# Signal recovery is based on the development of the Shannon sampling theorem?

One of the earliest extensions of this theorem was stated by Shannon himself in his 1949 paper, which says that if $$x(t)$$ and its first $$(M - 1)$$ derivatives are available, then uniformly spaced samples of these, taken at the reduced rate of $$M$$ times, are sufficient to reconstruct $$x(t)$$.

This result will be referred to as the derivative sampling theorem in this paper. I'm working with $$M=3$$ and having trouble designing a synthesis filter with an impulse response of the following form:

$$\text{sinc}(t)^3$$
$$t\cdot\text{sinc}(t)^3$$
$$t^2\cdot\text{sinc}(t)^3$$

Can you help?

• I have no idea why you want those strange-looking impulse responses. But if you have them already defined in the time domain, you're nearly done. You do have to window them, with a decent window (like Kaiser), to a finite length and to sample them with a high sample rate and make tables of coefficients outa that. Commented Apr 18 at 17:10
• My problem is applying these 3 filters in speech signal processing. The input is a speech signal, which is then sampled including the signal, the first derivative of the signal, and the second derivative of the signal. Can using the above 3 filters restore the original signal? Commented Apr 21 at 17:46
• I dunno. Appears to me that $$t^2 \mathrm{sinc}^3(t)$$ is the inverse Fourier Transform of a the second-derivative of the piecewise quadratic function that is continuous to the second-derivative. Commented Apr 22 at 3:48

## 1 Answer

You may find the paper Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples by Y. Eldar and Al Oppenheim, which discusses some approach to synthesize the filters, useful.

• Thank you very much. I have a question that you can answer for me: multichannel data acquisition (my problem is 3 channels including: signal sample, first derivative sample of the signal, second derivative sample of the signal) is considered uniform or non-uniform sampling? Commented Apr 20 at 9:41
• Being uniform or non-uniform depends on how the sampling points are distributed. Let's say that the sample instances are $t_{n-1},t_{n},t_{n+1}$. If $t_{n}-t_{n-1} = T_{s} = \text{const},~\forall n$, then the function (signal) is uniformly sampled. Otherwise, the resampling is done in a non-uniform fashion. Recall that reconstruction from the derivatives allows to resample the signal at sub-Nyquist. In the problem at your hand, you can reconstruct a signal sampled at rate $\Omega_{N}/3$.
– AHT
Commented Apr 21 at 15:05
• You're right. I read the paper but didn't understand much. Can you explain more clearly to me? If possible, to make it easier to understand, can you simulate it on a program like matlab? Commented Apr 22 at 8:25