5.7 Transversal equalizer

Transversal Equalizer

A training sequence used for equalization is often chosen to be a noise-like sequence which is needed to estimate the channel frequency response.

In the simplest sense, training sequence might be a single narrow pulse, but a pseudonoise (PN) signal is preferred in practise because the PN signal has larger average power and hence larger SNR for the same peak transmitted power.

Figure 1 Received pulse exhibiting distortion.

Consider that a single pulse was transmitted over a system designated to have a raised-cosine transfer function $H_{\text{RC}}(t)=H_t(f)\text{.}{H}_r(f)$ , also consider that the channel induces ISI, so that the received demodulated pulse exhibits distortion, as shown in Figure 1, such that the pulse sidelobes do not go through zero at sample times. To achieve the desired raised-cosine transfer function, the equalizing filter should have a frequency response

$H_e(f)=\frac{1}{H_c(f)}=\frac{1}{\mid H_c(f)\mid }{e}^{-{\mathrm{j\theta }}_c(f)}$ (1)

In other words, we would like the equalizing filter to generate a set of canceling echoes. The transversal filter, illustrated in Figure 2, is the most popular form of an easily adjustable equalizing filter consisting of a delay line with T-second taps (where T is the symbol duration). The tab weights could be chosen to force the system impulse response to zero at all but one of the sampling times, thus making $H_e(f)$ correspond exactly to the inverse of the channel transfer function $H_c(f)$

Figure 2 Transversal filter

Consider that there are $\mathrm{2N}+1$ taps with weights $c_{-N},c_{-N+1},\text{.}\text{.}\text{.}{c}_N$ . Output samples $z(k)$ are the convolution the input sample $x(k)$ and tap weights $c_n$ as follows:

$z(k)=\sum _{n=-N}^{N}x(k-n)c_n$ $k=-\mathrm{2N},\text{.}\text{.}\text{.}\mathrm{2N}$ (2)

By defining the vectors z and c and the matrix x as respectively,

$z=\left[\begin{array}{c}z(-\mathrm{2N})\\ ⋮\\ z(0)\\ ⋮\\ z(\mathrm{2N})\end{array}\right]$ $c=\left[\begin{array}{c}{c}_{-N})\\ ⋮\\ c_0\\ ⋮\\ c_N\end{array}\right]$ $x=\left[\begin{array}{cccccc}x(-N)& 0& 0& \dots & 0& 0\\ x(-N+1)& x(-N)& 0& \dots & \dots & \dots \\ ⋮& & & ⋮& & ⋮\\ x(N)& x(N-1)& x(N-2)& \dots & x(-N+1)& x(-N)\\ ⋮& & & ⋮& & ⋮\\ 0& 0& 0& \dots & x(N)& x(N-1)\\ 0& 0& 0& \dots & 0& x(N)\end{array}\right]$

We can describe the relationship among $z(k)$ , $x(k)$ and $c_n$ more compactly as

$z=x\text{.}c$ (3a)

Whenever the matrix x is square, we can find c by solving the following equation:

$c=x^{-1}z$ (3b)

Notice that the index k was arbitrarily chosen to allow for $\mathrm{4N}+1$ sample points. The vectors z and c have dimensions $\mathrm{4N}+1$ and $\mathrm{2N}+1$ . Such equations are referred to as an overdetermined set. This problem can be solved in deterministic way known as the zero-forcing solution, or, in a statistical way, known as the minimum mean-square error (MMSE) solution.

Zero-Forcing Solution

At first, by disposing top N rows and bottom N rows, matrix x is transformed into a square matrix of dimension $\mathrm{2N}+1$ by $\mathrm{2N}+1$ . Then equation $c=x^{-1}z$ is used to solve the $\mathrm{2N}+1$ simultaneous equations for the set of $\mathrm{2N}+1$ weights $c_n$ . This solution minimizes the peak ISI distortion by selecting the $C_n$ weight so that the equalizer output is forced to zero at N sample points on either side of the desired pulse.

$z(k)=\{\begin{array}{cc}1& k=0\\ 0& k=\pm \mathrm{1,}\pm \mathrm{2,}\pm 3\end{array}$ (4)

For such an equalizer with finite length, the peak distortion is guaranteed to be minimized only if the eye pattern is initially open. However, for high-speed transmission and channels introducing much ISI, the eye is often closed before equalization. Since the zero-forcing equalizer neglects the effect of noise, it is not always the best system solution.

Minimum MSE Solution

A more robust equalizer is obtained if the $c_n$ tap weights are chose to minimize the mean-square error (MSE) of all the ISI term plus the noise power at the out put of the equalizer. MSE is defined as the expected value of the squared difference between the desire data symbol and the estimated data symbol.

By multiplying both sides of equation (4) by $x^T$ , we have

$x^{T}z=x^T\text{xc}$ (5)

And

$R_{\text{xz}}=R_{\text{xx}}c$ (6)

Where $R_{\text{xz}}=x^{T}z$ is called the cross-correlation vector and $R_{\text{xx}}=x^{T}x$ is call the autocorrelation matrix of the input noisy signal. In practice, $R_{\text{xz}}$ and $R_{\text{xx}}$ are unknown, but they can be approximated by transmitting a test signal and using time average estimated to solve for the tap weights from equation (6) as follows:

$c=R_{\text{xx}}^{-1}{R}_{\text{xz}}$

Most high-speed telephone-line modems use an MSE weight criterion because it is superior to a zero-forcing criterion; it is more robust in the presence of noise and large ISI