We have seen the polynomial
interpolation will yield an infinitely differentiable continuous time interpolation of a finite set of data points, and we have introduced the
Lagrange polynomials as a very convenient way to find this interpolation
polynomial. Remember, the Lagrange polynomial well-defined by this formula, which is dependent on the length of the interpolation
interval in this case, 2N plus 1 points, and we have seen
these polynomials. We have plotted them.
They look like so. In this case, N is equal to two. So we have polynomials
of degree four. With Lagrange interpolation,
the continuous time interpolation is
defined by this formula, which is a linear combination of functions here that depend in highly nonlinear ways
both on the length of the interpolation interval and on the index of the
interpolation point. By contrast, if you use a local or Kernel-based
interpolation scheme, we have that
the interpolated signal is still a linear
combination of functions. But these functions are just shifted copies of the
same prototype function here. So with polynomial interpolation, the advantage is that we have a maximally smooth interpolation, but the drawback is that
interpolation kernel depend on the length of the
interpolation interval. Conversely, with
local interpolation, we have the same
interpolation kernel independently of
the length of the dataset, but the problem is that we
have a lack of smoothness. Now, here is probably the most remarkable result that you will see in this class. As the number of points in the dataset that we need to interpolate grows to infinity, all the Lagrange
interpolation polynomials end up converging to shifts
of the same function. So as big N goes to infinity, all Lagrange polynomials end
up having the same shape and they're just shifted versions
of the same function, and this function is
actually the sinc function. This result tells
us that we can keep the kernel-based interpolation
approach and yet achieve a maximally
smooth interpolator. The drawback is that as we know, the sinc is a two-sided
infinite support function is the output of an ideal system that we cannot
realize in practice. However, it tells us that we have an optimal result that we can try to approximate as
well as we can afford. The famous sinc
interpolation formula therefore is the following; is a mixed domain
convolution once again. Here we state the formula for an arbitrary interpolation
interval T_s, and it is a linear
combination of shifted and scaled versions of
the sinc function weighed by the
discrete-time samples. In practice, here is
a portion of our dataset. We place a sinc function at
each discrete time point and you see that
the sinc function naturally fulfills
the interpolation property, namely, it's equal to one
and zero and it's equal to zero at all other multiples of the interpolation interval. When we sum all the sinc
functions together, we obtain a maximally
smooth interpolator for our data points. Of course to be thorough, we should formally
prove the convergence of the Lagrange polynomials
to the sinc function. This is unfortunately
a bit technical and so the full proof is in the book. One very nonrigorous
way to develop a mathematical intuition as
to why this is plausible is the fact that sinc
of t minus n and the infinite degree
Lagrange polynomial share an infinite number of
zeros because they're both equal to zero for all integer values of
the argument since polynomials that shared
the same zeros are identical. By some hand-wavy extrapolation, we can say the sinc and the infinite degree Lagrange
polynomials are the same. We can also be very
pragmatic and look at the convergence from
a numerical point of view. So we take the definition of the arbitrary Lagrange
polynomial of order n. Let's take the Lagrange
polynomial number zero, and if we manipulate definition, it's very easy to obtain these representations of
product for k that goes from one to capital N of 1
minus t squared over k square. Now, this is very easy
to compute and so we can plot this function against
the sinc function. So here's the sinc function with time-based T_s equal to one, and here we plot the Lagrange
polynomial of order 100, which remember is
a polynomial of degree 200. As we increase the degree
of the polynomial, we can see that the convergence, the sinc function gets
better and better.