Finite impulse response

A filter having a finite impulse response (English finite impulse response filter, FIR filter, or sometimes referred to as transversal filter ) is a discrete, usually digitally implemented filter and is used in the field of digital signal processing.

• 5.1 average filter

Properties

The characteristic of FIR filters is that they have an impulse response with guaranteed finite length. This means that FIR filters can, no matter how the filter parameters are chosen, never become unstable or are encouraged to self- oscillation. FIR filters are usually non-recursive filter, in other words have no feedback or ribbons in their structure. There are also some special FIR filter structures with feedback, an example of this are cascaded - integrator -comb filter. Nevertheless, certain transmission functions can be implemented by FIR filters with discrete recursive filter structures.

Often FIR filters are implemented as digital filters. The implementation of analog FIR filter is more difficult and less common, mainly because of accuracy problems. Examples are electric bucket brigade circuits, BBD be used for analog delay analog N- path filters, as well as acoustic realizations ( Surface Acoustic Wave ) filters.

Theoretical foundations

As an essential property FIR filters are always stable. This follows mathematically from the necessity that the two direct forms the only n-fold pole of the transfer function always lies at the origin, and thus within the unit circle. With recursive FIR filters the zeros fall exactly together with the pole and lift it up. By this principle, guaranteed stability FIR filters are used for example in adaptive filtering as the basis for the filter structure. Also, quantization errors affect on the filter coefficients by the necessary restriction to a finite number of points only to a small extent.

Furthermore, the DC gain of an FIR filter of the normal form is equal to the sum of all filter coefficients and the impulse response h (k) delivering the sequence of filter coefficients. This coefficient sequence is in an FIR filter with order m always m 1 values ​​long, making a straight filter order always always has an even number of filter coefficients is an odd number of filter coefficients and for odd filter order.

In contrast to IIR filters significantly higher order m is necessary for realizing a predetermined filter transfer function using the FIR filter.

Basic types

In addition to the general, complex-valued FIR systems, especially real-valued FIR systems play an important role in the practical realization of filters. Real-valued FIR filters have coefficient than on real numbers and can be depending on whether an even or odd filter order or an even or odd symmetry of the impulse response is present, divided into four basic types. These four types differ from one another in the position of the fixed predetermined zeros in the magnitude frequency response | H ( j? ) |.

In the following table, the normalization in the literature, however, is not uniform, the exemplary impulse responses are the four basic types specified. In the transfer function H ( j? ), The parameter π Ω represent those on this normalized frequency Ω is equal to π at half the sampling frequency, which is the cutoff frequency of the time-discrete transmission systems. Ω = 2π corresponding to the sampling frequency. Since it is always to discrete-time systems, the magnitude frequency response | H ( j? ) | Periodicity of sampling frequency on which is different depending on the type.

Wherein the filter design, these special properties is essential. For example, results in the design of a high-pass filter with an FIR structure with an even filter order and odd symmetry (type 2) or odd filter order and even symmetry (type 3) no sense, since in both cases forcibly at Ω = π unwanted zero point is located and this contradicts the requirement of an open-topped frequency response with a high pass. Similarly, the realization of a low-pass filter with type 2 or type 4 gives problems because the forced zero Ω = 0; always suppresses the desired features of a low-pass DC component.

Forms of implementation

Non-recursive FIR filter can be implemented in a variety of basic shapes which are shown in more detail below.

Direct Normal Forms

The two direct normal forms 1 and 2, as a canonical normal form and in English as a Direct Form I ( DF1) and Direct Form II, designated ( DF2 ), the FIR filter indicate that with a minimum number of storage elements and the filter coefficients for implementing a given transfer function get along. There are two different forms, as in the two adjacent figures, it is shown an example of an FIR filter of 4th order.

Both normal forms are equivalent and can be converted into each other. Depending on the specific realization of the filter, for example in a digital signal processor by means of software of processed sequentially or in parallel by digital circuitry, data processing such as FPGAs that can be more efficient to implement one or the other shape. The normal form 1 for example, offers the possibility to combine the output side to a single adder adder with m 1 inputs, while the normal form 2 allows to sum same coefficients and thus reducing multipliers.

The difference equation for the filter response is identical for both forms and is for a system of order m in the time domain:

The factors occurring h (i ) are the filter coefficients dar. Rearranging the transfer function can be determined:

In real-valued filtering allows the complex parameter z by ejΩ, replace with Ω as normalized frequency. The elements are also referred to as Z taps, and provide for the realization of memory elements and time delays represents a FIR filter of the form has always zeros and m is an m- multiple pole in the origin at 0 in the Z- plane, with which the frequency response is determined solely by the zeros in the z-plane.

The determination of the constant time invariant FIR filter in the filter coefficient h (i ) is carried out within the dimensions of the filter, according to specific requirements. These specifications provide for represents a fundamental statements about the desired transfer function, such as the realization of a high-pass, low-pass, band-pass or band-stop filter and its cut-off frequency values ​​. There are also other parameters such as allowable deviations of the magnitude frequency response of an ideal shape and the slope in the transition region. The sizing is then made with numerical software packages such as MATLAB or Octave.

Optimization

By utilizing the even and odd symmetry of the impulse response of linear-phase FIR filters can be divided into two normal forms complex multiplier to reduce to half. The data from the respective memory elements are first added together, which are then multiplied with the same filter coefficients.

Cascaded form

FIR filters of higher orders can be in multiple, smaller FIR filter of lower order, usually 2nd order split, and another switch in series ( cascade ) to form as the sum of the transfer function of the higher-order filter. In the illustration alongside a cascaded FIR filter is shown 6th order, consisting of three individual FIR filters each 2nd order. In most cases in English literature these filter structures are also referred to as Second Order Structure, abbreviated as SOS or filter.

The individual elementary FIR filter can then contain occur in the normal form 1 or 2 normal form. The transfer of filter coefficients from the normal shape in a cascaded form requires a transformation of the filter coefficients.

The reason for this conversion lies in the implementation in digital systems in the quantization of the filter coefficients. By only finite precision with which those coefficients can be stored, and rounding errors may lead to inadmissible errors. In the cascaded filter structure, the distribution of said quantized filter coefficients is more evenly distributed than in the two normal forms.

The transfer function H ( z) for cascaded FIR filter of even order, as in the figure for the filter order m = 6 with the filter coefficients h, results to:

Polyphase form

The structure of polyphase FIR filters represents a parallel interconnection of individual FIR filter, which is also the adjective polyphas ( multi-phase ) is derived. In this single, basic FIR filter of low order are connected sequentially in time and summed up their respective output. To be filtered data sequence runs through parallel but offset in time, the individual elementary filter.

Polyphase forms offer several advantages for the optimization, in particular in the implementation of direct digital hardware circuits. Thus, individual filter element of the polyphase filter can be operated at a lower clock rate than the overall filter. Or it can be implemented in only a part of the filter in order to minimize hardware, the filter coefficient can be cyclically interchanged.

Other forms

In addition, there are other forms such as FIR Lattice Filter which occur both as an FIR filter as well as an IIR filter. These filters owing to their structure, especially in the area of ​​the prediction error spectral digital radio transmissions, for example, in digital mobile networks, the application.

Fast Convolution

The usual realization in one of the two normal forms, the direct execution of discrete, aperiodic convolution represents an alternative and functionally identical implementation option is the fast convolution. In this case, a fast Fourier transform (FFT ) and its inverse transformation is combined with the multiplication of the pulse response in the spectral range, which results in a cyclic convolution results. This operation, however, is not the same as the non-periodic convolution, which is due to the block-based processing of the FFT. Can be achieved of the discrete, non-periodic convolution identical filter implementation by the application of techniques such as the overlap-save method or the overlap-add method. Depending on the number format and type of implementation, from FIR filter orders of 40 upwards with the fast convolution efficient implementations can be achieved than by direct execution of the aperiodic convolution operation.

Examples

Average filter

The simplest FIR filter is the classical averaging by summing individual successive values ​​and dividing by their number. Performs one this not in blocks ( number 1 to number 10, number 10 to number 20, etc), but overlapping ( number 1 to number 10, number 2 to number 11, number 3 to number 12, etc ) by, so come one to the moving average. The result is a change over time slowly changing value, which can not follow rapid changes.

This low-pass filter has the inherent disadvantage that the first side peak in the amplitude response is not attenuated by more than -21 dB (no matter how wide the selected median filter ). This is shown by ringing. The disadvantage is overcome by example with Binomialfiltern.