2 Equalization and Inverse Modeling

Figure 2.8: Block diagram of the general adaptive inverse system identification (inverse modeling) problem.

The basic idea of inverse modeling, also known as deconvolution or equalization, is shown in Fig. 2.8. The input signal $u(n)$ is filtered through a physical system, which might be a communication channel for instance. The observed distorted signal $x(n)$ is filtered through an adaptive inverse model of the physical system such that the output $y(n)$ is as close as possible to the input signal $u(n)$. To achieve this goal, the coefficients of the adaptive filter are adjusted to minimize the difference between the filter output $y(n)$ and a delayed version of the input $u(n)$. The delay $\Delta$ is chosen to match the delay introduced by the combined physical system and the adaptive filter path. On convergence, the convolution of the adaptive filter response and the physical system response equals to a delayed impulse $\delta(n-\Delta)$. The frequency response of the adaptive filter $\vW (z)$ is then an approximation of the inverse of the frequency response of the physical system $\vH (z)$ such that $\vW (z) \simeq z^{-\Delta}/\vH (z)$. The adaptive filter in such applications basically tries to undo the distortion introduced by the physical system to restore the input signal as much as possible. Inverse modeling has found many practical applications in control systems and communication systems. The most widely used application of this technique is channel equalization, where the physical system is a communication channel and the adaptive filter is referred to as an adaptive channel equalizer filter. The input signal $u(n)$ in this case is the transmitted data (usually in the form of modulated pulses). The transmitted data is distorted by the communication channel in different ways. The most serious kind of distortion is the inter-symbol interference resulting from the fact that the channel response is never an impulse but one that is nonzero over many symbol periods. This results in interference between neighboring data symbols making symbol detection using a simple threshold detector unreliable and, therefore, increasing the detector symbol error rate. The adaptive equalizer is required to reduce the inter-symbol interference distortion while avoiding amplifying the additive noise usually present at the equalizer input. The problem in the above channel equalizer setup is that the reference signal $u(n)$ is not available during normal transmission at the receiver side to be used as the desired signal, which is necessary for updating the equalizer coefficients. This is solved by introducing a training session prior to transmission. In the training session, the transmitter sends a sequence of training symbols that are known at the receiver side. The training sequence is locally generated at the receiver and used to adjust the equalizer coefficients to minimize the symbol error rate. Once the optimal coefficients have been found, the detected symbols are similar to the transmitted symbols and can be used as the desired signal for further adaptation of the equalizer coefficients to track any further changes in the channel. This mode of operation is usually referred to as the decision directed mode and works well as long as the changes in the communication channel is slow enough and the adaptive algorithm can successfully track the changes. Channel equalizers are usually implemented as adaptive transversal FIR filters. The Adaptive Signal Processing Toolbox includes several inverse modeling applications such as equalizer_nlms (Section 10.19), and equalizer_rls (Section 10.20).