2.15 Eigenbases and lsi systems

Why is an eigenbasis so useful? It allows us to greatly simplify the computation of the output for a given input. For example, suppose that $X$ is a vector space and that $L:X\to X$ is a linear operator with eigenvectors ${\left\{v_k\right\}}_{k\in \Gamma }$ . If ${\left\{v_k\right\}}_{k\in \Gamma }$ form a basis for $X$ , then for any $x\in X$ we can write $x={\sum }_{k\in \Gamma }{c}_k{v}_k$ . In this case we have that

In the case of a DT, LSI system $H$ , we have that $\frac{1}{\sqrt{2\pi }}{e}^{-j\omega n}$ is an eigenvector of $H$ and for any $x\left[n\right]$ we can write

From the same line of reasoning as above, we have that

Whenever we have an eigenbasis, we can represent our operator as simply a diagonal operator when the input and output vectors are represented in theeigenbasis. The fact that convolution in time is equivalent to multiplication in the Fourier domain is just one instance of this phenomenon. Moreover, while we have been focusing primarily on the DTFT, it should nowbe clear that each Fourier representation forms an eigenbasis for a specific class of operators, each of which defines a particular kind of convolution.

• DTFT discrete-time convolution (infinite)
• CTFT continuous-time convolution (infinite)
  $$(f*g)\left(t\right)={\int }_{-\infty }^{\infty }f\left(\tau \right)g(t-\tau )d\tau$$
• DFT discrete-time circular convolution
  $$(x⊛y)\left[n\right]=\sum _{k=0}^{N-1}x\left[k\right]y_N[n-k]$$
• CTFS continuous-time circular convolution
  $$(f⊛g)\left(t\right)={\int }_0^{\tau }f\left(t\right)g_T(b-\tau )d\tau$$

This is the main reason why we have to care about circular convolution. It is something that one would almost never want to do – but if you multiply twoDFTs together you are doing it implicitly, so be careful and remember what it is doing.