EC-702-B Information Theory and Coding - Jun 2025 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aState and prove Shannon Hartley Theorem.
   How to answer

   • State Shannon-Hartley theorem: C = B log2(1+S/N)
   • State assumptions: bandlimited AWGN channel with bandwidth B, signal and noise powers
   • Outline proof using Nyquist sampling and Gaussian entropy maximization
   • Derive capacity expression and discuss limiting cases
2. 7 Marks
   bExplain decoder for cyclic code with the help of a block diagram.
   How to answer

   • Define cyclic decoder function: syndrome computation and error correction
   • Describe blocks: received buffer, syndrome calculator, error pattern detector, correction
   • Explain syndrome-based decoding steps
   • Present neat labelled block diagram
3. 7 Marks
   aWhat is the significance of the syndromes of a (6, 3) systematic cyclic code? Give a complete list codes and syndromes.
   How to answer

   • Define syndrome for cyclic codes and its role in error detection
   • Explain significance: zero vs non-zero syndrome, error pattern identification
   • List all (6,3) systematic codewords
   • List corresponding syndromes and interpretation
4. 7 Marks
   bDiscuss the procedure for generation of a systematic cyclic code. Draw and explain the systematic cyclic encoder circuit for a (15, 9) cyclic code with generator polynomial $g(X) = 1 + X^3 + X^4 + X^5 + X^6$.
   How to answer

   • Outline steps for systematic cyclic encoding using generator polynomial
   • Explain division: x^(n-k)m(x) divided by g(x) to get parity
   • Draw shift-register encoder for n=15, k=9 with $g(X)=1+X^3+X^4+X^5+X^6$
   • Explain operation of circuit stages
5. 7 Marks
   aA channel encoder uses a (7, 4) linear systematic cyclic code in the systematic form, generator polynomial being $X^3 + X + 1$. Determine the correct codeword transmitted if the received word is -

   1. i)1011011
   2. ii)1101111
   How to answer

   • Compute syndrome by dividing received polynomial by $X^3+X+1$
   • Identify error pattern from syndrome for each received word
   • Correct the error to obtain valid codeword
   • List transmitted codewords for i) 1011011 and ii) 1101111
6. 7 Marks
   bA zero memory source has a source alphabet. $S = \{s_1, s_2, s_3\}$ with $P = \{0.5, 0.3, 0.2\}$. Find the entropy of the source. Find the entropy of its second extension and verify.
   How to answer

   • Compute source entropy H(S) = -sum p_i log2 p_i
   • Compute second extension probabilities and entropy H(S^2)
   • Show H(S^2) = 2*H(S)
   • Verify numerically for P={0.5,0.3,0.2}
7. 7 Marks
   aWhat is a perfect code? Explain the features of (7, 4) Hamming code.
   How to answer

   • Define perfect code via Hamming bound with equality
   • Give parameters of (7,4) Hamming code: n=7,k=4,d=3
   • Explain single-error-correcting and features: parity checks, syndrome
   • Mention efficiency and sphere-packing
8. 7 Marks
   bExplain maximum likelihood decoding of convolutional codes.
   How to answer

   • Define maximum likelihood principle for convolutional decoding
   • Explain trellis search and path metrics / Hamming distance
   • Describe Viterbi algorithm steps for ML decision
   • Discuss error performance and complexity
9. 7 Marks
   aDerive the expression for channel capacity when bandwidth becomes infinite.
   How to answer

   • Start from Shannon-Hartley law C = B log2(1+S/(N0 B))
   • Take limit B -> infinity using ln(1+x) approximation
   • Derive C_infinity = (S/N0) log2 e
   • Interpret power-limited regime result
10. 7 Marks
    bShow how a 4 bit stage shift register can generate a convolution code for a input train 11001.
    How to answer

    • Describe 4-bit stage shift register convolutional encoder structure
    • Show shifting of input train 11001 step by step
    • Derive output coded bits using generator connections
    • Present resulting convolution code sequence
11. 7 Marks
    aExplain the structure of Block Codes and Systematic Block Codes.
    How to answer

    • Define (n,k) block code mapping
    • Explain systematic form with message plus parity structure and generator matrix
    • Compare non-systematic vs systematic with example
    • Mention parity-check matrix relation
12. 7 Marks
    bWhat is Parity Check Polynomial? Explain with a suitable examples.
    How to answer

    • Define parity check polynomial h(X) = (X^n+1)/g(X)
    • Explain its use in syndrome and error detection
    • Give suitable example with generator polynomial
    • Show encoding/decoding relation
13. 7 Marks
    aDefine the minimum distance of a code. How is it important in error detection and correction?
    How to answer

    • Define Hamming distance and minimum distance d_min
    • State error detection condition: d_min >= e+1
    • State error correction condition: d_min >= 2t+1
    • Explain importance with examples
14. 7 Marks
    bConsider a message ensembles $S = \{s_1, s_2, s_3, s_4, s_5, s_6, s_7\}$ with probabilities $P = \{0.45, 0.15, 0.12, 0.08, 0.08, 0.08, 0.04\}$, Construct a binary code and determine its efficiency using Shannon-Fano coding procedure.
    How to answer

    • Sort symbols by probabilities P={0.45,0.15,0.12,0.08,0.08,0.08,0.04}
    • Apply Shannon-Fano successive partitioning and assign 0/1
    • Compute average length L and entropy H(S)
    • Compute efficiency eta = H/L
15. 14 Marks
    Write short notes on (any four) :

    1. a)BCH codes
    2. b)Lempel Ziv. coding
    3. c)Viterbi decoding
    4. d)Error Control Strategy
    5. e)Trellis and state diagram
    How to answer

    • Define each chosen topic briefly
    • Give principle, diagram or steps where applicable
    • Mention applications / advantages
    • Cover any four of the five listed items