Error Checking - Pinpoint
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Now let’s talk about error checking. What do you think error checking means when we’re talking about sending and retrieving data? And why do you think it would be useful? Well, let’s take a look at an example. When we send data, such as this text, hello there, sometimes the data might get garbled in transmission. When you were a kid, if you played with radios or walkie talkies, or anything like that, you might have had trouble understanding what other people say, Heck, even it happens today, sometimes on cell phone calls. So if there was a way that we could confirm that the data we’re receiving is the data that was actually sent without adding too much overhead to the data, that might be a really powerful tool, a great intuition to think of is squeezing the data into a smaller space, and then using that to check and see if we got the right answer.
So if we start with the text, hello there, and we squeeze the kerning, between the characters really narrow, we get this image shown below, that we can use as a little check to make sure that the data that we received was the same thing. So on the receiving end, if instead of receiving Hello, there we see Hello, ghere. If we squeeze that together the same way, we noticed that the squeezed image does not look the same as the squeezed image above. And so by sending the squeezed image above, we would know that we didn’t get the right data. And that squeezed image doesn’t take up a whole lot of space. That’s really the intuition behind error checking, we’re trying to send a little bit more data along with a larger transmission. And we can use that little bit of extra data to make sure the transmission itself was received without errors. That’s the real intuition behind this.
So let’s take a look at an example. One of the best ways we could do this is simply adding together the numbers. So if we take the sentence Hello there, and we convert it to ASCII, we see that a capital H is 72, e is 101, L is 108, and so on and so forth. So if we take all of those numbers, we could just add them up, and send all of that ASCII text plus the number 1101. Pretty cool, isn’t it, then on the receiving end, if the T got replaced with a G, obviously, the sum we would get would be different. And we would know that somehow that message got corrupted. Of course, this might be a little bit complex, for example, 1101 doesn’t fit in an eight bit binary number, so we may have to truncate it a bit or use modulo, to reduce it down to a number that fits into an eight bit binary value. Now, of course, this is a very simple form of error checking, and it does have some flaws. So let’s talk about some of these different questions. For example, if we only have the sum that came along with that message, could we use that to recover the original? So if I just gave you 1101? Could you tell me what original message produced that sum? Probably not. It would be really, really difficult to do. So we can’t use it to recover the original.
How well does it detect errors? Well, if we change the T to a G, we would probably detect that. But what if we replaced an L with an M and another L with the letter right before L? How well does this detect errors? Well, let’s consider the case where one of the L’s in hello gets replaced with an M, and the other L gets replaced with a K. M is one greater than L. But K is one less than L. And so when we sum those together, we would still get the exact same sum. So it is possible for us to create an error that the sum itself does not catch, that’s not really good. So as we can see, are the areas that cannot detect absolutely, we could swap characters, we could rearrange the characters, they would all still sum up to the exact same value. There’s tons of ways that we could do this. And of course, we talked about this a little bit already, but 1101 is larger than eight bits. How should we handle that? Well, we could use modular division where we divide it by 256. And keep the remainder, which would be the modulo operation that you learn about in your programming courses that would allow that number to fit in just eight bits. But of course, the big question we always ask in this class, can we do better. And I would argue that there are ways that we can definitely do better than a simple addition error checking.
So let’s take a look at another example of error checking. In this case, it’s called pinpoint error checking. Here, we’ve received a number 483754. So very large number, and we’ve taken that number and we’ve arranged it in a four by four grid. And then we’ve added a column in a row to the end of the grid that we can use to calculate what we call our checks up, which is the sum that we’re going to create to make sure that these numbers are correct. To create that checksum, what we will do is we will sum up each row and each column and then we will keep just the ones place of that value in this call. So for exam We have 4 plus 8 is 12, plus 3 is 15 plus 7 is 22. So here we would put 2, here we have 5 + 4 is 9, + 3 is 12, + 3 is 18. So we would keep the 8. Likewise, we could go down, we have 4 + 5 is 9, + 2 is 11, + 3 is 14. So we would keep the 4, here we have 8 + 4 is 12, + 2 is 14, + 9 is 23. So we would keep 3. And we could do this all the way through until we got all of these numbers, which we see right here, we see 4306 along the bottom, and 2858 along the side, then when we send the message, we could simply add those numbers in where they go, we see down here are 4837 of our original number, plus this 2 for our checksum 5436, plus an 82256, plus a 53397, plus an 8. And then we have 4306, which is our bottom checksum. So we have a 16 digit number. And we’ve added 8 more digits to it to provide a checksum. So it’s gotten a little bit bigger, it’s about 50%, bigger than it was.
But what can we do with this checksum that’s very, very powerful. So here, we’ve received a message. And we have in gray, our original checksum. And then in red, we have the checksum that we calculated from the message that we received. So take a look at this and see if you can figure out exactly where the error is in this received message. All right, let’s see if we can find the error. So we look at our checksum for matches up. 3…, uh-oh, this one doesn’t match up. So we’re going to circle this one. Zero matches up, six matches up, two matches up, eight matches up, uh-oh, here, 5 and the 0 don’t match up, and then the 8 matches up. So if we look at the row and the column, we know that this one is incorrect. Now the big question is, can we use this to restore that value to what it originally was, we know here that this value should be eight larger, or should be five larger, and we know that this value should be there, five larger five smaller. So with this number, if we add five to it, and just keep the ones place, we know that this number should probably have been a two. So let’s look at our original. Uh huh. It was a two. So using this pinpoint checksum by adding a little bit more data to the data that we’re transmitting, not only can we detect an error, but we can actually correct an error. And there are some different ways we can measure this. For example, we can detect a lot of different errors, but we may not be able to correct as many errors as we can detect. And that’s one of the important things to remember about checksum such as this. So now that you’ve seen an example, we’ll have a little quiz after this where you can do another example on your own just to see how this works.