# THE AUTOCORRELATION FUNCTION

AI and Robotics

Nov 24, 2013 (4 years and 5 months ago)

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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

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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 . .

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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

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

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. . what a wonderful

book . . and
thin
, too.

nearly all fields, books

that are both profound

and thin.)

Pergamon Press Ltd.

London, 1953

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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

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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

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 . .

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

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
-

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!

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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

use of the “matched filter” for maximum

sensitivity and the achievement of

high resolution at the same time.

Block diagrams can be
very

complicated.

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. . well, our time is probably

running short, so this will suffice . .

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Signal processing discussion . .

a) reception as a convolution . . specifically

a convolution of the signal being

with the “impulse response function”

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

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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

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)

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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.

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Recall, phase angles

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

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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.

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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.

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Our signal being

convolved with a

matched filter . .

pulse compression

a)

b)

c)

a) ordinary square pulse,

no pulse compression . .

modulation bandwidth roughly

reciprocal pulse width,
not

, 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

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1990

Dr. Marv Cohen (Georgia Tech. Research Institute)

presentation at the IEEE International Radar Conference

-
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”

-
90”

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. . further from Dr. Cohen’s

-
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!

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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

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. . 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

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.

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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.

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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

Further, if that
auto
-

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