Thursday, May 26, 2011

Pulse Code Modulation: Quantization

The discrete signal i.e. sampled values produced as a result of sampling (discussed in the previous post) has to be quantized, hence the process called Quantization. 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 quantized in one of the fixed number of quantization levels.
In the figure below, the process has been explained where an input discrete signal s(t) has been quantized into a signal sq(t). The input signal s(t) moves between low peak amplitude AL to high peak amplitude AH . This range from AL to AH is divided into M intervals (an interval represent a quantization level, also referred as quantization interval) each of size L such that L =  (AH - AL)/M. In the figure, eight intervals have been shown i.e. M=8. Let As be the peak-to-peak amplitude of s(t)  i.e. total range between high and low peaks i.e. As = AH - AL hence L = As /M. The value of each quantization interval is taken as the center of the interval, in figure, shown as a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 . The range of values for each interval has been shown as l0, l1, l2 , l3 , l4 , l5 , l6 , l7 . Moreover, the midpoints between consecutive quantization intervals are shown by  A01 , A12 , A23 , A34 , A45 , A56 , A67 . Each of these midpoints is at a distance of L/2 from its corresponding intervals values e.g. A67 is the midpoint between two quantization intervals whose values are a6 and a7 and is located at a distance of L/2 from a6 and a7. The distance between two consecutive midpoints and also between two interval values is L. It is, in fact, these mid points which define when the value of the quantized signal sq(t) changes. It should also be noted that the peak values i.e. AH and AL are at a distance of L/2 from their corresponding quantization levels. This results in affecting the peak-to-peak amplitude of sq(t) shown as  Aq and has a value of (M-1)L.
As shown in the figure, when the signal being quantized i.e. s(t) is in the range of l7 , the quantized signal sq(t) maintains a constant value of a7 . As soon as s(t) gets in the range of l6 , sq(t) makes a jump to the value of a6 and maintains this value as long as s(t) assumes a value in the range of l6 . The sudden jump is made when s(t) crosses the midpoint  A67 . Similar changes occur to sq(t) through the rest of the process of quantization. Therefore, at any time, the quantized signal sq(t) holds a value from a0 , a1 , a2 , a3 , a4 , a5 , a6  and  a7 . That means, at any moment, sq(t) exits in one of the quantization intervals that is nearest to  s(t).
As the above description and the figure indicate that sq(t) is not same as s(t) rather it’s an approximation with a difference between the two. This difference is very important and is known as Quantization Error. Quantization Error’s magnitude is always less than or equal to 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.


Anonymous said...

it is obviously understood by every one

Anonymous said...

Really helpfull

waresha khan said...

the error occur after quantiztion,, how it is transmitted with the message signal ??