Consider the rate-1/3 turbo encoder depicted in Figure 2 . It is designed by parallel concatenation of two recursive and systematic convolutional (RSC) codes separated by a random interleaving function. Figure 2 3.a Draw the trellis diagram for the first RSC encoder which is the encoder generating the first part, \( M \) and \( P_{1} \), of a codeword. Also, draw the trellis diagram for the second RSC encoder which is the encoder generating the second part, \( P_{2} \), of a codeword. [5 marks] 3.b Determine the first three terms in the union bound of the turbo encoder shown in Figure 2. Is it a good turbo code? [10 marks] 3.c We have seen in class that turbo codes are decoded using an iterative decoding algorithm. This algorithm can perform very close to the maximum-likelihood decoding. How do we know that? In other words, what is the evidence behind that statement? [5 marks]
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a The trellis diagram for the first and second RSC encoders would be the same since they are identical encoders. The trellis diagram would depend on the specific RSC code used, which is not provided in the question. However, a typical trellis diagram for a rate-1/3 Show more…
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For this question, we consider the Hill cipher given in the textbook on an alphabet A consisting of 26 English characters (A-Z), 10 numeric characters (0-9) and the following special characters: : ; < = [ which corresponds to integers 0 to 40. Here the plaintext is processed successively in blocks of size m. The encryption algorithm takes a block with m plaintext digits and transforms into a cipher block of size m using a key matrix of size m x m by the linear transformation, which is given by: c1 = (k1,1p1 + k1,2p2 + ... + k1,mpm) mod 41 c2 = (k2,1p1 + k2,2p2 + ... + k2,mpm) mod 41 ... cm = (km,1p1 + km,2p2 + ... + km,mpm) mod 41 Note: For this question, correspondence between plaintext and number modulo 41 are as follows "A" <-> 0, "B" <-> 1, "C" <-> 2, ..., "Z" <-> 25, "0" <-> 26, "1" <-> 27, "2" <-> 28, ..., "9" <-> 35, ":" <-> 36, ";" <-> 37, "<" <-> 38, "=" <-> 39, "[" <-> 40 (a) The following is ciphertext where the encryption method described above has been employed: OANJASCDOP7A4R82YQR[N11Z;AXCJNV9<ROAZX UO[06;;2U4;ZXWKW:V2BMV:9264:DGOPJSB=9L9:EF where the key matrix is: 29 1 5 0 26 28 17 38 25 8 37 40 1 26 14 40 33 31 34 14 31 23 29 12 23 Find the corresponding plaintext. (b) How many different keys are possible in this system described by the Question? What if we disallowed the symbols "[", "=", "<" so as to only consider an alphabet with 38 characters? In other words, now considering a Hill cipher working mod 38, how many possible keys are there? (c) We return now to the full alphabet A. This cipher is easily broken with a known plaintext attack. An adversary discovers the following ciphertext is encrypted using this cipher with m = 5 (55 characters in total, no spaces): A8VS3XRDEON6JEVXGJID13C07L4C1R4Q965XWRA5DQGYWTNHYO4ND8Z If the following combination of plaintext and ciphertext is given (please replace both "?????" by the last five digits of your student number), decrypt the cipher by giving the plaintext as well as both encryption and decryption keys. Plaintext: XY[LER5HI;:LMOPQR?????SBJ Ciphertext: [WQ:9BOA?????CUQ<ANY1IMTD You need to show step-by-step details of your working. Make sure to include the details of any package, functions used, and/or programs developed. Simply showing the final result and/or a program would not receive marks.
Akash M.
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Consider a (7,4) code with generator matrix Do the following: (a) Find all codewords of the code; (b) Is this a systematic code? (c) What is dmin in this code? (d) Find the parity check matrix H of the code; (e) Establish a table of error vectors (with no more than 2 errors) and the corresponding syndromes. (f) Implement the decoder in Matlab (input: senseword; output: codeword). Draw the curve of codeword error rate with respect to the single bit error rate.
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