MY CAREER IN COMMUNICATION SYSTEMS
(THE EARLY YEARS)
SOLOMON W. GOLOMB
(VITERBI PROFESSOR OF
COMMUNICATIONS IN THE VITERBI
SCHOOL OF ENGINEERING)
CSSE
TUESDAY, OCTOBER 24, 2006
AT USC
MY CAREER IN COMMUNICATION SYSTEMS
(THE EARLY YEARS)
In June, 1951, having just celebrated my 19th
birthday and received a B.A. degree in
Mathematics from the Johns Hopkins
University, I started the first of four successive
summers at the Martin Co. (now part of
Lockheed-Martin) at its original location just
east of Baltimore.
1
I worked in the “systems engineering” division,
a term I hadn’t heard before, and whose
meaning I absorbed by osmosis. My first
summer was with the Controls Group, and
thereafter (while I was a grad student at
Harvard in “Pure Mathematics”) with the
Communications Group. In retrospect, I’ve
spent the past 50-plus years applying all the
areas of math that my professors assured me
couldn’t possibly have any applications, to
problems in communication technology.
2
Communications is a great place to learn
“systems thinking”. Every communications
system has a block diagram, usually containing
several subsystems. Sometimes a subsystem
will itself have a complex block diagram; and
the entire comm. system is usually part of a
much bigger system. The aerospace industry,
decades ago, created “pert charts” for
scheduling system development. My own
research focus has been to look for
techno/math hurdles, that if overcome, will allow
a superior system to be configured, and then
solve the tech/math problem.
3
By the summer of 1954, at the Martin Co., I was
developing the mathematical model for binary
linear feedback shift registers, which in a
“mysterious” way could sometimes produce
long “pseudo-random” sequences of 1’s and
0’s. Already in 1954, this was of interest to a
number of organizations for a variety of
applications:
• for secure telemetry to guide missiles (Martin
Co.; JPL)
• for cryptographic “key streams” (NSA)
• for coded radar signals (Lincoln Labs)
4
My work had come to the attention of Dr.
Eberhardt Rechtin at JPL, who actually came to
meet me at my Harvard office in Jan. 1955,
while I was continuing my work for Martin as a
consultant. I spent academic 1955-56 in
Norway (where I finished my Ph.D. thesis for
Harvard), and on my return to the U.S. I
interviewed at several organizations (including
Lincoln and NSA), but I decided to work for Eb
Rechtin in his telecommunications section at
JPL.
5
At JPL we concluded that signals using linear
shift register sequences could easily be broken
by a “sophisticated jammer”; so I turned my
attention to nonlinear sequences. This turned
out to be intimately related to properties of
boolean functions, and I developed the theory
of the cryptanalysis of sequences obtained
either from shift registers with nonlinear
feedback, or by the nonlinear modification of
linear sequences, or by the nonlinear
combination of two or more sequences.
6
My cryptanalytic approach involved multi-dimensional
correlations, and I developed a theory of “correlation
immunity” based on a set of invariants for boolean
functions of n binary variables. In 1957-1959 all this was
of course classified, even though the flow of information
between me and NSA was always one way (from me to
them). However, I presented a talk, “On the
Classification of Boolean Functions”, at a large IEEE
meeting at UCLA in 1959, which was printed in the
IEEE Trans. on Information Theory; and I included this
paper as a chapter in my 1967 book Shift Register
Sequences, in the section on nonlinear sequences! I
just didn’t mention the application, explicitly, that
motivated it.
7
On October 4, 1957, Sputnik I was launched. Less than
four months later, on Jan. 31, 1958, Explorer I was
successfully launched. Wernher von Braun’s Huntsville
group provided the first stage rocket, a modified
Redstone missile. JPL developed stages two, three,
four, and the payload, some of which was assembled in
my lab at JPL. This complicated structure had never
been tested, but it succeeded on the first try, only 88
days after the U.S. Army (which funded JPL and
Huntsville) was told it could proceed with a satellite
project.
8
Later in 1958, NASA was created, and became JPL’s
main source of funding. I turned my attention from
secure missile guidance to satellite and space
communication. There was interest at JPL in creating a
ranging system, for accurate determination of the
distance between a space probe and earth. I had a
number of discussions with Eb Rechtin about this.
9
A typical problem: We could probably develop a system
with precision to 100 nanoseconds. Light travels about
one foot per nanosecond, but over distances of millions
of miles, how accurately could we express the range in
meters, given the imprecision in our knowledge of the
propagation velocity? Also, the precise location of our
large tracking antenna at Goldstone hadn’t been
surveyed to tie it into any standard grid to better than
±100 meters.
10
Here’s how we “solved” these problems. We would
measure distance in “light-seconds”, and put the origin
of our coordinate system at our antenna at Goldstone.
Any future improvements in refining the propagation
velocity of our radio signal could be used to improve our
range measurement in meters. We knew exactly where
our antenna was if we defined it to be at the origin of
our coordinate system — we just didn’t know where
Pasadena was! Future surveying could reduce that
uncertainty.
11
Our ranging system would be based on binary
modulation of an RF-signal, which was already a JPL
signalling approach to missile guidance. The modulation
would be based on a very long, random-looking binary
signal, and we would correlate the incoming signal
against “all” the shifts of the binary-modulated signal,
and the correlation peak would indicate how long the
signal traveled, and hence the range. However, if we
used a periodic binary signal, there would be a bigrange-ambiguity corresponding to multiples of the
period.
12
Here is the solution that Eb and I came up with. We
would have a number of relatively short-period
sequences, of relatively prime period, where each short
sequence would have an impulse-like autocorrelation
function. We would combine these short sequences
using a boolean function which would be “worst
possible” for cryptographic purposes! Instead of
resisting a “correlation attack” to separate the short
“components sequences”, it would make it easy to
extract the component sequences by correlation.
13
Here is an example. Suppose we had 9 binary shortperiod sequences, all generated at the same “chip rate”,
and we combined them, term-by-term, using the
majority-decision boolean function, where maj(x1, x2, x3,
x4, x5, x6, x7, x8, x9) = 1 if at least 5 of the xi’s are 1, and
=0 otherwise. This majority-decision sequence is
positively correlated with each of its input sequences,
and its period is the product of the nine individual
(relatively prime) periods. If the “short” periods average
around 80, the combined period will be ≈1.44 x 1017,
easily enough to avoid a gross-ambiguity problem.
14
We used ranging on several groundbreaking
experiments. In April, 1961, we had a successful radar
(radio detection and ranging) contact with Venus, the
first with another planet. (The U.S. Army had bounced a
radar signal off the moon in 1946.) Several other groups
in the U.S. and abroad reported detecting Venus at the
same 1961 conjunction (or even, in the case of Lincoln
Labs, at a conjunction a few years earlier). The problem
was, Venus wasn’t where any of those other groups
claimed to have detected her!
15
The “astronomical unit” (A.U) is the average value of the
semi-major axis of the earth’s elliptical orbit around the
sun, and is the basic scale factor for the solar system.
The International Astronomical Union (the same folks
who recently decided that Pluto is no longer a planet)
had established an “official value” of the A.U., I guess by
majority vote, based on a collection of indirect
measurements. Our Venus radar measurement gave a
direct answer. The astronomers were wrong by one part
in 103, a huge error. Our measurement refined the A.U.
to better than one part in 106.
16
JPL was only authorized to do “engineering”, not
“science”. What were we doing in the planetary radar
business, and in refining the value of the A.U.? Eb
Rechtin had provided an irrefutable justification. JPL was
soon to launch a space probe (“Mariner 2”) to Venus,
and it was important to know where Venus really was! If
we had relied on the “official” value of the astronomical
unit, we could have missed Venus by 100,000 miles!
17
The JPL ranging system had another major scientific
achievement — the most accurate test, by far, up to that
time, of the predictions of general relativity. I explained
how this would work in a talk I gave in 1960, in
Washington, D.C., at a meeting of the International
Union of Radio Science (U.R.S.I.), titled “The Role of
Ranging in Space Exploration”. Here is how it works:
18
RF-Signal
Sun
E
SPACE
PROBE
When a space probe is on the other side of the Sun from
Earth, Einstein predicts that the RF-signal is bent by the
Sun’s gravitational field. Ordinary matter would
accelerate toward the Sun (as comets do), but since
photons are already travelling at the speed of light, when
they gain energy from a gravitational field, they can’t
speed up. Instead, their frequency increases.
19
The JPL ranging system is coherent, that is, it actually
counts RF-cycles. The two relativistic effects — bending
the propagation path, and increasing the frequency (I.e.,
shortening the wavelength) both increase the number of
RF-cycles, compared to a non-relativistic model. (An
accurate calculation involves both special- and generalrelativity effects.) When Mariner 9 was heading toward
Mars, around 1969, this experiment was conducted as I
had proposed it, and Einstein was verified to an
accuracy of about 99%.
20
In a paper I published in 1966 (some three years after I
joined USC full-time) titled “run-length encoding”, I
described a simple type of lossless data compression,
easy to implement and well-suited to situations where
you don’t know in advance the statistics of what you will
encounter. I was very pleased, a few years ago, to learn
that these “Golomb codes” were being used to send
back pictures from Mars on the two Mars Rovers.
21
One final note, about my study on nonlinear sequences.
Some 30 years after my work at JPL on boolean
functions, a paper was published by James L. Massey
(an American then living in Zurich) and Guo-Zhen Xiao
(of Xidi’an University in Xi’an, China) in the IEEE Trans.
on Information Theory, in which they “rediscovered” my
cryptanalysis by correlation of nonlinear sequences.
22
I know both Xiao and Massey quite well. Both have
copies of my Shift Register Sequences book. I am sure
they did not consciously plagiarize, but the similarities
are striking. When I enquired about getting my old JPL
reports declassified (so I could prove that my original
application was the same as theirs), I ran into a catch22. Those reports could only be declassified by the
agency that classified them originally, and that agency
no longer exists!
23