lecture11 - ECE 5670 Digital Communications Lecture 11...

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ECE 5670 : Digital Communications Lecture 11: Intersymbol Interference Management: Low SNR Regime 1 3/1/2011 Instructor: Salman Avestimehr Introduction So far, we have seen that the wireline channel is modeled as an FIR (finite impulse response) filter acting on the transmit voltage sequence. The main difference between the AWGN channel and wireline channel is the presence of inter-symbol interference (ISI), i.e., transmit voltages of the previous symbols also mix along with the additive Gaussian noise with the voltage of the current symbol. The main challenge in wireline channel is to handle noise and ISI simultaneously in the quest to achieve rate efficient reliable communication. Our approach to wireline channel will be to “simply” process the transmit and receive voltage sequences (at the transmitter and receiver, respectively) to mitigate ISI and harness the block codes developed for rate efficient reliable communication on the AWGN channel. In the next few lectures we study the receiver centric methods to deal with ISI. The transmitter will be more or less the same as that in the case of AWGN channel. In this lecture, we focus on the low SNR regime: in this scenario, the noise power dominates the total signal power. Thus noise dominates the ISI. A First Approach The received voltage is y [ m ] = L 1 summationdisplay l =0 h l x [ m l ] + n [ m ] (1) = h 0 x [ m ] + L 1 summationdisplay l =1 h l x [ m l ] + n [ m ] (2) = S [ m ] + I [ m ] + n [ m ] (3) where S [ m ] def = h 0 x [ m ] (4) is the “signal”; I [ m ] def = L 1 summationdisplay l =1 h l x [ m l ] (5) 1 Based on lecture notes of Professor Pramod Viswanath at UIUC.
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is interference. The implication of the low SNR regime is that E bracketleftbig ( I [ m ]) 2 bracketrightbig E bracketleftbig ( n [ m ]) 2 bracketrightbig . (6) In this regime, the natural approach is to ignore the interference completely and treat it as noise. In other words, the receiver could just do the nearest neighbor detection (even though it may not be optimal since the discrete interference definitely does not have Gaussian statistics). A natural question is: What reliable rate of communication is feasible with this approach? In an AWGN channel, SNR is the only parameter of interest: the capacity is a direct function of the SNR. If we are to use that intuition here, we could look at the SINR (signal to interference plus noise ratio) defined as: SINR def = E bracketleftbig ( S [ m ]) 2 bracketrightbig E bracketleftbig ( I [ m ]) 2 bracketrightbig + E bracketleftbig ( n [ m ]) 2 bracketrightbig (7) = h 2 0 E [ x [ m ] 2 ] E bracketleftbigg parenleftBig L 1 l =1 h l x [ m l ] parenrightBig 2 bracketrightbigg + σ 2 . (8) Here σ 2 is equal to E ( n [ m ] 2 ), the variance of the discrete-time noise. In deriving this expression, we implicitly used the statistical independence between transmitted voltages and the noise. Denoting E ( x [ m ] 2 ) = E (9) to be the signal power (the student is encouraged to think of binary modulation for con- creteness), the SINR can be written as SINR = h 2 0 E parenleftBig L 1 l =1 h 2 l parenrightBig E + σ 2 (10) = h 2 0 SNR 1 + parenleftBig L 1 l =1 h 2 l parenrightBig
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