title: Modulo-N code
text: # Modulo-N code
  
  Modulo-N code is a lossy compression algorithm used to compress correlated data sources using modular arithmetic.
  
  ## Compression
  
  When applied to two nodes in a network whose data are in close range of each other modulo-N code requires one node (say odd) to send the coded data value as the raw data $M_{o}=D_{o}$; the even node is required to send the coded data as the $M_{e}=D_{e}{\bmod {N}}$. Hence the name modulo-N code.
  
  Since at least $\log _{2}K$ bits are required to represent a number K in binary, the modulo coded data of the two nodes requires $\log _{2}M_{o}+\log _{2}M_{e}$ bits. As we can generally expect $\log _{2}M_{e}\leq \log _{2}M_{o}$ always, because $M_{e}\leq N$. This is the how compression is achieved.
  
  A compression ratio achieved is ${\text{C.R.}}={\frac {\log _{2}M_{o}+\log _{2}M_{e}}{2\log _{2}M_{o}}}.$
  
  ## Decompression
  
  At the receiver by joint decoding we may complete the process of extracting the data and rebuilding the original values. The code from the even node is reconstructed by the assumption that it must be close to the data from the odd node. Hence the decoding algorithm retrieves even node data as
  
      $\operatorname {CLOSEST} (M_{o},N.k+M_{e}).$
  
  The decoder essentially finds the closest match to $M_{o}\simeq N.k+M_{e}$ and the decoded value is declared as $N.k+M_{e}$
  
  ## Example
  
  For a mod-8 code, we have Encoder
  
  Decoder
  
  Modulo-N decoding is similar to phase unwrapping and has the same limitation: If the difference from one node to the next is more than N/2 (if the phase changes from one sample to the next more than $\pi$), then decoding leads to an incorrect value.
  
  