1
THE AUTOCORRELATION FUNCTION
Key to Woodward’s Resolution Theory
an evening with an fundamental concept
in signal processing that leads to a most
important property of nearly all sensors
R. T. Hill
IEEE Signal Processing Society
an IEEE Lecturer
Victorian Chapter, Australia
25 May 2009
2
Grandpa . . do you know about
A recent breakthrough in
inter

generational communication!!
This evening, then, we’ll deal with the importance of
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The signals we use and their processing . .
What qualities are important to us . .
. . in communication ?
. . in radar (and other sensors) ?
. . common to both !
Our emphasis . . RESOLUTION
A few general remarks:
. . sensitivity, accuracy, resolution, definition, registration . .
4
On the resolving power of a signal . .
Let’s consider “range resolution” in radar . .
“Everyone knows”
Δ
R =
c
τ
/ 2,
the shorter the pulse, the better
the resolution!
HOWEVER . . Woodward pointed out that it’s
not
the temporal
shortness itself of the pulse but rather the
bandwidth
of the pulse that gives us good range resolution:
Δ
R = c
τ
eff
/ 2 = c / 2B
i
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. . and in Doppler?
Doppler shift of our signal is indeed another
dimension in which we may well want to
resolve
radar echoes. In communication,
with moving participants, Doppler effects
may have other consequences.
Well, “everyone knows” that resolution in
frequency is limited by the
length of time
over which
the signals being received are sampled . . that is,
Δ
f
d
= 1 / T
cpi
Range (time) and range

rate (frequency) resolution
potentials
. . signal bandwidth and signal duration . .
but how might we
quantify
resolution?
Let’s turn again to Woodward
6
. . what a wonderful
book . . and
thin
, too.
(I admire greatly, in
nearly all fields, books
that are both profound
and thin.)
Pergamon Press Ltd.
London, 1953
7
Woodward’s assertion about
resolution
:
If we wish to distinguish at the receiver slight
time shifts (that is, to have fine range resolution),
then
“the signal waveform must have the
property of being as different from
its shifted self as possible.”
Simple and profound . . and our theme for this
evening’s lecture . .
the
difference
of functions
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How “different” are two functions?
Ah . . let’s measure that by
summing their difference
Sounds like a good idea, very
straightforward . .
. . hmmm . . on second
thought, might be better
to
square
that difference
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A pedantic view
of dealing with
the
square
of the
difference of two
functions.
Now, let’s expand that binomial square
and examine the
middle term
. .
Would you say
that
these
two
functions have
no difference??
Better
Summing just
the difference
10
Discussion . . how “telling”, that middle term:
The functions differ
greatly
? A small middle term.
They differ only
slightly
? A large middle term.
Functions
identical
? Maximum middle term (2f
2
in the integrand).
Functions totally “uncorrelated”? Middle term = 0.
In our brevity, we’ll just mention how this
“correlation integral”
describes both deterministic signals and random signals, and with
appropriate normalization, becomes the “correlation coefficient”
describing random variables.
So valuable, yet just the
middle term of a binomial squared.
Back to
and specifically the
significance of
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Since we speak often of the
correlation
function,
we might
ask, “Of what variable might
correlation be
a function
?”
Staying for the moment in the
time
domain, we go back to
Woodward’s assertion about resolution being a matter of how
much a signal differs from its own
time

shifted self.
The context here is, obviously, “range resolution” in radar

we
would expect
no
difference in two “echoes” (of the same signal)
if there were
no
time separation t’ between them, but we want
maximum difference for all t’
≠ 0. Of interest, then, is the signal’s
temporal
auto
correlation function, a
function of that separation t’
.
If we were interested in not being bothered or confused by the
presence of signals other than our own, we would be interested
in our signal’s temporal
cross
correlation with each, showing great
differences (minimum correlation) for all t’
–
an “orthogonality”.
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The temporal autocorrelation function . .
Of course, in radio work
there is an oscillatory
correlation associated
with each cycle of our
carrier frequency which
is not of interest to us
–
the conjugating of our signal in the convolution of it with its time

separated
replica eliminates the carrier

frequency operator e^j2
π
f
c
t.
A familiar exercise in elementary
calculus classes the world over
is to convolve a “square pulse”
with itself . . we’re reminded, then,
that such a convolution produces a
non

zero function of width
twice
that
of the original time

bounded function.
Recall, it is the
shape
of c(t’) that interests
us in our context. The shape here does
not connote much
resolution
.
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Woodward guides us to considering
resolution
in
both
dimensions of interest in
much
radar signal
processing
–
range and range

rate.
Here, only a few comments will remind us of these essential concepts

radial motion of scatterers (of interest or not)
produces a Doppler shift in the returned signals

a “coherent” radar is one designed to sense
and process (filter) such signals to an advantage

such Doppler processing is absolutely
essential
in airborne radar, to separate moving targets from
non

moving Earth surface “clutter”

such processing is the very key to
imaging
done
routinely these days in Synthetic Aperture Radar in
both airborne and orbiting space

based radars, Doppler
resolution leading directly to fine “cross

range” resolution

fine Doppler resolution contributes to much work in
“target recognition”
–
target Doppler “signature”
–
as well.
Clearly we need waveforms of good resolution in both range and range

rate!
14
The two

dimensional autocorrelation function of Woodward
Using the
spectrum
of our
signal, we would have
developed
κ
(“kappa”), the
frequency autocorrelation
function,
κ
(f’), much as we
did c(t’).
Then, Woodward presents the two dimensional
autocorrelation function (above), which we note
while making only passing reference to this
equivalency depending upon the
convolution
integral theorem
–
roughly, that the convolution
of two functions equals the integral of the product
of their Fourier Transforms.
On the left, we see Woodward’s diagram of
X
(t’,f’)
for a Gaussian

weighted series of Gaussian

shaped
pulses . . we’ll take a quick look, to established that
the evident failure to resolve constitutes
ambiguity
in our inference (the
measurements
we’re making).
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So, wideband modulation on our carrier
gives us fine
range
resolution, and a long
“coherent processing interval” gives us
fine
Doppler
resolution . .
. . how does a receiver actually effect the resolution?
To answer this question, let’s first revert to just one
dimension . . range . . and discuss methods of “pulse
compression” (to achieve fine range resolution) so widely
used in radar. Our treatment then of Doppler processing
will be
very
brief . . necessary, but brief!
To proceed, we need a block diagram
of a radar receiver so we can discuss
use of the “matched filter” for maximum
sensitivity and the achievement of
high resolution at the same time.
Block diagrams can be
very
complicated.
16
Generalized Receiver Block Diagram
. . well, our time is probably
running short, so this will suffice . .
17
Signal processing discussion . .
a) reception as a convolution . . specifically
a convolution of the signal being
received (our “echo” and noise)
with the “impulse response function”
of the receiver
b) the “matched filter” principle . . for maximum
sensitivity
to our signal . . says that the
IRF should be the complex conjugate of
our signal
c) with matched filter operation, then, the output
contains the ACF, affording good resolution
if a decent S/N be available
18
Pulse compression methods . .
First, we need to create the clever pulse
in our waveform generator and send it
over to the transmitter to be amplified and
transmitted, keeping a replica of it for our
reception (our Matched Filter).
Frequency modulation and
phase modulation (of radar’s
typically constant

amplitude
pulses) are the methods . .
here we see a “linear” frequency
modulation (an “up chirp” here)
and here a “binary” phase code
(a seven bit “Barker” code here)
19
first bit out
Binary phase coding
with a “tapped delay line”
Showing a 180
o
phase shift on one tap,
giving the binary code sequence “ +

+ + “ , one of the Barker codes.
On receive (after down conversion), the signal is sent through
its Matched Filter . . in this case, the same circuit with the taps reversed:
+ +

+
Clearly, pulse compression is a “convolution” process, and we see the “time” or “range” sidelobes
in the output which, for all the Barker binary codes, are never more than unity value, while the narrow
main peak is full value, the number of bits in the code. In this matched situation, the output is the
“autocorrelation function”, and low sidelobes is a very desirable attribute of a candidate code.
20
Recall, phase angles
add
when complex numbers (vectors) are
multiplied
–
that is, the
signal is “rotated back” in phase by the amount it might have been progressing in phase.
To the extent that such a component was
in
the input signal will we get an output in this
particular filter. We’ve built a Matched Filter for
that
frequency component alone.
Important
: Here, for simplicity, we show the “starting phase” (signal and reference) as zero.
Our use of I and Q video (from the “vector detector” previously described) and our representing
of the reference signal by its quadrature components, the
frequency
filtering process here is
made entirely independent of the entry or “starting” phase of the sampling.
Doppler filtering
–
“Matching” or “testing” for each frequency
View a single Doppler “filter” as a classic “Matched Filter”
–
that is, we
multiply
the samples of the input signal
with the conjugate of the signal being sought
.
sample #1 2 3 4
signal
x
reference
=
product
Adv2d09
21
Doppler filtering
Implementation
. . Performing a spectral analysis of the signal received
by a coherent radar using N pulses in a coherent dwell
sample and hold, and
A

D conversion
DFT
f
d1
the filtered
outputs
f
dN
I
Q
Discrete Fourier Transform: basically, the signal is divided into N parallel channels; in
each, a phase rotation from sample to sample
counter
to the spectral component
being tested for is introduced and the products accumulated. If the signal had N
pulses (samples) in the coherent processing interval (the coherent dwell), the
reference rotation in the “first” filter channel uses steps of 2
/ N, the second uses
steps of 4
/ N, and so on, the Nth channel using 2N
/ N steps, or no rotation at all,
testing for zero Doppler (or its ambiguities, integral number of cycles per sample period . .
Dopplers equaling multiples of the radar’s prf).
If N be a binary integer, the DFT can be performed by the convenient FFT algorithm.
22
All right, a pulse compression receiver
works as a “convolving” matched filter
(and Doppler filters are just matched
filters, too) . . so . .
. . what makes a modulating function a “good” one?
We’ll look at some phase code research
and then some interesting
hybrid
modulation
to shape cleverly the two

dimensional
auto

correlation function.
And THAT will bring us to the
end of this evening’s talk!!
I promise.
23
Our signal being
convolved with a
matched filter . .
as in a radar with
pulse compression
a)
b)
c)
a) ordinary square pulse,
no pulse compression . .
modulation bandwidth roughly
reciprocal pulse width,
not
particularly broadband
, not
particularly resolute.
b) linear fm or phase code
modulated . . same energy,
and we’re matched to the
modulation . . pretty good
code selection: low peak
(and low integrated) sidelobes
c) same
bandwidth
as b), but
not such a great code
Notice, sensitivity is the same . . it’s
resolution
we achieve by using (and
matching to) broadband modulated signals.
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1990
–
Dr. Marv Cohen (Georgia Tech. Research Institute)
presentation at the IEEE International Radar Conference
“RADAR

90” presented this table about binary sequences
that have minimum peak sidelobe levels, with an example
code in each case.

Do you see the “Barker codes” there?
(recall, maximum length Barker code
in just 13 bits . . )

Note the strange sequence of values of
numbers
of minimum PSL codes . .
. . curious

What are
other
attributes by which we
might measure “optimality”?
Well . .
integrated sidelobe level (ISL)
is one; behavior with Doppler
shifts during (long) code reception
is another . . Cohen illustrates
. . from Cohen, “Minimum Peak Sidelobe Pulse Compression Codes”
in IEEE 1990 International Radar Conference “Radar

90”
About “optimum” codes . .
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. . further from Dr. Cohen’s
Radar

90 code

search paper
Discussion
►
integrated sidelobe level

in a
sense, the least
number
of
sidelobes at that minimum
(among all the codes of that
length) peak level is a further
measure of merit
►
both the loss of ”processing gain”
and any increase in the sidelobe
level resulting from Doppler
phase progression during the
pulse are of interest . .
►
the search
has
continued today
beyond just 48 bits (
many
systems use
much
longer codes,
routinely . . without knowing
whether they are ”optimum”!
►
the search “strategy” is itself an
interesting part of Dr. Cohen’s
important paper!
26
Hybrid Modulation . .
shaping the ambiguity function . .
. . and matching to the entire coherent dwell
The alert student recognizes the application of the conjugate Matched Filter in
normal Doppler filtering, and he recognizes as well that the pulses of the coherent
dwell may well be modulated for pulse compression. Often we think of such processes
as two . . yet these researchers remind us that we have
one
signal that lasts for the
entire cpi
–
the modulation on each pulse need not be repeated! We have
one long
signal
to which we can match.
Levanon and Mozeson, in
IEEE AES Transactions, April 2003
27
. . from the Levanon and Mozeson 2003 paper: we see the modulation schemes for the eight

pulse dwell on
the left, for both the binary (top) and the polyphase coding (bottom), shown superimposed upon each of the
already linear frequency modulated pulses, as at the top, on the right.
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from the 2003 Levanon and Mozeson, op cit.
Here we see the
shaping of the ambiguity function
made
possible by this hybrid coding technique, for the case of the
binary coding. Again, each pulse of the eight is
uniquely
coded, all are fm up

chirped identically.
The single cell (of the ambiguity diagram) at the left top is
for comparison, the fm pulses having NO phase coding.
Notice the high “range sidelobes” (in black) along the zero

Doppler axis.
Below is shown the same cell (left) with the zero

Doppler axis more clearly shown (right) for the case
of the binary phase code set . . note the sidelobe

free
region on the range axis at zero Doppler . .
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. . and for the polyphase coding (again, unique to each pulse) we see this shaping and the regions
of virtually no range sidelobes (the zero

Doppler axis) more clearly on the right.
In the coherent processing (Doppler filtering) of the eight

pulse dwell, this experience is achieved in
each of the Doppler filters: no range sidelobes along
that filter’s
central Doppler axis.
One sees (obvious) the change of response to signals
elsewhere
in the ambiguity cell illustrated
compared to what it was in the reference LFM

only case. This hybrid technique with “orthogonal”
code sets employed permits a freedom in choosing where in the range

velocity space one wants
to suppress aliasing.
In this 2003 paper, these researchers
also explored using a “serrated”
frequency modulation on each
pulse, then the same superpositions
of either the binary or polyphase code sets.
etc.
30
Then, in 2004, these authors presented this variation: each pulse is frequency modulated
with the Costas sequence shown,
modified
by the linear ramping around each frequency
value, then each given its unique binary phase code (the matrix here), giving the peculiar
hybrid modulation in “Fig. 2” above, and the extraordinary ambiguity

function shaping shown.
Levanon and Mozeson, “Orthogonal Train of Modified Costas Pulses”, IEEE Radar Conference 2004.
31
There . . we’re done
The Autocorrelation Function . .
we’ve seen that the measure of the “correlation”
between two functions is, justifiably, just the
“middle term” of the sum of the square of their
difference . . very logical
and that this correlation might be a
function
of
the time displacement of the two, themselves
possibly identical radar pulses.
Further, if that
auto

correlation function had a
very narrow
central peak and low sidelobes to
boot, that might be very nice for
resolving
(in
that dimension) two closely spaced “echoes”.
That’s all there is to it. Now, good luck in ALL that you do . . and
thanks for coming to my talk this evening!
Bob Hill
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