ADAPTIVE CHANNEL EQUALIZATION

11.5 ADAPTIVE CHANNEL EQUALIZATION

The speed of data transmission over telephone channels is usually limited by channel distortion that causes intersymbol interference (ISI). At data rates below 2400 bits the ISI is relatively small and is usually not a prob- lem in the operation of a modem. However, at data rates above 2400 bits, an adaptive equalizer is employed in the modem to compensate for the channel distortion and thus to allow for highly reliable high-speed data

Chapter 11

APPLICATIONS IN ADAPTIVE FILTERING

FIGURE 11.5 Application of adaptive filtering to adaptive channel equalization

transmission. In telephone channels, filters are used throughout the sys- tem to separate signals in different frequency bands. These filters cause amplitude and phase distortion. The adaptive equalizer is basically an adaptive FIR filter with coefficients that are adjusted by means of the LMS algorithm to correct for the channel distortion.

A block diagram showing the basic elements of a modem transmit- ting data over a channel is given in Figure 11.5. Initially, the equalizer coefficients are adjusted by transmitting a short training sequence, usu- ally less than one second in duration. After the short training period, the transmitter begins to transmit the data sequence {a(n)}. To track the possible slow time variations in the channel, the equalizer coefficients must continue to be adjusted in an adaptive manner while receiving data. This is usually accomplished, as illustrated in Figure 11.5, by treating the decisions at the output of the decision device as correct and by using the decisions in place of the reference {d(n)} to generate the error signal. This approach works quite well when decision errors occur infrequently, such as less than one error in 100 data symbols. The occasional decision errors cause only a small misadjustment in the equalizer coefficients.

11.5.1 PROJECT 11.4: ADAPTIVE CHANNEL EQUALIZATION The objective of this project is to investigate the performance of an adap- tive equalizer for data transmission over a channel that causes intersym- bol interference. The basic configuration of the system to be simulated is shown in Figure 11.6. As we observe, five basic modules are required.

Note that we have avoided carrier modulation and demodulation, which is required in a telephone channel modem. This is done to simplify the simulation program. However, all processing involves complex arithmetic operations.

Adaptive Channel Equalization 605

FIGURE 11.6 Experiment for investigating the performance of an adaptive equalizer

The five modules are as follows:

1. The data generator module is used to generate a sequence of complex- valued information symbols {a(n)}. In particular, employ four equally probable symbols s + js, s − js, −s + js, and −s − js, where s is a scale factor that may be set to s = 1, or it can be an input parameter.

2. The channel filter module is an FIR filter with coefficients {c(n),

0 ≤ n ≤ K − 1} that simulates the channel distortion. For distortion- less transmission, set c(0) = 1 and c(n) = 0 for 1 ≤ n ≤ K − 1. The length K of the filter is an input parameter.

3. The noise generator module is used to generate additive noise that is usually present in any digital communication system. If we are model- ing noise that is generated by electronic devices, the noise distribution should be Gaussian with zero mean. Use the randu function.

4. The adaptive equalizer module is an FIR filter with tap coefficients {h(k), 0 < k < N − 1}, which are adjusted by the LMS algorithm. However, due to the use of complex arithmetic, the recursive equation in the LMS algorithm is slightly modified to

h n (k) = h n−1 (k) + △ e(n)x ∗ (n − k)

where the asterisk denotes the complex conjugate.

5. The decision device module takes the estimate ˆ a(n) and quantizes it to one of the four possible signal points on the basis of the following

Chapter 11

APPLICATIONS IN ADAPTIVE FILTERING

decision rule:

Re [ˆ a(n)] > 0 and Im [ˆ a(n)] > 0 −→ 1+j Re [ˆ a(n)] > 0 and Im [ˆ a(n)] < 0 −→ 1−j Re [ˆ a(n)] < 0 and Im [ˆ a(n)] > 0 −→ −1 + j Re [ˆ a(n)] < 0 and Im [ˆ a(n)] < 0 −→ −1 − j

The effectiveness of the equalizer in suppressing the ISI introduced by the channel filter may be seen by plotting the following relevant sequences in a two-dimensional (real–imaginary) display. The data generator out- put {a(n)} should consist of four points with values ±1 ± j. The effect of channel distortion and additive noise may be viewed by displaying the sequence {x(n)} at the input to the equalizer. The effectiveness of the adaptive equalizer may be assessed by plotting its output {ˆ a(n)} af- ter convergence of its coefficients. The short-time average squared error ASE(n) may also be used to monitor the convergence characteristics of the LMS algorithm. Note that a delay must be introduced into the output of the data generator to compensate for the delays that the signal encoun- ters due to the channel filter and the adaptive equalizer. For example, this delay may be set to the largest integer closest to (N + K)/2. Finally, an error counter may be used to count the number of symbol errors in the received data sequence, and the ratio for the number of errors to the total number of symbols (error rate) may be displayed. The error rate may be varied by changing the level of the ISI and the level of the additive noise.

It is suggested that simulations be performed for the following three channel conditions:

a. No ISI:

c(0) = 1, c(n) = 0, 1 ≤ n ≤ K − 1

b. Mild ISI: c(0) = 1, c(1) = 0.2, c(2) = −0.2, c(n) = 0, 3 ≤ n ≤ K − 1

c. Strong ISI: c(0) = 1, c(1) = 0.5, c(2) = 0.5, c(n) = 0, 3 ≤ n ≤ K − 1 The measured error rate may be plotted as a function of the signal-