Optimum Timing Phase for an Infinite Equalizer

We consider a noisy, random, baseband pulse train, b(t) = Z anx{t - nT) -f n(t), (1) where x it) is the channel (or channel front-end filter) impulse response and the a n are independent binary data which take values ±1 with equal probability. The additive zero mean gaussian noise process is denoted by n(t). Under very ideal conditions, the pulse x{t) would be of the form (sin irt/T)/(irt/T), and sampling (1) at time t = kT would yield the quantity a k + n(kT). Under more realistic conditions, the brick-wall shape of the above pulse is difficult to approximate in practice, and a smoother characteristic in the frequency domain is taken as the ideal. Using some excess bandwidth (i.e., the Fourier transform extends beyond ir/T rad/s) still, in principle, allows the ideal set of 189 samples expressed by x(0) = 1, x{kT) = 0 , k * 0; but this ideal set is never exactly attained in practice because of unknown channel distortion. One possible measure of distortion could be £ xkT) k* 0 2 x (0) Actually, since there is now nothing special about choosing t = 0 as the sampling instant for x{t), an even more appropriate measure would be £ z2(r + kT) The above measure allows us to choose a best sampling epoch r before declaring how badly the signal has been distorted. However, any direct consideration of the above type of criterion seems to result in considerable mathematical difficulty. Also, in practice, one is more interested in the situation where the set of digital samples {X(T + kT)}, or rather their noisy versions {6(r + kT)}, are linearly combined by an equalizer attempting to undo the channel distortion.