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So today we are going to talk
about another process of lumping

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or another process of
discretization what will lead to

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the digital abstraction.
So today's lecture is titled

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"Go Digital".
So let me begin with a usual

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review.
And so in the first lecture we

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started out by looking at
elements and lumping them.

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For example,
we took an element and said for

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the purpose of analyzing
electrical properties let's lump

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this element into a single
lumped value called a resistor,

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R.
And this led to the lumped

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circuit abstraction.
The lumped circuit abstraction

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says let's take these elements,
connect them with wires and

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analyze the properties of these
using a sort of analysis

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technique.

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So a set of a methods.
We've looked at the KVL,

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KCL method.
Another example of a method we

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looked at was the node method.
And of this category there is

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one method you should remember,
which you can apply to every

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single circuit and it will
simply work, is the node method.

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For linear circuits other
methods also apply,

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and these include
superposition,

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Thevenin method,
and in recitation or in your

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course notes you would have
looked at the Norton method.

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So that's what we did so far.
So this is a toolkit.

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So now you have a utility belt
with a bunch of tools in it,

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and you can draw from those
tools.

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And, just like any good
carpenter, you know,

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the carpenter has to cut a
piece of wood.

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He could use a chisel.
He could use a saw.

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He could use an electric saw.
And the reason you pay

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carpenters $80 an hour in the
Boston region is because they

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know which tool to use for what
job.

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So what we'll learn today is,
so this was one process of

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discretization.
We discretized matter.

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This gave us the discipline
here that we decided to follow,

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lumped matter discipline,
that moved us from Maxwell's

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equations into this new
playground called EECS.

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Where all elements looked like
these rinky-dinky little values

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like resistors and voltage
sources and so on.

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What we'll do today,
if that wasn't simple enough,

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let's simplify our lives even
further.

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What we're going to do is lump
some more.

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So what else can we lump?
We've lumped matter,

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so all matter is taken care of.
So what can we lump to make

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life even easier?
When in doubt,

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if things are complicated,
discretize it or lump it,

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right?
So what do you think?

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What we will do today is lump
signal values.

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So we'll just deal with lumped
values.

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And this will lead to the
digital abstraction.

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And the related reading is
Chapter 5 of the course notes.

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So before we do this kind of
lumping, let me motivate why we

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do this.
One reason is to simplify our

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lives, but there is no need to
just go around simplifying

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things just because we can.
Let's try to see if there are

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other reasons motivating the
digital abstraction.

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So what I would like to start
with is a simple example of a

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analog processing circuit that
you should now be able to

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analyze.
So I'm going to be motivating

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digital.
So let's start with an analog

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circuit that looks like this,
two resistors,

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R1 and R2.
And what I'm going to do is

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apply a voltage source here,
V1, apply another one here,

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V2, and make this connection.
And let me call this voltage V

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nought and call this my output.
This voltage with respect to

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ground node, rather than drawing
this wire here,

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I often times draw a ground
here and simply throw ground

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wherever I want.
This symbol simply refers to

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the fact that the other terminal
is taken at the ground node.

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So here is my V nought.
Now, let's go and analyze this

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and see what it gives us.
In this example,

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V1 and V2 may be outputs of two
sensors, maybe heat sensors or

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something like that.
This is a heat sensor on that

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side of the room and this is a
heat sensor on this side of the

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room.
And I pass their signals

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through two resistors and I look
at the voltage there.

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So by now you should be able to
write the answer V nought,

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or the value V nought almost by
inspection.

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Just to show you,
let me use superposition.

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When you see multiple sources,
the first thing you should

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think about is can I use
superposition to simplify my

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life?
And let me do that.

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V nought here is the sum of two
voltages, one due to V1 acting

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alone and one due to V2 acting
alone.

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So what's the voltage here due
to V1 acting alone?

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To find out that I short this
voltage, I zero out this voltage

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and look at the effect of V1.
So the effect of V1,

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if this were shorted out,
is simply V1 x R2 / R1 + R2.

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This is now a voltage divider,
right?

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A voltage V applied across two
resistors and the output taken

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across one resistor.
So that's this value.

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Then I could do the second
part.

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To look at the effect of V2,
what I will do is short this

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voltage and look at the effect
of this.

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Now, this voltage is across
this resistor divider.

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And so I get R1 / (R1 + R2)
here.

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So you'll notice that for
something like this,

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if I had applied KVL and KCL of
the node method I would have

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gotten a bunch of equations,
but here I wrote it just by

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inspection.
You should be able to look at

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circuit patterns like this and
write the answers down very

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quickly.
Let's say if I chose R1 to be

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equal to R2 then V nought would
simply be (V1 + V2) / 2.

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So if these two values were
equal, I simply get the output,

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the average of the two
voltages.

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So this guy is an adder
circuit.

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It adds up these two voltages.
But more precisely it's an

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averaging circuit.
It takes two voltages and gives

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me the average value.
Now, if you have two sensors in

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the room, you might think of why
you want to take that average

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value to control the temperature
of the room.

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But suffice it to say that V
nought is the average of the two

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values.
So let me show you a quick demo

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of this example and then look at
what the problems are with this

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example.
So let's say,

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as one example,
I applied a square wave at V1,

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which is the top curve,
the green curve,

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and I applied a triangular wave
at V2, that's the second one.

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As you expect,
the output is going to be the

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sum of the two voltages scaled
appropriately.

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So notice that I have a square
wave with a superimposed

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triangular wave on top.
And I can play around.

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What I could do is change the
amplitude of my wave form here.

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And, as you notice,
the amplitude of the output

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component also changes
accordingly.

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So this is one simple example
of an adder circuit,

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and the two wave forms get
summed up and I get the output.

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So I'll switch to Page 3.
Let me just draw a little

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sketch for you here.
Here, what I showed you was I

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had a triangular wave coming on
one of these inputs and I had a

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square wave on the other one,
and the output looks something

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like this.

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OK?
No surprise here.

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This is a simple analog signal
processing circuit which gives

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me the average of two wave
forms.

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Now, let me do the following.
Often times I may need to look

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at this value some distance
away.

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So let's say this person here
wants to look at the value.

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So I bring this wire here.
And I also bring the ground

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connection and I look at it.
I look at this value here.

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And when I have a long wire I
can get noise added onto the

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circuit.
So let's say a bunch of noise

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gets added into the signal
there.

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And what I end up seeing here
is not something that looks like

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this but something that looks
like that.

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That's not unusual.
And the problem with this is

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now when I look at this,
if I'm looking to distinguish

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between, say,
a 3.9 and a 3.8,

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it's really hard to do that
because my noise is overwhelming

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my signal.
I have a real problem,

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a real problem here.
Noise is a fact of life.

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So what do we do?
This is so fundamental.

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Large bodies of courses in
electrical engineering are

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devoted to how do I carefully
analyze signals in the presence

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of noise?
You'll take courses in speech

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processing that look at clever
techniques to recognize speech

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in the presence of noise and so
on and so forth.

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One technique we adopt that
we'll talk about here,

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which is fundamental to EECS,
is using the digital

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abstraction.
Let me show you how it can

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really help with the noise
problem.

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So the idea is value lumping or
value discretization.

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Much like we lumped matter,
we've discretized matter into

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discrete chunks,
let's discretize value into two

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chunks.
Let's simply say that now I'm

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going to deal with two values
and I can, say,

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call them high,
low.

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I have a bunch of choices here.
I may call it 5 volts and 0

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volts.
I may call it true and false.

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What I'm doing is I'm just
restricting my universe to deal

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with just two values,
zero and one.

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This is like dealing with a
number system with only two

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digits.
And these are zero and one.

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So what I've now done is I'm
saying that rather than dealing

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with all possible continuous
values, 0.1, 3.9999 recurring

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and so on and so forth,
what I'm going to do is simply

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deal with a high and a low.
Dealing with this whole

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continuum of numbers is really
complicated.

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Let me simplify my life and
just postulate that I am going

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to be looking at high and low.
Whenever I see something I'll

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look at it and say high or low,
is it black or white,

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period.
There's no choice here,

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just two individual values.
So that sounds simple,

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and nice and so on,
but what's the point?

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What do we get by doing that?
Let's take our example.

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Let's take what might be a
digital system.

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Let's take a digital system and
let's say I have a sender.

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Much like I sent a signal value
a long distance,

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let me have a sender,
and I have a ground as well and

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here is a receiver.
This symbol simply says that

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both of them share a ground
wire.

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So the sender and a receiver.
And what I'm interested in

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doing, the sender is interested
in sending a signal to the

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receiver.
And in the digital system,

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the way I would send a digital
signal is all I can use is ones

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and zeros, OK?
So let's say the sender sends

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something like this.
The sender wants to send a

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value.
This is my time axis and this

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is 2.5 volts,
this is 0 volts and this is 5

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volts.
My sender has some agreement

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with the receiver and says I'm
just going to be sending to you

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low values and high values.
And this signal here would

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correspond to "0" "1" "0".
It's a symbol.

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That's why I have input zero in
quotes there.

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We'll go into this in much more
detail later,

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but for now suffice it to say
that I'm sending a set of

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signals here "0" "1" "0".
This simplistic scheme will not

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work in many situations but go
along with this for a few

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seconds.
So I send the signal sequence

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"0" "1" "0" out here.
And notice that there is a high

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and a low.
And the agreement the sender

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and the receiver have is that,
look, if you see a value that's

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higher than 2.5 volts that's a
high.

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00:15:48 --> 00:15:53
If you see a value below 2.5
volts in the wire that's a low.

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00:15:53 --> 00:15:57
And I'm going to send a 0 volt
and a 5 volt from here.

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So now at the sending site
let's say I don't have any noise

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in this system.
Let's say this is my Vn,

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some noise being added.
And let's say Vn is 0.

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Then in that case I will
receive exactly what is sent "0"

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"0" 5, 2.5, 0 volts.
And this is time.

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Nothing fancy here,
right?

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My receiver receives a "0" "1"
"0".

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Now, the beauty of this is that
now suppose I were to impose

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noise much like I had noise out
there and Vn was not 0.

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Rather Vn was some noise
voltage, let's say 0.2 volts

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peak to peak.
Let's say that simply got

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superposed on the signal.
In which case what do I get?

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What I end up here with is a
signal that looks like this.

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So the receiver gets that
signal because a noise is added

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into my signal and that's what I
get.

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But guess what?
No problem.

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The receiver says oh,
yeah, this is a 0 because the

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values are less than 2.5,
this is a 1 and this is a 0.

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00:17:21 --> 00:17:24
"0" "1" "0".
So here my receiver was able to

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receive the signal and correctly
interpret it without any

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problems.
So because I used this value

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discretization and because I had
this agreement with the

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receiver, I had better noise
immunity.

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00:17:39 --> 00:17:45


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00:17:45 --> 00:17:48
Consequently,
I had what is called a noise

246
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margin.
Noise margin says how much

247
00:17:51 --> 00:17:56
noise can I tolerate?
And in this situation,

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00:17:56 --> 00:18:00
because the sender sends 5
volts and 0 volts,

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the 5 volts can creep all the
way down to 2.5,

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I'll still be OK.
Similarly, 0 could go all the

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00:18:10 --> 00:18:14
way up to 2.5,
I'd still be OK.

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00:18:14 --> 00:18:21
So in this case I have a noise
margin of 2.5 volts for a 1 and

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similarly 2.5 volts for a 0,
because there are 2.5 volts

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00:18:28 --> 00:18:34
between a 0 volt and 2.5.
So notice that I have a nice

255
00:18:34 --> 00:18:37
little noise margin here,
which simply is the English

256
00:18:37 --> 00:18:41
meaning of the term there is a
margin for noise.

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00:18:41 --> 00:18:45
And even though I can change
the signal value by up to 2.5

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00:18:45 --> 00:18:50
volts, the receiver will still
correctly interpret the signal.

259
00:18:50 --> 00:18:54
So I've decided to discretize
values into highs and lows.

260
00:18:54 --> 00:18:57
And because of that,
if all I wanted to do in life

261
00:18:57 --> 00:19:03
is send highs and lows I can
send them very effectively.

262
00:19:03 --> 00:19:08
There are many complications,
but if all I care about is

263
00:19:08 --> 00:19:14
sending highs and lows I can
send it with a lot of tolerance

264
00:19:14 --> 00:19:17
to noise.
So many of you are saying but

265
00:19:17 --> 00:19:21
what about this,
but what about that?

266
00:19:21 --> 00:19:27
There are lots of buts here.
And let's take a look at some

267
00:19:27 --> 00:19:31
of them.
If you look up there.

268
00:19:31 --> 00:19:38
What I ended up doing was
creating a design space that

269
00:19:38 --> 00:19:42
looked like this.
This is on Page 6.

270
00:19:42 --> 00:19:50
What I did was I said with a
range of values from 0 to 5,

271
00:19:50 --> 00:19:58
what I'm going to do is at 2.5
I drew a line and I said as a

272
00:19:58 --> 00:20:05
sender if you wanted to send a 0
then you would send a value

273
00:20:05 --> 00:20:10
here.
And if you wanted to send a 1

274
00:20:10 --> 00:20:14
you would send a value here.
Similarly, for a receiver.

275
00:20:14 --> 00:20:18
And if the sender sent a value
all the way up in 5 volts that

276
00:20:18 --> 00:20:22
was the best thing,
but technically the sender

277
00:20:22 --> 00:20:25
could send any value between 2.5
and 5.

278
00:20:25 --> 00:20:29
And if there was no noise then
the receiver could correctly

279
00:20:29 --> 00:20:35
interpret a 1 if it was above
this and 0 if it was below this.

280
00:20:35 --> 00:20:39
The problem with this approach
really is that if I allow the

281
00:20:39 --> 00:20:44
sender to send any value above
2.5 all the way to 5 then there

282
00:20:44 --> 00:20:48
really is no noise margin in
this situation.

283
00:20:48 --> 00:20:53


284
00:20:53 --> 00:20:55
OK?
Because if I allowed the sender

285
00:20:55 --> 00:21:00
to send any value between 2.5
and 5 then what if I have a

286
00:21:00 --> 00:21:04
value 2.5 for a 1?
Then I may end up getting very

287
00:21:04 --> 00:21:08
little noise margin on the other
side.

288
00:21:08 --> 00:21:11
Worse yet, what if I get a
value 2.5?

289
00:21:11 --> 00:21:16
That's a much worse situation.
What if the receiver receives a

290
00:21:16 --> 00:21:18
value of 2.5?
Now what?

291
00:21:18 --> 00:21:23
What does the receiver do?
The receiver cannot tell

292
00:21:23 --> 00:21:28
whether it's a 1 or a 0.
The receiver gets hopelessly

293
00:21:28 --> 00:21:33
confused.
So to deal with that,

294
00:21:33 --> 00:21:40
I'm going to fix this,
what I'm going to do is the

295
00:21:40 --> 00:21:45
following.
Switch to Page 7.

296
00:21:45 --> 00:21:53
What I'll do here is to prevent
the receiver from getting

297
00:21:53 --> 00:22:02
confused, if the receiver saw
2.5, what I'm going to do is

298
00:22:02 --> 00:22:10
define what is called "no man's
land".

299
00:22:10 --> 00:22:15
I'm going to define the region
of my voltage space called the

300
00:22:15 --> 00:22:19
forbidden region.
And what I'm going to do is,

301
00:22:19 --> 00:22:24
say, let's say I defined it as
2 volts, 3 volts and 5 volts,

302
00:22:24 --> 00:22:28
0, 2, 3 and 5.
With my forbidden region,

303
00:22:28 --> 00:22:33
if I have a sender then I tell
the sender you can send any

304
00:22:33 --> 00:22:39
value between 3 and 5 for a 1.
And you can send any value

305
00:22:39 --> 00:22:44
between 2 and 0 for a 0.
To send the symbol 0,

306
00:22:44 --> 00:22:50
I can send any voltage between
0 and 2, and similarly for 1.

307
00:22:50 --> 00:22:55
At the receiving side,
if I see any value between 3

308
00:22:55 --> 00:23:00
and 5, I read that as a 0,
and any value between 0 and 2 I

309
00:23:00 --> 00:23:06
read that as 2 volts.
So I may label this value VH

310
00:23:06 --> 00:23:11
and label this threshold VL,
so there's a high threshold and

311
00:23:11 --> 00:23:15
a low threshold.
So this solves one problem.

312
00:23:15 --> 00:23:20
Now the receiver can never see
a value in the forbidden region.

313
00:23:20 --> 00:23:24
Now, I can stand her and
pontificate and say,

314
00:23:24 --> 00:23:30
oops, that's a forbidden
region, thou shalt not go there.

315
00:23:30 --> 00:23:33
But what if I get some noise
and a value goes in there?

316
00:23:33 --> 00:23:36
In real systems values may
enter there.

317
00:23:36 --> 00:23:39
But what I'm saying,
so this is the beauty of using

318
00:23:39 --> 00:23:41
a discipline.
Let me use my playground

319
00:23:41 --> 00:23:43
analogy.
This is my playground.

320
00:23:43 --> 00:23:47
We got into this playground
using the discrete matter of

321
00:23:47 --> 00:23:51
discipline, the playground of
EECS, but in that playground

322
00:23:51 --> 00:23:55
some region of that playground
deals with just high and low

323
00:23:55 --> 00:23:57
values.
I further restrict the

324
00:23:57 --> 00:24:00
playground and I say I'm only
going to focus on that

325
00:24:00 --> 00:24:06
playground in which all signal
values have a forbidden region.

326
00:24:06 --> 00:24:09
All senders and receivers
adhere to a forbidden region.

327
00:24:09 --> 00:24:14
And if there is any signal in
this space, in the forbidden

328
00:24:14 --> 00:24:16
space then my behavior is
undefined.

329
00:24:16 --> 00:24:19
I don't care.
You want to go there?

330
00:24:19 --> 00:24:21
Sure.
I don't know what's going to

331
00:24:21 --> 00:24:24
happen to you.
Now, we're engineers,

332
00:24:24 --> 00:24:26
right?
So we've disciplined ourselves

333
00:24:26 --> 00:24:32
to play in this playground.
It's like I tell my 9-year-old,

334
00:24:32 --> 00:24:33
don't go there,
right?

335
00:24:33 --> 00:24:36
And of course he wants to go
there.

336
00:24:36 --> 00:24:39
He says what will happen if I
go there?

337
00:24:39 --> 00:24:42
And the answer here will be
undefined, OK?

338
00:24:42 --> 00:24:45
Something really bad could
happen to you.

339
00:24:45 --> 00:24:48
I don't know what it is but
something really bad,

340
00:24:48 --> 00:24:52
you know, a lightening bolt or
who knows what,

341
00:24:52 --> 00:24:55
but something really bad.
And you as a designer of a

342
00:24:55 --> 00:25:00
circuit can, let's say you were
Intel.

343
00:25:00 --> 00:25:02
Intel designs its chips.
And let's say Intel decides to

344
00:25:02 --> 00:25:05
play in this playground and
there is a forbidden region.

345
00:25:05 --> 00:25:08
So Intel says oh,
it's really easy for me if in

346
00:25:08 --> 00:25:11
the forbidden region the chip
simply burns up and catches

347
00:25:11 --> 00:25:13
fire, we'll sell more chips.
That's fine.

348
00:25:13 --> 00:25:15
Whatever you want.
The key here is that all I'm

349
00:25:15 --> 00:25:19
saying is that I am going to
discipline myself into playing

350
00:25:19 --> 00:25:22
in this playground and that's
where I will define my rules,

351
00:25:22 --> 00:25:24
and you stay within the
boundaries and all the rules

352
00:25:24 --> 00:25:28
will apply.
It's called a "discipline."

353
00:25:28 --> 00:25:31
You're disciplining yourselves
to stay within it.

354
00:25:31 --> 00:25:34
There's no logic to it.
It's just a discipline.

355
00:25:34 --> 00:25:37
Just do it and you'll be OK.
When we look at practical

356
00:25:37 --> 00:25:40
circuits and so on,
we have to address the issue of

357
00:25:40 --> 00:25:43
what happens when things go in
there.

358
00:25:43 --> 00:25:45
But let's postpone that
discussion.

359
00:25:45 --> 00:25:48
For now I've solved one of my
problems, which is,

360
00:25:48 --> 00:25:52
the previous problem was what
does a receiver do if it saw a

361
00:25:52 --> 00:25:53
2.5?
Now it can't see a 2.5.

362
00:25:53 --> 00:25:57
But then the receiver asks,
Agarwal, but what if I see a

363
00:25:57 --> 00:26:00
2.5?
I can tell the receiver you can

364
00:26:00 --> 00:26:03
do whatever you want to do.
You can stomp it.

365
00:26:03 --> 00:26:05
You can squish it.
You can burn it.

366
00:26:05 --> 00:26:07
You can chuck it.
Whatever you want.

367
00:26:07 --> 00:26:10
It's up to you.
Do whatever you want.

368
00:26:10 --> 00:26:13
You won't see a value.
If you do, do whatever you

369
00:26:13 --> 00:26:14
want.
It's undefined.

370
00:26:14 --> 00:26:16
That works.
So you, as the receiver

371
00:26:16 --> 00:26:19
designer can do whatever you
want when you see a 2.5.

372
00:26:19 --> 00:26:22
You can say yeah,
I'll just put out a 1 if I see

373
00:26:22 --> 00:26:25
a 2.5 or a 2.6.
I'll just do something.

374
00:26:25 --> 00:26:27
No one cares.
So this is pretty good.

375
00:26:27 --> 00:26:32
This is pretty good.
We still have a problem,

376
00:26:32 --> 00:26:35
though.
Do people see the problem here?

377
00:26:35 --> 00:26:40
This still doesn't quite work.
If Intel did this,

378
00:26:40 --> 00:26:46
instead of your laptops failing
and blue-screening every hour

379
00:26:46 --> 00:26:49
they'd be doing it every
millisecond.

380
00:26:49 --> 00:26:54
So the problem is this
discipline have allowed the

381
00:26:54 --> 00:27:00
sender to send any value between
3 and 5 as a 1.

382
00:27:00 --> 00:27:05
And any value between 3 and 5
at the receiver is treated as a

383
00:27:05 --> 1.


384
1. --> 00:27:08
Do you see where the problem

385
00:27:08 --> 00:27:09
is?
Yes?

386
00:27:09 --> 00:27:14
The sender sends a 1.99 and the
noise pumps it into forbidden

387
00:27:14 --> 00:27:16
region.
Exactly.

388
00:27:16 --> 00:27:20
So the sender says it's
legitimate, I'm Intel.

389
00:27:20 --> 2.
They've told me stick to 0 and

390
2. --> 00:27:23


391
00:27:23 --> 00:27:30
And Intel parts will be sending
to values between 0 and 2.

392
00:27:30 --> 00:27:32
And Motorola parts,
which are receivers,

393
00:27:32 --> 00:27:35
you know they have to receive 0
and 2.

394
00:27:35 --> 2.
So Intel can send the value,

395
2. --> 00:27:37


396
00:27:37 --> 00:27:39
They can because it's 1.9 out
of 2.

397
00:27:39 --> 00:27:42
It's legal.
This way I can make really

398
00:27:42 --> 00:27:45
cheap parts.
But now the problem is that

399
00:27:45 --> 00:27:48
even the smallest amount of
noise will bump it into the

400
00:27:48 --> 00:27:52
forbidden region,
and so therefore this one has a

401
00:27:52 --> 00:27:54
problem.
And the problem is that this

402
00:27:54 --> 00:28:00
one offers zero noise margin.
There is no noise margin.

403
00:28:00 --> 00:28:05
There is no margin for noise in
the discipline.

404
00:28:05 --> 00:28:10
All right, back to the drawing
board, folks.

405
00:28:10 --> 00:28:15
Switch to Page 8.
Let's get rid of all this stuff

406
00:28:15 --> 00:28:20
and go back to the drawing
board.

407
00:28:20 --> 00:28:31


408
00:28:31 --> 00:28:35
OK, so what do we do now?
How about the following?

409
00:28:35 --> 00:28:39
How, about as before I say,
as a receiver,

410
00:28:39 --> 00:28:44
if you see a value between 3
and 5 you treat that as a 1 and

411
00:28:44 --> 00:28:49
a value between 0 and 2 you
treat that as a 0.

412
00:28:49 --> 00:28:52
No difference.
So as a receiver same as

413
00:28:52 --> 00:28:56
before.
But now what I do is I hold the

414
00:28:56 --> 00:29:03
sender to tougher standards.
I hold the feet of the sender

415
00:29:03 --> 00:29:09
to the fire and say you have to
adhere to tougher standards.

416
00:29:09 --> 00:29:16
So what I'm going to do is hold
the sender to tougher standards,

417
00:29:16 --> 00:29:21
maybe four walls.
That is tell the sender that if

418
00:29:21 --> 00:29:27
you want to send to 0 or a 1,
for a 1 you have to send a

419
00:29:27 --> 00:29:33
value between 4 and 5,
and for a 0 a value between 0

420
00:29:33 --> 00:29:37
and 1.
Sender is now held to tougher

421
00:29:37 --> 00:29:41
standards.
This is what my chart looks

422
00:29:41 --> 00:29:44
like.
So now I do have some noise

423
00:29:44 --> 00:29:48
margin.
Can someone tell me what is the

424
00:29:48 --> 00:29:51
noise margin here for a 1?
1 volt.

425
00:29:51 --> 00:29:57
And the reason is that the
lowest voltage a sender can send

426
00:29:57 --> 00:30:02
is 4 volts, OK?
If the 4 leaks down to 2.99

427
00:30:02 --> 00:30:05
that's in the forbidden region,
I'm in trouble.

428
00:30:05 --> 2.99.


429
2.99. --> 00:30:08
This is my forbidden region

430
00:30:08 --> 00:30:10
here.
And 2.99 is in the forbidden

431
00:30:10 --> 00:30:12
region.
I'm in trouble.

432
00:30:12 --> 00:30:17
So notice that the lowest value
that the receiver can receive is

433
00:30:17 --> 00:30:19
3 volts.
So if I sent the 4 and sent

434
00:30:19 --> 00:30:24
this over a long cable to you,
the value can be beaten up by

435
00:30:24 --> 00:30:28
noise to such an extent that you
may begin receiving 3s but

436
00:30:28 --> 00:30:35
nothing lower than a 3.
So this is a noise margin,

437
00:30:35 --> 00:30:39
1 volt.
Similarly, for a 0 the noise

438
00:30:39 --> 00:30:44
margin is also 1 volt.
So let me label these.

439
00:30:44 --> 00:30:48
There are four important
thresholds here.

440
00:30:48 --> 00:30:53
This threshold is called VOL.
V output low.

441
00:30:53 --> 00:31:01
These have special meanings.
This threshold here is called

442
00:31:01 --> 00:31:06
VOH, V output high.
This threshold here is called V

443
00:31:06 --> 00:31:12
input high and this threshold
here is called V input low.

444
00:31:12 --> 00:31:19
So VOH simply says that senders
must send voltages higher than

445
00:31:19 --> 00:31:23
VOH.
Receivers must receive values

446
00:31:23 --> 00:31:28
higher than VIH as a 1.
So these four thresholds

447
00:31:28 --> 00:31:33
together give you your
threshold.

448
00:31:33 --> 00:31:46


449
00:31:46 --> 00:31:50
For the sender gets 2.5,
what does sender do?

450
00:31:50 --> 00:31:53
It could do that.
So, in that case,

451
00:31:53 --> 00:31:57
you can do that.
If all you want to do is have

452
00:31:57 --> 00:32:03
one value here then what you
have is an infinitesimal value

453
00:32:03 --> 00:32:07
here for the forbidden region.
That's fine.

454
00:32:07 --> 00:32:09
It's up to you to design it
that way.

455
00:32:09 --> 00:32:12
You can.
But it turns out that when you

456
00:32:12 --> 00:32:14
design circuits,
when we see some examples in

457
00:32:14 --> 00:32:18
the next lecture it turns out to
be fairly practical and easy to

458
00:32:18 --> 00:32:21
do it this way.
But, again, these are design

459
00:32:21 --> 00:32:22
choices.
If I'm Intel,

460
00:32:22 --> 00:32:25
Intel wants all its parts to
work together.

461
00:32:25 --> 00:32:28
So parts that follow a common
discipline can work together,

462
00:32:28 --> 00:32:33
right?
Because senders will send

463
00:32:33 --> 00:32:38
values, receivers will receive
these values here,

464
00:32:38 --> 00:32:45
so it will simply work.
So the noise margin for a 1

465
00:32:45 --> 00:32:52
here is simply VOH minus VIH and
the noise margin for a 0 is VIL

466
00:32:52 --> 00:32:57
minus VOL.
VIL minus VOL is the noise

467
00:32:57 --> 00:33:02
margin for a 0.
So what do we have here?

468
00:33:02 --> 00:33:06
What we have here is a
discipline that we've agreed to

469
00:33:06 --> 00:33:11
follow where senders are held to
a tough standard and receivers

470
00:33:11 --> 00:33:16
are held to a different standard
so that I allow myself some

471
00:33:16 --> 00:33:20
margin for error.
And it's up to you as a

472
00:33:20 --> 00:33:24
designer to choose ranges for
the forbidden region.

473
00:33:24 --> 00:33:29
Now, you may say that I want to
make my forbidden region as

474
00:33:29 --> 00:33:34
small as possible.
But you will see in practical

475
00:33:34 --> 00:33:37
circuits it's very hard to
achieve that.

476
00:33:37 --> 00:33:41
Practical devices that you get,
they have a natural region that

477
00:33:41 --> 00:33:45
gets very, very hard to break
apart, and that tends to

478
00:33:45 --> 00:33:48
establish what that region looks
like.

479
00:33:48 --> 00:33:53
So to continue with an example
here, I may have the following

480
00:33:53 --> 00:33:57
voltage wave form for a sender.
So I have some sender,

481
00:33:57 --> 00:34:00
I have a sender here.

482
00:34:00 --> 00:34:08


483
00:34:08 --> 00:34:13
I have VOL, VIL,
VIH, VOH and some other high

484
00:34:13 --> 00:34:17
voltage.
And then, as a sender,

485
00:34:17 --> 00:34:22
if I want to send a "0" "1" "0"
then I send a 0.

486
00:34:22 --> 00:34:29
I have to be within this band.
And then for a 1 I have to be

487
00:34:29 --> 00:34:35
within this band.
So this is an example of,

488
00:34:35 --> 00:34:41
say, "0" "1" "0" "1".
And at the receiver --

489
00:34:41 --> 00:34:51


490
00:34:51 --> 00:34:55
Let's have VOL,
VIL, VIH, VOH.

491
00:34:55 --> 00:35:02
So at the receiver,
I interpret any signal below

492
00:35:02 --> 00:35:06
VIL as a 0.
So I may get some signal that

493
00:35:06 --> 00:35:08
looks like this.

494
00:35:08 --> 00:35:16


495
00:35:16 --> 00:35:21
And I'll still interpret that
as a "0" "1" "0" "1".

496
00:35:21 --> 00:35:26
So to summarize here,
this discipline that forms the

497
00:35:26 --> 00:35:34
foundations of digital systems
is called "a static discipline".

498
00:35:34 --> 00:35:41


499
00:35:41 --> 00:36:03
The static discipline says if
inputs meet input thresholds --

500
00:36:03 --> 00:36:08
So if an input to a digital
system meets the input

501
00:36:08 --> 00:36:14
thresholds then outputs will
meet, or the digital system

502
00:36:14 --> 00:36:19
should ensure that the outputs
--

503
00:36:19 --> 00:36:24


504
00:36:24 --> 00:36:29
Output thresholds.
So this means that if I have a

505
00:36:29 --> 00:36:35
system like this then if I give
it good inputs.

506
00:36:35 --> 00:36:40
And by giving it good inputs I
mean for 1s I have signal values

507
00:36:40 --> 00:36:45
that are greater than VIH and
for 0s signal values which are

508
00:36:45 --> 00:36:48
less than VIL.
These are valid inputs.

509
00:36:48 --> 00:36:53
So if my inputs are valid,
that is below VIL for a 0 and

510
00:36:53 --> 00:36:57
above VIH for a 1 then this
digital system D will produce

511
00:36:57 --> 00:37:03
corresponding outputs that
follow output thresholds.

512
00:37:03 --> 00:37:07
For a 1 it will produce outputs
that are greater than VOH and if

513
00:37:07 --> 00:37:12
it needs to produce a 0 it will
produce outputs that are less

514
00:37:12 --> 00:37:15
than VOL.
So notice that there is this

515
00:37:15 --> 00:37:19
tough requirement in digital
systems that for the inputs,

516
00:37:19 --> 00:37:23
I should recognize as a 1
anything higher than a VIH.

517
00:37:23 --> 00:37:27
But if I want to produce a 1,
I have to produce a tough 1

518
00:37:27 --> 00:37:31
like a 4-volt 1.
So there is a discipline that

519
00:37:31 --> 00:37:35
all my digital systems must
follow, and that discipline is

520
00:37:35 --> 00:37:39
called a static discipline.
So static discipline encodes

521
00:37:39 --> 00:37:42
the thresholds,
encodes four thresholds that

522
00:37:42 --> 00:37:46
all digital systems must follow
so that they can talk to each

523
00:37:46 --> 00:37:49
other.
So if Intel and Motorola want

524
00:37:49 --> 00:37:51
to make parts that are
compatible with,

525
00:37:51 --> 00:37:55
say, Pentium 4 devices then
they will all talk over the

526
00:37:55 --> 00:38:00
phone or something and agree on
a static discipline.

527
00:38:00 --> 00:38:02
We will say that,
all right, all my peripherals

528
00:38:02 --> 00:38:05
will follow a static discipline
with the following volted

529
00:38:05 --> 00:38:07
thresholds.
And this way parts made by

530
00:38:07 --> 00:38:10
different manufacturers can
interoperate and still provide

531
00:38:10 --> 00:38:11
immunity to noise.
Yes.

532
00:38:11 --> 00:38:13
Question?

533
00:38:13 --> 00:38:19


534
00:38:19 --> 00:38:21
Absolutely.
There are many constraints on

535
00:38:21 --> 00:38:24
how you as a designer choose the
noise margin.

536
00:38:24 --> 00:38:28
As a designer you want to make
your noise margin as large as

537
00:38:28 --> 00:38:32
possible.
The larger the noise margin the

538
00:38:32 --> 00:38:35
better you can tolerate noise
which is why,

539
00:38:35 --> 00:38:40
how many people have heard of
some devices called rad hard

540
00:38:40 --> 00:38:44
devices, radiation hard devices?
Some of you have.

541
00:38:44 --> 00:38:48
There are a bunch of devices.
Different manufacturers make

542
00:38:48 --> 00:38:52
different kinds of devices for
different markets.

543
00:38:52 --> 00:38:57
For consumer markets they use
parts which may have relatively

544
00:38:57 --> 00:39:01
poor noise margins because
consumers can tolerate more

545
00:39:01 --> 00:39:05
faults.
But if you're building devices

546
00:39:05 --> 00:39:09
for, say, the medical industry
or for spaceships and so on,

547
00:39:09 --> 00:39:13
you need to be held to a much,
much tougher standard.

548
00:39:13 --> 00:39:16
So for those devices you may
end up having much,

549
00:39:16 --> 00:39:20
much tighter bands in which you
have to operate so you have a

550
00:39:20 --> 00:39:23
tougher noise margin.
So that leads us to,

551
00:39:23 --> 00:39:28
given these sort of voltage
thresholds, we now move into the

552
00:39:28 --> 00:39:31
digital world.
And in the digital world we can

553
00:39:31 --> 00:39:33
build a bunch of digital
devices.

554
00:39:33 --> 00:39:36
The first device we will look
at is called a combinational

555
00:39:36 --> 00:39:37
gate.

556
00:39:37 --> 00:39:42


557
00:39:42 --> 00:39:46
A combinational gate is a
device that adheres to the

558
00:39:46 --> 00:39:51
static discipline,
Page 11, and this is a device

559
00:39:51 --> 00:39:56
whose outputs are a function of
inputs alone.

560
00:39:56 --> 00:40:03


561
00:40:03 --> 00:40:07
So I can build little boxes
which take some inputs,

562
00:40:07 --> 00:40:13
produces an output where the
outputs are a function of the

563
00:40:13 --> 00:40:17
existing inputs.
And this kind of a device is

564
00:40:17 --> 00:40:22
called a combinational gate.
And I can analyze such devices

565
00:40:22 --> 00:40:28
for the kinds of things that I
would like to do.

566
00:40:28 --> 00:40:34
Before I go into the kinds of
devices I'd like to build,

567
00:40:34 --> 00:40:39
let's spend a few minutes
talking about how to process

568
00:40:39 --> 00:40:44
signals.
How to process digital signals,

569
00:40:44 --> 00:40:48
Page 10.
So notice that you have two

570
00:40:48 --> 00:40:52
values, 0 and a 1.
So devices like my

571
00:40:52 --> 00:40:55
combinational gate,
for example,

572
00:40:55 --> 00:41:02
can only deal with 0s and 1s.
So I have to come up with some

573
00:41:02 --> 00:41:06
kind of a mathematics or some
kind of a set of processing that

574
00:41:06 --> 00:41:10
can work with 0,1 values.
So 0,1 map completely natural

575
00:41:10 --> 00:41:15
to the logic true and false.
So I can borrow from logic and

576
00:41:15 --> 00:41:19
use true and false to do my
processing of signals.

577
00:41:19 --> 00:41:23
So if all I care about is
processing logic values,

578
00:41:23 --> 00:41:29
0s and 1s, trues and falses
then that's all I need.

579
00:41:29 --> 00:41:33
I can also use numbers.
How do I represent a number?

580
00:41:33 --> 00:41:38
3.9 which is 0s and 1s.
It turns out that this is a

581
00:41:38 --> 00:41:43
whole field in itself.
You'll hear more about this in

582
00:41:43 --> 00:41:46
recitation.
Let me also point you to the

583
00:41:46 --> 00:41:52
last section of the course
notes, Chapter 5.6 I believe,

584
00:41:52 --> 00:41:57
that talks about how to
represent numbers.

585
00:41:57 --> 00:42:01
The basic insight is much like
you can represent arbitrary long

586
00:42:01 --> 00:42:05
numbers with the digits 0
through 9 in the same way,

587
00:42:05 --> 00:42:09
but concatenating digits you
can represent arbitrary long

588
00:42:09 --> 00:42:11
numbers with 0-1-1-1-0-0 and so
on.

589
00:42:11 --> 00:42:15
So you can have a whole
sequence of digits and you can

590
00:42:15 --> 00:42:19
build a binary number system.
So you can read A&L Section

591
00:42:19 --> 00:42:22
5.6, I believe.
It's the last section for

592
00:42:22 --> 00:42:24
numbers.
And you will also discuss this

593
00:42:24 --> 00:42:31
in your recitation tomorrow.
Let me spend some more time

594
00:42:31 --> 00:42:36
talking about Boolean logic,
two-valued logic,

595
00:42:36 --> 00:42:40
and how to process these
systems.

596
00:42:40 --> 00:42:46
So one way of processing it is
using logic statements of the

597
00:42:46 --> 00:42:52
following form.
If X is true and Y is true then

598
00:42:52 --> 00:43:00
Z is true, else is Z false.
So this is a logic statement.

599
00:43:00 --> 00:43:04
It says if X is true and Y is
true then Z is true,

600
00:43:04 --> 00:43:08
else Z is false.
So I can process this with 0s

601
00:43:08 --> 00:43:13
and 1s, trues and falses.
And I do this all the time so I

602
00:43:13 --> 00:43:16
have a succinct notation for
this.

603
00:43:16 --> 00:43:20
I express this as Z is X anded
with Y.

604
00:43:20 --> 00:43:24
X and Y is Z.
So Z is true if X is true and Y

605
00:43:24 --> 00:43:27
is true.
A shorthand notation for this

606
00:43:27 --> 00:43:33
is just a dot.
And a circuit notation for this

607
00:43:33 --> 00:43:38
is called an "AND gate".
That's a little circuit.

608
00:43:38 --> 00:43:41
I haven't told you what's
inside it.

609
00:43:41 --> 00:43:47
It's an abstract little device
called an AND gate which takes

610
00:43:47 --> 00:43:54
two inputs, produces one output
Z where the output is related to

611
00:43:54 --> 00:43:59
the inputs in the following
manner.

612
00:43:59 --> 00:44:03
That's a little device called
an AND gate.

613
00:44:03 --> 00:44:07
I could also represent logic in
truth tables.

614
00:44:07 --> 00:44:12
And truth tables simply
enumerate all the values and the

615
00:44:12 --> 00:44:17
corresponding outputs.
Inputs can be 0-0-0-1-1-0 or

616
00:44:17 --> 00:44:21
1-1.
For an AND system output is 1,

617
00:44:21 --> 00:44:25
only if both are ones,
it's a 0 otherwise.

618
00:44:25 --> 00:44:30
So that's a truth table for AND
gate.

619
00:44:30 --> 00:44:36
So from 0s and 1s we deal with
logic and we create devices like

620
00:44:36 --> 00:44:40
the AND gate to process digital
signals.

621
00:44:40 --> 00:44:46
And what we will do is look at
a whole bunch of little symbols

622
00:44:46 --> 00:44:51
like this, like the AND gate to
process our input signals.

623
00:44:51 --> 00:44:57
And these devices might look
like other functions like OR

624
00:44:57 --> 00:45:03
gates and so on.
Let me show you a quick demo.

625
00:45:03 --> 00:45:09
What I'm going to show you is a
signal feeding an AND gate.

626
00:45:09 --> 00:45:15
And one signal is going to look
like this, and my signal Y is

627
00:45:15 --> 00:45:21
going to look like this.
So you expect a processed

628
00:45:21 --> 00:45:23
output.
So 1-0-1-0-1-0-1.

629
00:45:23 --> 00:45:29
And the output is simply going
to be --

630
00:45:29 --> 00:45:31
This is my time axis going this
way.

631
00:45:31 --> 00:45:35
It is going to be an AND-ing of
these two signal values like so.

632
00:45:35 --> 00:45:39
What I'm also going to show you
is I'm going to superimpose

633
00:45:39 --> 00:45:43
noise on this wire.
I'm going to superimpose noise

634
00:45:43 --> 00:45:47
on the wire, and what I want you
to observe is the output of this

635
00:45:47 --> 00:45:50
digital gate.
The output will stay exactly

636
00:45:50 --> 00:45:52
like this, even though I impose
noise.

637
00:45:52 --> 00:45:55
The ultimate test.
So stay right there.

638
00:45:55 --> 00:46:00
Let's do this demo.
Give me a couple of seconds.

639
00:46:00 --> 00:47:42


640
00:47:42 --> 00:47:45
If you look at the signal up
there, look at the middle wave

641
00:47:45 --> 00:47:49
form, and I'm imposing let's
have a digital system in a noisy

642
00:47:49 --> 00:47:52
environment like a lumberyard,
for example,

643
00:47:52 --> 00:47:56
or chopping a bunch of trees in
my backyard and building digital

644
00:47:56 --> 00:48:00
systems on the side.
And if I have my buddies

645
00:48:00 --> 00:48:04
revving up chainsaws
superimposing noise on my second

646
00:48:04 --> 00:48:10
input, but look at the output.
And just to show that I'm not

647
00:48:10 --> 00:48:13
bluffing here,
what I'll do is I'll pass the

648
00:48:13 --> 00:48:17
noise through and make the noise
larger.

649
00:48:17 --> 00:48:22
And you'll notice that when the
noise begins to surpass the

650
00:48:22 --> 00:48:26
noise margins the output begins
to go berserk.

651
00:48:26 --> 00:48:31
Watch.
Can you increase it gradually?

652
00:48:31 --> 00:48:35
Notice that as I put in a lot
more noise then the output

653
00:48:35 --> 00:48:40
begins to go berserk,
but as long as my input is

654
00:48:40 --> 00:48:45
within the noise margin my
output stays perfectly stable.

655
00:48:45 --> 00:48:48
So that's the "Intro to Digital
Systems".

656
00:48:48 --> 00:48:51
You'll see numbers in
recitation.

657
00:48:51 --> 48:54
And we'll see you at lecture on
Tuesday.