EE 2200 / Spring 2009: Assignment One
1
ECE 2200 / Spring 2009
CORRELATION
ASSIGNMENT ONE
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1. In a number of applications, the location of a particular pattern of values in a
streaming sequence is a critical task. For example, terrestrial broadcast digital televi
sion uses a particular brief sequence from its 8element discrete alphabet of (e±ectively)
{±
1
,
±
3
,
±
5
,
±
7
}
inserted in the transmitted data to assist necessary frame synchroniza
tion at the receiver. Consider the following idealization of such a task.
A transmitted string of sampled binary data, i.e. a sequence of plus and minus ones,
has a marker sequence, drawn from the same alphabet, embedded to indicate the start
of a new frame of data. Consider a received subsequence
{
...
+ 1
,

1
,

1
,
+1
,
+1
,
+1
,

1
,
+1
,

1
,

1
,
+1
,

1
,
+1
, ...
}
In this illustrative example, the triplet of plus ones is the marker and the seventh entry,
a minus one, is the start of the new frame. One way to compare the 3term marker of
plus ones to each sequential triplet in the received sequence is to form the correlation
sum, i.e. a sum of an elementbyelement product. For the ²rst three entries this is
c
= (+1)
*
(+1) + (

1)
*
(+1) + (

1)
*
(+1)
where the data value in each product pair is listed ²rst. For this binary
{±
1
}
sequence,
when an element of the data matches its corresponding element of the marker, their
product will be +1. If they di±er, it will be

1. If all match, the sum is equal to the
number of elements in the marker. If one pairup di±ers, the sum will be equal to the
number of elements minus 2. If all pairups are opposite in sign, then the sum is equal
to the negative of the number of marker elements.
The sum of products, also known as a crosscorrelation, for the 11 various fully over
lapping alignments can be written as
c
(
k
) =
2
s
j
=0
b
(
j
)
a
(
k
+
j
)
for
k
= 0 to 10 with
a
(
i
) =
{
+1
,

1
,

1
,
+1
,
+1
,
+1
,

1
,
+1
,

1
,

1
,
+1
,

1
,
+1
}
for
i
= 0 to 12, and
b
(
j
) =
{
+1
,
+1
,
+1
}
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Updated January 20, 2009.