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B–7 Examples of Important Distributions 703
1.0
0.5
10 1
1
2pz
e z2 /2
10 2
Q (z)
10 3
10 4
10 5
10 6
10 7
10 8
0
Figure B–7
Sinusoidal Distribution
1 2 3 4 5 6
The function Q(z) and an overbound,
z
1
12p z e-z2 >2 .
THEOREM. If x = A sin c, where y has the uniform distribution,
f c (c) = L
1
2p ,
then the PDF for the sinusoid is given by
|c| … p
0, elsewhere
0, x 6 -A
1
f x (x) = e
p2A 2 - x , |x| … A
2
(B–66)
(B–67)
0, x 7 A
A proof of this theorem will be given in Sec. B–8.
A sketch of the PDF for a sinusoid is given in Table B–1, along with equations for the
CDF and the variance. Note that the standard deviation, which is equivalent to the RMS value
as discussed in Chapter 6, is s = A> 12. This should not be a surprising result.
The sinusoidal distribution can be used to model observed phenomena. For example,
x might represent an oscillator voltage where c = v 0 t + u 0 . Here the frequency of oscillation
is f 0 , and ω 0 and t are assumed to be deterministic values. u 0 represents the random
startup phase of the oscillator. (When the power of an unsynchronized oscillator is turned on,