# Signal and system convolution

In the convolution equation $y(n)=x(n)\circledast h(n)$ if two things are given like $y(n)$ and $x(n)$ then how can find the $h(n)$?

• please kindly tell me about continoues signal Nov 19 '16 at 11:14
• Answer upvotes and better answer validation are required for this question Jul 28 '19 at 14:46

This is known as deconvolution. A typical approach would be to apply a Fourier transform to both sides $$y = x \circledast h \overset{\mathscr{F}}{\Leftrightarrow} Y=XH$$ and divide both sides by the known $X$ to obtain $H=Y/X$. Then apply an inverse Fourier transform to both sides, yielding $h$.

The linked wikipedia article goes into more detail on variations of this approach.

While the ⊛ operator suggests some symmetry in the operation, $x$ and $y$ are often variable inputs and outputs, whereas $h$ stands for the impulse responses of a "steady", linear, time-invariant system. Hence, I would could the problem (linear) system identification. Since you know $x$ and $y$, it can be called myopic (or informed) deconvolution. In some cases, when $h$ is known to smooth the input, it can be called blur identification.
As explained by @PeterK, the most straightforward method uses a transform that converts a convolution into a product. The Fourier transform is a candidate, and the issue with the vanishing of the denominator can be tackled with solutions of the form $Y/(X+\epsilon)$ or $YX/(X^2+\epsilon)$, potentially depending on the nature of $x$ and $y$ (stochastic, deterministic). There are alternatives, such as in Short-Time Fourier Analysis Techniques for FIR System Identification and Power Spectrum Estimation, 1979.
A larger field of research (restoration) resorts to finding approximate estimations $\hat{h}$ for $h$, using additional hypotheses (FIR, positivity, causality), or taking computational errors, noises or system imperfections into account.
• @sohaibsajid The formulation I used can be discrete. Just think about $X$ as $X[n]$ Nov 26 '16 at 9:42