The next operator we examine is convolution. It arises in many applications; and it could be derived in many ways. A basic derivation is from the multiplication of two polynomials, say X(Z) = x1 + x2 Z + x3 Z2 + x4 Z3 + x5 Z4 + x6 Z5 times B(Z) = b1 + b2 Z + b3 Z2 + b4 Z3. Identifying the k-th power of Z in the product Y(Z)=B(Z)X(Z) gives the k-th row of the convolution transformation (4).
Equation (4) could be rewritten as
The adjoint of (5) crosscorrelates a fixed portion of filter input across a variable portion of filter output.
Module tcai1 is used for and module tcaf1 is used for .tcai1transient convolution tcaf1transient convolution
The polynomials X(Z), B(Z), and Y(Z) are called Z transforms. An important fact in real life (but not important here) is that the Z transforms are Fourier transforms in disguise. Each polynomial is a sum of terms and the sum amounts to a Fourier sum when we take .The very expression Y(Z)=B(Z)X(Z) says that a product in the frequency domain (Z has a numerical value) is a convolution in the time domain (that's how we multipy polynomials, convolve their coefficients).