Reference no: EM132225738

Assignment Tasks -

Assume that the generator matrix G has dimension of K x N (i.e., K input bits and N output bits; see the table in attached file for your specific values of K and N). Suppose that we can transmit information (a bit sequence) with/without channel coding. The K bits (b₁, · · ·, b_K), where b_k ∈ {0, 1} with k (1, 2, · · ·, K), can be transmitted without channel coding and the received signal becomes

or in the compact form: r = s + w

where the probability density function (pdf) of the noise, w_k (where k = 1,2, · · ·, K), is given by f(w) in Table 1 (attached). Note that (2b_k - 1) = -1 when b_k = 0 and (2b_k - 1) = 1 when b_k = 1.

We can also transmit the K message bits (b₁, b₂, · · · ,b_K) by using a channel linear block code which has the generator matrix G given in Table 1. After encoding, a codeword of N bits c = (c₁, c₂, · · · , c_N) is transmitted and the received signal is written as

where the pdf of w_n (n = 1, 2, · · ·, N) is the same as in (1).

1. Theory:

(a) When the maximum likelihood (ML) detection is employed for the uncoded case in (1), find the bit error probability P_b and the symbol error probability P_s. Hint: Note that the noise is either Laplacian or Gaussian depending on your group.

(b) Construct the ML decoding rule after the detection (hard-decision) for the coded signals in (2).

Hint: Note that there are 2^N possible received binary signal vectors after taking hard-decision on each r_n where n = 1, 2, · · · , N. For each received signal vector, the ML decoding should be made for a decision. In some cases, two codewords can be chosen as they have the same likelihood values.

(c) Now, a specific codeword c^- is selected for your study (see Table 1). In the ML decoding, find the error probability when this specific codeword is transmitted. That is, find the probability that the ML decoding fails to make a correct decision when t is transmitted.

Hint: In the ML decoding, if two codewords have the same Hamming distance, we assume that one of them is randomly chosen with a half probability.

2. Matlab:

(a) Using Monte Carlo simulation to find the bit error rate and symbol error rate of the uncoded system in (1) and of the coded system in (2) and compare with the theoretical results found in Part (1.a) and Part (1.c).

(b) Write your program for a general case of noise variance and plot performance against the usual E_b/N₀ = 0 : 10 (dB) with hard decoding.

Attachment:- Assignment Files.rar