Determine minimum euclidean distance between merging paths

Question

Solve the following problem:

Sketch the trellis for the Viterbi detector of the equalized signal in Problem 1 and label all the states. Also, determine the minimum Euclidean distance between merging paths.

Problem 1:  In a magnetic recording channel, where the read back pulse resulting from a positive transition in the write current has the form

p(t) = [1 + (2t/T₅₀)²]^-1

a linear equalizer is used to equalize the pulse to a partial response. The parameter T₅₀ is defined as the width of the pulse at the 50 percent amplitude level. The bit rate is 1/Tb and the ratio of T₅₀/T_b = ? is the normalized density of the recording. Suppose the pulse is equalized to the partial-response values

x(nT) = {1    n=-1,1
            {2    n=0
            {0    otherwise

Where x(t) represents the equalized pulse shape.

a. Determine the spectrum X( f ) of the band-limited equalized pulse.

b. Determine the possible output levels at the detector, assuming that successive transitions can occur at the rate 1/T_b.

c. Determine the error rate performance of the symbol-by-symbol detector for this signal, assuming that the additive noise is zero-mean Gaussian with variance σ².