Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist. Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist One can use the system response directly to determine how any particular frequency component is handled by a system with that Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency , where , we obtain |H(s)| which is the system gain for frequency f. The relative phase shift between the output and input for that frequency component is likewise given by arg(H(s)). Examples Important system properties Some of the most important properties of a system are causality and stability. Causality is a necessity for a physical system whose independent variable is time, however this restriction is not present in other cases such as image processing. Causality A system is causal if the output depends only on present and past, but not future inputs. A necessary and sufficient condition for causality is where is the impulse response. It is not possible in general to determine causality from the two-sided Laplace transform. However when working in the time domain one normally uses the one-sided Laplace transform which requires causality. Stability A system is bounded-input, bounded-output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if every input satisfying leads to an output satisfying (that is, a finite maximum absolute value of implies a finite maximum absolute value of ), then the system is stable. A necessary and sufficient condition is that , the impulse response, is in L1 (has a finite L1 norm): In the frequency domain, the region of convergence must contain the imaginary axis . As an example, the ideal low-pass filter with impulse response equal to a sinc function is not BIBO stable, because the sinc function does not have a finite L1 norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for and equal to a sinusoid at the cut-off frequency for , then the output will be unbounded for all times other than the zero crossings. Discrete-time systems Almost everything in continuous-time systems has a counterpart in discrete-time systems. Discrete-time systems from continuous-time systems