Linear interpolation

Linear interpolation is definitely the most popular and most widely used reconstruction method. The reasons for this are that it is simple and pretty straightforward to implement and the results are usually not so bad. Linear interpolation in one dimension results in simply connecting sampling points with straight lines.

To compare linear interpolation with ideal interpolation in frequency domain we note first that linear interpolation corresponds to convolution with a tent function, which is defined by

$$\mathrm{tent_T}(x) = \left\{ \begin{array}{cl} 1 - \frac{\... ...ert x\vert < T$}\\ 0 & \textrm{else}\\ \end{array} \right.$$ (4.4)

where T is the sampling distance. The frequency response of the tent function can be computed with a simple trick, because the tent function is the convolution of two box functions, i.e.,

$$\mathrm{tent_T}(x) = \Pi_T(x) \ast \Pi_T(x)$$ (4.5)

so that, following the convolution theorem, the frequency response of the tent function is the squared frequency response of the box function

$$\mathrm{tent_T}(x) \rightleftharpoons T^2 \cdot \mathrm{sinc}^2(x)$$ (4.6)

The tent function is depicted in Fig. 4.4 together with its frequency response, which is nevertheless far from ideal.

Figure 4.4: On top the box and tent functions are depicted (which correspond to nearest neighbor and linear interpolation, respectively),below their frequency responses together with the ideal frequency response.