Assignment1 - EE 2200 / Spring 2009: Assignment One 1 ECE...

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EE 2200 / Spring 2009: Assignment One 1 ECE 2200 / Spring 2009 CORRELATION ASSIGNMENT ONE 1 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 8-element 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 3-term marker of plus ones to each sequential triplet in the received sequence is to form the correlation sum, i.e. a sum of an element-by-element 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 pair-up di±ers, the sum will be equal to the number of elements minus 2. If all pair-ups 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 cross-correlation, 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 } 1 Updated January 20, 2009.
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EE 2200 / Spring 2009: Assignment One 2 for j = 0 to 2, which produces c ( k ) = [ - 1 , - 1 , +1 , +3 , +1 , +1 , - 1 , - 1 , - 1 , - 1 , +1] The peak at location 4, indicating a complete match, implies (correctly) that 3 locations
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This note was uploaded on 03/02/2011 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell University (Engineering School).

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Assignment1 - EE 2200 / Spring 2009: Assignment One 1 ECE...

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