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Folded Spectrum and ISI-Free Transmission in Software Integrated barcode 3 of 9 in Software Folded Spectrum and ISI-Free Transmission




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11.3 Folded Spectrum and ISI-Free Transmission generate, create none none in none projects Bar code to 2D Code Equalizers a none none re typically implemented digitally. Figure 11.3 shows a block diagram of an end-to-end system with a digital equalizer.

The input symbol d k is passed through a pulse shape lter g(t) and then transmitted over the ISI channel with impulse response c(t). We de ne the equivalent channel impulse response h(t) = g(t) c(t), and the transmitted signal is thus given by d(t) g(t) c(t) for d(t) = k dk (t kTs ) the train of information symbols. The pulse shape g(t) improves the spectral properties of the transmitted signal, as described in 5.

5. This pulse shape is under the control of the system designer, whereas the channel c(t) is introduced by nature and is outside the designer s control. At the receiver front end white Gaussian noise n(t) is added to the received signal for a resulting signal w(t).

This signal is passed through an analog matched lter g m ( t) to obtain output y(t), which is then sampled via an A/D converter. The purpose of the matched lter is to maximize the SNR of the signal before sampling and 329. Equalizers Types Linear Nonlinear MLSE Structures Transversal Lattice Transversal Lattice Transversal Channel Estimator Tap Update Algorithms LMS Gradient RLS RLS Fast RLS Square Root RLS LMS Gradient RLS RLS Fast RLS Square Root RLS LMS RLS Fast RLS Square Root RLS Figure 11.2: none none Equalizer Types, Structures, and Algorithms. subsequent processing1 .

Recall from 5.1 that in AWGN the SNR of the received signal is maximized prior to sampling by using a lter that is matched to the pulse shape. This result indicates that for the system shown in Figure 11.

3, SNR prior to sampling is maximized by passing w(t) through a lter matched to h(t), so ideally we would have gm (t) = h(t). However, since the channel impulse response c(t) is time-varying and analog lters are not easily tuneable, it is generally not possible to have g m (t) = h(t). Thus, part of the art of equalizer design is to chose gm (t) to get good performance.

Often gm (t) is matched to the pulse shape g(t), which is the optimal pulse shape when c(t) = (t), but this design is clearly suboptimal when c(t) = (t). The fact that g m (t) cannot be matched to h(t) can result in signi cant performance degradation and also makes the receiver extremely sensitive to timing error. These problems are somewhat mitigated by sampling y(t) at a rate much faster than the symbol rate and designing the equalizer for this oversampled signal.

This process is called fractionally-spaced equalization [1]. The equalizer output then provides an estimate of the transmitted symbol. This estimate is then passed through a decision device that rounds the equalizer output to a symbol in the alphabet of possible transmitted symbols.

During training the equalizer output is passed to the tap update algorithm to update the tap values such that the equalizer output matches the known training sequence. During tracking, the round-off error associated with the symbol decision is used to adjust the equalizer coef cients. Let f (t) denote the combined baseband impulse response of the transmitter, channel, and matched lter:.

f (t) = g( t) c(t) gm ( t).. (11.2). While the ma tched lter could be more ef ciently implemented digitally, the analog implementation before the sampler allows for a smaller dynamic range in the sampler, which signi cantly reduces cost.. Then the mat none for none ched lter output is given by y(t) = d(t) f (t) + ng (t) = dk f (t kT ) + ng (t), (11.3). where ng ( t) = n(t) gm ( t) is the equivalent baseband noise at the equalizer input and T is the symbol time. If we let f [n] = f (nTs ) denote samples of f (t) every Ts seconds then sampling y(t) every Ts seconds yields the discrete time signal y[n] = y(nTs ) given by . y[n] =. k= . dk f (nTs none for none kTs ) + ng (nTs ) dk f [n k] + [n]. k= .
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