1. FSMs, Multiplication, and Division
CS/COE 0447
Jarrett Billingsley

2. Class announcements
what IS project 2
idfk
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3. Finite State Machines
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4. What's an FSM?
a Finite State Machine is a machine with a finite number of states.
:^) lol jk
it's a way of thinking about a sequential process where:
there are inputs which can change the state
the system has a state (memory)
there are outputs based on the state (and maybe on the inputs)
should panic?
bears seen: 3
bears seen: 2
bears seen: 0
bears seen: 1
yeah ok
nah
- FSMs come up all the time in programming and hardware design
- they're great for controlling simple multiple-step procedures
- FSMs are easy to reason about and transform (take CS1511)
- by breaking up more complex things into interdependent FSMs you can make it easier to think about your program/circuit
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5. Are flip-flops FSMs? I guess?
they're like the simplest possible ones...
but it's not really important :B
- I just made this slide cause I almost always get this question…
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6. A very simple example off high med low
what states can a ceiling fan be in?
high, medium, low, and off
what are the input and output?
the chain and the motor
when you pull the chain, it changes state
pull
pull
pull
off
high
med
low
pull
- I don't really know why they go high, medium, low, but they do.
- unless it's the one in our bedroom. then it goes medium, medium, low, off. something's broken…
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7. Missing some arrows off high med low
at any point in time, which of 2 choices can the input (chain) be?
pulled or not-pulled
when you don't pull the chain, what happens?
off
high
med
low
pull no
pull no
pull no
pull no
pull
this is the full state transition diagram
- sometimes we just leave those "stay-in-the-same-state" arrows off, though, to say "if none of the conditions are satisfied, don't change"
- but that can get you into trouble
- did you mean to stay in the same state? or did you just forget a possibility?
every state should be able to handle every input!
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8. Table-a-fying it S In Snext off 1 hi med low off hi hi med med low low
we can represent this diagram with a table: S In
Snext
off 1 hi
med
low
off
high
med
low
pull no
off hi hi
med
for the input, let's say:
0 means not pulling
1 means pulling
med
low
low
off
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9. State off hi med low Motor power 0% 100% 50% 25%
What about the outputs?
our fan controller has to control the motor after all
we can make a table showing the output(s) for each state
note that the input isn't needed at all in this case.
State
off hi
med
low
Motor power 0%
100%
50%
25%
this is a Moore Machine.
Mealy Machines let the output depend on the state and inputs.
the outputs only depend on the current state!
- we'll mostly stick with Moore machines, but sometimes a Mealy machine is more elegant. depends on the situation!
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10. Making a circuit out of it
this is a sequential circuit – the state changes over time
but the state transition and output tables are combinational
here's the general organization of any Moore FSM circuit:
state feeds back into transition logic
outputs based on state
transition logic
inputs
output logic
outputs D Q
state register
- this pattern is gonna come up over and over as we design circuits…
- this is basically how all sequential and combinational circuits come together!
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11. Kinds of FSMs
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12. Counters 0 bears 1 bear 2 bears ≥3 bears
counters just count how many of something they've "seen"
usually the states form a ring (like the ceiling fan)
sometimes they make a line:
0 bears
1 bear
2 bears
≥3 bears
bear or
cat
in this machine, there's no way to "get back" to an earlier state.
- that can get real annoying real fast, if you want to count up to large numbers.
- but sometimes you need to keep track of a number, and it doesn't have to be part of the FSM itself…
there are also as many states as numbers you want to represent…
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13. this is kind of a generator-counter combo.
Generators
sometimes you have no inputs
other than some signal to tell you to go to the next step (what's that signal called?)
these are useful for generating a sequence of values. 4 1 2 5 7 8 6 3 9
this is kind of a generator-counter combo.
I've used these for sequencing steps in a more complex process (another FSM!).
- that would be the clock.
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14. these arrows on the right are unconditional transitions
Controllers
let's say you want to make a simple vending machine
it accepts nickels and dimes and lets you buy 25¢ items 0¢ 5¢
10¢
15¢
20¢
25¢
30¢
nickel
dime
dispense
give nickel
- unconditional transitions happen… unconditionally…
- basically, when the clock signal happens.
these arrows on the right are unconditional transitions
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15. Multiplication and Division practical FSMs
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16. Remember this? we used this algorithm to do multiplication:
how might we make a circuit to count the steps?
for(n bits in multiplier) {
if((multiplier & 1) != 0)
product += multiplicand
multiplicand <<= 1
multiplier >>= 1 }
what do we use to make a decision?
what do we use to store values?
- an FSM to count the steps and sequence the actions…
- a MUX or write enable to make the decision…
- registers to store the values…
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17. + 1 A very common pattern D Q have a look at simple_counter.circ
this is a very simple FSM. every clock tick, it adds 1 to the value in the register.
this is the hardware equivalent of "x = x + 1".
this could really be anything though. like… shifting?
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18. So… what are we gonna need
we've got 3 variables (+1 for the loop counter)
we're doing 2 shifts and (maybe) an add on each loop D Q +
product
kindaaa…? D Q
<< 1
multiplicand
how do we decide when to add to the product? D Q
>> 1
multiplier D Q + 1
loop step
how do we know when to start and stop?
- well, gee whiz, this feels like a GREAT topic for a LAB doesn't it
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19. Fast multiplication as a circuit
remember there were two phases:
calculating the partial products
summing them
calculating the partial products is easy
it's just shifts and ANDs
in hardware, shifting can be done by just rerouting wires!
summing the partial products is easy
we use a binary tree of adders!
the product just pops out the end.
of course, this uses a lot of silicon…
have a look at fast_mult_4x4.circ
Fun Stuff
Partial Products +
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20. subtraction and shifting…
Deja vu
We used this division algorithm:
divisor <<= n (bits in dividend)
remainder = dividend
quotient = 0
for(n bits in dividend) {
divisor >>= 1
if(divisor > remainder) {
append 0 to quotient
} else {
remainder -= divisor
append 1 to quotient }
another for loop…
another if-else…
subtraction and shifting…
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21. - >> << + Feels pretty familiar 1 1 1
we've got 3 variables (+1 for the loop counter)
we're doing 2 shifts and (maybe) a subtract on each loop D Q -
remainder
kindaaa…? D Q
>> 1
divisor
how do we "append a 0 or 1" to the quotient? D Q
<< 1
quotient D Q + 1
loop step
how do we know when to start and stop?
- again, labtime
- "appending a 0 or 1" is just setting the low bit after shifting left
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22. Their similarity is no coincidence
in MIPS, both mul and div use the HI and LO registers
this is because in the original design, both algorithms used the same registers, just with different controllers
they do some fancy stuff to save bits
check out the book – it goes into way too much detail on this, but it's kinda neat
your lab will do something similar!
hohohoho!
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