lect3 - 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 Communications - Spring 2007 Lecture 3 2 Symbol Detection and Sequence Estimation In this Section of the Course we consider symbol reception. Topics correspond to Sections 5-1, 5-2 and 5-3 of the Course Text. We assume that each symbol sent is received without distortion (i.e. with the same shape, although in general delayed and attenuated) in additive Gaussian bandlimited noise. The objective of the receiver is to process the noisy symbols and make decisions as to which symbols were sent. We will see that, under the assumptions stated above, the optimum receiver for a memoryless modulation scheme is fairly simple, whereas the optimum receiver for a system with memory can be substantially more involved. Review: 1. Signal Representation: Real-valued passband signal: s ( t ) CTFT ⇐⇒ S ( f ) Equivalent lowpass signal: s l ( t ) CTFT ⇐⇒ S l ( f ) s ( t ) = Re { s l ( t ) e j 2 πf c t } (1) S ( f ) = 1 2 [ S l ( f- f c ) + S * l (- f- f c )] (2) 2. Signal Space Concept: For N-dimensional linear modulations schemes, s m ( t ) = N summationdisplay k =1 s mk f k ( t ) = f ( t ) s m m = 1 , 2 , ··· , M . (3) (note that the s m are again column vectors). For memoryless linear modulation schemes, the signal space representation of symbols will lead to simple optimum symbol detection algorithms. 3. Trellis Diagram Sequence Representation: For modulation schemes with memory, a transmitted symbol sequence can be represented as a path through a trellis diagram. This representation will lead to efficient oprimum sequence estimation algorithms. 4. Inner Product: The inner product between two real-valued signals r ( t ) and f ( t ), each limited in duration to 0 ≤ t ≤ T , is < r ( t ) , f ( t ) > = integraldisplay T r ( t ) f ( t ) dt = Re { 1 2 integraldisplay T r l ( t ) f * l ( t ) dt } (4) Kevin Buckley - 2007 2 5. Digital Communication System: Figure 1 is an illustration of a typical communication system under consideration. I n is a discrete-time sequence representing the symbol sequence. The forms of I n and s ( t ) depend on the modulation scheme. channel encoder mapping to symbols transmit filter channel c(t) a k I n s(t) n(t) r(t) Figure 1: Digital communication channel block diagram. 2.1 Correlation Demodulator & Matched Filter for Symbol Detec- tion This Section addresses symbol detection for linear, memoryless modulation schemes. Consider a set of symbols s m ( t ); m = 1 , 2 , ··· , M ; 0 ≤ t ≤ T , with s m ( n ) received at symbol time n in Additive White Gaussian Noise (AWGN). The assumption that the noise is white, with spectral level N 2 , effectively means that it has a much broader bandwidth than the receiver (which has bandwidth dictated by the symbols), and it has a flat PSD of level N 2 over this receiver bandwidth. Given the received signal r ( t ) = s m ( n ) ( t ) + n ( t ) ; ≤ t ≤ T (5) the symbol detection objective is to decide which of the M symbols was transmitted.symbols was transmitted....
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lect3 - Kevin Buckley - 2007 1 ECE 8770 Topics in Digital...

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