Transient convolution

convolution convolution ! transient

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).  

$$\bold y \eq \left[ \begin{array} {c} y_1 \\ y_2 \\ y... ..._4 \\ x_5 \\ x_6 \end{array} \right] \eq \bold B \bold x$$ (4)

Notice that columns of equation (4) all contain the same signal, but with different shifts. This signal is called the filter's impulse response.

Equation (4) could be rewritten as  

$$\bold y \eq \left[ \begin{array} {c} y_1 \\ y_2 \\ y... ..._1 \\ b_2 \\ b_3 \end{array} \right] \eq \bold X \bold b$$ (5)

In applications we can choose between $\bold y = \bold X \bold b$and $\bold y=\bold B\bold x$.In one case the output $\bold y$is dual to the filter $\bold b$,and in the other case the output $\bold y$is dual to the input $\bold x$.Sometimes we must solve for $\bold b$ and sometimes for $\bold x$;so sometimes we use equation (5) and sometimes (4). Such solutions begin from the adjoints. The adjoint of (4) is  

$$\left[ \begin{array} {c} \hat x_1 \\ \hat x_2 \\ \hat x... ...y_4 \\ y_5 \\ y_6 \\ y_7 \\ y_8 \end{array} \right]$$ (6)

The adjoint crosscorrelates with the filter instead of convolving with it (because the filter is backwards). Notice that each row in equation (6) contains all the filter coefficients and there are no rows where the filter somehow uses zero values off the ends of the data as we saw earlier. In some applications it is important not to assume zero values beyond the interval where inputs are given.

The adjoint of (5) crosscorrelates a fixed portion of filter input across a variable portion of filter output.  

$$\left[ \begin{array} {c} \hat b_1 \\ \hat b_2 \\ \hat b... ..._4 \\ y_5 \\ y_6 \\ y_7 \\ y_8 \end{array} \right]$$ (7)

Module tcai1 is used for $\bold y=\bold B\bold x$and module tcaf1 is used for $\bold y = \bold X \bold b$.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 $Z=e^{i\omega\Delta t}$.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).