Сызықтық жауапты сүзгілер

Filter that has a linear response

Linear filters process time varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using LTI ("linear time invariant") system theory revealing their transfer functions in the frequency domain and their impulse responses in the time domain. Real time implementations of such linear signal processing filters in the time domain are inevitably causal, an additional constraint on their transfer functions. An analog electronic circuit consisting only of linear components (resistors, capacitors, inductors, and linear amplifiers) will necessarily fall in this category, as will comparable mechanical systems or digital signal processing systems containing only linear elements. Since linear time invariant filters can be completely characterized by their response to sinusoids of different frequencies (their frequency response), they are sometimes known as frequency filters. Non real time implementations of linear time invariant filters need not be causal. Filters of more than one dimension are also used such as in image processing. The general concept of linear filtering also extends into other fields and technologies such as statistics, data analysis, and mechanical engineering.

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. For instance, consider a damped harmonic oscillator such as a pendulum, or a resonant L C tank circuit. If the pendulum has been at rest and we were to strike it with a hammer (the "impulse"), setting it in motion, it would swing back and forth ("resonate"), say, with an amplitude of 10 cm. After 10 minutes, say, the pendulum would still be swinging but the amplitude would have decreased to 5 cm, half of its original amplitude. After another 10 minutes its amplitude would be only 2.5 cm, then 1.25 cm, etc. However it would never come to a complete rest, and we therefore call that response to the impulse (striking it with a hammer) "infinite" in duration. The complexity of such a system is specified by its order N. N is often a constraint on the design of a transfer function since it specifies the number of reactive components in an analog circuit; in a digital IIR filter the number of computations required is proportional to N.

Classical analog filters are IIR filters, and classical filter theory centers on the determination of transfer functions given by low order rational functions, which can be synthesized using the same small number of reactive components. Using digital computers, on the other hand, both FIR and IIR filters are straightforward to implement in software. A digital IIR filter can generally approximate a desired filter response using less computing power than a FIR filter, however this advantage is more often unneeded given the increasing power of digital processors. The ease of designing and characterizing FIR filters makes them preferable to the filter designer (programmer) when ample computing power is available. Another advantage of FIR filters is that their impulse response can be made symmetric, which implies a response in the frequency domain that has zero phase at all frequencies (not considering a finite delay), which is absolutely impossible with any IIR filter.

The frequency response or transfer function of a filter can be obtained if the impulse response is known, or directly through analysis using Laplace transforms, or in discrete time systems the Z transform. The frequency response also includes the phase as a function of frequency, however in many cases the phase response is of little or no interest. FIR filters can be made to have zero phase, but with IIR filters that is generally impossible. With most IIR transfer functions there are related transfer functions having a frequency response with the same magnitude but a different phase; in most cases the so called minimum phase transfer function is preferred. Filters in the time domain are most often requested to follow a specified frequency response. Then, a mathematical procedure finds a filter transfer function that can be realized (within some constraints), and approximates the desired response to within some criterion. Common filter response specifications are described as follows:

A low pass filter passes low frequencies while blocking higher frequencies. A high pass filter passes high frequencies. A band pass filter passes a band (range) of frequencies. A band stop filter passes high and low frequencies outside of a specified band. A notch filter has a null response at a particular frequency. This function may be combined with one of the above responses. An all pass filter passes all frequencies equally well, but alters the phase relationship among them. An equalization filter is not designed to fully pass or block any frequency, but instead to gradually vary the amplitude response as a function of frequency: filters used as pre emphasis filters, equalizers, or tone controls are good examples.

Meeting a frequency response requirement with an FIR filter uses relatively straightforward procedures. In the most basic form, the desired frequency response itself can be sampled with a resolution of and Fourier transformed to the time domain. This obtains the filter coefficients hi, which implements a zero phase FIR filter that matches the frequency response at the sampled frequencies used. To better match a desired response, must be reduced. However the duration of the filter's impulse response, and the number of terms that must be summed for each output value (according to the above discrete time convolution) is given by where T is the sampling period of the discrete time system (N 1 is also termed the order of an FIR filter). Thus the complexity of a digital filter and the computing time involved, grows inversely with , placing a higher cost on filter functions that better approximate the desired behavior. For the same reason, filter functions whose critical response is at lower frequencies (compared to the sampling frequency 1/T) require a higher order, more computationally intensive FIR filter. An IIR filter can thus be much more efficient in such cases. Elsewhere the reader may find further discussion of design methods for practical FIR filter design.

Since classical analog filters are IIR filters, there has been a long history of studying the range of possible transfer functions implementing various of the above desired filter responses in continuous time systems. Using transforms it is possible to convert these continuous time frequency responses to ones that are implemented in discrete time, for use in digital IIR filters. The complexity of any such filter is given by the order N, which describes the order of the rational function describing the frequency response. The order N is of particular importance in analog filters, because an Nth order electronic filter requires N reactive elements (capacitors and/or inductors) to implement. If a filter is implemented using, for instance, biquad stages using op amps, N/2 stages are needed. In a digital implementation, the number of computations performed per sample is proportional to N. Thus the mathematical problem is to obtain the best approximation (in some sense) to the desired response using a smaller N, as we shall now illustrate. Below are the frequency responses of several standard filter functions that approximate a desired response, optimized according to some criterion. These are all fifth order low pass filters, designed for a cutoff frequency of .5 in normalized units. Frequency responses are shown for the Butterworth, Chebyshev, inverse Chebyshev, and elliptic filters. As is clear from the image, the elliptic filter is sharper than the others, but at the expense of ripples in both its passband and stopband. The Butterworth filter has the poorest transition but has a more even response, avoiding ripples in either the passband or stopband. A Bessel filter (not shown) has an even poorer transition in the frequency domain, but maintains the best phase fidelity of a waveform. Different applications emphasize different design requirements, leading to different choices among these (and other) optimizations, or requiring a filter of a higher order.

A popular circuit implementing a second order active R C filter is the Sallen Key design, whose schematic diagram is shown here. This topology can be adapted to produce low pass, band pass, and high pass filters. An Nth order FIR filter can be implemented in a discrete time system using a computer program or specialized hardware in which the input signal is subject to N delay stages. The output of the filter is formed as the weighted sum of those delayed signals, as is depicted in the accompanying signal flow diagram. The response of the filter depends on the weighting coefficients denoted b0, b1, bN. For instance, if all of the coefficients were equal to unity, a so called boxcar function, then it would implement a low pass filter with a low frequency gain of N+1 and a frequency response given by the sinc function. Superior shapes for the frequency response can be obtained using coefficients derived from a more sophisticated design procedure.