Hi and welcome to Module 7.3, which will
wrap up the module on Quantization.
So, in this module, we're going to look at
what happens when we go back and forth
from the analog to the digital world in
practice.
So, for instance, look at this box.
This box is a high quality A to D and D to
A, analog to digital and digital to analog
converter, that I can use to record my
bass or my guitar.
Now, what's inside this box?
How does this box work?
And so, what we will do, is look at very
simple circuits that illustrate how
digital to analog and analog to digital
converter work.
Now, of course, we cannot go too deep into
the description because this is not an
electronics class, but even the simple
overview will give you a pretty good idea
of how things work.
And you will see that it's not too
complicated.
Hi and welcome to Module 7.3 of Digital
Signal Processing.
In this module we will talk about analog
to digital and digital to analog
conversion.
Try to understand how these things are
done in practice.
We have seen that sampling discretizes
time and quantization discretizes
amplitude.
But how is it done in, In practice?
The answer is cheaply and in a very
compact way.
This photograph shows you the few discrete
electronic components that you have to put
together to build an analog to digital
converter.
And if you did so, you would probably end
up with one of the largest digital to
analog converters in existence because in
most cases, like in your cell phone, the
whole circuit would be integrated and
miniaturized.
At the heart of both digital to analog and
analog to digital converters is a circuit
called the operational amplifier or op-amp
for short.
The op-amp has two inputs, the inverting
input and the non inverting input, and one
output.
And the input-output relationship is that
the voltage, at the output is equal to the
difference between the voltages at the
inverting and non inverting input, times a
gain factor.
We idealize the op-amp by saying that it
has infinite input gain and zero input
current.
And although these are, of course,
idealizations, practical implementations
of operational amplifiers are remarkably
close to this ideal.
And not very complex either, because this,
for instance, is the schematics for an
operational amplifier and you can see that
it only requires six transistors and one
resistor.
The operational amplifier can be used in
two configurations, the open loop and
closed loop.
In the open loop configuration, the op-amp
works as a comparator.
You put a reference voltage at the
inverting input and a test voltage at the
non-inverting input.
As soon as the difference between these 2
voltages is non-zero, the output due to
the infinite gain will shoot up or down to
the maximum or minimum allowed output
voltage, which coincides with the voltage
of the power supply that powers the
op-amp.
In the closed loop configuration, we have
a feedback path that links the output to
the inverting input.
So, what happens when we apply a voltage
to the non-invertant input?
Well, if the voltage at the inverting
input is less than x, then the difference
between inverting and non-inverting would
drive the output up.
Conversely, if the voltage here was larger
than x, the difference would be negative
and this would drive the output down.
So, the only equilibrium point for this
system is when the voltage at the
inverting input is exactly equal to x,
which of course means the output voltage
is equal to x as well.
So, the input-output relationship of the
closed loop is output equal to the input.
This configuration is known as a buffer
stage because although the output has the
same voltage as the input.
The second property of the ideal op-amp
prevents any current from flowing into the
amplifier.
Therefore there is a separation between
this side and that side of the amplifier.
This is very useful for taking
measurements without perturbing the
measured quantity.
We can complexify the closed loop a little
bit by adding a couple of resistors.
In this case, we have what is called an
inverting amplifier.
This is always likely more complex than
the simple closed loop we saw before.
Again, the system will stabilize when the
difference between inverting and
non-inverting inputs is zero.
So, we can say that the voltage here,
let's call it V0, is equal to zero,
because the non-inverting input is
connected to ground.
That means that the current flow in, in
the first resistor will be according to
Ohm's Law i0 equal to x over R1.
This current will not be able to flow into
the operational amplifier because of the
second law.
So, we'll have to flow into this resistor
here.
And therefore, the output voltage will be
just the voltage drop over R2, caused by
i0.
And so y is equal to minus R2 over i0,
which gives us the final input-output
relationship for the inverting amplifier.
The output is a fraction of the input,
with a change of sign.
We're now ready to look at an A to D
converter.
The device starts with a sample and hold
circuit that performs the sampling.
So we have our analog input here.
Here we have an op-amp in buffer
configuration.
And here we have a MOSFET which is really
like a switch.
And here we have a train of pulses at the
sampling frequency.
When a pulse reaches the gate at the
MOSFET, the MOSFET closes very briefly and
allows the first buffer to charge this
capacitor to the instantaneous level of
the input signal.
The capacitor is connected to another
buffer stage that will put out the volume
measured by the capacitor for the duration
of the interval between pulses.
We put a buffer stage here so that we can
put out a constant voltage without
discharging the capacitor, between sample
in instance.
An analog to digital converter needs not
only to sample the input signal, but also
to quantize it.
So, the sample and hold provides a stable
voltage level between sample and times,
and now we need a circuit that converts
that to binary format.
Here we have a simple diagram of a two-bit
quantizer.
What this quantizer does is take a maximum
positive voltage, V0, and a minimum
negative voltage minus V0.
This could be the A and B extrema that we
used in our quantization example.
And then, uses a series of four resistors
of equal value to produce intermediate
boundary levels.
The iK's in our quantization example that
will be used to define the quantization
regions.
Here, you have plus 0.5 Volts.
Here, you have zero Volts.
Here, you have minus 0.5 Volts.
So, we're dividing the interval from V0 to
minus V0 into four equally spaced
intervals.
These reference voltages here are used
with a bank of comparators so, operational
amplifiers in open loop, to determine
which quantization interval the current
sample value belongs to.
So, let's work out an example with an
input voltage of 0.2 V0.
So, what have here is that the first
comparator will have 0.2 Volts at the
non-inverting input with the reference of
0.5 Volts.
So, the reference is higher.
So, here the output will be a negative
voltage.
This comparator will compare 0.2 Volts to
0 Volts.
And so, this comparator will output a
positive voltage and similarly this one
will compare 0.2 to minus 0.5.
So, again the output will be positive.
Next, we have a logic network that will
encode this comparison levels into a
binary value.
These are Exclusive OR gates and the truth
table for this gates Is the following.
So, the output will be positive only if
the inputs are of different sign.
So, this gate will output a positive level
and this gate will output a negative
level.
Now, we have a network of diodes that acts
as a parallel to serial converter.
So, we have a negative voltage here that
will be blocked by these two diodes so
nothing happens here.
A positive voltage here that will go
through, and will indicate the most
significant bit out of the quantizer.
Since this is positive, the most
significant bit will be one.
And here, the negative voltage will not go
through the diode so the least significant
bit will be connected to the ground and it
will be equal to zero.
So, the voltage of 0.2 V0 will be encoded
by the binary notation 1 0.
Finally, let's look at a D to A converter.
Suppose we have a sample in binary
notation, so x of n is equal to a set of
bits from zero, b1 to bR minus 1, so we
use R bits per sample.
We use this binary representation to drive
a bank of voltage generators at voltage
V0.
So, in the previous example, we had a 2
bit quantizer that gave us a
representation for our current value of 1
0.
So, in that case, if we were to apply that
value to this backup of generators, this
voltage generator would be on and this
voltage generator would be off.
These generator are connected to a
resistors structure that is known under
the name of R2R ladder.
Now the details of this structure are a
little bit tedious to work out.
But if you're patient and you apply
Thevenin's Theorem to this ladder, you
will see that the system seen by this
point in the circuit, is equivalent to a
resistor of value R, connected to a
voltage source of value V equal to the sum
for K that goes from zero to R minus one
of V0 divided by two to the K times bK.
So, in other words, each bit in the minor
representation of the sample contributes a
voltage source of value V0 over 2K where K
is the position of the bit in the binary
representation.
So, if this is equivalent to a voltage
source of value K, V0 over 2K b0, plus a
resistor of value R, this is simply an
inverting amplifier and the value out here
will be minus V0 times x, where x is our
binary representation of the input.
The resulting signal will be a piecewise
constant signal where the transitions
between levels will be dictated by the
speed which new samples appear at the
voltage generator bank.