Linear Interpolation as a Convolution

Equivalent to filtering the continuous-time weighted impulse train

$$\sum_{n=-\infty}^\infty y(nT)\delta(t-nT)$$

with the continuous-time ``triangular pulse'' FIR filter

$$h_l(t) = \left\{\begin{array}{ll} 1-\left\vert t/T\right\vert, & ... ...\vert t\right\vert\leq T \\ [5pt] 0, & \hbox{otherwise} \\ \end{array}\right.$$

followed by sampling at the desired phase

Replacing $h_l(t)$ by $h_s(t)\mathrel{\stackrel{\mathrm{\Delta}}{=}}$sinc$\left(\frac{t}{T}\right)$ converts linear interpolation to ideal bandlimited interpolation (to be discussed later)