{"id":2916,"date":"2024-08-16T18:28:50","date_gmt":"2024-08-16T12:58:50","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=2916"},"modified":"2025-07-16T15:05:45","modified_gmt":"2025-07-16T09:35:45","slug":"additive-white-gaussian-noise-channel","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ec\/communication-systems\/additive-white-gaussian-noise-channel","title":{"rendered":"Additive White Gaussian Noise Channel (AWGN)"},"content":{"rendered":"

If the channel is affected by White Noise then it is called as AWGN channel. White Noise is basically thermal noise.<\/p>\n

In an additive white Gaussian noise channel, the channel output y is given by<\/p>\n

y = x + n<\/p>\n

where, x is channel input and n is an additive band limited white Gaussian noise with zero mean and variance \u03c32<\/sup>.<\/p>\n

Capacity Cs<\/sub> of an AWGN channel is given by\"AWGN<\/p>\n

It is known as Shannon Hartley law.
\nHence, it is easier to increase the information capacity of a continuous communication channel by expanding its bandwidth than by increasing the transmitted power for a prescribed noise variance.<\/p>\n

Source Coding\"Source<\/h3>\n

A conversion of the output of a discrete memoryless source into a sequence of symbols (i.e. binary code word) is called source coding.
\nObjective:<\/strong> An Objective: objective of source coding is to minimize the average bit rate to achieve the efficient representation of data generated by source by removing redundancy.<\/p>\n

Important Terms<\/strong><\/h4>\n

Let \u2018X \u2019 be a DMS with finite entropy H(X) and an alphabet {x1<\/sub>, x2<\/sub> \u22c5 \u22c5 \u22c5 xm<\/sub>} with corresponding probabilities of occurrence P(xi<\/sub>) (i = 1, 2 \u22c5 \u22c5 \u22c5 m). Let the binary code word assigned to symbol xi<\/sub> by the encoder have length ni<\/sub> , measured in bits. The length of a codeword length of a codeword is the number of binary digits in the codeword. The average code word code word length L, per source symbol is given by\"Code<\/p>\n

Error Control Coding<\/h3>\n

When a signal is transmitted through channel, the probability of error for a particular signalling scheme is function of signal to noise ratio at receiver input and information rate. In practical systems, maximum signal power, the bandwidth of channel and noise power spectral density No\/2 are fixed for particular environment.
\nUsing signalling scheme, it is not possible often to achieve acceptable probability of error. Therefore, error control coding is used to achieve required minimum probability of error.
\nMethods of error correction are:<\/p>\n\n\n\n\n
ARQ<\/th>\nFEC<\/th>\n<\/tr>\n
\n
    \n
  • ARQ stands for Automatic Repeat request<\/li>\n
  • When an error is detected, a request is sent to transmitter from receiver to retransmit the signal.<\/li>\n
  • It needs duplex arrangement.<\/li>\n
  • Time delays are involved due to request and repeat signals, therefore it cannot be used in real time signals.<\/li>\n<\/ul>\n<\/td>\n
\n
    \n
  • FEC stands for forward error correction.<\/li>\n
  • In automatic mechanism of error correction in the form of error correction code is employed and hence no retransmission occurs<\/li>\n
  • It needs simplex arrangement.<\/li>\n
  • No delays are involved. Hence, it can be used in real time signals.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    \"Duplex<\/p>\n

    Parity Check Code<\/h4>\n

    The parity check is done by adding an extra bit, called parity bit, to the data to make the number of 1s either even or odd depending upon the type of parity. The parity check is suitable for single bit error detection only.<\/p>\n

    When the number of check bits is one, it is known as parity check code. It is of two types:<\/p>\n

      \n
    • Even Parity Check Code<\/strong>
      \nIf check bit is such that total number of 1\u2019s in the code word is even, it is even parity check code.<\/li>\n
    • Odd Parity Check Code<\/strong>
      \nWhen the check bit is such that total number of 1\u2019s in code word is odd, it is called odd parity check code.<\/li>\n<\/ul>\n\n\n\n\n\n
      Message<\/th>\nCode for even parity Check bit<\/th>\nCode for odd parity Check bit<\/th>\n<\/tr>\n
      \n

      010011<\/p>\n<\/td>\n

      		1<\/td>\n\n

      0<\/p>\n<\/td>\n<\/tr>\n

      101110<\/td>\n0<\/td>\n\n

      1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      If single error occurs in received message, it can be easily detected as it changes the number of 1\u2019s from even to odd or from odd to even.<\/p>\n

      Important Definitions<\/h4>\n

      Code word
      \n<\/strong>
      \n\u2022 It is the block of bits at the output of channel encoder<\/p>\n

      Code rate<\/strong><\/p>\n

      r = Number of bits in message block\/ Number of bits in encoded codeword = k\/n \u2264 1<\/p>\n

      Rb<\/sub> = Bit rate before channel encoding = 1\/Tb<\/sub><\/p>\n

      Ro<\/sub> = Bit rate (or channel data rate) after channel encoding = 1\/To<\/sub><\/p>\n

      kTb<\/sub> = nTo<\/sub><\/p>\n

      \"<\/p>\n

      Code Efficiency<\/strong><\/p>\n

      Code efficiency = k\/n = Code rate<\/p>\n

      Hamming Weight of a Code word W(x)<\/strong><\/p>\n

      W(x) = Number of non zero elements in the given code word.
      \neg: Hamming weight of a binary code word \u201c11010001\u201d is \u201c4\u201d.<\/p>\n

      Hamming Distance
      \nThe Hamming distance, between two code words, is the number of locations in which their respective
      \nelements differ.
      \ne.g.: Hamming distance between the code words \u201c1011010\u201d and \u201c0111000\u201d is 3<\/p>\n

      Hamming Distance<\/strong><\/p>\n

      The Hamming distance, between two code words, is the number of locations in which their respective
      \nelements differ.
      \ne.g.: Hamming distance between the code words \u201c1011010\u201d and \u201c0111000\u201d is 3\"At these three locations bits in two given code words are different. So the Hamming distance between the two given code words is \u201c3\u201d<\/p>\n

      Hamming distance between two given code words also can be defined as the Hamming weight of the code word obtained by the addition (Exclusive-OR) of the given two code words. weight of the code word obtained by the addition (Exclusive-OR) of the given two code words.
      \ne.g.: Hamming distance between the code words \u201c1011010\u201d and \u201c0111000\u201d is given by,
      \n(1011010) \u2295 (0111000) = (1100010)<\/p>\n

      Some properties of Hamming distance between two code words<\/strong>
      \nd(x, y) = Hamming distance between the code words x and y. Then<\/p>\n

      d(x, y) \u2265 0 \u2192 with equality if x = y
      \nd(x, y) = d(y, x) \u2192 Symmetry
      \nd(x, y) \u2264 d(x, z) + d(z, y) \u2192 Triangle inequality<\/p>\n

      Minimum Distance (dmin<\/sub>)<\/strong><\/p>\n