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Let $s(t)$ be a signal that can be approximated by a uniform spline function of order $K$ (say $K=2$): $$s(t)\approx\sum_{n\in \mathbb{Z}}c_n\beta_+^{(K)}(t-n) $$

Suppose that we know the values of $s(t)$ at integer locations $s(n)=x[n]$. Develop and implement a filter-based algorithm to recover sequence $c_n$ from $x_n$.

With the coefficient sequence $c_n$ we can easily compute the sampled version of the derivative or fractional translation of $s(t)$, i.e. $s(n)$ and $s(n+\tau)$ where $\tau$ is a fractional number. Test your results with function $s(t)=\sin(2\pi \cdot 0.001\cdot t)$for $t\in [0,100]$.

Check the attached problem please. I am a beginner in spline fitting and have a few questions:

1) How to find the coefficients $c[n]$. Is it by DTFT?

2) I understand how to find the derivative but not sure about how to deal with fractional translation?

3) What's the right function in Matlab to obtain the (discrete) samples of a uniform spline function

Thanks!

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  • $\begingroup$ uhm, what is "$\beta_+^{(K)}$"? is it some kind of spline basis function? is that well-defined somewhere? $\endgroup$ – robert bristow-johnson Oct 29 '14 at 18:56
  • $\begingroup$ @robert It is Causal elementary B-splines. A precise definition could be found in Page 542, "Foundations of Signal Processing" fourierandwavelets.org/FSP_v1.1_2014.pdf $\endgroup$ – Bing Li Oct 29 '14 at 23:01

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