Page 484 - Invited Paper Session (IPS) - Volume 2

IPS355 Georg Lindgren
Among Steve Rice’s many deep contributions to communication theory,
David Slepian mentions three as particularly influential. His 1950 paper on
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“Communication in the presence of noise” was the first to evaluate explicit
bounds for the error probability in information systems. “Reflection of
electromagnetic waves from slightly rough surfaces”, from 1951, was
fundamental for the understanding of radar returns from the ocean. It was an
early example of the Fourier model (1) generalized to space-time. The third
example given by Slepian is the 1963 analysis of “Noise in FM receivers”, an
ingenious analysis of a mysterious phenomenon.

4. Rice’s formula and its ramifications
The modern version of Rice’s formula for the expected number of
upcrossings of a given level per time unit (=“rate of upcrossings”) by a
differentiable stationary process reads

+
+
= (#{ ∈ [0,1]; () = , }) = (0) ()(′(0) |(0) = ) (2)

We’ll discuss this formula for a stationary processes with spectral density ()
and covariance function () = ∫ exp()(), and spectral moments
2
2 = ∫ ().
a) The origin of Rice’s formula for the rate of level crossings motivates the
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title of the present paper – “inspired by random noise, inspiring statistics
research”. The root of the formula is in a study 12 from 1939 on the
distribution of the height of local maxima of a random function, stating: “If
suitable conditions are satisfied the probability that the random curve =
() has a maximum in the rectangle ( , + dx ; y , y + dy )[… ] is
0
0
0
0
0
0
0
( , )dx dy where ( , ) = − ∫ −∞ ( , 0, ).” Here ( , 0, ) is
0
0
0
0
0
0
0
0
the joint density of the curve and its first and second derivative, taken at
the point = .
0
In the paper Rice motivates his interest: “Here the distribution of the
maxima of [...] random curves is studied. Although this problem is of some
physical interest I have been unable to find references to any earlier work.
Problems of this nature occur in the investigation of the current reflected
by small random irregularities along telephone transmission lines.”
Integrating out in the maximum formula Rice obtains the rate of
0
local maxima regardless of height, i.e. the rate of zero downcrossings by
the derivative of the process. He also rewrites the formula to give the
average number of downcrossings. With modern notation his formula
reads,

1
−
′
′′
′ (0) ( () | () = 0)ds. (3)
(#{ ∈ [0,1]; () = 0, downcrossing}) = ∫ ′
()
0
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