shuffling file names around
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# lec10
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# lec11
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## Half-adder
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At this point I'l mention that just reading isn't going to get you anywhere, you have to try things, and give it a real earnest attempt.
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This will be the building block for adding bit-strings together later on however, for now we are going to just add two singular bits.
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To accomplish this we'll build a half adder.
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__ALU:__ Arithmetic Logic Unit
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This means our logic circuit must adhere to the following logic table.
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If both inputs are 0 then our result is 0 and we don't have to carry anything out.
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If only one input A/B is 1 then our result will clearly be 1 and our carry will be 0.
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Finally if both inputs are 0 then since we can't fit 2 in a single bit it means we have to carry-out a 1, and our result will be 0.
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## Building a 1-bit ALU
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With all of this in mind we have a table to guide how we will implement our logic circuit.
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I __highly__ suggest that you try to build a logic circuit on your own first as most of the content is best learned through practice.
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| A | B | Carry-out | Result |
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|---|---|---|---|
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| 0 | 0 | 0 | 0 |
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| 0 | 1 | 0 | 1 |
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| 1 | 0 | 0 | 1 |
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| 1 | 1 | 1 | 0 |
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First we'll create an example _ALU_ which implements choosing between an `and`, `or`, `xor`, or `add`.
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Whether or not our amazing _ALU_ is useful doesn't matter so we'll go one function at a time(besides `and/or`).
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First recognize that we need to choose between `and` or `or` against our two inputs A/B.
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This means we have two inputs and/or, and we need to select between them.
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_Try to do this on your own first!_
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## Full Adder
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If we only want to add single-bit's then a half-adder works fine but if we want to add multiple bits say `1011 + 0010` then we need to consider that we will likely have to chain these together.
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The full-adder has 1 main difference from the half-adder, it has 3 inputs, 2 main inputs and 1 input for the carry bit.
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The carry bit will propagate along the operation now if we chain these together, _just like real addition!_
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Next we'll add on the `xor`.
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AGAIN: try to do this on your own, the main hint I'll give here is: the current mux needs to be changed.
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With this we can start chaining together multiple Full-adders we can start adding multiple bits at the same time since the carry now propagates along the chain.
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Finally we'll add the ability to add and subtract.
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You may have also noted that we can subtract two things to see if they are the same dhowever, we can also `not` the result of the `xor` and get the same result.
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## Ripple Adders
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An N-bit adder is really just made up of Full adders chained together.
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Each adder is chained to the next by the carry-out line which then acts as the next adder's carry-in line.
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If we have say a 4-bit ripple adder, then each bit in the bit strings will go to a different adder.
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For now the initial carry in bit will be fed a 0 everytime.
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At this point our _ALU_ can `and`, `or`, `xor`, and `add`/`sub`.
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The mux will choose one which logic block to use; the carry-in line will tell the `add` logic block whether to add or subtract.
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Finally the A-invert and B-invert line allow us to determine if we want to invert either A or B (inputs).
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## N-bit ALU
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Here we see that our 4-bit input A & B have the values `1111` & `0000` respectively.
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The 0th bit goes to Adder0, the 1th goes to Adder1, and so on.
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You should have also noticed that each one also takes a carry-in, which for now is 0 but it does mean we have to let the comutation happen one adder at a time.
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None of the sequential adders can do anything if the previous have done anything yet either.
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At the end of it all we get some bit results, and a carry-out.
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We can then simply reassemble the results back into one bit-string to assemble our final result.
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For sanity we'll use the following block for our new ALU.
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Note that we are chaining the carry-in's to the carry-out's just like a ripple adder.
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also each ALU just works with `1` bit from our given 4-bit input.
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@ -1,41 +0,0 @@
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# lec11
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At this point I'l mention that just reading isn't going to get you anywhere, you have to try things, and give it a real earnest attempt.
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__ALU:__ Arithmetic Logic Unit
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## Building a 1-bit ALU
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First we'll create an example _ALU_ which implements choosing between an `and`, `or`, `xor`, or `add`.
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Whether or not our amazing _ALU_ is useful doesn't matter so we'll go one function at a time(besides `and/or`).
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First recognize that we need to choose between `and` or `or` against our two inputs A/B.
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This means we have two inputs and/or, and we need to select between them.
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_Try to do this on your own first!_
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Next we'll add on the `xor`.
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AGAIN: try to do this on your own, the main hint I'll give here is: the current mux needs to be changed.
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Finally we'll add the ability to add and subtract.
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You may have also noted that we can subtract two things to see if they are the same dhowever, we can also `not` the result of the `xor` and get the same result.
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At this point our _ALU_ can `and`, `or`, `xor`, and `add`/`sub`.
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The mux will choose one which logic block to use; the carry-in line will tell the `add` logic block whether to add or subtract.
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Finally the A-invert and B-invert line allow us to determine if we want to invert either A or B (inputs).
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## N-bit ALU
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For sanity we'll use the following block for our new ALU.
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Note that we are chaining the carry-in's to the carry-out's just like a ripple adder.
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also each ALU just works with `1` bit from our given 4-bit input.
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118
cst337/lec/lec8.md
Normal file
118
cst337/lec/lec8.md
Normal file
@ -0,0 +1,118 @@
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# lec9
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This lecture has a corresponding activity found in `lab/` it is called `combinational-logic.md`.
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It is more useful to practice combinational logic as opposed to read about it so the sub section here will be minimal in information.
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It's recommended that you try as many of the problems in the activity until you understand the concept, _don't bother doing them all_.
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## Combinational Logic
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### OR
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`a+b` is equivalent to saying `a` or `b`.
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### AND
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`ab` is equivalent to saying `a` and `b`.
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Note that this syntax is simlar to multiplication so `a*b` is equivalent to the above.
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### NOT
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`!a` is equivalent to saying not `a`.
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We can also denote it with a bar over the expression we want to _not_.
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### Big AND
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Behavior is the same as an `and` but instead of two inputs we can have many more inputs.
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It will only ever return a 1 if all inputs are 1.
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### Big OR
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Again we are mimicing the behvior of the normal or gate but this time we can have multiple inputs as opposed to just two.
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If only one of the many inputs is 1 then we return a 1 for the output of the Big OR.
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## Decoders
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Here we'll learn by doing
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```
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Selector = 2 Bits
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Output = 4 Bits
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```
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As a challenge you can try using the combinational logic gates from above to try and tackle this yourself
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|s1 |s2 |o3 |o2 |o1 |o0 |
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|---|---|---|---|---|---|
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| 0 | 0 | 0 | 0 | 0 | 1 |
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| 0 | 1 | 0 | 0 | 1 | 0 |
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| 1 | 0 | 0 | 1 | 0 | 0 |
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| 1 | 1 | 1 | 0 | 0 | 0 |
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## Multiplexor
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Typically we'll refer to multiplexors by their size.
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> what does it do?
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It takes a signal as `2^n` inputs and out puts out `n` signals as output.
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Example: We have a selector(s0), two inputs[in0 & in1], and one output `out`.
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The selector will select an input and we will generate some output in `out`.
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|s0 | i1 | i0 | out|
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|---|---|---|---|
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|0 | 0 | 0 | 0|
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|0 | 0 | 1 | 1|
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|0 | 1 | 0 | 0|
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|0 | 1 | 1 | 1|
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|1 | 0 | 0 | 0|
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|1 | 0 | 1 | 0|
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|1 | 1 | 0 | 1|
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|1 | 1 | 1 | 1|
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This ultimately lets us pick data out of memory given some address.
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## Half Adder
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For now we'll take two inputs and get 1 output, with a carry-output.
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Let's add 2 bits
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|a |b |out|
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|---|---|---|
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|0 |0 |0 |
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|0 |1 |1 |
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|1 |0 |1 |
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|1 |1 |0 |
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What about the carry bit however? What would _it_ look like given the preivous operations?
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|a |b |carryout|
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|---|---|---|
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|0 |0 |0 |
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|0 |1 |0 |
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|1 |0 |0 |
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|1 |1 |1 |
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Before what this implies note that the result of the carryout resembles
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## Full Adder
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Two inputs, One output, One carry-out, One carry-in
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Here we'll add up `a & b`(inputs) and `c` carry-in
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|c|a|b |output|
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|---|---|---|---|
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|0|0|0 |0|
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|0|0|1 |1|
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|0|1|0 |1|
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|0|1|1 |0|
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|1|0|0 |1|
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|1|0|1 |0|
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|1|1|0 |0|
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|1|1|1 |1|
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@ -1,118 +1,49 @@
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# lec9
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# lec10
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This lecture has a corresponding activity found in `lab/` it is called `combinational-logic.md`.
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It is more useful to practice combinational logic as opposed to read about it so the sub section here will be minimal in information.
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It's recommended that you try as many of the problems in the activity until you understand the concept, _don't bother doing them all_.
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## Half-adder
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## Combinational Logic
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This will be the building block for adding bit-strings together later on however, for now we are going to just add two singular bits.
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To accomplish this we'll build a half adder.
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|
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### OR
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This means our logic circuit must adhere to the following logic table.
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If both inputs are 0 then our result is 0 and we don't have to carry anything out.
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If only one input A/B is 1 then our result will clearly be 1 and our carry will be 0.
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Finally if both inputs are 0 then since we can't fit 2 in a single bit it means we have to carry-out a 1, and our result will be 0.
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`a+b` is equivalent to saying `a` or `b`.
|
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With all of this in mind we have a table to guide how we will implement our logic circuit.
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I __highly__ suggest that you try to build a logic circuit on your own first as most of the content is best learned through practice.
|
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### AND
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`ab` is equivalent to saying `a` and `b`.
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|
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Note that this syntax is simlar to multiplication so `a*b` is equivalent to the above.
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### NOT
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`!a` is equivalent to saying not `a`.
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We can also denote it with a bar over the expression we want to _not_.
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### Big AND
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Behavior is the same as an `and` but instead of two inputs we can have many more inputs.
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It will only ever return a 1 if all inputs are 1.
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### Big OR
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Again we are mimicing the behvior of the normal or gate but this time we can have multiple inputs as opposed to just two.
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If only one of the many inputs is 1 then we return a 1 for the output of the Big OR.
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## Decoders
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Here we'll learn by doing
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```
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Selector = 2 Bits
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Output = 4 Bits
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```
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As a challenge you can try using the combinational logic gates from above to try and tackle this yourself
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|s1 |s2 |o3 |o2 |o1 |o0 |
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|---|---|---|---|---|---|
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| 0 | 0 | 0 | 0 | 0 | 1 |
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| 0 | 1 | 0 | 0 | 1 | 0 |
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| 1 | 0 | 0 | 1 | 0 | 0 |
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| 1 | 1 | 1 | 0 | 0 | 0 |
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## Multiplexor
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Typically we'll refer to multiplexors by their size.
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> what does it do?
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It takes a signal as `2^n` inputs and out puts out `n` signals as output.
|
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Example: We have a selector(s0), two inputs[in0 & in1], and one output `out`.
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The selector will select an input and we will generate some output in `out`.
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|s0 | i1 | i0 | out|
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| A | B | Carry-out | Result |
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|---|---|---|---|
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|0 | 0 | 0 | 0|
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|0 | 0 | 1 | 1|
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|0 | 1 | 0 | 0|
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|0 | 1 | 1 | 1|
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|1 | 0 | 0 | 0|
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|1 | 0 | 1 | 0|
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|1 | 1 | 0 | 1|
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|1 | 1 | 1 | 1|
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| 0 | 0 | 0 | 0 |
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| 0 | 1 | 0 | 1 |
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| 1 | 0 | 0 | 1 |
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| 1 | 1 | 1 | 0 |
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This ultimately lets us pick data out of memory given some address.
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## Half Adder
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For now we'll take two inputs and get 1 output, with a carry-output.
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Let's add 2 bits
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|a |b |out|
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|---|---|---|
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|0 |0 |0 |
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|0 |1 |1 |
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|1 |0 |1 |
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|1 |1 |0 |
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What about the carry bit however? What would _it_ look like given the preivous operations?
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|a |b |carryout|
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|---|---|---|
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|0 |0 |0 |
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|0 |1 |0 |
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|1 |0 |0 |
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|1 |1 |1 |
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Before what this implies note that the result of the carryout resembles
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## Full Adder
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Two inputs, One output, One carry-out, One carry-in
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If we only want to add single-bit's then a half-adder works fine but if we want to add multiple bits say `1011 + 0010` then we need to consider that we will likely have to chain these together.
|
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The full-adder has 1 main difference from the half-adder, it has 3 inputs, 2 main inputs and 1 input for the carry bit.
|
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The carry bit will propagate along the operation now if we chain these together, _just like real addition!_
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|
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Here we'll add up `a & b`(inputs) and `c` carry-in
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|c|a|b |output|
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|---|---|---|---|
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|0|0|0 |0|
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|0|0|1 |1|
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|0|1|0 |1|
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|0|1|1 |0|
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|1|0|0 |1|
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|1|0|1 |0|
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|1|1|0 |0|
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|1|1|1 |1|
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With this we can start chaining together multiple Full-adders we can start adding multiple bits at the same time since the carry now propagates along the chain.
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## Ripple Adders
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An N-bit adder is really just made up of Full adders chained together.
|
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Each adder is chained to the next by the carry-out line which then acts as the next adder's carry-in line.
|
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If we have say a 4-bit ripple adder, then each bit in the bit strings will go to a different adder.
|
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For now the initial carry in bit will be fed a 0 everytime.
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|
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|
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|
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Here we see that our 4-bit input A & B have the values `1111` & `0000` respectively.
|
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The 0th bit goes to Adder0, the 1th goes to Adder1, and so on.
|
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You should have also noticed that each one also takes a carry-in, which for now is 0 but it does mean we have to let the comutation happen one adder at a time.
|
||||
None of the sequential adders can do anything if the previous have done anything yet either.
|
||||
At the end of it all we get some bit results, and a carry-out.
|
||||
We can then simply reassemble the results back into one bit-string to assemble our final result.
|
||||
|
||||
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Block a user