Welcome to Module Six of Digital Signal
Processing.
This module is concerned with one of the
central topics of signal processing which
is sampling an interpolation.
Sampling an interpolation are at the heart
of interacting between the continuous time
world and the discreet time world.
And we are going to review all the basic
concepts and prove a central theorem which
is sampling.
First, what I do think about interpolation
now interpolation is about as old as
computation itself and we can see an old
analog device which is called a slide
rule.
We have the picture here of a slide rule
that was used to compute trigonometric
function and other functions that are not
easy to calculate by hand.
Now why is interpolation important?
Because, if you make a table of a
function, and here we see an example of a
table of a function.
You want to know how many values you have
to actually tabulate so as to have a good
approximation of the function.
And that's the whole theory of
interpolations.
That we are going to review in detail for
passing from a sequence to a continuous
time function.
Interpolation raises two interesting
questions.
The first one, if you have the values of a
function at a number of points how should
you interpolate between these 2 point.
Should you linearly interpolate between
two values?
That's probably not good enough when the
function is complex.
The second interesting question is, is
there a minimum set of values you need to
measure the function at so that you can
perfectly constructed.
That result is known as a sampling
theorem.
It's a very simple and very powerful
result.
Namely if a function has a maximum
frequency F0 then it is sufficient to
sample the function that twice F0.
From these samples it is possible to
perfectly reconstruct, so it's a function.
That's a cornerstone result of signal
processing.
It allows to go from the continuous time
world to the discreet time world, and go
back perfectly.
Assume you have just two samples, then
it's simple enough.
There is a straight line that goes
perfectly through these two samples.
If you have three samples examples same
story.
You have a parabola so close to three
samples.
If you have many samples, you also can try
it when interpolate, and of course you
have samples which can see this is a
trickier issue, some what we are down with
two or three samples.
If we have a function we can take, any
samples.
Let's take equally spaced samples.
This will generate the sequence xn, which
in this case is equal to x of n times T.
T is the space thing between two samples.
So, given a sequence, we have seen how to
obtain continuous time function through a
process called interpolation, so going
from the sequence to, say continuous time
function, with interpolation.
We had also seen how to start with a
continuous time function, and get samples
through a process called sampling.
The key question is, when is there a one
to one relationship between sequences and
continuous time functions.
When can we go back and forth sampling and
interpolation and these two are exact
images of each other, and that's exactly
the topic of the sampling theorem.
We study the question of that one to one
relationship between a band limited
function and its samples in great detail
in this module.
And to arise at these results, where again
you can use all the tools in the box, that
we have developed so far.
So Albert's space his projections
filtering thing functions and so on.
Ans so every thing comes together here to
give [inaudible] his profile and very
useful result.
The big message of this module on sampling
an interpolation can be summarized in the
following points.
Continuous time is the physical world so
it is a reality.
It is a bit trickier than the reality of
sequences and discreet time that we have
seen so far.
The space of band limited functions can be
very well said in a Albert's space
framework it is a sub space of general
functions namely one set have no
frequencies beyond certain limit.
So we can use Albert space geometry to
define an atom of basis for the subspace
of band limited functions.
That [inaudible] basis is going to be
given by the sync function, and it's
shifts by integral multiples of the
sampling interval.
And to sample a general signal, one that
is not band limited.
What we do is that we first projected onto
the subspace of band limited functions.
We do this by low pass filtering, and then
we can represent perfectly that function
on this subspace.
So we see that we need all the traits that
we have seen so far, we put them together.
And we are able to understand, in detail,
sampling and interpolation.
The space of band limited function is a
subspace.
So in our two, would be a line going to a
zero region lets call it BL for band
limited.
If particular signal or a function is band
limited, it will belong to this subspace.
That leaves here on the subspace, and we
are going to prove the sampling theorem,
which has such a vector x can be written
as a linear combination of n log to z of
the weighting coefficients alpha n times
pi n, where pi n are the basis functions
for this band limited subspace.
And we're going to see the pi n are, so
string functions so that's a
representation theorem for band limited
function.
Now if we have the general vector that is
not band limited then we can not represent
it using a linear combination of string
functions so we have the first project it
down to band limited function.
So we use a projection theorem, we find
its projection x hat.
And x hat now can be represented as linear
combination of sinc function.
So a general function can always be band
limited by e projected to the subspace of
band limited functions and then
represented perfect.
Before moving to the heart of the topic
let us briefly review its history.
The Shannon Sampling Theorem, has a very
interesting history which goes back well
before Shannon.
So numerical analysis people were
concerned about interpolating tables of
functions and the first one to prove the
version of the sampling theorem was
Whittaker in England in 1915.
Harry Nyquist of Bell Labs came up with
the Nyquist criterion.
Namely, that the function that had the
maximum frequency, F0 could be sampled at
2 times F0.
In the Soviet Union, Kotelnikov proved the
sampling theorem.
The son of the first Whittaker, another
Whittaker.
Further proved results on the sampling
theorem.
Then Herbert Raabe in Berlin in 1938 wrote
his PhD thesis about the sampling theorem
but wrong time, wrong city.
He got 0 credit for it.
Dennis Gabor worked on the version of the
sampling theorem in the mid 1940s.
Then Claude Shannon inventor of
information theory wrote, beautiful papers
that is in the further reading for this
class.
Where the Shannon sampling theorem really
appears in the forms that we use today.
Last but no least, in 1949, Someya in
Japan also proved the sampling theorem.
You can see that's a very valid history,
it's a very fundamental result.
So many different people independently,
actually came up with this result.