The diagram shows how the unknown signal can be identified. It is the reason that DSP processors can do multiplications and additions in parallel.
Convolution is used for digital filtering .
The reason convolution is preferred to correlation for filtering has to do with how the frequency spectra of the two signals interact. Convolving two signals is equivalent to multiplying the frequency spectra of the two signals together - which is easily understood, and is what we mean by filtering . Correlation is equivalent to multiplying the complex conjugate of the frequency spectrum of one signal by the frequency spectrum of the other. Complex conjugation is not so easily understood and so convolution is used for digital filtering . Convolving by multiplying frequency spectra is called fast convolution .