 2 -a
~ (- g)
dv
These functions are plotted in Fig. E.2-1.
This table is given in three sections:
E. 1 Gaussian-input RIDFs
E.2 Gaussian-plus-bias-input RIDFs
E.3 Gaussian-plus-bias-plus-sinusoid-inputRIDFs
E.1
GAUSSIAN-INPUT RlDFs
x(t)
= r(t)
an unbiased Gaussian process
TABLE O F R A N D O M - I N P U T DESCRIBING F U N C T I O N S (RIDFs) (Continued)
--
Nonlinearity
16. Linear gain
Comments
TABLE O F RAN DOM-INPUT DESCRIBING F U N C T I O N S (RIDFs) (Continued)
-
Nonlinearity
-
Comments
See Sec. 7.2
n=2,4,6,
...
See Sec. 7.2
y
-
=
4 x
(x 2 0)
=
-4-x
(x < 0)
26. Odd square root
y
=~
See Fig. E.l-3
1 1 %
27. Cube root characteristic
r ( x ) is gamma function
574
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.1-2
RIDFs for litniter and tlrr.esholrlcharacteristics.
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.1-3 RZDFfor the simple polynonlial nonlinearityy
o r y = c,xn-' 1x1 (n even).
= c,xn
(n odd)
575 R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.1-4 Harmonic nonlinearity RIDF.
E.2 GAUSSIAN-PLUS-BIAS-INPUT RlDFs
x(t) = r(t)
+B
The gain to the gaussian input component is given by:
and the corresponding gain to the bias input component is:
1
N,(a,B)
=
jm
-
~ Z O By@+
B)exp
This section uses the additional function G ( x ) = xPI(x)
(- $)
+ PF(x).
The functions PF(x), PI(x), and G(x) are plotted in Fig. E.2-1.
dr
TABLE O F RANDOM-INPUT DESCRIBING F U N C T I O N S (RIDFs) (Continued)
Nonlinearity
7. Sharp saturation or limiter
8. Dead zone or threshold
9. Gain-changing nonlinearity
Comments
TABLE OF R A N D O M - I N P U T DESCRIBING F U N C T I O N S (RIDFs) (Continued)
Nonlinearity
y=x
16. Linear gain
Comments
TABLE O F R A N D O M - I N P U T DESCRIBING F U N C T I O N S (RIDFs) (Continued)
Nonlinearity
56. Biased ideal relay
Comments
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.2-I
Graphs of PF(x), PZ(x), and G(x).
587 E.3 GAUSSIAN-PLUS-BIAS-PLUS-SIN U S O I D - I N P U T RlDFs
x(t) = r(t)
+ B + A sin ( o t + 8)
The gain to the gaussian input component is given by
the gain to the bias input component is
dry(r
+ B + A sin 0 ) exp (-
and the corresponding gain to the sinusoid input component is
2)
TABLE O F RANDOM-INPUT DESCRIBING F U N C T I O N S (RIDFs) (Continued)
Nonlinearity
Comments
y = xS
18. Cubic characteristic
y
= Msinmx
J, and J, are the Bessel
functions of orders 0 and
1 , respectively.
NR = Mm cos mB exp
M
NB = - sin mB exp
B
?)J,,(mA)
(- (-
2M
Nd = A con mB exp
29. Harmonic nonlinearity
(-
my)~o(m~)
mSuz
i)l,(mA)
590
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-I
Three-input RZDFs for the ideal-relay nonlinearity.
In 3 parts
Figure E.3-Ia
Gain to the gaussian input component. (ideal relay)
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
591
1.o
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.09
0.08
I
Figure E.3-lb
2
3
4
5
6
7
4
0
Gain to the bias input component. (ideal relay)
1.o
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
I I I I I
0
1
2
3
4
5
6
7
A
u Figure E.3-lc
Gain to the sinusoid input cotnponent.
(ideal relay)
592
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-2
Three-input RIDFs for the limiter nonlinearity.
In 1 1 parts
Figure E.3-2a
Gain to the gaussian and bias input components. (limiter, IB1/6
= 0)
0
Figure E.3-26
Figure E.3-2c
1
2
3
4
5
6 Gain to the sinusoid input component. (limiter, IB)/6 = 0)
Gain to thegaussian input component. (limiter, IBI/S
=
0.5)
6
Figure E.3-2d Gain to the bias input component. (limiter, IBI/S
= 0.5)
6
596
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-2h
Gain to the sinusoid input component. (limiter, 1B1/6
=
I)
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-2i Gain to theguussiun input component. (limiter, IBI/S
=
2)
597 6
598
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-2j
Gain to the bias input component. (limiter, lB1/6 = 2 )
6
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-2k
Gain to the siriusoid input conlponent. (limiter, 1Bj/6 = 2 )
599 600
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3
Three-input RZDFs for the relay with dead zone nonlinearity.
In 1 1 parts
6
0
NB
1
0.8 0.6 0.5 0.4 0.3 0.2 0.1 0.08 0.06 0.05 0.04 0.03 0.02 0.01 0.008 0.006 0.005 0.004 0.003 0.002 o.OO11
0
1
1
2
3
4
5
6
7
4
6
Figure E.3-3a
I W = 0)
Gain to the gaussian and bias input compotlents.
(relay with dead zofie,
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-36
Gain to the sinusoid input component.
601 (relay with dead zone, [BI/S = 0 )
601
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
6
Figure E.3-3c Gain to the gaussian input component. (relay with dead zone, IB)/6 = 0.5)
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3d
Gain to the bias input component. (relay with dead zone, IB1/6 = 0.5)
603 604
RANDOM-INPUT DESCRIBING FUNCTIONS
Figure E.3-3e
Gain to the sinusoid input component. (relay with dead zone, ( B ( / 6= 0.5)
605
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
10
9
8
7
6
5
4
3
2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.09
0.08
0.07
0
Figure E.3-3f
I
2
3
4
Gain to thegaussian input component.
5
6
(relay with dead zone, lBl/6
'
=
A
6
I)
606
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3g
Gain fo the bias input component. (relay with dead zone, IB1/6 = I )
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3h Gain to the sinusoid input component. (relay with dead zone, IB1/6
607 =
I)
608
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3i
Gaitr to the gaussian input component. (relay with dead zone, 1B1/6 = 2 )
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3j Gain to the bias input component. (relay with dead zone, IB1/6
=2)
609 610
R A N D O M - I N P U T DESCRIBING F U N C T I O N S
Figure E.3-3k
Gain to the sinusoid input component. (relay with dead zone, IB1/6
= 2)