Deriving the Convolution Kernel of the Inverse of a Signal Let $y$ be the inverse (in the sense of convolution) of $x$, i.e. 

$$x \star y = \delta$$

Context: $x[n]$ is a discrete signal defined for $n = 0,\ldots, N$. We can assume $x[n] = 0$ if $n \not\in [0,\ldots, N]$.

How to compute the inverse of $x$ (for the convolution)?



\begin{align}(x\star y)[0]&amp;=x[0] y[0] = 1\implies y[0] = \frac{1}{x[0]}\\  
(x\star y)[1]&amp;=x[1] y[0] + x[0] y[1] = 0\implies y[1] = -\frac{x[1]}{x[0]^2}\\
(x\star y)[2]&amp;=x[2] y[0] + x[1] y[1] + x[0] y[2] = 0\implies y[2] = -\frac{x[2]}{x[0]^2} + \frac{x[1]^2}{x[0]^3}
\end{align}

I tried to find $y[3]$ manually, but I don't see a general formula appearing...
Since Convolution is a Linear Operation the easiest way to derive the coefficients is using a Matrix Representation of the problem.

Take the kernel $ y $ and build from it the Convolution Matrix.
Then write the vector representation of the Delta Funciton $ d $ and solve:

$$ Y x = d $$

In case the problem is solvable (Namely there is an inverse operator of $ Y $) you'll get you answer.
Of course you can always use Least Squares Solution.

Full code for building Convolution Matrix was given in Estimating a Signal Given a Noisy Measurement of the Signal and Its Derivative (Denoising) and Deconvolution of Synthetic 1D Signals - How To?