# lect6 - Kevin Buckley 2007 1 ECE 8770 Topics in Digital...

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Kevin Buckley - 2007 1 ECE 8770 Topics in Digital Communications - Sp. 2007 Lecture 6 3 MLSE with Intersymbol Interference (ISI) This Section of the course notes corresponds to Section 10.1 of the Course Text. For back- ground information on ISI, read Section 9-1 and Subsections 9-2-1 through 9-2-3 of the Text. In this Section of the Course we deal with ISI caused by memory in the channel. This mem- ory, which my be the result of multipath propagation, negates the use of symbol-by-symbol detection for optimum reception. Here we consider a MLSE approach to dealing with the ISI. We have already studied MLSE for modulation schemes with memory (i.e. DPSK, PRS, CPM). We will see that the same approaches that applied then, also apply now. Although our focus will be on MLSE, note that as with reception for digital modulation schemes with memory, MAP sequence estimation and symbol-by-symbol MAP are alternatives approaches. In Section 4 of the Course we will consider an alternative approach – channel equalization. 3.1 Basics of Digital Communications with ISI This Subsection corresponds to Subsections 10-1-1 and 10-1-2 of the Course Text. The goal here is to develop an ISI model that will allow us to directly apply the techniques described previously in Subsections 2.3-5. We will focus on MLSE for N = 1 and N = 2 dimensional linear modulation schemes. The approach easily extends to higher dimensional and nonlinear schemes. Consider QAM, for which PAM and PSK can be consider special cases. In Subsec- tion 1.5.3 we established the following equivalent lowpass representation of symbols s m ( t ); m - 1 , 2 , · · · , M : s ml ( t ) = V m e m g ( t ) ; 0 t T m = 1 , 2 , · · · , M , (1) where g ( t ) is a real-valued pulse shape. For symbol time n and transmitted symbol m = m ( n ), we can represent the transmitted symbol as s m ( n ) ( t - nT ) = V m ( n ) e m ( n ) δ ( t - nT ) * g ( t ) (2) where 1 T is the symbol rate and δ ( t - nT ) is the impulse function delayed to time nT . Let I n = I n ( n ) = V m ( n ) e m ( n ) . With this representation, the real part of I n corresponds to the cosine basis function term on the signal space representation, while the imaginary part corresponds to the sine term. For PAM or 2-PSK, there would be no sine term (i.e. these are N = 1 dimensional modulation schemes). { I n } is the random information sequence, for each symbol time n representing K = log 2 ( M ) bits.

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Kevin Buckley - 2007 2 Consider the digital communications channel illustrated as a lowpass equivalent in Figure 1(a). Using the I n representation of symbols, this illustration represents PAM, PSK and QAM where, respectively, I n is of the form: I n = A m (3) I n = e j 2 π ( m - 1) /M I n = V m e m . Mathematically, we can think of the modulator in this figure as effectively forming I ( t ) = summationdisplay n I n δ ( t - nT ) (4) and then processing it with an LTI filter with impulse response g ( t ). The lowpass equivalent transmitted signal is v ( t ) = summationdisplay n I n g ( t - nT ) . (5) (b) (a) I n c( t ) LTI Channel r (t) l z (t) ( g(t) ) Modulator v (t) r (t) l z (t) I(t) LTI Channel h( t ) Figure 1: Representations of a digital communication LTI ISI channel.
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