ABSTRACT
This chapter shows linear systems that are optimal in some sense for some problems of practical importance. A linear time-invariant (LTI) system is intended to be a differentiator; that is, its output is the differentiation of its input. If the input is noisy, however, the output could be extremely noisy. The chapter covers the matched filter, and the wiener filter. It also discusses signal-to-noise ratio. Maximization of output signal-to-noise ratio (SNR): The ratio of output signal power to output noise power at some specified time is maximized. Minimization of the mean-square error (MSE): The mean-square value of the difference between the system's output and the desirable input signal is minimized. This is typically used in e.g. speech signal processing when the noiseless input signal is random but with known characteristics, such as to estimate the value of a signal that is known to be present.
