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And we'll roll forward and kind of see what floating point gives us. So let me just start by saying, well, what's the range of 32-bit two's complement? You guys know it, right? Everyone knew that? If I say, OK, write down on a piece of paper? OK, good, I'm glad you did. I usually write it this way. I don't remember | [
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And we'll roll forward and kind of see what floating point gives us. So let me just start by saying, well, what's the range of 32-bit two's complement? You guys know it, right? Everyone knew that? If I say, OK, write down on a piece of paper? OK, good, I'm glad you did. I usually write it this way. I don't remember. I remember 4 billion. The rest is a blur. But I think, I'm sorry, 2 billion. | [
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The rest is a blur. But I think, I'm sorry, 2 billion. Yeah, 4 billion for unsigned. So let's write some banking software. So forget all this real number stuff. Let's just do banking software. And I say, OK, I'm going to store one 32-bit two's complement number to represent pennies in your account. Anyone have more? No? All right. So we're done. Got our banking software. Let's say you have a friend who has more. | [
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Got our banking software. Let's say you have a friend who has more. Well, OK, 64 bits. There's not that much money in the world. So we're still done. So what about chemistry? Anyone in chemistry? OK, good. So you got chemistry homework, right? You probably want to use a computer instead of going to the lab and being dangerous with chemicals. What's Avogadro's number? 6.0. OK, that's what I had, too. | [
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0. OK, that's what I had, too. But well, so 10 to the third is 2 to the 10th. So that's about 80 bits. OK, so who's got Avogadro's number to 80 bits? That's 24 significant figures. The ones in chemistry, you know this one, right? No? What are we going to do? What are we going to do? OK, we've got to give you some homework. I should let you go early, because this is going to take you quite some time. | [
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The ones in chemistry, you know this one, right? No? What are we going to do? What are we going to do? OK, we've got to give you some homework. I should let you go early, because this is going to take you quite some time. Maybe we can just be close. Maybe we can just make up some bits for the other lower bits. Who's going to tell you you're wrong? Not the chemists, right? All right, so what about quantum mechanics? Anyone have quantum mechanics? OK, so some of you do. | [
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Maybe we can just make up some bits for the other lower bits. Who's going to tell you you're wrong? Not the chemists, right? All right, so what about quantum mechanics? Anyone have quantum mechanics? OK, so some of you do. What about Planck's constant? Anyone have that one yet? OK. You should use Ergs, because Ergs are cool, and all the physicists use them. So yeah, those have to go after the binary point. | [
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You should use Ergs, because Ergs are cool, and all the physicists use them. So yeah, those have to go after the binary point. It's a small number, right? So we'll need 90 bits after the binary point. Oh, that's 24 digits for Avogadro's number. So do we really need 170 bits of precision? Do we need, what is it, 51 significant figures of Avogadro's number to do our chemistry? Probably not, right? But it would be nice if we could represent both of those numbers. | [
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Oh, that's 24 digits for Avogadro's number. So do we really need 170 bits of precision? Do we need, what is it, 51 significant figures of Avogadro's number to do our chemistry? Probably not, right? But it would be nice if we could represent both of those numbers. So I think people mentioned a bigger range of numbers, maybe with the same number of bits, maybe 32. So small numbers and big numbers. | [
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So I think people mentioned a bigger range of numbers, maybe with the same number of bits, maybe 32. So small numbers and big numbers. So what can we sacrifice to do that? Well, let's think about what we do as humans. I mean, we've already done it a few times in these slides. We have this notion of scientific notation. So we say, well, what we care about is maybe 5, 6, 7 significant figures. | [
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We have this notion of scientific notation. So we say, well, what we care about is maybe 5, 6, 7 significant figures. And we have an exponent. But we don't just write all the rest of the digits just because it's a big number. We don't write all the zeros just because it's a small number. We put it in this handy scientific notation, which has three parts. There's a sine | [
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] |
I mean, we've already done it a few times in these slides. We have this notion of scientific notation. So we say, well, what we care about is maybe 5, 6, 7 significant figures. And we have an exponent. But we don't just write all the rest of the digits just because it's a big number. We don't write all the zeros just because it's a small number. We put it in this handy scientific notation, which has three parts. | [
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We don't write all the zeros just because it's a small number. We put it in this handy scientific notation, which has three parts. There's a sine. There's these significant figures, which I'll call them antissa. It refers to the precision, right? More digits means we know the answer better. And then we have this exponent. So let's make a representation that has those three pieces. So that's what we call floating point. | [
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So let's make a representation that has those three pieces. So that's what we call floating point. So Belval Kahan at Berkeley really is the author of this floating point representation. Prior to the standardization, there were lots of different floating point methods in different computers. And they had lots of different numerical problems. So actually, IEEE floating point is really well thought out. | [
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And they had lots of different numerical problems. So actually, IEEE floating point is really well thought out. And that's mostly due to Kahan. So pretty much every modern computer implements this. Well, any modern computer that supports floating point will implement IEEE 754 standard floating point. So this is what single precision looks like. It's got 32 bits. It has a sine, which of course is one bit. | [
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] |
It's got 32 bits. It has a sine, which of course is one bit. It has an exponent, which it uses eight bits. It has an antissa. Remember, those are the significant figures. You've got 23 bits for that. It's about six digits. So what does a bit value represent? So let me ask you a question first. So what are the values of that digit? Can it be anything? Maybe 0 to 4, 1 to 7? What can that digit be? Remember, canonical form is something it's not supposed to be, right? It's not supposed to be 0, because you can change the exponent. | [
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] |
So what does a bit value represent? So let me ask you a question first. So what are the values of that digit? Can it be anything? Maybe 0 to 4, 1 to 7? What can that digit be? Remember, canonical form is something it's not supposed to be, right? It's not supposed to be 0, because you can change the exponent. You can change the exponent. You can change the exponent. It's not supposed to be 0, because you can change the exponent. | [
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You can change the exponent. It's not supposed to be 0, because you can change the exponent. So remember, you're supposed to not have 0. You can have anything 1 to 9. So you can change the exponent if you had a 0 there. Just move it around so that you've got canonical form, unique form. All right, so what about in binary? Same question. Now binary. I'll give you the answer. Not 0 is the answer. | [
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] |
I'll give you the answer. Not 0 is the answer. But that means it's just a 1. OK, so now how many things do I need to represent one value? How many bits? 0. It's always a 1. I don't need any bits to represent that 1. So the leading one is implicit. So now we can go on and look at how we interpret this number. So if I have these bits, I can plug it into this formula, and that will tell me the number. | [
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] |
So now we can go on and look at how we interpret this number. So if I have these bits, I can plug it into this formula, and that will tell me the number. So negative 1 to the sine. So if sine is a 0, that means it's positive. If sine is a 1, that means it's negative. This implicit 1 here, remember it's implicit. It doesn't take any actual bits in the number, because implicitly there's going to be a 1 in scientific notation in binary. | [
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This implicit 1 here, remember it's implicit. It doesn't take any actual bits in the number, because implicitly there's going to be a 1 in scientific notation in binary. Then the mantissa bits, all 23 of them, multiplied by 2 to the bits of the exponent interpreted as unsigned minus 127. So we can go all the way from about negative 120 to about positive 120. 2 to the negative 120, 2 to the positive. | [
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So we can go all the way from about negative 120 to about positive 120. 2 to the negative 120, 2 to the positive. It's about 10 to the 38. Plus or minus. Yeah. Can you explain that again? Sure. Ah, yeah, yeah | [
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] |
It doesn't take any actual bits in the number, because implicitly there's going to be a 1 in scientific notation in binary. Then the mantissa bits, all 23 of them, multiplied by 2 to the bits of the exponent interpreted as unsigned minus 127. So we can go all the way from about negative 120 to about positive 120. 2 to the negative 120, 2 to the positive. It's about 10 to the 38. Plus or minus. | [
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It's about 10 to the 38. Plus or minus. Yeah. Can you explain that again? Sure. Ah, yeah, yeah. So here in decimal, the answer was not 0. So we changed the exponent to make it be not 0. The answer is also not 0 in any other base, because you can always shift the exponent. But in binary, that has a further implication. If it's not 0, it must be 1. And if there's only one thing to choose from, you don't need to store any bits to represent what that leading digit is. | [
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If it's not 0, it must be 1. And if there's only one thing to choose from, you don't need to store any bits to represent what that leading digit is. It's always 1. There's always a 1 there. Now, there's one caveat that I'll show in the next slide. But the general interpretation is, yes, there's always a 1 there. Yeah. Why is it precocious? It's a biased offset. It's so that you can have big and small exponents. | [
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Why is it precocious? It's a biased offset. It's so that you can have big and small exponents. So if you want something like Planck's constant, you need a very small exponent. If you want something like Avogadro's number, you need a big one. So you want to be able to represent big and small. So this just means my 8-bit unsigned value in that exponent field is going to represent big exponents and little ones. | [
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So you want to be able to represent big and small. So this just means my 8-bit unsigned value in that exponent field is going to represent big exponents and little ones. And the choice of 127 is to make that balanced. That's why. Yeah. What's the name of that exponent? It's unsigned. It's unsigned. It's actually, there's a different name for it. So if you go read the standard, it's called a biased offset or something. | [
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It's actually, there's a different name for it. So if you go read the standard, it's called a biased offset or something. But if you look at it as 8 bits and you translate it into decimal as an unsigned number, then you can subtract 127. And that'll give you the right answer. It's not two's complement. So yeah. Yeah. For doubles versus quotas, what are doubles basically? I know that you have 14 points of precision. | [
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Yeah. For doubles versus quotas, what are doubles basically? I know that you have 14 points of precision. I don't remember. I'd have to look it up. So if the double, you have 64 bits in total. One is the signed bit, obviously. And then the exponent and the mantissa split the other 63. I don't remember exactly how. And then there's actually now a standard quad, which is 128 bits. Yeah. | [
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And then there's actually now a standard quad, which is 128 bits. Yeah. Yeah. How would a standard quad work in terms of like? Yeah. So which bit corresponds to a half depends on your exponents. But the first bit of the mantissa in order appears after the implicit one. And so you would just take the bits here and write them out. And that would give you the binary scientific notation form of the number. | [
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And so you would just take the bits here and write them out. And that would give you the binary scientific notation form of the number. Yeah. So going back and forth on paper is relatively easy. If you can put something in binary, and we'll do some examples. If you can put it in binary scientific notation, you basically just copy the bits into the 32. Yeah. OK. All right. So the one caveat, well, a couple. | [
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All right. So the one caveat, well, a couple. IEEE supports infinity, not a number. You don't need to know most of this stuff. The one thing you do need to know, so it's here in the notes if you want to look at it. You don't need to use it, though. The one thing you do need to know is that, of course, we need a 0 | [
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If you can put it in binary scientific notation, you basically just copy the bits into the 32. Yeah. OK. All right. So the one caveat, well, a couple. IEEE supports infinity, not a number. You don't need to know most of this stuff. The one thing you do need to know, so it's here in the notes if you want to look at it. You don't need to use it, though. The one thing you do need to know is that, of course, we need a 0. | [
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You don't need to use it, though. The one thing you do need to know is that, of course, we need a 0. And 0 does not start with an implicit 1. So we need to be able to represent the number 0. That's the bit pattern of all 0's. And that was not accidental. So if you have a floating point number that has all 0's in it, it's the same 0 representation as unsigned in 2's complement. It's all 0's. | [
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So if you have a floating point number that has all 0's in it, it's the same 0 representation as unsigned in 2's complement. It's all 0's. But the denormalization is kind of beyond the scope of the class. So I'll put it in the slides for you. And yeah. It's still the implicit. No, no, no. I mean, 0 can't have an implicit 1. If you write the number 0 in scientific notation, you have to put a 0. | [
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I mean, 0 can't have an implicit 1. If you write the number 0 in scientific notation, you have to put a 0. So that's the exception to the rule. So the all 0 bit pattern is 0 in floating point. So how do you get the 0? The representation is well-defined. So we've just agreed on it. So now, if you were to go out and build hardware, you know what to do. Yeah, yeah. So that's a good question. | [
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Yeah, yeah. So that's a good question. But all representations that computers are going to use have to be defined in advance and agreed upon. So this is the one exception. I mean, there's also denormalized numbers. But that's beyond the scope of the class. All right. OK. So conversion. So you can do this in the tool, too, if you want to practice it. But it's not actually too hard. You convert to binary, change to scientific notation, and encode each of the three parts. | [
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But it's not actually too hard. You convert to binary, change to scientific notation, and encode each of the three parts. So we can do an example. The integer part, you know how to do. We've done that already. The fractional part is actually equally easy. So you can write the polynomial in the other direction. So it would be these inverse powers of 2 is our fractional part. And then you think, well, if I multiply f by 2 and I multiply this side by 2, the only way this side can be bigger than 1 is if this term is 1. | [
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So it would be these inverse powers of 2 is our fractional part. And then you think, well, if I multiply f by 2 and I multiply this side by 2, the only way this side can be bigger than 1 is if this term is 1. Because all of these, there's a quarter, there's an eighth, and so forth. So the only way for the right side to be bigger than 1 is for a negative 1 to be 1. So all you have to do is multiply both sides, multiply your number by 2. | [
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] |
So the only way for the right side to be bigger than 1 is for a negative 1 to be 1. So all you have to do is multiply both sides, multiply your number by 2. And every time you get a 1, that's a 1 bit. And every time you have something less than 1, that's a 0 bit. So let me walk you through an example. So let's say this number, it looks intimidating, but I promise we won't have too many bits. | [
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So let me walk you through an example. So let's say this number, it looks intimidating, but I promise we won't have too many bits. So we have that number. First, we write 5 in binary. We get 1, 0, 1. Now we need to convert that fraction. So multiply it by 2. We get something less than 1. So a negative 1 is 0. Subtract 0. Multiply it by 2. Still less than 1. Another 0. Subtract 0. Multiply it by 2. | [
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] |
Subtract 0. Multiply it by 2. Still less than 1. Same thing. Keep going. Eventually, we will get something bigger than 1. So that tells us a minus 5 is 1 | [
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So we have that number. First, we write 5 in binary. We get 1, 0, 1. Now we need to convert that fraction. So multiply it by 2. We get something less than 1. So a negative 1 is 0. Subtract 0. Multiply it by 2. Still less than 1. Another 0. Subtract 0. Multiply it by 2. Still less than 1. Same thing. Keep going. Eventually, we will get something bigger than 1. So that tells us a minus 5 is 1. | [
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Eventually, we will get something bigger than 1. So that tells us a minus 5 is 1. And then we'll subtract the 1 off of both sides. So we'll get half left. And then we'll multiply by 2 and get 1. So a negative 6 is also 1. And then we're done. So that's our fraction in binary. So we would give you shorter problems on exams if we were to give you something like this. And we certainly wouldn't give you something where you had to look at all 23. | [
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So we would give you shorter problems on exams if we were to give you something like this. And we certainly wouldn't give you something where you had to look at all 23. That would just be mean. All right? You don't want to do that by hand. You want to teach your computer to do that. And I promise the tool is also like 4 or 5 bits. So even if they look intimidating, if you work it out on paper, it's just a few bits. | [
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And I promise the tool is also like 4 or 5 bits. So even if they look intimidating, if you work it out on paper, it's just a few bits. All right. So one last step. So we've got it in binary. So change it to binary scientific notation. And now this 1 here is implicit. These are the bits of the mantissa. Then you've got a bunch of implicit 0's after that. So the other remaining bits are 0's. | [
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Then you've got a bunch of implicit 0's after that. So the other remaining bits are 0's. This is our exponent. So into the field, we have to put 129. So then 129 minus 127 is the 2 that we want. And there's our mantissa with 15 more 0's. So that's it. Yeah. This one I think I'll save, because I want to make sure you understand it. Floating point's a little weird. It's not associative. | [
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Floating point's a little weird. It's not associative. So we'll talk about that on Wednesday. Thanks. 1 to 3 tomorrow. Oh, I stole my mic live. It doesn't matter. 1 to 3 tomorrow. Oh, I stole my mic live. It doesn't matter. It doesn't matter. | [
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] |
So we're going to do another review session. Same rules as last time, so hopefully people have thought about what they want to talk about. Anyone want to throw out the first suggestion? Eric? Okay, sure. One comment quickly on that is that essential prime implicants are not, that name is not required knowledge. And I know it showed up on some of the previous midterms, but you don't need to know what that means. | [
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One comment quickly on that is that essential prime implicants are not, that name is not required knowledge. And I know it showed up on some of the previous midterms, but you don't need to know what that means. We do ask sometimes whether the solution to a particular KMAP is unique or not. And so we can talk about that. But knowing what's an essential prime implicant is not necessary. | [
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And so we can talk about that. But knowing what's an essential prime implicant is not necessary. Mohamed? Okay. Okay. And, yeah, Daniel? I'm sorry? Decoders. Okay. And then? Actually, let me. Adders. Okay. Some of these are pretty specific, so they won't take much time. Anything else people want to throw up there before we vote? See if there's anyone. Okay, we'll give both of you one last chance. | [
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Anything else people want to throw up there before we vote? See if there's anyone. Okay, we'll give both of you one last chance. Go ahead. Okay, Mohamed, what were you going to add? You mean the canonical ones? Okay. All right, so this is a long list, so let's do voting quickly so we don't spend too much time on that. So you can vote for as many as you want. Essential prime implicants? Two. | [
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So you can vote for as many as you want. Essential prime implicants? Two. N-type MOSFETs? About 20. Decoders? Oh, lots. Oh, my goodness. Okay, 50. Timing diagrams? A lot. 40, let's say. DeMorgans law? Maybe about 15. Matters? 15. Functions of MUXes? About 30 to 40. Analyzing sequential feedback circuits? Lots again. 45, I think. Canonical SOP, POS? Three. Okay, so let's start off with decoders then. | [
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Canonical SOP, POS? Three. Okay, so let's start off with decoders then. So, decoder basically gives you minterms of the input. So, the way it works, you would put in some number of wires. Let me draw it the way they've been drawing it in other sections. The way we drew it was like this. These are the same. So, basically, you're taking in some number of select wires, and those select signals you can think about as having some kind of coded value. | [
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These are the same. So, basically, you're taking in some number of select wires, and those select signals you can think about as having some kind of coded value. And you want to know which one of those coded values is held, and so you create an output for each possible bit pattern. So, for two bits, you've got four possible bit patterns. So, each of those wires on the outside, on the right side, carries a one if and only if the bit pattern coming in matches that particular number. | [
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] |
So, for two bits, you've got four possible bit patterns. So, each of those wires on the outside, on the right side, carries a one if and only if the bit pattern coming in matches that particular number. So, 0, 1, 2, or 3 are the four possible bit patterns interpreted as unsigned. The enable then just enables the decoder to output a one at all. So, if enable is zero, the decoder outputs all zeros. | [
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The enable then just enables the decoder to output a one at all. So, if enable is zero, the decoder outputs all zeros. So, enable equals zero means that all outputs are zero. And enable equals one means exactly one output is a one. Because the bit pattern can only match one of the four possible bit patterns. So, this is a decoder's useful primarily in implementing logic where we need to know which of those four bit patterns was being carried. | [
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Because the bit pattern can only match one of the four possible bit patterns. So, this is a decoder's useful primarily in implementing logic where we need to know which of those four bit patterns was being carried. For a three-day decoder, for example, which of the eight possible bit patterns was being carried by the inputs. So, a lot of things like vending machines or memories will make use of them. | [
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For a three-day decoder, for example, which of the eight possible bit patterns was being carried by the inputs. So, a lot of things like vending machines or memories will make use of them. We can also use them for implementing arbitrary logic functions by just using OR gates to OR together the minterms | [
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] |
So, this is a decoder's useful primarily in implementing logic where we need to know which of those four bit patterns was being carried. For a three-day decoder, for example, which of the eight possible bit patterns was being carried by the inputs. So, a lot of things like vending machines or memories will make use of them. We can also use them for implementing arbitrary logic functions by just using OR gates to OR together the minterms. | [
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So, a lot of things like vending machines or memories will make use of them. We can also use them for implementing arbitrary logic functions by just using OR gates to OR together the minterms. Remember, these are minterms, right? So, matching a particular bit pattern is the same as a minterm. So, each of these outputs correspond to a minterm. If you wanted to know how to build them, then you can just build the AND gates corresponding to the minterms. | [
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So, each of these outputs correspond to a minterm. If you wanted to know how to build them, then you can just build the AND gates corresponding to the minterms. So, you can say, well, here's an AND gate that takes enable along with S1 prime and S0 prime. And so, this would be the zero output. And then the AND gate that takes, I'll just draw enable linked up. The others I'll leave along with S1 prime and S0. | [
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And then the AND gate that takes, I'll just draw enable linked up. The others I'll leave along with S1 prime and S0. This would be the one and hook up enable to all four of them. This would be S1, S0 prime. That would be the two. And then S1, S0. That would be the three. And so, that's how you would implement a decoder. You would basically generate all the minterms with AND gates. If there were an enable input, AND it together with all of the minterms. | [
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You would basically generate all the minterms with AND gates. If there were an enable input, AND it together with all of the minterms. And then each of those AND gates would give us one output. Yeah. Yeah, so the outputs, the labeling of outputs on a decoder is usually using the unsigned representation of the input values. So, usually the inputs we would name something like select and it'd be S1, S0. | [
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Yeah, so the outputs, the labeling of outputs on a decoder is usually using the unsigned representation of the input values. So, usually the inputs we would name something like select and it'd be S1, S0. Or for five bits, it would be S4 down through S0. And then the output labels would be the unsigned values. So, if you had a 5 to 32 decoder, they'd be labeled from 0 to 31. And each of those outputs would correspond to the five bit pattern represented by S4 through S0 as an unsigned number. | [
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So, if you had a 5 to 32 decoder, they'd be labeled from 0 to 31. And each of those outputs would correspond to the five bit pattern represented by S4 through S0 as an unsigned number. If you then wanted to make different functions, it's not terribly interesting with a 2 to 4 decoder. But if you wanted to make XOR, for example, how would I do that with, say, another OR gate down here? But which minterms does XOR have? Yeah, 1 and 3, right? So, I'd pull off the 1 and 3 outputs and OR those together and that would give me XOR. | [
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If you then wanted to make different functions, it's not terribly interesting with a 2 to 4 decoder. But if you wanted to make XOR, for example, how would I do that with, say, another OR gate down here? But which minterms does XOR have? Yeah, 1 and 3, right? So, I'd pull off the 1 and 3 outputs and OR those together and that would give me XOR. Yeah, 1, 2, sorry. What about OR? Yeah, so OR should be true for this minterm, this minterm, and this minterm. | [
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Yeah, 1, 2, sorry. What about OR? Yeah, so OR should be true for this minterm, this minterm, and this minterm. So, again, 2 to 4 decoder is not terribly interesting. It would be easier to just pull S1 and S0 directly. You can do any function using, by the way, the canonical SOP version, right? Because the canonical SOP is the sum of minterms. So, if you write any function of S1, S0 as a sum of minterms, then you go find those minterms or the outputs of the decoder together that will give you the function. | [
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You can do any function using, by the way, the canonical SOP version, right? Because the canonical SOP is the sum of minterms. So, if you write any function of S1, S0 as a sum of minterms, then you go find those minterms or the outputs of the decoder together that will give you the function. Anything else you want to talk about with decoders? Yeah, this is no longer really used that much in practice. | [
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] |
So, if you write any function of S1, S0 as a sum of minterms, then you go find those minterms or the outputs of the decoder together that will give you the function. Anything else you want to talk about with decoders? Yeah, this is no longer really used that much in practice. You know, if you were building something out of TTL and you had a decoder and you wanted an arbitrary function, maybe you'd use it. | [
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] |
Anything else you want to talk about with decoders? Yeah, this is no longer really used that much in practice. You know, if you were building something out of TTL and you had a decoder and you wanted an arbitrary function, maybe you'd use it | [
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] |
So, if you write any function of S1, S0 as a sum of minterms, then you go find those minterms or the outputs of the decoder together that will give you the function. Anything else you want to talk about with decoders? Yeah, this is no longer really used that much in practice. You know, if you were building something out of TTL and you had a decoder and you wanted an arbitrary function, maybe you'd use it. | [
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] |
Anything else you want to talk about with decoders? Yeah, this is no longer really used that much in practice. You know, if you were building something out of TTL and you had a decoder and you wanted an arbitrary function, maybe you'd use it. But these days, it's easier to just build a function. Yeah, Mohammed? This is the decoder. Yeah, this is how the decoder is built, these four AND gates. | [
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Yeah, Mohammed? This is the decoder. Yeah, this is how the decoder is built, these four AND gates. I didn't draw the NOT gates to complement these, but this is the decoder. So, let me put a box around it. Sorry, I added more logic. That's the decoder. I zoom this. So, those AND gates are the decoder. All right, so let me do... It was not on the topic, so let me see what happens timing-wise. | [
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. It was not on the topic, so let me see what happens timing-wise. Okay, so next up then was supposed to be analyzing sequential feedback circuits, but let me instead, because timing diagrams I think will be brief, just talk a little bit about timing diagrams. I mean, really, I know there have been a couple of questions on timing diagrams. Oops, these are not all... There have been a couple of questions on timing diagrams on some of the previous exams. | [
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. There have been a couple of questions on timing diagrams on some of the previous exams. You need to know what a timing diagram is, so what it's, you know, you should be able to recognize one, for example. Most of the stuff we're going to do, you wouldn't draw any significant timing, certainly in this part of the class, because we're assuming clock synchronous sequential circuits, so everything operates on a clock signal. | [
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You need to know what a timing diagram is, so what it's, you know, you should be able to recognize one, for example. Most of the stuff we're going to do, you wouldn't draw any significant timing, certainly in this part of the class, because we're assuming clock synchronous sequential circuits, so everything operates on a clock signal. And I don't think we've ever... I mean, I'd have to go back and look through the old exams, but usually we don't ask you to reason about gate delays and things like that on a timing diagram. | [
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. I mean, I'd have to go back and look through the old exams, but usually we don't ask you to reason about gate delays and things like that on a timing diagram. So it would really just be mapping out the kind of thing you did in the homework where you would draw the clock cycles, right? And you would represent what's going on in the different signals from clock cycle to clock cycle without more information than that in the timing diagram. | [
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I mean, I'd have to go back and look through the old exams, but usually we don't ask you to reason about gate delays and things like that on a timing diagram. So it would really just be mapping out the kind of thing you did in the homework where you would draw the clock cycles, right? And you would represent what's going on in the different signals from clock cycle to clock cycle without more information than that in the timing diagram. So, you know, it's the kind of thing we also did in lecture where, for example, if you had a binary counter. | [
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So it would really just be mapping out the kind of thing you did in the homework where you would draw the clock cycles, right? And you would represent what's going on in the different signals from clock cycle to clock cycle without more information than that in the timing diagram. So, you know, it's the kind of thing we also did in lecture where, for example, if you had a binary counter.. | [
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So, you know, it's the kind of thing we also did in lecture where, for example, if you had a binary counter... Let's just make it easy, so we'll give Z1 and Z0 outputs. And of course, there's a clock input, so here's the clock. Then the clock should be a square wave, at least as best I can draw a square wave. And so then if your counter is starting at zero, you might say, okay, so, you know, when do the value. | [
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Then the clock should be a square wave, at least as best I can draw a square wave. And so then if your counter is starting at zero, you might say, okay, so, you know, when do the value... You know, draw the rest of Z1 and Z0 for a two-bit binary counter. So when should Z1 and Z0 change? Yeah, the rising edges, right? So the first thing would be to go mark the rising edges. Here's one, here's another one, and so what should change at this first rising edge? Z0, right? Because it's a two-bit binary counter, so it should go from zero to one. | [
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So when should Z1 and Z0 change? Yeah, the rising edges, right? So the first thing would be to go mark the rising edges. Here's one, here's another one, and so what should change at this first rising edge? Z0, right? Because it's a two-bit binary counter, so it should go from zero to one. So Z1 will stay fixed, Z0 will go up to one | [
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So when should Z1 and Z0 change? Yeah, the rising edges, right? So the first thing would be to go mark the rising edges. Here's one, here's another one, and so what should change at this first rising edge? Z0, right? Because it's a two-bit binary counter, so it should go from zero to one. So Z1 will stay fixed, Z0 will go up to one. Okay, what happens at the next rising edge? Yeah, so we'll go to the one-zero state, right? So that'll go up, this one will go down. | [
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So Z1 will stay fixed, Z0 will go up to one. Okay, what happens at the next rising edge? Yeah, so we'll go to the one-zero state, right? So that'll go up, this one will go down. What about here? Z0 goes up, Z1 stays up. And what about... go back to zero. And now we're back in the original state, right? And so this is just going through the four counting states. So this is zero, zero, zero, one, one, zero, one, one, and then we go back to zero, zero. | [
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And now we're back in the original state, right? And so this is just going through the four counting states. So this is zero, zero, zero, one, one, zero, one, one, and then we go back to zero, zero. And then we go, I guess, to zero, one, right? Skating flip-flops. You mean of the ripple counter sort? You mean something like that? To D or to clock? Okay, so yes, there's certainly... It's certainly possible because we talked about registers and things like that for you to have flip-flops running on a common clock. | [
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. It's certainly possible because we talked about registers and things like that for you to have flip-flops running on a common clock. So these flip-flops inside of this counter, for example, are running on a common clock, right? So all of these transitions, if I were to draw the inside of that binary counter, those are two flip-flops running off the same clock. And then, of course, the D inputs are going to be expressions of the counter. | [
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So these flip-flops inside of this counter, for example, are running on a common clock, right? So all of these transitions, if I were to draw the inside of that binary counter, those are two flip-flops running off the same clock. And then, of course, the D inputs are going to be expressions of the counter. Now, this is finite state machine design, right? So it's later. But we did do registers, right? So we did shift registers, we did registers with parallel load. | [
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Now, this is finite state machine design, right? So it's later. But we did do registers, right? So we did shift registers, we did registers with parallel load. So those topics can be on the exam, topics of finite state machines. Was there anything else people want to talk about with timing diagrams or can we move on? Okay. All right. So let's look a little bit at analyzing sequential feedback circuits. | [
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All right. So let's look a little bit at analyzing sequential feedback circuits. So... What is a sequential feedback circuit? So if I have a... If without using a flip-flop, I take the output of one of my circuits, and I link it back and make it into an input as well, we call that a sequential feedback circuit. So, for example, a latch is a sequential feedback circuit. So... If your output's connected back to your input and there's no clock signal in the design, then that's a sequential feedback circuit. | [
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. If your output's connected back to your input and there's no clock signal in the design, then that's a sequential feedback circuit. So the one we looked at in class, what we started with, was the R-bar S-bar latch, which looked like this. And then we built that up and all the way up through a gated D-latch, and then we went to a D-flip-flop based on this storing a bit design. You can also build SR latches out of NOR gates. | [
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] |
And then we built that up and all the way up through a gated D-latch, and then we went to a D-flip-flop based on this storing a bit design. You can also build SR latches out of NOR gates. These are the common designs in CMOS. Instead of NAND, you use NOR. So this is an SR latch. So if you think about the functionality here... I'll label it Q-bar, although if you turn both of those on, they'll both be zero. | [
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You can also build SR latches out of NOR gates. These are the common designs in CMOS. Instead of NAND, you use NOR. So this is an SR latch. So if you think about the functionality here... I'll label it Q-bar, although if you turn both of those on, they'll both be zero. So here, these are active high inputs, right? So since it's a NOR gate, if I put a one in on the R input, that will force the NOR gate output to zero, so Q will be zero. | [
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I'll label it Q-bar, although if you turn both of those on, they'll both be zero. So here, these are active high inputs, right? So since it's a NOR gate, if I put a one in on the R input, that will force the NOR gate output to zero, so Q will be zero. So they have the opposite input sense from the R-bar S-bar latch on the left. So this was our Q, and let's see which way did this work. | [
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So they have the opposite input sense from the R-bar S-bar latch on the left. So this was our Q, and let's see which way did this work. So if I put this as zero... Would both latches give the same outputs? Both of these can store a bit, but the input sense... These are active low inputs, these are active high inputs, and you'll notice they're swapped relative to Q. Active high means that in order to take the human action represented by the name, so if you want to set the output bit, which refers to Q, on the left, you make the S-bar input to zero to set Q to one. | [
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] |
These are active low inputs, these are active high inputs, and you'll notice they're swapped relative to Q. Active high means that in order to take the human action represented by the name, so if you want to set the output bit, which refers to Q, on the left, you make the S-bar input to zero to set Q to one. On the right, you make the S input equal to one, which means active high, to set Q to one. | [
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Active high means that in order to take the human action represented by the name, so if you want to set the output bit, which refers to Q, on the left, you make the S-bar input to zero to set Q to one. On the right, you make the S input equal to one, which means active high, to set Q to one. All right, so really all we want you to be able to do is analyze stable states. So there are a couple ways you can do it. | [
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All right, so really all we want you to be able to do is analyze stable states. So there are a couple ways you can do it. There's the simple way and the harder way. So since we already did the circuit on the left, let's do the one on the right. So you can say, okay, so for R and S, I can look at what is Q. Let's give this name P, actually. So if I set R and S to zero and zero, does that tell me anything about Q or P? No, right, because those are NOR gates. | [
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Let's give this name P, actually. So if I set R and S to zero and zero, does that tell me anything about Q or P? No, right, because those are NOR gates. So if I put a zero, I mean, think of it as OR followed by NOT. If I put a zero into an OR gate, I need to know what the other input is, right? Zero or whatever is whatever. So for both of those zero inputs to the NOR gates, I don't change anything in my circuit. | [
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