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ch4 - Wong Lok Theory of Digital Communications 4 ISI...

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Wong & Lok: Theory of Digital Communications 4. ISI & Equalization Chapter 4 Intersymbol Interference and Equalization The all-pass assumption made in the AWGN (or non-dispersive) channel model is rarely practical. Due to the scarcity of the frequency spectrum, we usually filter the transmitted signal to limit its bandwidth so that efficient sharing of the frequency resource can be achieved. Moreover, many practical channels are bandpass and, in fact, they often respond differently to inputs with different frequency components, i.e., they are dispersive . We have to refine the simple AWGN (or non-dispersive) model to accurately represent this type of practical channels. One such commonly employed refinement is the dispersive channel model 1 : (4.1) where is the transmitted signal, is the impulse response of the channel, and is AWGN with power spectral density . In essence, we model the dispersive characteristic of the channel by the linear filter . The simplest dispersive channel is the bandlimited channel for which the channel impulse response is that of an ideal lowpass filter. This lowpass filtering smears the transmitted signal in time causing the effect of a symbol to spread to adjacent symbols when a sequence of symbols are transmitted. The resulting interference, intersymbol interference (ISI) , degrades the error performance of the communication system. There are two major ways to mitigate the detrimental effect of ISI. The first method is to design bandlimited transmission pulses which minimize the the 1 For simplicity, all the signals considered in this chapter are real baseband signals. All the developments here can, of course, be generalized to bandpass signals using either the real bandpass representation or the complex baseband represen- tation. 4.1
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Wong & Lok: Theory of Digital Communications 4. ISI & Equalization effect of ISI. We will describe such a design for the simple case of bandlimited channels. The ISI- free pulses obtained are called the Nyquist pulses . The second method is to filter the received signal to cancel the ISI introduced by the channel impulse response. This approach is generally known as equalization . 4.1 Intersymbol Interference To understand what ISI is, let us consider the transmission of a sequence of symbols with the basic waveform . To send the th symbol , we send , where is the symbol interval. Therefore, the transmitted signal is (4.2) Based on the dispersive channel model, the received signal is given by (4.3) where is the received waveform for a symbol. If a single symbol, say the symbol , is transmitted, the optimal demodulator is the one that employs the matched filter, i.e., we can pass the received signal through the matched filter and then sample the matched filter output at time to obtain the decision statistic. When a sequence of symbols are transmitted, we can still employ this matched filter to perform demodulation. A reasonable strategy is to sample the matched filter output at time to obtain the decision statistic for the symbol . At , the output of the matched filter is (4.4) where
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