Introduction to Sampling Theorem Video Lecture | Signals and Systems - Electrical Engineering (EE)

this video will introduce the sampling
theorem conceptually and a subsequent
video will go through the mathematics
involved in deriving it so quite often
we are confronted with a situation that
is illustrated by the picture that I
have on the screen and that situation is
the following we have a time wave form
which I've labeled X of T this might be
typically it's a signal from the real
world so for example it might be the
output of the sensor it might be music
or speech that we've acquired through a
microphone and turned into a voltage but
at some continuous-time signal and what
we typically want to do with this is we
want to sample it the benefits of
sampling the signal is that we can then
represent it in a computer and in the
computer we can do all sorts of
wonderful things to it so that's one
application another reason why we might
want to do this is to be able to store
the data so for example when you think
about a music CD you have a real signal
which is the music you sample that and
for a music CD that sampling occurs at
44100 Hertz and those samples are stored
on the CD and then when you play the CD
what you want to do is reconstruct the
original signal the original X of T so
you go from your samples back to the
original X of T the sampling theorem
answers the question which is what
sample rate
do I need to do this okay
the idea is that clearly if I choose
maybe one sample here and another sample
here and so on there's a good chance
that I will not be able to reconstruct
my signal exactly we want to reduce the
number of samples to the extent that we
can because every sample takes up space
to to store or to process and so the
question is how many samples per second
do I need to take and it turns out the
answer is for those of you that are
impatient that my sample frequency that
is the rate the number of samples per
second has to be greater than twice the
bandwidth of the signal and in just a
minute we'll explain what bandwidth is
so again for those of you that are
impatient this is the answer you find
the bandwidth of your signal you double
that and then make sure that your sample
frequency is larger than that and
typically in real life you want it to be
at least two and a half to three times
the bandwidth so let's give you a brief
explanation about why this is the case
and this explanation requires that we
look at things in the frequency domain
so if I take my signal X of T and
compute its Fourier transform which I
would graph in this case as a magnitude
spectrum typically it's going to look
something like this like some pile of
frequencies and sometimes actually not
very often but we do stuff to make this
happen the signal will go to zero at
some frequency and a
kind of screwed this up add some
frequency which I'll call two pi B so
between negative two PI B and two pi B
there the signal spectrum is non zero
for values of Omega greater than 2 pi B
or less than 2 pi B the signal spectrum
is 0 ok and again this doesn't happen
that often in real signals but typically
you'll filter them to make this happen
when I sample so again we'll represent
this in the frequency domain what
happens is that I have my original
spectrum so let's see we'll call this
perhaps the magnitude spectrum of X sub
s to indicate that this is a sampled
signal so my original spectrum of X of s
this guy up here basically shows up down
here again and that's nice because that
means we can eventually recover it if we
need to which is usually why we sample
in the first place but the interesting
thing is that this spectrum also is
replicated at different frequencies so
at a frequency that is minus Omega s and
Omega s in this case would be 2 pi times
F sub s where F sub s is the number of
samples per second I get a copy of X of
Omega this guy not only does he come
down here he ends up here and at a
frequency of
plus Omega s I get another copy of the
spectrum and it turns out this actually
happens over and over again so out here
it to Omega s I get another copy of the
spectrum and so the idea is at every
integer multiple of Omega s I have a
copy of the spectrum of X now let's do
this kind of backwards from the way you
typically see this done the way that I
recover this that I recover X of Omega
from X of s if you look at the picture
if I have a low-pass filter that has a
transfer function that looks like this
so we've got some Omega C which is the
cutoff and minus Omega C and it goes to
a height of 1 if I have a low-pass
filter like this this component of the
spectrum which is the component that I
want comes through the low-pass filter
without any change but these components
out at Omega s and 2 Omega s get
destroyed they go away because my
low-pass filter has a transfer function
of zero out in these frequencies so it
turns out that I can reconstruct my
sampled signal by running it through a
low-pass filter this is actually an
ideal low-pass filter but suppose that
Omega s let's see let's go to suppose
that Omega s is not large enough that
this spectrum and the spectrum are
separated suppose that Omega s is small
enough so suppose that we had an Omega s
here
so that I get a copy of the spectrum
that's centered here at this Omega s and
then this would be to Omega s and so on
so if Omega s is not big enough then I
get these copies of the spectrum and
each copy if they overlap essentially
add together so my sum might look
something like this where the parts that
overlap get larger and if that happens
then there's no way I can get a low-pass
filter that's going to separate out
Omega or X of Omega from this mess
because of these overlapping parts so
when I try to reconstruct my signal my
original let's see let's get a even
uglier color my original X of Omega gets
corrupted by all this stuff that's been
added in and I have another chunk out
here so this process of corrupting the
signal is called aliasing and aliasing
is obviously bad so in order for
aliasing not to occur you can see that
we need the condition that two PI B or
again B is the bandwidth of the signal
is less than Omega s over two because if
I have Omega s out here I need to be
able to get this half of the of the
extra spectrum plus this half of the
original spectrum that I want into the
space without them overlapping and so
this condition here would work and
typically you'll see this expressed well
as Omega s is greater than
two times two PI B and quite often
you'll see it expressed not in radial
frequency but in terms of just a
frequency so if I divide both sides by
two pi I get F sub S which is the
sampling frequency is greater than 2 B
okay so this is the result this is the
statement of the sampling theorem and
hopefully this messy diagram has given
you some understanding of the concepts
involved there and in a subsequent video
we'll go through the mathematics in more
detail for those of you that really want
to understand it and that are skeptical
of my pictures