The discrete signal i.e. sampled values produced as a result of sampling (discussed in the previous post) has to be

**, hence the process called***quantized***. In simple words, this is the process in which a value from a given set of values is assigned to each sample of the discrete signal. The number of values in the set actually is the number of quantization levels to which samples of discrete signals are assigned to. Digital communication is based on bits and bytes, the number of bits used identify the number of quantization levels, hence, in this case when the samples are binary encoded that means they are essentially being***Quantization***in one of the fixed number of quantization levels.***quantized*In the figure below, the process has been explained where an input discrete signal

*s(t)*has been quantized into a signal*s*. The input signal_{q}(t)*s(t)*moves between low peak amplitude*A*to high peak amplitude_{L}*A*. This range from_{H}*A*to_{L}*A*is divided into_{H }*M*intervals (an interval represent a quantization level, also referred as quantization interval) each of size*L*such that*L =*(*A*In the figure, eight intervals have been shown i.e. M=8. Let A_{H}- A_{L})/M._{s}*be the peak-to-peak amplitude of**s(t)*i.e. total range between high and low peaks i.e. A_{s}*= A*hence_{H}- A_{L }*L =*A_{s}*/M.*The value of each quantization interval is taken as the center of the interval, in figure, shown as*a*_{0}, a_{1},*a*_{2}, a_{3}, a_{4}, a_{5},*a*The range of values for each interval has been shown as_{6}, a_{7}.*l*_{0}, l_{1},*l*_{2}, l_{3}, l_{4}, l_{5},*l*Moreover, the midpoints between consecutive quantization intervals are shown by_{6}, l_{7}.*A*. Each of these midpoints is at a distance of_{01}, A_{12}, A_{23}, A_{34}, A_{45}, A_{56}, A_{67}*L/2*from its corresponding intervals values e.g.*A*is the midpoint between two quantization intervals whose values are_{67 }*a*and_{6 }*a*and is located at a distance of_{7 }*L/2*from*a*and_{6 }*a*. The distance between two consecutive midpoints and also between two interval values is_{7}*L.*It is, in fact, these mid points which define when the value of the quantized signal*s*changes. It should also be noted that the peak values i.e._{q}(t)*A*and_{H}*A*are at a distance of_{L}*L/2*from their corresponding quantization levels. This results in affecting the peak-to-peak amplitude of*s*shown as_{q}(t)*A*and has a value of_{q}*(M-1)L.*As shown in the figure, when the signal being quantized i.e.

*s(t)*is in the range of*l*, the quantized signal_{7}*s*maintains a constant value of_{q}(t)*a*. As soon as_{7}*s(t)*gets in the range of*l*,_{6}*s*makes a jump to the value of_{q}(t)*a*and maintains this value as long as_{6 }*s(t)*assumes a value in the range of*l*. The sudden jump is made when_{6}*s(t)*crosses the midpoint*A*. Similar changes occur to_{67 }*s*through the rest of the process of quantization. Therefore, at any time, the quantized signal_{q}(t)*s*holds a value from_{q}(t)*a*_{0}, a_{1},*a*_{2}, a_{3}, a_{4}, a_{5},*a*and_{6}*a*. That means, at any moment,_{7}*s*exits in one of the quantization intervals that is nearest to_{q}(t)*s(t).*As the above description and the figure indicate that

*s*is not same as_{q}(t)*s(t)*rather it’s an approximation with a difference between the two. This difference is very important and is known as**. Quantization Error’s magnitude is always less than or equal to***Quantization Error**L/2*(We’ll discuss Quantization Error later). The approximation can be improved by making the interval size smaller, hence increasing the number of intervals (quantization levels). In digital communication, high number of intervals requires high number of bits which itself, in turn, require high transmission bandwidth. On the other hand, if interval size is increased using lesser number of quantization levels, approximation is poor. Therefore, there has to be a tradeoff between approximation and bandwidth.