Infinite impulse response
A filter having an infinite impulse response (English infinite impulse response filter, IIR filter or IIR system) is a specific type filter in the signal processing. He called called a linear shift-invariant filters, and system LSI (Linear Shift - Invariant ). Depending on the specific choice of the filter parameters, this filter type, in contrast to finite impulse response filters provide with an infinitely long duration impulse response.
Motivation
Filters are used, for example, to recover from a composite signal, as it is used for example when broadcasting or wideband data transmission individual partial signals. A frequency-selective filter that is continuously used to filter out from the totality of electromagnetic vibrations in a specific radio station. In the broadband technology (see OFDM), discrete filter (banks ) can be used to bring together a plurality of digital signals so that each signal occupies one subchannel available in the frequency channel, and these signals after the transmission to separate again.
Most analog, ie continuous-time systems, have an infinitely long impulse response. This is always the case for example if, in the circuit, a capacitor or a coil is contained. The need for an infinitely long filter arises from the fact that one often wants to reverse the effect last long filter. Thus, signal distortion can be modeled as a finite digital filter in the transmission and measured accordingly by test signals. To undo this disorder should - ideally - be used an infinitely long digital filter or a good approximation thereof.
Other names for " filters " are system or, " old-fashioned " operator. A "filter " is usually a frequency selective system LSI. In mathematical terms, a stable system LSI is a bounded linear convolution operator on a normed sequence space, such as the Hilbert space of quadratsummierbaren consequences or the Banach space of bounded sequences term by term.
In the continuous case, a filter with a betragsintegrable function S ( t) is defined, wherein each of the continuous-time signal x ( t) is the signal y (t ) = (s * x ) is assigned (t ), i.e.
The discrete-time case, a digital filter is defined by a betragssummierbare sequence S [ n], where each of the discrete-time signal x [ n] is a signal y [n ] = (s * x) [n] is assigned, ie
The S ( t) function or the sequence S [n ] is the impulse response of the system and as such can be measured directly. Is the internal structure of the system is known, the impulse response can be derived through combination of the impulse responses of the elemental components of the system.
In general, the evaluation of a digital IIR filter infinite number of calculation steps for each element yn required. In the special case of a recursive system, there is, however, a finite representation which would, however, require the execution of the calculation, and an endless - A decay phase.
The system function is the Laplace transform of the function S ( t) and the Z- transform of the sequence S [n]. In case of an IIR digital filter, the z-transform is the Laurent series
This defines a continuous function on the unit circle of the complex plane, and z by the parameterization = eiω a continuous, periodic function
This is the Fourier series and the transfer function of the system S. The system is an ideal frequency-selective filter, if the transfer function of only the values 0 and 1. This is subject to the above conditions are not possible, but can be approximated with arbitrary precision.
Recursive or rational systems
Recursive systems that are practically feasible IIR systems, the theory of linear inhomogeneous recurrences with solutions in these standardized areas is significant.
Is there a sequence, so that the convolution product also yields a finite sequence, then one speaks of a causal recursive system. This can be realized by a finite algorithm or a signal circuit containing feedback, ie to resort to other, already calculated elements of the output signal. There is always a real first element, a transient is to be considered in practice.
Applying a recursive filter to a signal x [n], y [n] = (s * x) [n] is then restricted to the finite linear recurrence solution
Or
Which component-wise and in normal form, the calculation formula
Results.
Corresponds Here
- X [ n] is the input signal,
- Y [ n] the (filtered ) output signal
- B [ k] is the filter coefficient of the input signal (with filter order M )
- A [ L] the filter coefficients of the feedback output signal ( with filter order N) and
- S (z) of the transfer function in the frequency domain.
The system functions can then be written as a fraction:
Generally, the coefficients are normalized so that. Here, however, for the sake of completeness it should not be accepted.
Such a fraction can be converted by partial fractions and careful use of the geometric series again in a formal Laurent series or in its coefficient sequence S. The result is a causal system, that is, a sequence with values zero to negative indices, if and only if the denominator has Laurent polynomial b ( Z) only zeros inside the unit circle.
Structure of IIR filters
For rational IIR systems, there are various ways to realize them as a network of addition, multiplication and delay elements. The actual implementation can take place depending on the application in digital signal processors or in digital hardware such as FPGAs or ASICs.
In principle various types of transfer functions can be realized in all of IIR filter structures. For practical reasons, one tries in the IIR filter realization to comply with existing, proven analog filter functions and to win by suitable transformations from the digital IIR filter coefficients. The most important continuous-time filter transfer functions that can be implemented by means of the bilinear transformation in the discrete-time IIR structures include Butterworth filters, Bessel filters, Chebyshev and Cauer filter. The choice of a specific filter transfer function has nothing to do with the selection of an appropriate IIR structure and is largely independent of it. Thus, a particular Bessel filter can be realized both in the form shown below as DF1 in the SOS structure, with virtually identical filter transfer function.
Direct Form 1 (DF1 )
The direct - form one has the advantage to be relatively easily derived on the difference equation. It is also the shape that requires that the IIR filter with the lowest possible number of carried out at a given transmission in the form of multiplication stages, and an accumulator for summing only the individual partial results. A minor disadvantage of this structure is, however, that the implementation of a relatively large number of delay elements (T- elements, locations, engl. Taps ) are required for implementation.
A far greater disadvantage of IIR structure is the sensitivity to quantization of the fixed filter coefficients. Especially in fixed-point implementations this can lead to so-called limit cycles. Under certain circumstances, the accuracy of the filter coefficients must be significantly higher than the bit width of the input signal, which increases the implementation complexity greatly under certain circumstances.
Direct Form 2 ( DF2 )
Another structure for implementing an IIR system is the canonical form of direct or Direct Form 2 ( DF2 ). In contrast to the network structure shown above, it requires only half the number of delay elements, for two accumulators are needed in the implementation. Disadvantages such as extreme sensitivity of the filter coefficients may for quantization is the same as the DF1. It is further noted that the dynamic behavior ( change in the filter coefficients during runtime) of DF2 from the filters DF1 and filter may vary only after a transient again provides identical results.
Cascaded IIR ( SOS)
In practical implementations of higher order IIR filters are often constituted by a serial concatenation (cascading) of IIR second-order filters the DF1 DF2 or. In the English literature, these systems are then used as Second Order Structure, abbreviated as SOS, respectively. SOS IIR filter to avoid the unfavorable distribution of the one-sided and pole-zero filter coefficients in the complex plane and are much more tolerant of quantization in the filter coefficient. Especially with implementations of IIR filters of higher order in fixed-point DSPs, the SOS structure in preference to the direct- form 1 and 2 should always be given.
We can also design at the SOS form the stability analysis much easier, since it only successively and independently each IIR filter 2nd order stability must be checked and if all elementary filter structures are stable on its own, and the overall filter order is higher stable. With appropriate numerical mathematics packages such as MATLAB, the determined filter coefficients of the DF1 or DF2 form can be relatively straightforward to forward to the appropriate SOS form.
Combinations of parallel IIR filters
Both the DF1 and DF2 and derived forms SOS structures can also be implemented in parallel depending on the application. This is especially the case of direct hardware implementations in FPGAs as an advantage. The latency of a particular filter assembly can be reduced, although usually associated with it a much higher circuit need arises.
Lattice filters
Another particular structural form are lattice filters which occur both as a recursive form as well as a non- recursive form. The transfer function of this filter structure includes a special feature on only poles and no zeros. With appropriate mathematics packets the filter coefficients for a given transfer function with relatively little effort, can be determined.
Wave digital filters
In addition, the IIR filter structures can also be realized by the direct reproduction of analogue and usually discrete passive filter circuits in the form of wave- digital filters. The concrete IIR structure is based directly on the electronic circuit. The advantage of those filter structures is the high insensitivity to quantization of the filter coefficients. The disadvantage is the significantly higher amount of Multipliziereren or addition stages.
Comparison of different filter structures
The following is a comparison of the different IIR structures is given at about the structures across functionally identical realization of a Cauer 8th order filter with the same dynamics. The factor of W and M represents a cost factor for the realization in hardware is: The higher the value, the more circuitry to be operated for the implementation.