Assignment:
The trellis for a four-state code used as a component code within a turbo code is shown in Figure 8.25b. The code rate is z and the branch labeling. u v represents the output, branch word (code bits) for that branch. where a is a data bit (systematic code) and v is a parity bit. A block of N = 1024 samples are received from a demodulator. Assume that the first signals in the block arrive at time k = 1. and at each time k, a noisy data bit and parity bit is received. At time k = 1023. the received signals have noisy u, v values of 1.3, -0.8, and at time k = 1024. the values are -1.4. -0.9. Assume that the a priori probability for the data bit being a 1 or 0 is equally likely and the encoder ends in a state a = 00 at termination time k = 1025. Also, assume that the noise variance is equal to 2.5.
a) Calculate the branch metrics for time k = 1023 and k = 1024.
b) Calculate the reverse state metrics for time k= 1023, 1024, and 1025.
c) The values of the forward state metrics at time k = 1023 and k = 1024 are given below in Table for each valid state. Based on the values in the table and the values calculated in parts a) and b) calculate the values for the likelihood ratio associated with the data bits at time k = 1023 and k = 1024, and using the MAP decision rule, find the most likely data bit sequence that was transmitted.
Provide complete and step by step solution for the question and show calculations and use formulas.