and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing filter banks and MIMO systems, it is often useful to consider vectors of signals. A linear system that is not time-invariant can be solved using other approaches such as the Green function method. Continuous-time systems Impulse response and convolution The behavior of a linear, continuous-time, time-invariant system with input signal x(t) and output signal y(t) is described by the convolution integral: {| | | |- | | (using commutativity) |} where is the system's response to an impulse: . is therefore proportional to a weighted average of the input function . The weighting function is , simply shifted by amount . As changes, the weighting function emphasizes different parts of the input function. When is zero for all negative , depends only on values of prior to time , and the system is said to be causal. To understand why the convolution produces the output of an LTI system, let the notation represent the function with variable and constant . And let the shorter notation represent . Then a continuous-time system transforms an input function, into an output function, . And in general, every value of the output can depend on every value of the input. This concept is represented by: where is the transformation operator for time . In a typical system, depends most heavily on the values of that occurred near time . Unless the transform itself changes with , the output function is just constant, and the system is uninteresting. For a linear system, must satisfy : And the time-invariance requirement is: In this notation, we can write the impulse response as Similarly: {| | | |- | | (using ) |} Substituting this result into the convolution integral: which has the form of the right side of for the case and then allows this continuation: In summary, the input function, , can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at . The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse responses, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral. The mathematical operations above have a simple graphical simulation. Exponentials as eigenfunctions An eigenfunction is a function for which the output of the operator is a scaled version