Interpolation
=============

Interpolation is the process of reconstructing a CT signal $x(t)$ from
its samples $x[n]=x(n T_s)$.

We start with the ideal interpolation given the correct sampling of a
bandlimited signal. This will result in *sinc interpolation*. Next we
consider interpolation methods that are more often used in practice:
nearest neighbor (and zero-order hold) interpolation, linear
interpolation and spline interpolation.


Sinc Interpolation
------------------

Consider a signal $x(t)$ that is bandlimited such that $|X(\omega)|=0$
for $|\omega|>\omega_b$. We have seen in a previous that in case we
sample with frequency $\omega_s>2\omega_b$ the shifted copies of
$X(\omega)$ do not overlap. So multiplying $X_s(\omega)$ with a block
of width $\omega_s/2$ we get the Fourier transform of the original
signal.

Because multiplication with a block in the Fourier domain corresponds
with convolution in the time domain with the inverse Fourier transform
of the block. In a previous chapter we have seen that the inverse
Fourier transform of a block is the sinc function:

.. math::
   alsdklasdk


and so:

.. math::
   x(t) = (x_s \ast \phi_{\mbox{sinc}})(t)

Remember that:

.. math::
   x_s(t) = \sum