B.TECH - Semester 7 information theory and coding Question Paper 2020 (sep)
Practice authentic previous year university questions for better exam preparation.
- What are the properties of cyclic codes?
- (a) A $(7,4)$ cyclic codes has a generator polynomial : $g(X)=X^{3}+X+1$.
- (a) Derive the expression for the channel capacity of a binary symmetric channel (BEC).
- For a linear $(n, k)$ block code, prove that $\mathrm{CH}^{\top}=0$ where H is the parity check matrix and $C$ the code matrix.
- Explain standard array using an example.
- (i) Find the Generator matrix (ii) List all the code words
- (i) Draw the block diagram of encoder and syndrome calculator (ii) Find the generator and parity matrices in systematic form.
- (a) Explain the convolution encoder with suitable block diagram also explain the following terms
- (b) State and derive Shannon-Hartley theorem.
- Construct a convolutional encoder with parameter $(2,1,3)$ and generator sequence (1011) and (1101). Determine the encoder output produced by the message sequence 10100 using trellis structure. ( $\mathbf{2} \boldsymbol{\times} \mathbf{1 0} \boldsymbol{=} \mathbf{2 0}$ Marks)
- (b) Construct a convolutional encoder with parameter (2, I, 3) and generator sequence (1011) and (1101). Determine the encoder output produced by the message sequence 10100 using trellis structure.
- (b) Write short notes on :
- With the help of neat diagram, explain DES algorithm.
- Describe the viterbi algorithm for convolution codes also write the procedure for decoding a code word using Viterbi decoding algorithm.
- Determine the corrected code if a 7-bit Hamming code is received as 1111101. Assume even parity.
- Define free distance and free length of convolutional codes.
- Write the G-matrix for $(2,1,3)$ convolutional encoder $\mathrm{g}(1)=(1011) \mathrm{g}(2)=(1111)$. Each question carries 20 marks.
- (b) Explain the error correction and detection capabilities of linear block codes. 10
- (i) Reed-Solomon codes (ii) BCH codes
- (a) Explain viterbi encoder in detail. Determine the decoded data bits by applying viterbi decoding algorithm, if received code word $\mathrm{m}=1010101101$.
- The parity check matrix of a particular $(7,4)$ linear code is given as : $$ H=\left[\begin{array}{lllllll} 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 \end{array}\right] $$ (a) Find the Generator Matrix (G)
- The parity check matrix of a particular $(7,4)$ linear code is given as : $$ H=\left[\begin{array}{lllllll} 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 \end{array}\right] $$ (b) List all the code vectors.
- (a) The parity check matrix of a particular (7,4) linear code is given as $$ H=\left[\begin{array}{lllllll} 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 \end{array}\right] $$
- (b) Define mutual information and prove that $I(x, y)=H(x)+H(y)-H(x, y)$.
- With neat block diagram, explain the encoding and decoding methods of cyclic codes.
- (a) Define Marginal, Conditional, and Joint entropies. Also derive the relation between them.
- (b) Consider two sources $S_{1}$ and $S_{2}$ emits messages $x_{1}, x_{2}, x_{3}, x_{4}$ and $y_{1}, y_{2} y_{3}, y_{4}$ with joint probability $P(X, Y)$. Find $H(X), H(Y), H(X, Y)$ and $H(X / Y), H(Y / X)$ and $I(X ; Y) 10$ | X | 1 | 2 | 3 | | :--- | :--- | :--- | :--- | :--- | | 1 | $1 / 8$ | $1...
- (a) A DMS has five symbols with probabilities of occurrence $0.4,0.16,0.19$, 0.15 , and 0.1 . Construct Huffman code and also find its coding efficiency and redundancy.
- (a) A voice-grade channel of the telephone network has a bandwidth of 3.4 KHz .
- (i) Calculate channel capacity of the telephone channel for a signal-to-noise ratio of 30 dB . 4 (ii) Calculate the minimum signal-to-noise ratio required to support information transmission through the telephone channel at the rate of $4800 \mathrm{bits} / \mathrm{sec}$.
- (b) A Gaussian channel has a bandwidth of 4 KHz and a two-sided noise power spectral density $(\eta / 2)$ of $10^{\wedge}-14 \mathrm{watts} / \mathrm{Hz}$. The signal power at the receiver has to be maintained at a level less than or equal to 0.1 milli watt. Calculate the capacity of the channel.
- (c) Discuss Band width - SNR trade off and Shannon's limit.
- (i) Time domain approach (ii) Frequency domain approach.
- (b) A convolutional encoder with 3 shift register stages and rate $1 / 2$ is represented by the following generator polynomial : $g_{1}(x)=1+X+X^{2}$; $g_{2}(x)=X+X^{2}$. Draw the state and trellis diagram for the encoder.
- Calculate the capacity of an ideal AWGN channel with infinite bandwidth.
- Write a short notes on (a) Coding Efficiency (b) Code Redundancy.
- Find the signal to noise ratio to put T1 carriers ( 1.554 Mbps ) on a 50 KHz line.
- A line has a band width of 3100 Hz . A typical value of $\mathrm{S} / \mathrm{N}$ ratio is 30 dB . Calculate the channel capacity.
- What is a perfect code? Is a $(7,4)$ code a perfect code?
- Define Hamming distance and show that minimum Hamming distance of a linear block code is equal to the minimum weight of the code words.
- We need a data word of atleast 7 bits. Calculate values of $k$ and $n$ that satisfy this requirement in Hamming code.
- Briefly explain maximum likelihood decoding of convolutional codes.
- Differentiate public key cryptography and secret key cryptography.
- The impulse response of a convolutional encoder is 100111 . Determine the output sequence when input is 101. ( $\mathbf{1 0} \boldsymbol{\times} \mathbf{4} \boldsymbol{=} \mathbf{4 0}$ Marks) PART - B Module - I
- The source symbols and associated probabilities of a five symbol DMS source is as follows | Symbols | X 1 | X 2 | X 3 | X 4 | X 5 | | :--- | :---: | :---: | :---: | :---: | :---: | | Probabilities | 0.1 | 0.15 | 0.16 | 0.19 | 0.4 | Construct a binary Huffman code and determine the code efficiency,...
- (a) Derive the expression for the chain rule of the conditional entropy. 5
- (b) Prove that $I(X, Y)=H(X)+H(Y)-H(X, Y)$. 5
- State Shannon-Hartley theorem. Derive the trade-off relation between bandwidth and signal-to-Noise ratio. 10 ( $\mathbf{2} \boldsymbol{\times} \mathbf{1 0} \boldsymbol{=} \mathbf{2 0}$ Marks) .
- Obtain the relation between Entropy and Mutual information.
- Write the properties of entropy.
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