of the same function. That is, where f is the eigenfunction and is the eigenvalue, a constant. The exponential functions , where , are eigenfunctions of a linear, time-invariant operator. A simple proof illustrates this concept. Suppose the input is . The output of the system with impulse response is then which, by the commutative property of convolution, is equivalent to where the scalar is dependent only on the parameter s. So the system's response is a scaled version of the input. In particular, for any , the system output is the product of the input and the constant . Hence, is an eigenfunction of an LTI system, and the corresponding eigenvalue is . Direct proof It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems. Let's set some complex exponential and a time-shifted version of it. by linearity with respect to the constant . by time invariance of . So . Setting and renaming we get: i.e. that a complex exponential as input will give a complex exponential of same frequency as output. Fourier and Laplace transforms The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided Laplace transform is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids (i.e., exponential functions of the form where and ). The Fourier transform gives the eigenvalues for pure complex sinusoids. Both of and are called the system function, system response, or transfer function. The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of t less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the bilateral Laplace transform). The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the