Linear Filter

Linear Filter

Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity. This results from systems composed solely of components (or digital algorithms) classified as having a linear response.

Most filters implemented in analog electronics, in digital signal processing, or in mechanical systems are classified as causal, time invariant, and linear. However the general concept of linear filtering is broader, also used in statistics, data analysis, and mechanical engineering among other fields and technologies. This includes noncausal filters and filters in more than one dimension such as would be used in image processing; those filters are subject to different constraints leading to different design methods, which are discussed elsewhere.

A linear time-invariant (LTI) filter can be uniquely specified by its impulse response h, and the output of any filter is mathematically expressed as the convolution of the input with that impulse response. The frequency response, given by the filter's transfer function, is an alternative characterization of the filter. The frequency response may be tailored to, for instance, eliminate unwanted frequency components from an input signal, or to limit an amplifier to signals within a particular band of frequencies. There are a number of particularly desirable or useful filter transfer functions, of which this article will present an overview.

Among the time-domain filters we here consider, there are two general classes of filter transfer functions that can approximate a desired frequency response. Very different mathematical treatments apply to the design of filters termed infinite impulse response (IIR) filters, characteristic of mechanical and analog electronics systems, and finite impulse response (FIR) filters, which can be implemented by discrete time systems such as computers (then termed digital signal processing).