Determine the surviving sequences

Question

Solve the following problem:

Consider a four-level PAM system with possible transmitted levels, 3, 1, -1, and -3. The channel through which the data is transmitted introduces inter symbol interference over two successive symbols. The equivalent discrete-time channel model is shown in Figure. {η_k } is a sequence of real-valued independent zero-mean Gaussian noise variables with variance σ² = N₀. The received sequence is

a. Sketch the tree structure, showing the possible signal sequences for the received signals y₁, y₂, and y₃.

b. Suppose the Viterbi algorithm is used to detect the information sequence. How many probabilities must be computed at each stage of the algorithm?

c. How many surviving sequences are there in the Viterbi algorithm for this channel?

d. Suppose that the received signals are

y₁ = 0.5, y₂ = 2.0, y₃ = -1.0

Determine the surviving sequences through stage y₃ and the corresponding metrics.

e. Give a tight upper bound for the probability of error for four-level PAM transmitted over this channel.