5.2.2. 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.

5.2.2.1. 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:

\[alsdklasdk\]

and so:

\[x(t) = (x_s \ast \phi_{\mbox{sinc}})(t)\]

Remember that:

\[x_s(t) = \sum\]