Convolution Interpretation

Linearly interpolated fractional delay is equivalent to filtering and resampling an impulse train carrying the signal samples with a continuous-time filter having the simple triangular impulse response

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

Convolution of the impulse train with $h_l(t)$ produces a continuous-time linearly interpolated signal

$$x(t) = \sum_{n=-\infty}^{\infty} x(nT) h_l(t-nT). \protect$$ (I.6)

This continuous result can then be resampled at the desired fractional delay.

In discrete time processing, the operation Eq. (I.6) can be approximated arbitrarily closely by digital upsampling by a large integer factor $M$, delaying by $K$ samples (an integer), then finally downsampling by $M$, as depicted in Fig. I.13 [90]. The integers $K$ and $M$ are chosen so that $\eta \approx K/M$, where $\eta$ the desired fractional delay.

Figure I.13: Linear interpolation as a convolution.

The convolution interpretation of linear interpolation, Lagrange interpolation, and others, is discussed in [380].