Note: The following system model description in section 1.1 is extracted from [1] with slight modification.

1.1 System Model

In the analysis, I will assume an FIR channel model with a fractionally-spaced equalizer (FSE) with sampling interval T/2 (where T is the symbol period). The source symbols are chosen from a finite M-ary real alphabet of zero mean where all symbols are equally probable. The system model is shown graphically in Figure 1, where s_n is the baud-spaced source symbol at sample index n, $\ensuremath{\mathbf{c}}$ is the vector representing the fractionally-spaced channel impulse response, is additive white Gaussian channel noise, is the vector containing the fractionally-spaced equalizer coefficients, and y_n is the baud-spaced equalizer output. The number of coefficients in the channel and equalizer response vectors are N_c and N_f, respectively. The system output can be expressed as $y_n=\ensuremath{\mathbf{s}}^T(n)\ensuremath{\mathbf{Cf}}+\ensuremath{\mathbf{w}}^T(m)\ensuremath{\mathbf{f}}$ where $\ensuremath{\mathbf{s}}(n)=\left[s_n,s_{n-1},\ldots,s_{n-N_s+1}\right]^T$ is the length $N_s=\lfloor(N_c+N_f-1)/2\rfloor$ vector of baud-spaced source symbols, $\ensuremath{\mathbf{w}}(m)=\left[w_m,w_{m-1},\ldots,w_{m-N_f+1}\right]^T$ is the vector of additive zero-mean white Gaussian noise with variance $\sigma_w$, and $\ensuremath{\mathbf{C}}$ is the $N_s\times N_f$ decimated channel convolution matrix given by

$$\ensuremath{\mathbf{C}} = \left[\begin{array} {ccccccc} ... ...s \ & & & & &c_{N_c-1}&c_{N_c-2}\ \end{array}\right]$$

  

Figure 1: T/2-spaced multirate system model with added noise.