Linear filter

Linear filters 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 signal processing filters. Linear filters 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 signal processing filters. The general concept of linear filtering is also used in statistics, data analysis, and mechanical engineering among other fields and technologies. This includes non-causal filters and filters in more than one dimension such as those used in image processing; those filters are subject to different constraints leading to different design methods. 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 H ( ω ) {displaystyle H(omega )} , is an alternative characterization of the filter. Typical filter design goals are to realize a particular frequency response, that is, the magnitude of the transfer function | H ( ω ) | {displaystyle |H(omega )|} ; the importance of the phase of the transfer function varies according to the application, inasmuch as the shape of a waveform can be distorted to a greater or lesser extent in the process of achieving a desired (amplitude) response in the frequency domain. 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. The impulse response h of a linear time-invariant causal filter specifies the output that the filter would produce if it were to receive an input consisting of a single impulse at time 0. An 'impulse' in a continuous time filter means a Dirac delta function; in a discrete time filter the Kronecker delta function would apply. The impulse response completely characterizes the response of any such filter, inasmuch as any possible input signal can be expressed as a (possibly infinite) combination of weighted delta functions. Multiplying the impulse response shifted in time according to the arrival of each of these delta functions by the amplitude of each delta function, and summing these responses together (according to the superposition principle, applicable to all linear systems) yields the output waveform. Mathematically this is described as the convolution of a time-varying input signal x(t) with the filter's impulse response h, defined as: The first form is the continuous-time form, which describes mechanical and analog electronic systems, for instance. The second equation is a discrete-time version used, for example, by digital filters implemented in software, so-called digital signal processing. The impulse response h completely characterizes any linear time-invariant (or shift-invariant in the discrete-time case) filter. The input x is said to be 'convolved' with the impulse response h having a (possibly infinite) duration of time T (or of N sampling periods). Filter design consists of finding a possible transfer function that can be implemented within certain practical constraints dictated by the technology or desired complexity of the system, followed by a practical design that realizes that transfer function using the chosen technology. The complexity of a filter may be specified according to the order of the filter. 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). Consider a physical system that acts as a linear filter, such as a system of springs and masses, or an analog electronic circuit that includes capacitors and/or inductors (along with other linear components such as resistors and amplifiers). When such a system is subject to an impulse (or any signal of finite duration) it responds with an output waveform that lasts past the duration of the input, eventually decaying exponentially in one or another manner, but never completely settling to zero (mathematically speaking). Such a system is said to have an infinite impulse response (IIR). The convolution integral (or summation) above extends over all time: T (or N) must be set to infinity.