# How to shift with spline interpolation?

I have a sampled pulse shape: $h = [1, 0.5]$ and I do not know what is its real underlying continuous-time pulse. I want to compute the samples of $h(t-\Delta t)$.

If I write the continuous pulse with $\text{sinc}$ basis functions, I have: $$h(t) = \text{sinc}(t) + 0.5\times \text{sinc}(t-1)$$ Therefore, here, I can compute the new samples (samples of the time-shifted continuous pulse shape) with $\text{sinc}$ interpolation i.e. convolving with a $\text{sinc}$: $$h'_k = g_k(\Delta t)* h_k$$ where $g_k = \text{sinc}(kT+\Delta t)$ and $h_k = h(kT)$ where $T$ is the sampling period.

Now, the question is that If I write $h(t)$ in terms of another basis, e.g. a spline, how can I apply the shift then? convolving with the same but shifted spline?

• This has to be clarified. For instance I interpret your situation as: h(0)=1,h(1)=.5 and that is all the information you have. Is this correct? Now backing off a little more you mention "pulse" so obviously you know a little more that is not stated; beginning time, ending time, etc... I presume you don't know the shape between the beginning and end? There are a lot of "splines" in the literature; the most popular uses three points and two slopes. You don't have the information to fill in these gaps. Your sinc interpolation must have another parameter also. Why not linear interpolation? – rrogers Jul 28 '15 at 20:12