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You have to be careful because if you have more than one overlapping set here in your rows with different outputs, then you have an ill-defined truth table. So you've got to be careful how you merge them. We're getting there. Sorry. A multiplexer is what implements this function. So this output Q is the result of a multiplexer. But I'll show you logic diagrams in a bit. We'll derive them. | [
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But I'll show you logic diagrams in a bit. We'll derive them. So it's the thing that allows us to answer this question down here | [
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] |
You have to be careful because if you have more than one overlapping set here in your rows with different outputs, then you have an ill-defined truth table. So you've got to be careful how you merge them. We're getting there. Sorry. A multiplexer is what implements this function. So this output Q is the result of a multiplexer. But I'll show you logic diagrams in a bit. We'll derive them. | [
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But I'll show you logic diagrams in a bit. We'll derive them. So it's the thing that allows us to answer this question down here. Use one control signal to pick between two arbitrary signals. So the answer, we call a multiplexer. I debated whether I should put it in the titles, but that's what we're building towards. Yes. Although I call them d1 or d0. So here in this abbreviated one, what I was telling you, unselected inputs don't matter. | [
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Although I call them d1 or d0. So here in this abbreviated one, what I was telling you, unselected inputs don't matter. So remember, s is select. So the ones that are not selected, if s is 0, then d1. If s is 1, then d0. We'll mark those with don't cares. And furthermore, I could even, in this case, I could write the truth table in a really compact way. So I could say, well, if s is 0, the output is d0. | [
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And furthermore, I could even, in this case, I could write the truth table in a really compact way. So I could say, well, if s is 0, the output is d0. So whatever, this is now an input. But I'll say, well, that output corresponds to this input. And I just have two rows. So as we go forward in the class, it'll be more and more necessary as we build bigger systems to start writing truth tables where we take advantage of these shortcut notations. | [
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And I just have two rows. So as we go forward in the class, it'll be more and more necessary as we build bigger systems to start writing truth tables where we take advantage of these shortcut notations. So I just wanted to introduce a couple of them. So let's go ahead and solve this problem with the KMAP. So first, we'll copy. So I put d1 and d0 on the top. So we're going to go across. | [
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So I put d1 and d0 on the top. So we're going to go across. And then we'll just copy from q into this KMAP. So we'll start with this one and then go downwards. So that's 0, 1, another 0. But remember, we need to hop over when we copy. So 0, 1. And then 0, 0, 1, 1. So 0, 0, 1, 1. So what are the loops? So we could do this one. But if we look at this one, for example, we have to have this loop, right? Yeah. | [
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So what are the loops? So we could do this one. But if we look at this one, for example, we have to have this loop, right? Yeah. So remember, this was the example or similar to the one we used for consensus. So we don't need to have that one. But those two loops were done. So yeah. It doesn't actually matter, right? They're equivalent in this case. So you're right that I probably should take a look at that and make sure that I think that I'm, well, visually solve them and see that I think one is better than the other. | [
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It doesn't actually matter, right? They're equivalent in this case. So you're right that I probably should take a look at that and make sure that I think that I'm, well, visually solve them and see that I think one is better than the other. If it's close, I may solve them exactly to compare them. In this case, they're symmetric. So it's going to be identical. Yeah. So the question was SOP versus POS. | [
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Yeah. So the question was SOP versus POS. And they're the same here in terms of complexity. So we'll solve this one with SOP. So we'll write it down. It's S prime d0 and SD1. So I could have done that a little differently. I could have started with my small truth table and said, well, when I have S0, I get d0. So S prime, that's the min term for S. S equals 0, sorry. S prime anded with d0 gives me this row. | [
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S equals 0, sorry. S prime anded with d0 gives me this row. And then S, this min term, anded with d1 gives me this row | [
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So we'll write it down. It's S prime d0 and SD1. So I could have done that a little differently. I could have started with my small truth table and said, well, when I have S0, I get d0. So S prime, that's the min term for S. S equals 0, sorry. S prime anded with d0 gives me this row. And then S, this min term, anded with d1 gives me this row. And I OR those together. That's my function. | [
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And I OR those together. That's my function. And in that form, it's maybe a little easier to see that, well, all this is letting me do is when S is 0, I pick d0. And when S is 1, I pick d1. So this is a device, which we call a multiplexer, that allows us to pick between two signals, whatever they might be. And it will forward one of those signals, depending on our choice, which is S, to the output Q. | [
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So this is a device, which we call a multiplexer, that allows us to pick between two signals, whatever they might be. And it will forward one of those signals, depending on our choice, which is S, to the output Q. So here's what it looks like. All I did here is implemented the logic that we had on the last slide. So this is how a 2 to 1 MUX is implemented. This is how we draw it as a trapezoid with data inputs 1 and 0. | [
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So this is how a 2 to 1 MUX is implemented. This is how we draw it as a trapezoid with data inputs 1 and 0. And then which of those gets forwarded to Q depends on S. Yeah. Yeah. So the question is what are we going to use this for? And the answer is, well, when we want a piece of hardware that does more than one thing, we have to tell it which thing we want to do. And so we can control which inputs it looks at using something like a multiplexer. | [
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So the question is what are we going to use this for? And the answer is, well, when we want a piece of hardware that does more than one thing, we have to tell it which thing we want to do. And so we can control which inputs it looks at using something like a multiplexer. So I'll come back and actually put that in front of our devices. But yeah. Good question. Like this? Good question. | [
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Good question. Like this? Good question. What if we have four expressions, for example? So we have 3, 2, 1, and 0. What should we do? So yeah, one answer is to use hierarchical MUXes, right? So we can start by using one 2 to 1 MUX. We'll control it with a signal. Let's call it S1 for now. And let's decide between D3 or D2 will go into this input. And D1 or D0 will go into this input. | [
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And let's decide between D3 or D2 will go into this input. And D1 or D0 will go into this input. No. D3 or D2, the way I've numbered these. Yeah. Eventually. Yeah. So the next question then is, well, how do we get D3 or D2 to go into this input? Well, we're going to use a MUX, right? So how are we going to deliver two expressions? We use a MUX controlled by S0. So here on the top, we pick between D3 or D2 using S0. | [
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So the next question then is, well, how do we get D3 or D2 to go into this input? Well, we're going to use a MUX, right? So how are we going to deliver two expressions? We use a MUX controlled by S0. So here on the top, we pick between D3 or D2 using S0. One of those two will go into the input to this second level of MUXes, which is just one more MUX. And here we're picking between D1 or D0 also based on S0. | [
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One of those two will go into the input to this second level of MUXes, which is just one more MUX. And here we're picking between D1 or D0 also based on S0. This choice here is based on S1. So if you think about what ends up happening, I'll look at it in more detail on the next slide, but S1, S0 is basically a two bit unsigned number that chooses between input 3, input 2, input 1, and input 0. | [
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This choice here is based on S1. So if you think about what ends up happening, I'll look at it in more detail on the next slide, but S1, S0 is basically a two bit unsigned number that chooses between input 3, input 2, input 1, and input 0. Sorry? Yeah. Yeah. I really like it. No, no, no. It's great when you guys are predicting the slides. That means you're really on top of things. I like it. | [
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That means you're really on top of things. I like it. OK. Yes, you can do it with four AND gates | [
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So if you think about what ends up happening, I'll look at it in more detail on the next slide, but S1, S0 is basically a two bit unsigned number that chooses between input 3, input 2, input 1, and input 0. Sorry? Yeah. Yeah. I really like it. No, no, no. It's great when you guys are predicting the slides. That means you're really on top of things. I like it. OK. Yes, you can do it with four AND gates. | [
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OK. Yes, you can do it with four AND gates. And each of these is a minterm, right? So if you think about what's going on here, each of these four AND gates is producing a minterm ANDed with one of the D inputs. So let's go through those. So this top one is D3 ANDed with S1, S0, which is the minterm which you could think about as number 3. It's the 1, 1 minterm. So those get ANDed together and produced by this AND gate. | [
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It's the 1, 1 minterm. So those get ANDed together and produced by this AND gate. The next AND gate is D2 with S1, S0 prime. And that's 1, 0. That says, OK, I want input number 2. Down here, D1 ANDed with S1 prime, S0. That's 0, 1. So that gives you the D1 input. And then down here is D0 ANDed with S1 prime, S0 prime. So that gives you the D0 input. The outputs of all four of those, only one of these four minterms can be on. | [
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So that gives you the D0 input. The outputs of all four of those, only one of these four minterms can be on. S1, S0 has to have one of those four bit patterns. Only one of these four AND gates will produce a 1 ever. So then you OR them together. That gives you Q. Yeah. I thought it'd be easier for you to understand if I draw it this way. I'm sorry. I forgot to include that slide. Yeah, OK. | [
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I forgot to include that slide. Yeah, OK. So I mean, you should know, I know actually a few people had some trouble with this on the homework. So any time you have two-level SOP, you can just literally go replace both levels of gates with NAND and it'll be the same. So I mean, if you want them to be NAND, just draw NANDs in place of these. But I didn't do it. Sorry. Yeah, sorry. So that's an interesting question. | [
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Yeah, sorry. So that's an interesting question. So the question is, well, could you just connect these four outputs together? Remember that these are what we call active logic. And so if you think about the way these gates are working, you are actually connecting one of them to high voltage and the other, or maybe to high voltage if it outputs a 1. The other three are connected to ground. | [
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And so if you think about the way these gates are working, you are actually connecting one of them to high voltage and the other, or maybe to high voltage if it outputs a 1. The other three are connected to ground. So if you connect the outputs, you've created a path from high voltage to ground. I will show you later in the class how to do what you're saying, where you actually electrically disconnect some of the outputs from a wire, and then you can use multiple outputs to drive the same wire. | [
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] |
So if you connect the outputs, you've created a path from high voltage to ground. I will show you later in the class how to do what you're saying, where you actually electrically disconnect some of the outputs from a wire, and then you can use multiple outputs to drive the same wire. But you have to have electrical disconnection, not connection to ground, which is what you get out of the gates. | [
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I will show you later in the class how to do what you're saying, where you actually electrically disconnect some of the outputs from a wire, and then you can use multiple outputs to drive the same wire. But you have to have electrical disconnection, not connection to ground, which is what you get out of the gates. Good thinking, though. Okay. All right. So here's the symbolic form. Bigger trapezoid labeled with the four different values, two bits of select input, so crosshatch with the two. | [
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So here's the symbolic form. Bigger trapezoid labeled with the four different values, two bits of select input, so crosshatch with the two. And of course, we could then further generalize, right? So you can build 8-to-1 MOXs. You could build them out of 4-to-1 and 2-to-1 | [
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Good thinking, though. Okay. All right. So here's the symbolic form. Bigger trapezoid labeled with the four different values, two bits of select input, so crosshatch with the two. And of course, we could then further generalize, right? So you can build 8-to-1 MOXs. You could build them out of 4-to-1 and 2-to-1. You could build them out of AND gates. At some point, your OR gate at the end gets too big, so you'd really have to have multiple levels there. | [
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You could build them out of AND gates. At some point, your OR gate at the end gets too big, so you'd really have to have multiple levels there. But conceptually, it's the same. You can build 16-to-1, 32-to-1, and so forth. So anytime you want to select from 2-to-the-n, where I'll name that p to make the notation down here easier, with dp-1 down to d0 inputs, we need n bits of select. | [
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You can build 16-to-1, 32-to-1, and so forth. So anytime you want to select from 2-to-the-n, where I'll name that p to make the notation down here easier, with dp-1 down to d0 inputs, we need n bits of select. The log of p, log base 2 of p. Yeah. I mean, in theory, it's infinite. In practice, usually at about 4, you'll want to put more than. It depends on the semiconductor process, but usually it's around 4. | [
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In practice, usually at about 4, you'll want to put more than. It depends on the semiconductor process, but usually it's around 4. Yeah. You mean how do the CAD tools do? Yeah, so the CAD tools are actually, so there's a set of engineers who are working on the fabrication and will go and measure the fabrication capabilities and the speeds and the timings of the transistors and things like that, as well as the capacitive load of different wires. | [
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Yeah. You mean how do the CAD tools do? Yeah, so the CAD tools are actually, so there's a set of engineers who are working on the fabrication and will go and measure the fabrication capabilities and the speeds and the timings of the transistors and things like that, as well as the capacitive load of different wires. And then all of that gets fed into the CAD tools with process, semiconductor process-specific information that the CAD tools then use to optimize. | [
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You mean how do the CAD tools do? Yeah, so the CAD tools are actually, so there's a set of engineers who are working on the fabrication and will go and measure the fabrication capabilities and the speeds and the timings of the transistors and things like that, as well as the capacitive load of different wires. And then all of that gets fed into the CAD tools with process, semiconductor process-specific information that the CAD tools then use to optimize. So there's a lot of information that's not even available to the public. | [
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And then all of that gets fed into the CAD tools with process, semiconductor process-specific information that the CAD tools then use to optimize. So there's a lot of information that's not even available to the public. So unless you're contracting with that particular fabrication facility, they won't even give you those numbers. Those are proprietary numbers that you would get out of them. | [
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So unless you're contracting with that particular fabrication facility, they won't even give you those numbers. Those are proprietary numbers that you would get out of them. So the CAD tools are obviously designed to use that kind of information. And sometimes you'll see some of it if you're working on hardware. The rules of thumb have been sort of similar for a while, but the details are all hidden. | [
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And sometimes you'll see some of it if you're working on hardware. The rules of thumb have been sort of similar for a while, but the details are all hidden. All right. So we can use sets of MUXes also. Another way to generalize a multiplexer instead of just thinking, well, we could have bigger sets, we can use multiple, many copies of the same kind of multiplexer using the same select bits to switch between groups of bits. | [
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So we can use sets of MUXes also. Another way to generalize a multiplexer instead of just thinking, well, we could have bigger sets, we can use multiple, many copies of the same kind of multiplexer using the same select bits to switch between groups of bits. So for example, well, let me generalize it first. We can have an end-to-end multiplexer, which is actually M copies of N over M to 1 MUXes, each with log base 2 of N over M select bits. | [
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So for example, well, let me generalize it first. We can have an end-to-end multiplexer, which is actually M copies of N over M to 1 MUXes, each with log base 2 of N over M select bits. And typically, N over M is some power of 2. So for example, when we did our subtractor design, we said, well, we want to take B or complement B, one's complement of B. And we did some optimization there. | [
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So for example, when we did our subtractor design, we said, well, we want to take B or complement B, one's complement of B. And we did some optimization there. We could have instead just used MUXes and put B into one input of the MUX, B prime into the other input, but we'd have N copies of a 2 to 1 MUX to do that. And they would all be controlled by the same select bit. Either we want all of the bits complemented or none of the bits. | [
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And they would all be controlled by the same select bit. Either we want all of the bits complemented or none of the bits. None for addition, all for subtraction. Now, when we talked about it, we actually did a little bit of optimization for the MUX implementation | [
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We could have instead just used MUXes and put B into one input of the MUX, B prime into the other input, but we'd have N copies of a 2 to 1 MUX to do that. And they would all be controlled by the same select bit. Either we want all of the bits complemented or none of the bits. None for addition, all for subtraction. Now, when we talked about it, we actually did a little bit of optimization for the MUX implementation. | [
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None for addition, all for subtraction. Now, when we talked about it, we actually did a little bit of optimization for the MUX implementation. We said, well, because we're putting B and B prime in, we just need an XOR gate. We don't need a MUX. But we could have just put a MUX down. And actually, the CAD tool probably would have figured out to replace that with an XOR gate anyway. It's simple enough logic that the tool could figure it out. | [
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And actually, the CAD tool probably would have figured out to replace that with an XOR gate anyway. It's simple enough logic that the tool could figure it out. All right. So let's then back up a step and think about our ASCII character checker. So let's say now, instead of just upper and lowercase, I want to check for four different kinds of comparison. So I want to say, well, let me check for control characters. | [
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So let's say now, instead of just upper and lowercase, I want to check for four different kinds of comparison. So I want to say, well, let me check for control characters. Those happen to be the range 0 to 1 f hex. Check for lowercase, so 41 to 5a. 41, not 40, sorry. 41 to 5a hex. Uppercase 61 to 7a hex. And digits, 30 to 39 hex. So I want to check all four types with one piece of logic. | [
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And digits, 30 to 39 hex. So I want to check all four types with one piece of logic. So what should I do? Yeah, MUXes, what kind? Yeah, so a set of 4 to 1 MUXes, right? A 28 to 7, where each of the ASCII characters is compared against one of four possible lower range values, 0, 41, 61, or 30. And one upper range, 1f, 5a, 7a, and 39. Which of those four I put in is actually just controlled by the same two bits, for all 14 of those 4 to 1 MUXes. | [
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And one upper range, 1f, 5a, 7a, and 39. Which of those four I put in is actually just controlled by the same two bits, for all 14 of those 4 to 1 MUXes. So the same two bits make my choice for me between the four different possibilities. 0 to 1f, which gives me control characters. 41 to 5a, which gives me uppercase letters. 61 through 7a, which gives me lowercase letters. And 30 to 39, which gives me digits. | [
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61 through 7a, which gives me lowercase letters. And 30 to 39, which gives me digits. And then I just look down here at the output to see for the range I chose, was the answer yes or no? And I can use the same hardware. Make sense? Yeah. What are you, we're relying to give you what? I'm sorry. Ah, so the S input, right now we've only done combinational logic, right? So this is just like any other input to our logic, right? So somehow we need to tell this piece of hardware which operation we want to do. | [
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What are you, we're relying to give you what? I'm sorry. Ah, so the S input, right now we've only done combinational logic, right? So this is just like any other input to our logic, right? So somehow we need to tell this piece of hardware which operation we want to do. So later we'll see how we can store state, actually we'll start it shortly. But eventually we'll build full computers, and then it'll be up to us to make sure that the answer to your question gets routed to these two input bits at the right time, so that we perform the correct comparison. | [
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So later we'll see how we can store state, actually we'll start it shortly. But eventually we'll build full computers, and then it'll be up to us to make sure that the answer to your question gets routed to these two input bits at the right time, so that we perform the correct comparison. At this stage, probably not too much, right? At this stage, we're designing something that is usable, right? That the human can arrange to have the right two input bits. | [
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But eventually we'll build full computers, and then it'll be up to us to make sure that the answer to your question gets routed to these two input bits at the right time, so that we perform the correct comparison. At this stage, probably not too much, right? At this stage, we're designing something that is usable, right? That the human can arrange to have the right two input bits. I mean, humans can make mistakes everywhere, right? So you can design an adder and they put the wrong bits into the adder, they get the wrong answer out. | [
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At this stage, probably not too much, right? At this stage, we're designing something that is usable, right? That the human can arrange to have the right two input bits. I mean, humans can make mistakes everywhere, right? So you can design an adder and they put the wrong bits into the adder, they get the wrong answer out | [
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At this stage, probably not too much, right? At this stage, we're designing something that is usable, right? That the human can arrange to have the right two input bits. I mean, humans can make mistakes everywhere, right? So you can design an adder and they put the wrong bits into the adder, they get the wrong answer out. So in all stages of all kinds of engineering design, you have to think about, did a human make a mistake somewhere? There's sometimes other sources of error too, but humans are responsible for most mistakes. | [
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I mean, humans can make mistakes everywhere, right? So you can design an adder and they put the wrong bits into the adder, they get the wrong answer out. So in all stages of all kinds of engineering design, you have to think about, did a human make a mistake somewhere? There's sometimes other sources of error too, but humans are responsible for most mistakes. Yeah, when it seems like a computer is responsible for most mistakes, it's actually the programmer. | [
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So in all stages of all kinds of engineering design, you have to think about, did a human make a mistake somewhere? There's sometimes other sources of error too, but humans are responsible for most mistakes. Yeah, when it seems like a computer is responsible for most mistakes, it's actually the programmer. So yeah, occasionally there are things like cosmic ray strikes, right? That'll cause bit flips that will, so there are hardware, software slash errors that come up because of nature. | [
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Yeah, when it seems like a computer is responsible for most mistakes, it's actually the programmer. So yeah, occasionally there are things like cosmic ray strikes, right? That'll cause bit flips that will, so there are hardware, software slash errors that come up because of nature. Most bugs are because of humans at some point. All right. So one more piece, actually, before we start sequential logic, so maybe we won't get to sequential logic today. | [
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All right. So one more piece, actually, before we start sequential logic, so maybe we won't get to sequential logic today. So decoders, so it's going to be a little bit abstract. This will make more sense when we get to memory. So what if we have designed a representation with n bits? I mean, we did a bunch of them when we did bit slicing, right? But what if we have an n bit representation of something and we want to know, and we have a value in that representation, n bits, we want to know which of the values is it? So I want one signal that tells me, okay, this was the mango ice cream or this was the lychee or this was the pistachio or whatever, one signal each. | [
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This will make more sense when we get to memory. So what if we have designed a representation with n bits? I mean, we did a bunch of them when we did bit slicing, right? But what if we have an n bit representation of something and we want to know, and we have a value in that representation, n bits, we want to know which of the values is it? So I want one signal that tells me, okay, this was the mango ice cream or this was the lychee or this was the pistachio or whatever, one signal each. So what am I going to do with that? Well, naming sets of bits. | [
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So what if we have designed a representation with n bits? I mean, we did a bunch of them when we did bit slicing, right? But what if we have an n bit representation of something and we want to know, and we have a value in that representation, n bits, we want to know which of the values is it? So I want one signal that tells me, okay, this was the mango ice cream or this was the lychee or this was the pistachio or whatever, one signal each. So what am I going to do with that? Well, naming sets of bits. So again, when you receive memories or register files, probably three to four weeks out, we'll need this, right? This will be one of the primary components in the construction of a computer memory. | [
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So what am I going to do with that? Well, naming sets of bits. So again, when you receive memories or register files, probably three to four weeks out, we'll need this, right? This will be one of the primary components in the construction of a computer memory. So we will need to have decoders. Another one is actually generating arbitrary functions, logic functions dynamically. So this was used in old reconfigurable logic, so called programmable logic arrays, basically. | [
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Another one is actually generating arbitrary functions, logic functions dynamically. So this was used in old reconfigurable logic, so called programmable logic arrays, basically. So I'll show you how that works. But for many years, that was how we built dynamically reconfigurable hardware and did hardware prototyping quickly. So in other words, given a set of n bits, I want to generate a signal for every possible combination. | [
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But for many years, that was how we built dynamically reconfigurable hardware and did hardware prototyping quickly. So in other words, given a set of n bits, I want to generate a signal for every possible combination. So those signals then of course, correspond to the min terms. And so for every min term, I'll have a signal and exactly one of those will have the value one. So decoder generates all the min terms, and one of them will be one. | [
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And so for every min term, I'll have a signal and exactly one of those will have the value one. So decoder generates all the min terms, and one of them will be one. So you might remember, we saw that in the mux, right? So there's a similar structure. So here's a decoder, two to four decoder. So it takes two input bits and generates all the min terms. Here I've added an enable signal. | [
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So it takes two input bits and generates all the min terms. Here I've added an enable signal. So in fact, for this decoder, if enable is zero, enable goes to all four of these AND gates, and so all the outputs are zero, right? If enable is one, each of these becomes a min term of S1, S0, and so one of the outputs will be a one and the others will all be zero. So this is decoder | [
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So it takes two input bits and generates all the min terms. Here I've added an enable signal. So in fact, for this decoder, if enable is zero, enable goes to all four of these AND gates, and so all the outputs are zero, right? If enable is one, each of these becomes a min term of S1, S0, and so one of the outputs will be a one and the others will all be zero. So this is decoder. And if you were then to be mean and ask me, well, this is not NAND and NOR and it's not too level, I'd say, yeah, it's not quite accurate, right? You're going to need extra inverters to make this a real circuit for CMOS. | [
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] |
So this is decoder. And if you were then to be mean and ask me, well, this is not NAND and NOR and it's not too level, I'd say, yeah, it's not quite accurate, right? You're going to need extra inverters to make this a real circuit for CMOS. Because if you used a Morgan's law, you'll get extra inverters on this side, which you can easily just swap the inverted and non-inverted lines here, but I guess you'll get a bonus inverter and enable. | [
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And if you were then to be mean and ask me, well, this is not NAND and NOR and it's not too level, I'd say, yeah, it's not quite accurate, right? You're going to need extra inverters to make this a real circuit for CMOS. Because if you used a Morgan's law, you'll get extra inverters on this side, which you can easily just swap the inverted and non-inverted lines here, but I guess you'll get a bonus inverter and enable. Yeah. | [
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Because if you used a Morgan's law, you'll get extra inverters on this side, which you can easily just swap the inverted and non-inverted lines here, but I guess you'll get a bonus inverter and enable. Yeah. I'm not sure if I understand what you mean. I mean, a one input NAND gate or NOR gate is an inverter, right? If you think about the parallel versus series, they scale it down to one transistor, that's then an inverter. | [
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I'm not sure if I understand what you mean. I mean, a one input NAND gate or NOR gate is an inverter, right? If you think about the parallel versus series, they scale it down to one transistor, that's then an inverter. But unless you have two in a row, they don't just cancel. So when... Yeah. Yeah. Yeah. Okay. Yes, but I'm... Yeah, I'm not sure. I'm not sure if I understand. I mean, I can use a two input NAND and connect both inputs to the same thing. | [
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I'm not sure if I understand. I mean, I can use a two input NAND and connect both inputs to the same thing. I see. But I can do the inverter with one input. I mean, I can also have three inputs or 10 inputs. It's all the same. It's all an inverter if I connect them all to the same input. Yeah. But what I can't do is when I change this to a NOR gate in order to make it CMOS, I get inverters here. | [
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Yeah. But what I can't do is when I change this to a NOR gate in order to make it CMOS, I get inverters here. So all of these will be inverted. So these two inverters will be flipped. So that's fine. It's the same cost, but I'll still have one inverter that I need for the enable. So I'll have one extra inverter in the NAND NOR design if you work that out. Okay. So this is how we draw a decoder to the trapezoid going the opposite way. | [
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Okay. So this is how we draw a decoder to the trapezoid going the opposite way. So the NBIT signal usually comes in from the smaller side, not the narrow ends, but the smaller side of the trapezoid. And the outputs then come out from the other side. If the decoder has an enable signal, not all of them do, but it's pretty common. It'll come in from the narrow side. So this is a symbolic form of the decoder. | [
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It'll come in from the narrow side. So this is a symbolic form of the decoder. I do want you to notice that this structure is very similar to the MUX. So in the MUX, we then ANDed each of these min terms with one of the data inputs, and then we ORed all the results together. But we have the same AND gates to generate all the min terms. So they're very similar in that sense. The decoder allows these min terms to be used separately. | [
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So they're very similar in that sense. The decoder allows these min terms to be used separately. So for example, if you were building a vending machine and you wanted a bunch of products coded in some representation and each of these outputs could then control the mechanical release for that product. You'd release exactly one product based on the product that the vending machine said, OK, time to release a product. | [
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So for example, if you were building a vending machine and you wanted a bunch of products coded in some representation and each of these outputs could then control the mechanical release for that product. You'd release exactly one product based on the product that the vending machine said, OK, time to release a product | [
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But we have the same AND gates to generate all the min terms. So they're very similar in that sense. The decoder allows these min terms to be used separately. So for example, if you were building a vending machine and you wanted a bunch of products coded in some representation and each of these outputs could then control the mechanical release for that product. You'd release exactly one product based on the product that the vending machine said, OK, time to release a product. | [
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So for example, if you were building a vending machine and you wanted a bunch of products coded in some representation and each of these outputs could then control the mechanical release for that product. You'd release exactly one product based on the product that the vending machine said, OK, time to release a product. And you probably want to release one at a time rather than having your vending machine drop a random subset or something. | [
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You'd release exactly one product based on the product that the vending machine said, OK, time to release a product. And you probably want to release one at a time rather than having your vending machine drop a random subset or something. Well so there are no 2 to the n data inputs where n is the number of select bits. So there's nothing actually getting ANDed in with the min terms other than the enable signal. | [
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Well so there are no 2 to the n data inputs where n is the number of select bits. So there's nothing actually getting ANDed in with the min terms other than the enable signal. So that's one difference. And the min terms are not being ORed together. So in a practical sense, the way that's used is we can compose arbitrary functions by ORing together the right set of min terms. So it's not very interesting for 2 bits, but if you built a bigger decoder like 3 to 8 or 4 to 16, you could then pick out the right min terms, OR them together, and compose an arbitrary logic function. | [
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So in a practical sense, the way that's used is we can compose arbitrary functions by ORing together the right set of min terms. So it's not very interesting for 2 bits, but if you built a bigger decoder like 3 to 8 or 4 to 16, you could then pick out the right min terms, OR them together, and compose an arbitrary logic function. So that's how the old reconfigurable logic was built, by composing logic functions out of min terms. | [
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] |
So it's not very interesting for 2 bits, but if you built a bigger decoder like 3 to 8 or 4 to 16, you could then pick out the right min terms, OR them together, and compose an arbitrary logic function. So that's how the old reconfigurable logic was built, by composing logic functions out of min terms. And you can actually do that. So let me finish this and I'll get to your question. | [
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And you can actually do that. So let me finish this and I'll get to your question. You can actually do that for more than one function, because you have these min term wires. So you can have one function by just ORing the right set together, and then a different function by ORing a different set together, and a third function, and a fourth function, and so forth. Whereas with a MUX, those min terms are all ORed together into one wire. | [
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So you can have one function by just ORing the right set together, and then a different function by ORing a different set together, and a third function, and a fourth function, and so forth. Whereas with a MUX, those min terms are all ORed together into one wire. So you can't then use that to construct arbitrary logic. You can use it by putting 0s and 1s to the input, but one MUX will get you one function. | [
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So you can't then use that to construct arbitrary logic. You can use it by putting 0s and 1s to the input, but one MUX will get you one function. Whereas a decoder, you can then add OR gates outside the decoder to get you any number of functions you want. So the primary application, other than memory, was the programmable logic arrays. I'll show you, when we get in about two and a half weeks, I will show you an application which is a vending machine, which is, you pick a product and then there's internal logic that decides, am I going to actually deliver that product to you? Have you put enough money in? And so that goes to the enable of the decoder. | [
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So the primary application, other than memory, was the programmable logic arrays. I'll show you, when we get in about two and a half weeks, I will show you an application which is a vending machine, which is, you pick a product and then there's internal logic that decides, am I going to actually deliver that product to you? Have you put enough money in? And so that goes to the enable of the decoder. So if the answer is no, you didn't give me enough money for that product, then nothing gets released. | [
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I'll show you, when we get in about two and a half weeks, I will show you an application which is a vending machine, which is, you pick a product and then there's internal logic that decides, am I going to actually deliver that product to you? Have you put enough money in? And so that goes to the enable of the decoder. So if the answer is no, you didn't give me enough money for that product, then nothing gets released. And if the answer is yes, then we decode the signal, and then we take each of these and send it to a mechanical release. | [
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So if the answer is no, you didn't give me enough money for that product, then nothing gets released. And if the answer is yes, then we decode the signal, and then we take each of these and send it to a mechanical release. Or you can use it for that kind of application. The primary use, though, outside of memories, which we'll see in a few weeks, was these programmable logic arrays, which have now been superseded by FPGAs. | [
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Or you can use it for that kind of application. The primary use, though, outside of memories, which we'll see in a few weeks, was these programmable logic arrays, which have now been superseded by FPGAs. FPGAs use a different mechanism in terms of lookup tables. Oops, really? That was it? OK. We have a few minutes left, so let me give you a little start on any questions on this before I switch slide decks. | [
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Oops, really? That was it? OK. We have a few minutes left, so let me give you a little start on any questions on this before I switch slide decks. Yes, so based on S1 and S0, there will be four different minterms, and it will generate exactly one one for one of those minterms. Yeah. Yeah. The enable. Yeah | [
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FPGAs use a different mechanism in terms of lookup tables. Oops, really? That was it? OK. We have a few minutes left, so let me give you a little start on any questions on this before I switch slide decks. Yes, so based on S1 and S0, there will be four different minterms, and it will generate exactly one one for one of those minterms. Yeah. Yeah. The enable. Yeah. Yeah, the S1, S2 would be the product choice. | [
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Yeah. Yeah, the S1, S2 would be the product choice. That's right. And then each of these lines out would control whether a different one of the four products got dropped out. So a MUX is typically used when you need to choose between different functions. So for example, when we did the add or subtract, or when we did unsigned or two's complement comparator, you can use a MUX. When you want to compare different ranges of ASCII, you can use a set of MUXs to choose which type of character you're looking for. | [
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] |
So for example, when we did the add or subtract, or when we did unsigned or two's complement comparator, you can use a MUX. When you want to compare different ranges of ASCII, you can use a set of MUXs to choose which type of character you're looking for. So there will be many, many cases where we want to have the ability to choose between two things or four things, and we'll use MUXs in all of those cases. | [
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When you want to compare different ranges of ASCII, you can use a set of MUXs to choose which type of character you're looking for. So there will be many, many cases where we want to have the ability to choose between two things or four things, and we'll use MUXs in all of those cases. In fact, we'll see it probably Monday or Wednesday when we design registers. I see. So in a different sense. | [
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I see. So in a different sense. So this is giving me one signal for each pattern, whereas the MUX is unifying many signals under one wire. Right. So do I want to add or subtract? I want the answer to come out in the same place. Right. So when you want to combine things into one output, use a MUX. When you want to split something that's already been coded into individual wires, use a decoder. | [
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So when you want to combine things into one output, use a MUX. When you want to split something that's already been coded into individual wires, use a decoder. Yeah, it's trying to give you the ability to tell which of the patterns here was the one that was actually present on the bits. So that you can then do things with each of those individually. Go ahead. Okay. You mean if you wanted to build a bigger decoder from a smaller one? Yeah, then you have to. | [
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