So in this video, we're finally going to 
get the signal into the computer. 
We've talked about how numbers are 
represented by computers, but now we have 
to figure out how we actually convert an 
analog signal, like a voltage into a 
number that goes into a computer. 
This is called Analog-to-Digital 
Conversion. 
And I think the name is pretty obvious. 
We'll go through the details of that. 
It actually goes through two stages. 
One is to acquire values of the signal at 
various times. 
we'll learn that there's something called 
the sampling theorem, very important 
result in this business in which you can 
sample a signal without error. 
Amplitude quantization is another story. 
Now, we have to convert those continuous 
amplitude values, characteristic of 
analog signals, into a set of discreet 
values. 
That incurs error, as we'll see, and 
we'll learn how to analyze and control 
those errors. 
Alright. So, let's go over the definition 
again of an analog an a digital signal. 
analog signals are functions of a 
continuous variable, like time and I show 
an a, a typical analog signal here 
wiggles around and has a continuous range 
of values for the amplitude and its 
functions of continuous time. 
Digital signals are discreet value 
functions of the integers. 
So, I show as a bubble plot. 
values of a signal here, they're 
isolated, they only occur at the 
integers. 
And you'll notice these various fixed set 
of amplitudes and those are the only 
amplitudes that are allowed because it's 
a discrete-valued function. 
So, I want turn now to this is that 
signal gone through an A to D converter 
and you may wonder, well, 
that doesn't look anything like it. 
How in the world can you go back from the 
digital signal to the analog one? I'm 
going to show you you can't. 
but it's a, a very interesting story when 
you get into all the details. 
Alright. 
Well, first thing we're going to do is 
acquire individual values of the signal. 
And we can describe that in the following 
very simple way. 
we're going to start with our, our analog 
signal. 
There it is. 
And here is our old friend, 
the periodic pulse train. 
So, the separation between the pulses is 
T sub s. 
s means sampling, 
and T sub s is going to be the sampling 
interval. 
The width of each pulse is very narrow. 
Delta is going to be very small for us. 
Well, if you just multiply those two 
signals together, the analog signals 
times the periodic pulses, 
you get this waveform as shown in the 
bottom here. 
And, we're going to make delta so small 
that all that really results is that we 
get the value of the signal at the center 
of the pulse. 
And we'll assume that these pulses are 
really, really very narrow. 
Now, the qeustion is going ot be for us, 
can we, from these values, 
can we connect the dots? Can we connect 
the dots filling in what is zero so that 
looks like the original signal? 
And the asnwer is, we're going to be able 
to do that with no error as long as we 
play our cards right. 
Let's see how that works. 
Well, how do you figure that out? you got 
to look in the frequency domain. 
that's why we went. One of the reasons we 
went through all this effort to learn 
about signals in both time and frequency. 
So, what is its Fourier transform? 
So, what I'm going to do is I'm going to 
express the periodic pulse sequence in 
its Fourier series. 
And we know that that formula corresponds 
to the coeffcient values. 
So, let's do that multiplication by 
substituting in that the Fourier series. 
Rearrange things a little bit, so we're 
going to put all the functions that 
depend on time together. 
The questions going to be, what is the 
Fourier transform of that? Once I know 
that, because the Fourier is linear, I 
know it's just going to be a weighted 
linear combination of those individual 
spectra. 
Well, this is an old Fourier property. 
You multiply by a complex exponential and 
the time domain, and that corresponds to 
a shift in frequency, 
a delay in frequency in the frequency 
domain. 
So we get a readily simple result from 
our Fourier properties that we're going 
to have, each terms has a Fourier 
transform which corresponds to the 
original signal spectrum, capital S, 
shifted over by k/Ts because that is the 
frequency of that complex exponential. 
And then, we're going to get all added 
up. 
So, it's pretty easy actually to find the 
Fourier transform of our signal, 
our sample signal. 
It's just a rated linear combination of 
shifted spectra. 
Well, let's look at that in more detail 
and plot it. 
So, I'm going to use for my example a a 
rooftop spectrum, which I like to use. 
This is known as bandlimited. 
It's a very important consideration for 
sampling. 
And what that means it, it's spectrum is 
zero beyond some frequency. 
Here, that frequency is W. 
It also turns out that W is its 
bandwidth, that's the amount of the 
positive frequency axis where the 
spectrum is nonzero. 
But, 
what's important for us is what the cut 
off limit is for the bandlimited part of 
the signal. 
So, that's what the individual spectra 
look like. 
Now, when I delay and 
add them up with constants, 
this might be what I get. 
So, the, you can see I have taken the 
original when k=0, here's the original 
spectrum, multiply it by C0. 
And then, 
I shift it over by 1/Ts, multiply it by 
C1 I get that one, shift it over 2/Ts. 
I handle the negative indices the same 
way. 
So, this goes on forever and ever, and 
ever, as you might imagine. 
Now, the question's going to be, can I 
get back the original spectrum? Can I go 
from this and somehow get back? 
That would not bear. 
And the problems going to be these 
overlap regions. 
This is called aliasing, it's a technical 
term. 
And what happens in here is that we get 
these two pieces are added together, they 
overlap with each other and they're going 
to add up, 
that's what this formula says. And once 
they're added, I can't tell you what the 
individual components are. 
If two numbers add up to 3.2, I can't 
tell you which two unique numbers were 
added together to get 3.2. It's 
impossible. 
So, once you get into this area of 
overlap, I can't tell you what the 
original signal spectrum was in that 
region. 
And here, I can, but not in there. 
I'm in big trouble. 
So, what is the criterion by which you 
don't overlap? Separation between these 
is 1/Ts. 
And that separation has to be greater 
than the amount that comes from the part 
of the origin plus the amount that comes 
over from that direction. 
Well, each of those amounts is W. 
So, that has to be greater than 2W, 
and I can write that the other way by 
turning this upside down. 
The T sub s has to be less than 1/2W. 
If we meet this criterion, these 
individual rooftop spectra will not 
overlap. 
And at least we have a change of getting 
back to the original signal spectrum. 
Well, in this case, what I did is 
explicitly draw so that the spectra did 
overlap and that's what I get for a 
spectrum here. 
I get these overlapping areas, aliasing 
as a curve because I didn't sample quick 
enough, 
Ts needs to be smaller. 
Here's what happens when you do Ts to be 
smaller. 
And now, the individual pieces, the 
individual parts, do not overlap. 
I can see them all very clearly and 
that's because I've obeyed the rule that 
I draw up. 
Now, can I go from this back to this? 
How would I do that? And I think the 
answer is fairly obvious. 
What I would do is filter. 
I would apply an, what's called Ideal 
Low-Pass Filter, having a cut-off 
frequency of W that would cut-out all 
these higher shifted versions of the 
spectrum. 
And sure enough, I get back my original 
spectrum multiplied by C0. 
C0 is just a gain and I could easily 
compensate for that a little bit. 
What's much more important is to get rid 
of these frequency overlaps. 
We do not want aliasing. 
So, now the criterion for the sampling 
theorem, which I'm going to point out 
here, here's the criterion. So, we have 
to have two things. 
One is, this signal has to be 
bandlimited, and we're going to assume 
it's W Hertz. And the sampling, 
the pulse sequence, has to have a period, 
sampling interval, that's less than 1/2W. 
And what we've shown, just by example, is 
that if you obey those two rules, then 
you get a picture that looks like this. 
By simply applying a low-pass filter, I 
can get back the original signal. 
So, if I take my signal values, it's all 
I have is just the values. 
Every so often, every Ts, if I multiply 
that times my original pulse sequence. 
So, I take every pulse amplitude and make 
it s of nTs accordingly. 
And then, pass that through a low-pass 
filter, 
low-pass filter is this filter, 
I get back the original signal with no 
error. 
That is really very nice. 
And when it got, by the way, remember 
this is a frequency domain way of looking 
at it. 
It wasn't in the frequency domain, we 
wouldn't be able to say one way or the 
other. 
Now, this criterion is usually not 
specified in terms of the sampling 
interval. 
Rather, it's usually specified in terms 
of what we call the sampling or rate. 
So, the rate of course, is just one over 
the interval. 
And so, our criterion for obeying the 
sampling theorem is that the sampling 
rate has to be twice the highest 
frequency in the signal s. 
So, this low-pass filter here, by the 
way, is here to make sure that the signal 
is bandlimited. 
there are some A to D cards out there for 
computers that are pretty sloppy. 
And they don't have this first low-pass 
filter. 
And so, it is possible that you can send 
in a signal and that would be aliased 
once you've sampled it. 
But high quality A to D converters, 
analog to digital converters, have this 
front end filter which is called an 
anti-aliasing filter, for obvious 
reasons. It prevents aliasing. 
Then, the sampling rate inside has to be 
greater than twice the bandwidth of the 
front end filter. 
And if you do that, I can, from these 
sample values of the signal, get back the 
original reform in continuous time, I 
emphasize that. 
Just from a few values, I can get back 
the original signal very, very nice. 
Well, that's the first phase. 
we have a discrete time signal. 
Discrete time means we are in discrete 
time we only have the signal values of 
every T sub s seconds. 
It's not digital yet because the values 
coming out of here are continuous, in 
general. You put in a sign wave you get 
kinds of different values. 
It's not a discrete set of amplitudes, 
and so you have to do that phase next. 
And that's the nice part of a real world 
A to D converter. 
So, here is the sampling part to be 
created in order. 
And, of course, we can get back the 
original signal with that error if we 
just stop there. 
But now, we have to convert it to a set 
of discreet values, and that's called 
Amplitude Quantization. 
And I denote the amplitude quantization 
function iq. 
And what it looks like is what's called a 
staircase function. 
So, this is a function so any value here 
in that range gets called a 5, 
as you can see. 
And that's how we convert from a 
continuous set of values into a set of 
discrete ones. 
Here, I'm going from something that goes 
from -1 to 1 and from something that has 
integer value 0 to 7. 
Well, I think you can see this may be a 
function going this way. 
But coming back, if I tell you it's 5, I 
can't tell you precisely what the 
original signal value was. 
And that's the error that's inherent in 
amplitude quantization. 
Amplitude quantization introduces error. 
So, here's the, their, the signal at 
various stages in our A to D converter. 
Here's our regional signal, here it is 
after it's been sampled. 
And if the signal is being limited than I 
can from my little discrete values in 
time, get back the original signal which 
I showed is this ghostly dotted line 
underneath. 
I can connect the dots in exactly the 
right way. 
But then, once I go through the amplitude 
quantization part, that's now I only have 
eight possible values for the amplitude 
corresponding with the integer 0 to 7, 
and you can see the layering. 
And since these values do not correspond 
exactly to the values over here, there's 
an error. 
I cannot get back the original signal. 
So, we'd better appre, get some 
understanding of how big these errors can 
be. 
That's going to be important. So, and can 
we control those errors. 
So here's my quantization function. 
And let's blow up one of those intervals, 
let's look at it in some detail. 
So, every value in this interval is 
called a 4, 
right? So here is my original signal 
value. And, since it's in this interval, 
it's called 4. 
Well, once I get a value of 4, how do I 
translate that back into some amplitude 
value? 
There's going to be an error, which I 
call epsilon. 
But the question is, what do I use to, 
for the value to call it? And I think it 
makes a lot of sense that the value on a 
side tube the amplitude is the value in 
the center of the interval. 
If you picked over here, there's a huge 
error involved. 
If you picked over here, well, the error 
may be smaller for this signal value but 
the one over here is going to incur a 
large one. 
It seems to me a good compromise. 
It's just to pick the middle. 
And you can show that, that is the right 
one to pick. 
Well, the question is, how big is that 
error sort of on average? 
And what I'm going to calculate is the 
RMS value of the error. 
So, same calculation we've done before, 
we take the squared error. we're going to 
integrate it over the width of the 
interval. 
Since we're assuming we're in the center, 
we go from 2 plus delta over 2 down to -2 
delta over 2 relevant to that center, and 
divide by the total width. Take the 
square root, do the integral, and divide 
by delta, you'll get delta squared over 
12. 
It's a classic answer. 
So the RMS is the square root of delta 
square over 12. 
Now, little important fact is what's 
delta. Now, I have assumed that the 
signal is amplitude limited, which is not 
a bad assumption for most real world 
signals, goes between -1 and 1. 
In general, the signal goes between plus 
and minus its maximum magnitude, I think 
is pretty obvious. 
So, the total range across here is 2 
times the maximum, value. 
The number of quantization intervals is 
2^B. 
So, I show here 8 quantization intervals, 
that's what I've used in this example. 
What is B that corresponds to 8 
quantization intervals? Well, I'm sure 
that you know, since you know your powers 
of 2, 
that 8 is 2^3. 
So, this is what's known as a 3-bit 
converter. 
It turns out you can't buy a 3-bit 
converter that's way too few, and use it 
here, using it here as an example so we 
can see what's going on. 
Alright. 
What I need to do, though, is have some 
measure of quality of the amplitude 
quantized signal. 
And the typical thing that we use is 
called signal-to-noise ratio. 
So, it's defined to be the ratio of the 
signal power to the noise power. 
And noise in quote in here because it's 
the amplitude quantization here. 
So, the bigger the SNR, the better, the 
happier I am. 
That means, relative to the signal, the 
error is small. 
So, we have a signal plus error, and this 
is the original signal value. 
And so, the bigger this is relative to 
that, relative to the, relative to the 
error, the bigger the SNR. So, let's go 
through the calculation. 
So, what's the power in the signal? Well, 
I'm going to assume that for this 
calculation, that my signal is a sinusoid 
whose amplitude is A. And we know from 
what we've already done that the power in 
a sinusoid is the amplitude squared over 
2. 
Epsilon, as is determined by delta, the 
power in the error is delta squared over 
12 And now, delta is going to be the 
twice the amplitude of our sine wave, 
2A/2B. 
And so, you simply do that calculations 
to get in. 
And notice that the amplitudes cancel so 
our result is going to apply no matter 
how big the signal was, or how small, as 
long as the A to D converter takes into 
account that range of variation the SNR 
is fixed, it doesn't matter. 
And what you get is a somewhat weird 
answer. 
It's 3/2*2^2B. 
Now, 
the important part here is the SNR 
increases exponentially with B. 
So, the more bits you use, the finer the 
quantization interval, the smaller it is, 
and the SNR goes up exponentially. 
So, the way to reduce quantization error 
is to have more bits. 
it turns out that more bits could be a 
very expensive, A to D converter to buy. 
But no matter how many bits you can use 
in your A to D converter, there's still 
going to be some error. 
We cannot get back the original signal 
exactly, but the error can be reduced by 
controlling the number of bits. 
Now, it turns out that SNR is not usually 
stated this way. 
It's stated in a quantity called 
decibels. 
And here is a complete video on decibels 
that you may want to look at if you don't 
know about the decibel scale. 
It's a logarithmic scale. 
And since this, since this is a power 
ratio, x in dB, is always written 
lowercase d uppercase B, is equal to 10 
log 10 of x divided by some reference 
number. 
That's how you convert from, or number x 
that you may calculate and, and express 
it in dB. 
Well, the reference here is 1. And after 
I take the log and take care of it, this 
is what I get for the SNR expressed in 
decibels. 
This term, And I think it's pretty clear 
it comes from the 3/2. 
And I'm going to let you figure out that 
6b amounts to 2^2B expressed in decibels 
if you want to figure that out for 
yourself. 
So, what's very common is, is to have a 
8-bit converter. 
Most computers have at least 8-bit 
converters. 
And that means the signal to noise ratio 
is 48 plus a little bit. 
Well, 48 is almost 50. 
And because the definition of decibels, 
that's almost 5 orders of magnitude. 
So, with a 8-bit converter, we're almost 
guaranteed that the signal in general has 
a power that's 10^5 times the power of 
the quantization error. 
You may say, wow, that really is pretty 
good. I don't think I need to get much 
better than that. 
It turns out that there are people out 
there who think, they claim to be able to 
hear a noise that's that small. 
That's with the SNR is 50db, they claim 
they can hear that. 
And I think that's pretty close to the 
threshold of hearing. 
So, 8-bits may suffice for some 
applications for high quality audio more 
bits are used. 
In fact, I want you to go and look up 
what the CD sampling rate is and how many 
bits are used in CDs, in the original 
CDs. 
I think you'll be pretty surprised at 
what the answer is. 
And furthermore, once you know what the 
sampling rate is, I want you to tell me 
what the upper limit for the signal is 
assumed to be. 
What's the bandlimited quantity w for the 
signal for a given sampling rate used for 
CDs. 
Okay. 
So, this is the story of 
Analog-to-Digital conversion. 
the sampling part, which we see here 
occurs with no error. 
No error at all. 
It's all in the amplitude quantization 
part. 
It reduces error. 
It's inherent, we cannot avoid 
quantization here. 
But, we can figure out how big it is, and 
we can control it by using more bits. 
The more bits we use, the smaller the 
error.