, such that for , with the extreme limits defined by and . All the inputs that fall in a given interval range are associated with the same quantization index . Reconstruction Each interval is represented by a reconstruction value which implements the mapping . These two stages together comprise the mathematical operation of . Entropy coding techniques can be applied to communicate the quantization indices from a source encoder that performs the classification stage to a decoder that performs the reconstruction stage. One way to do this is to associate each quantization index with a binary codeword . An important consideration is the number of bits used for each codeword, denoted here by . As a result, the design of an -level quantizer and an associated set of codewords for communicating its index values requires finding the values of , and which optimally satisfy a selected set of design constraints such as the bit rate and distortion . Assuming that an information source produces random variables with an associated PDF , the probability that the random variable falls within a particular quantization interval is given by: . The resulting bit rate , in units of average bits per quantized value, for this quantizer can be derived as follows: . If it is assumed that distortion is measured by mean squared error, the distortion D, is given by: . A key observation is that rate depends on the decision boundaries and the codeword lengths , whereas the distortion depends on the decision boundaries and the reconstruction levels . After defining these two performance metrics for the quantizer, a typical rate–distortion formulation for a quantizer design problem can be expressed in one of two ways: Given a maximum distortion constraint , minimize the bit rate Given a maximum bit rate constraint , minimize the distortion Often the solution to these problems can be equivalently (or approximately) expressed and solved by converting the formulation to the unconstrained problem where the Lagrange multiplier is a non-negative constant that establishes the appropriate balance between rate and distortion. Solving the unconstrained problem is equivalent to finding a point on the convex hull of the family of solutions to an equivalent constrained formulation of the problem. However, finding a solution – especially a closed-form solution – to any of these three problem formulations can be difficult. Solutions that do not require multi-dimensional iterative optimization techniques have