 Hello everyone, today we are discussing on Quantization. At the end of this session, what we expect from students that students will be able to define Quantization, describe types of Quantization. They should be able to explain Quantization process in detail. We are going to cover these topics, definition, types of Quantization, Quantization process and Quantization error. First of all, we will discuss about the basic definition of Quantization. Quantization is the process of constraining an input from a continuous or otherwise large set of values such as the real numbers to a discrete set such as the integers. Now, here what is the actual set of values means what the real sampling values, those are converted into a complete integer values. The real numbers means what your voltage can be actual real time voltage can be say for example, 2.1 volt, 3.2 volts, etc. But when they are converted to discrete or quantized value, those are converted into complete integers, those are represented to say 2 volts, 3 volts, those are the known standard levels. The analog to digital converter performs this type of function to create series of digital values out of the given analog signal. This signal to get converted into digital has to undergo sampling and quantizing. The quantizing of an analog signal is done by discretizing the signal with a minimum number of quantization levels. Quantization is representing the sampled values of the amplitude by finite set of levels which means converting continuous amplitude sample into a discrete time signal. So, what are the different types of quantization? Broadly there are two types of quantization. One is uniform quantization also known as linear quantization, the other is non-uniform quantization also known as non-linear quantization. The type of quantization in which the quantization levels are uniformly spaced is termed as uniform quantization. Here what we observe in uniform quantization that the difference between the two quantization levels it is almost fixed, the difference between two quantization levels is known as step size. So, here in uniform quantization levels, uniform quantization the step size remains constant whereas the type of quantization in which the quantization levels are unequal and mostly the relation between them is logarithmic is termed as non-uniform quantization. Here we in the non-uniform quantization we observe that step size are going to vary continuously depending on the input signal requirement or depending on the quantization requirement. There are two types of uniform quantization, they are mid-rise, the other is mid-trade. You can observe these two figures, what change that you observe? It looks at first instant you observe that they are similar but they are not same. At the center you can see here this step is observed over here whereas in the mid-trade it is observed over here. So in the this is the mid-rise the step size is rise in between the two equal it is symmetric about two accesses, here also it is in the horizontal line there it is symmetric. So there are two types of uniform quantization one is known as mid-rise the other is known as mid-trade. Now so that is what we are discussed the mid-rise type is so called because the original lies in the middle of a raising part of the staircase like graph the quantization levels in this type are even in number. The mid-trade type is so called because the original lies in the middle of trade of the staircase like graph the quantization levels in this type are odd in numbers. Both the mid-trades and mid-trade type uniform quantizers are symmetric about the origin. Now let us try to understand the overall quantization process. For this you are going to follow this diagram here you can see that this is the example that I have shown you here this is the output and this is your timing access. So these are the input signals this is what the input signal sample here these are the sampling instances T1, T2, T3 and these are the sampling instances sampling instances that all depends on the sampling frequency the sampling frequency must be at least the twice of maximum input signal frequency that it has to follow. Whatever I have shown with the yellow lines those are the quantization levels the difference between two quantization levels is the step size. So how we are going to try to understand the basic process of the quantization here you can observe here here T1 the sample that sample at time instant T1 you observe that the sampling value is 1.8 volts T2 is 2 volts T3 is 3.1 volt and T4 is 4.4 volts. These are the actual sample values these are the actual sample values. Now what exactly happens in the quantization now it observes that for example the first case that is at the time instant T1 since the real value is 1.8 volt so it is approximate to 2 volts why it is because the 1.8 volt it is in between standard value 1 and 2. Now here the comparison take place so one it observes that 1.8 volt is more closer to 2 volt than the 1 volt so it is approximate to 2 volts like that for 2 volt there is no problem it is taken as it is for 3.1 volt what happened 3.1 volt it is in between 3 and 4 volts so it is more closer to 3 volts so it is approximate to 3 so like that for 4.4 it is approximate to 4 volts. So this is how the actual sample value those are quantized approximate it to no standard values so these are the real time number real values and these are complete integer values. So here what we can see here it is a linear quantization or uniform quantization where we observe that step sizes are fixed step sizes are fixed so the variation between the relation between input and output it is linear in nature. So this is what is the process of quantization. Now let us this is an example that details have been shown this again the input signal and this is what is the quantization take place. Now when this quantize the quantize output signal will look like this so this is actually the quantize signal. So now here when we say that when we are going to quantize there is always the difference between the actual sample value and quantize value and the difference is known as quantization error. So for any system during its functioning there is always a difference in the values of its input and output the processing of the system results in an error which is the difference of those values. So the difference between an input value and its quantize value is called quantization error. A quantized quantizer is a logarithmic function that performs quantization that is the rounding of the value. So EDC is the one example of the quantizer. Now this is what is the example that has been shown this is your input signal when it is quantized so it is going it will look like this and the difference between these two that is actually the quantization error. So output quantization noise signal can be it will appear as like this. Now what is non-uniform quantization? Non-uniform quantization again it is what we observe that the step size is going to vary. So here in the non-uniform quantization what you can see here is that the relation between the input and the output that has been shown. This is the case of the linear quantization. So what exactly happened that at the transmitter some non-linearity has been observed here and it will look like this. So this technique is known as a compression technique. Generally this technique is adopted in PCM system to minimize the quantization error and exactly the reverse procedure is performed at the receiver and that is known as expansion. So this is a compression and this is an expansion. There are two commonly used non-uniform quantizations are there one is known as a law the other is known as mu law. So both are non-linear quantizing characteristics you can see here. So for a law for the value of when a is equal to 1 it is the case of linear or uniform quantization the value can go on increase. So accordingly the characteristics of this compression that also changes and the maximum value of that is 85.5 whereas for mu law it ranges from 0 to 255 for the value 0 it appears as a linear and when it is 255 it is a maximum value. So here what we can observe the steps are steps sizes are going to vary continuously. So this is what all about non-uniform quantization. So here the word compounding that is what is used in PCM it is a combination of compressing and expanding which means that it does work. This is a non-linear technique used in PCM which compresses the data at the transmitter and expand the same data at the receiver. The effects of noise and crosstalk are reduced by using this technique. So there are two types of compounding just now we have studied one is a law and mu law. Uniform quantization for a law when a is equal to 1 the characteristic curve is linear and no compression is done a law has mid-rise at the origin hence it contains non-zero value. A law compounding is mostly used in PCM telephony system. For mu law compounding technique uniform quantization is achieved at value mu is equal to 0 where the characteristic curve is linear and no compression is done. Mu law has mid-raid at the origin hence it contains a zero value. Mu law compounding is used for speech and music signals.