lect4 - Kevin Buckley - 2007 1 ECE 8770 Topics in Digital...

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Unformatted text preview: Kevin Buckley - 2007 1 ECE 8770 Topics in Digital Comm. - Spring 2007 Lecture 4 2 Symbol Detection and Sequence Estimation Again, this Section of the Course corresponds to topics from Chapter 5 of the Course Text. 2.2 Optimum Symbol Detector 2.2.4 Decoding DPSK - a Suboptimum Symbol Detector We introduced binary DPSK as an example of a modulation scheme with memory and noted that one advantage of it is that it facilitates decoding without knowledge of the carrier phase i.e. carrier nonsynchronous reception is possible by simply detecting the change in initial phase from symbol to symbol. Here we describe a DPSK receiver that does not require carrier synchronization. As we shall see later, in Subsection 2.3.2 of the Course Notes, since DPSK is a modulation scheme with memory, optimum (ML or MAP) estimation of a symbol sequence requires the joint processing of all the symbols in the sequence and all the observed data over the extent of the sequence. In other words, decoupled symbol-by-symbol detection as described in this Subsection is not optimum. This discussion corresponds to Subsection 5.2.8 of the course text. Consider binary DPSK, where the transmitted signal is observed at the receiver with unknown phase in AWGN. The received signal over the k th symbol duration is r ( t ) = g ( t ) cos(2 f c t + k + ) + n ( t ) kT < t ( k + 1) T (1) where k is 0 or depending on what the k th symbol is and what all the previous symbols were. Consider DPSK and the demodulator depicted in Figure 1. k T ( k + 1 ) T ( . ) d t k T ( k + 1 ) T ( . ) d t 2 E g g ( t ) c o s ( 2 f t ) c 2 E g c g ( t ) s i n ( 2 f t ) r k , r r k , i T r r ( t ) r r k k - 1 ( k + 1 ) T Figure 1: A nonsynchronous binary DPSK receiver. Kevin Buckley - 2007 2 Two correlators are used instead of the one normally required for binary PS because, with unknown phase at the receiver, g ( t ) cos(2 f c t + k + ) is two dimensional over 0 < 2 (i.e. two basis functions are required to represent it over the range of unknown ). The 2-dimensional observation vector for symbol k is r k = [ r k,r , r k,i ] = [ radicalBig E s cos( k- ) + n k,r , radicalBig E s sin( k- ) + n k,i ] (2) where E s is the symbol energy (i.e. for binary modulation schemes E s is the same as the bit energy E b ). Eq. (2) can also be conveniently thought of as the complex- valued observation r k = r k,r + jr k,i = radicalBig E s e j cos( k- ) + n k , (3) where n k = n k,r + jn k,i . If, as shown in Figure 1, we form the produce r k r * k- 1 , then as explained in Sub- section 5.2.8 of the Course Text, for binary DPSK r k r * k- 1 E s x k + jy k (4) where x k and y k are real-valued (i.e. x k is the real part of r k r * k- 1 ) and x k = radicalBig E s + Re { n k + n * k- 1 } (5) y k = Im { n k + n * k- 1 } . (6) The noise in y k is statistically independent of that in x k , so that only x k need be processed. In x k , a E s indicates a 0 bit while a...
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lect4 - Kevin Buckley - 2007 1 ECE 8770 Topics in Digital...

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