Equivalently, the system's response to an impulse at n=0 is a "time" reversed copy of the unshifted weighting function. When h[k] is zero for all negative k, the system is said to be causal. Exponentials as eigenfunctions An eigenfunction is a function for which the output of the operator is the same function, scaled by some constant. In symbols, where f is the eigenfunction and is the eigenvalue, a constant. The exponential functions , where , are eigenfunctions of a linear, time-invariant operator. is the sampling interval, and . A simple proof illustrates this concept. Suppose the input is . The output of the system with impulse response is then which is equivalent to the following by the commutative property of convolution where is dependent only on the parameter z. So is an eigenfunction of an LTI system because the system response is the same as the input times the constant . Z and discrete-time Fourier transforms The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Z transform is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids; i.e. exponentials of the form , where . These can also be written as with . The discrete-time Fourier transform (DTFT) gives the eigenvalues of pure sinusoids. Both of and are called the system function, system response, or transfer function. Like the one-sided Laplace transform, the Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for t<0. The discrete-time Fourier transform Fourier series may be used for analyzing periodic signals. 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. That is, Just as with the Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easier to analyze systems and gain insight into their behavior. Examples Important system properties The input-output characteristics of discrete-time LTI system are completely described by its impulse response . Two of the most important properties of a system are causality and stability. Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time. Unstable systems can also be analyzed and built, but are only useful as part of a larger system whose overall transfer function