In the last presentation we completed analog and discrete time signal. In this lecture we will study digital signal. In digital signals we discretize both time and magnitude.
We have to discretize both time and magnitude axis. If you remember the discrete time signals we discretize the time axis but not magnitude. But in case of digital signals we have to discretize the magnitude axis as well. By discretization I mean we have to divide the time axis in equal intervals.
If delta t is the interval then we can find out this interval delta t by t1 minus t0 or we can have t2 minus t1 in the same way. Tn minus Tn minus 1. This is how we can find the interval. Now the next thing that we have to do is to discretize the magnitude axis also.
This is a very simple example in which we are trying to measure the temperature of a city. Capital T is the temperature and this is in degree Celsius. Small t is the time in seconds and we are measuring the temperature at T1, T2, T3, T4 and T5.
If I consider the case. of discrete time signals then let's see what we have. The temperature at time T1 is equal to 9 degree Celsius. You can clearly see it is equal to 9 degree Celsius. We have not discretized the magnitude axis.
So the temperature can take any value from 0 to 45. Every value is allowed from 0 to 45. So 9 degree Celsius is absolutely allowed. For T2, for T2 we have 38 degree Celsius. For T3 we have 24 degree Celsius.
For T4 we have 15 degree Celsius. And for T5 we have 45 degree Celsius. And this is for discrete time signal right.
Now we will consider the case of digital signal and we have to discretize this magnitude axis also. So let's do it. I am going to discretize this and I will have the next level. equal to 15 degree Celsius and another level equal to 30 degree Celsius. So we have 0, 15, 30, 45 as the allowed values for the temperature capital T.
This temperature can take values equal to 0, 15, 30, 30 and 45 only. We divide the magnitude axis into fixed number of levels and the signal can take value equal to this levels only. This line is very important. important, the most important part that you have to remember in digital signals. This signal can take value equal to this levels only.
Right! Now we will consider the same case, the same temperatures for the same time and we have to find out what is the value for temperature capital T at this times. We are considering the digital signal in this case. So let's start.
Temperature T at time T1. At T1 we have T1 is equal to T2. So we we have 9 degree Celsius but 9 is not allowed it is between 0 and 15. It is between 0 and 15. Now what we have to take 0 or 15. The difference between 0 and 9 is 9 degree Celsius and the difference between 9 and 15 is 6 degree Celsius. So this 9 degree Celsius is near to 15 degree Celsius. So at first sight it seems we have to consider 15 degree Celsius but this is not true.
To minimize the the error we have to take the lower value. We have to take 0 degree Celsius. This is the key point that you should remember. We don't select the higher value, we select the lower value. So the temperature T at time T1 is equal to 0 degree Celsius.
And the temperature T at time T2 is 38, but 38 is also not allowed. It is between 3030 and 45. We again have to take 3030 because this is the lower level. So at T2 we have 30 degree Celsius. Temperature T at T3 is equal to at T3 we have 24. 24 is near to 30 but again we have to consider the lower level so 15 degree Celsius. is the answer.
At T4 we have 15. 15 is definitely allowed so we have 15 degree Celsius. At T5 we have 45 and 45 is allowed so 45 degree Celsius. Celsius and this is the values in case of digital signal. So you can clearly see the difference between discrete time signals and digital signals.
We can have any value of the temperature within 0 to 45 but in this case we have the value for temperature equal to this levels only. Now we will see one more example in which we will consider the voltage. This is the time axis. Capital V is the voltage. We are considering the digital signal and we have 0 volts and 5 volts as the two values that are allowed.
Let's say at any time t1 the voltage is equal to 2 volts then voltage at T1 is equal to 0 volts because I have already explained you we have to consider the lower value so I have taken 0 volts. Now you can see we have error of 2. volts because the observed voltage was 2 volts but we are getting 0 volts. So how to overcome this error? We can overcome this error by increasing the number of levels.
If we increase the number of levels Error will reduce. This is very very important point. Very very important point.
On increasing the number of levels error will reduce. Let's see how. I am going to divide 0 to 5 in 4 equal parts. So I will have 1.25 as the next level. 2.5.
and 3.75 okay these are the levels so we now have 0, 1.25, 2.5, 3.75 and 5 as the allowed values for this voltage V. So voltage at time T1 is equal to 1.25 volts because we have to consider the lower value and 2 volt is between 1.25 and 2.5 now we can easily take 1.25 instead of 0. So the error is reduced and now we have error of 0.75 volts only. Earlier the error was 2 volts but now the error is 0.75. If we increase the number of levels more, for example if I have levels equal to 0, 1, 2, 3, 4 and 5 then this 2 volt can clearly be measured as 2 volts. So voltage at T1 in this case is equal to 2 volts and error of 0 volts is there.
So we have reduced the error of 2 volts to 0 volts by increasing the number of levels. This is all for this presentation. But there is one question if we were already having analog and discrete time signals then what is the need of digital signal? What is the need of digital signal? Signal, this is the question and in the next lecture we will discuss this need of digital signal.
If you know the answer of this question go ahead and post your answer in the comment section. This is all for this presentation. See you in the next one.