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D ( z ) = ( z 2 + ( 2 cos x ) z + 1 ) K

The special case for a = 2 is not of lowest order. It can be factored into [link] squared. Any length-4 even-symmetric filter can be factored intoproducts of terms of the form of [link] and [link] .

The fourth case is of an even-symmetric length-5 filter of the form

D ( z ) = z 4 + a z 3 + b z 2 + a z + 1

For a 2 < 4 ( b - 2 ) and b > 2 , the zeros are neither real nor on the unit circle; therefore, they must have complex conjugatesand have images about the unit circle. The form of the transfer function is

D ( z ) = { z 4 + [ ( 2 ( r 2 + 1 ) / r ) cos x ] z 3 + [ r 2 + 1 / r 2 + 4 cos 2 x ] z 2 + [ ( 2 ( r 2 + 1 ) / r ) cos x ] z + 1 } K

If one of the zeros of a length-5 filter is on the real axis or on the unit circle, D ( z ) can be factored into a product of lower order terms of the forms in [link] , [link] , and [link] and, therefore, is not of lowest order. The odd symmetric filters of [link] are described by the above factors plus the basic length-2 filter described by

D ( z ) = ( z - 1 ) K

The zero locations for the four basic cases of Type 1 and 2 FIR filters are shown in [link] . The locations for the Type 3 and 4 odd-symmetric cases of [link] are the same, plus the zero at one from [link] .

This figure consist of 5 different Cartesian graphs each containing a different circle. For each of these graphs the x axis is labeled real part of z and the y axis is labeled Imaginary part of z. The top left graph is labeled Length-2, Even Symmetric h(n). It contains a circle centered at the origin and on the far left there is a single hollow circle on the border. this circle is about at point (-1,0). The top right graph is labeled Length-2, Odd Symmetric h(n) and contains a circle centered around the origin. On the far right border of the circle there is a small circle. This circle exist near the point (1,0). The left middle graph is labeled Length-3, Even Symmetric h(n). On the large circle centered around the origin were two smaller circles on points on the border of the circle. The small circles are on the left half of the circle at points close to (-.75,.75) and (-.75,-.75). The middle right graph is labeled Length-3, Even Symmetric h(n). The graph contains a circle centered around the origin. On the left side of the circle there are two small circles. The small circles are not on the border of the circle. One was slightly to the right of the left most area of the circle a point near (.5,0). The other small circle is to the left of the left most border of the circle at a point near (-1.5,0). The bottom graph is labeled Length-5, Even Symmetric h(n), and contains a large circle centered around the origin. There are also four small circles near the left most border of the larger circle. There are two smaller circles on the inside of the circle. They are at two points near (-.5,-.5) and (-.5,.5). There are also to points to the left of the circle. These circles are near points (-1.25,.75) and (-1.25,-.75).
Zero Locations for the Basic Linear-Phase FIR Filter

From this analysis, it can be concluded that all linear-phase FIR filters have zeros either on the unit circle or in thereciprocal symmetry of [link] or [link] about the unit circle, and their transfer functionscan always be factored into products of terms with these four basic forms. This factored form can beused in implementing a filter by cascading short filters to realize a long filter. Knowledge of the locations of the transferfunction zeros helps in developing filter design and analysis programs. Notice how these zero locations are consistent withthe amplitude responses illustrated in [link] and [link] .

Section summary

In this section the basic characteristics of the FIR filter have been derived. For the linear-phase case, the frequencyresponse can be calculated very easily. The effects of the linear phase can be separated so that the amplitude can be approximatedas a real-valued function. This is a very useful property for filter design. It was shown that there are four basic types oflinear-phase FIR filters, each with characteristics that are also important for design. The frequency response can be calculatedby application of the DFT to the filter coefficients or, for greater resolution, to the N filter coefficients with zeros added to increase the length. A very efficient calculation of the DFTuses the Fast Fourier Transform (FFT). The frequency response can also be calculated by special formulas that include the effectsof linear phase.

Because of the linear-phase requirements, the zeros of the transfer function must lie on the unit circle in the z plane oroccur in reciprocal pairs around the unit circle. This gives insight into the effects of the zero locations on the frequencyresponse and can be used in the implementation of the filter.

The FIR filter is very attractive from several points of view. It alone can achieve exactly linear phase. It is easilydesigned using methods that are linear. The filter cannot be unstable. The implementation or realization in hardware or on acomputer is basically the calculation of an inner product, which can be accomplished very efficiently. On the negative side, theFIR filter may require a rather long length to achieve certain frequency responses. This means a large number of arithmeticoperations per output value and a large number of coefficients that have to be stored. The linear-phase characteristic makes thetime delay of the filter equal to half its length, which may be large.

How the FIR filter is implemented and whether it is chosen over alternatives depends strongly on the hardware or computer to beused. If an array processor is used, an FFT implementation [link] would probably be selected. If a fixed point TMS320 signal processor is used, a direct calculation of the inner product is probably best. Ifa floating point DSP or microprocessor with floating-point arithmetic is used, an IIR filter may be chosen over the FIR, or theimplementation of the FIR might take into account the symmetries of the filter coefficients to reduce arithmetic. To make these choices,the characteristics developed in this chapter, together with the results developed later in these notes, must be considered.

Fir digital filter design

A central characteristic of engineering is design. Basic to DSP is the design of digital filters. In many cases, the specifications ofa design is given in the frequency domain and the evaluation of the design is often done in the frequency domain. A typical sequence ofsteps in design might be:

  1. From an application, choose a desired ideal response, typically described in the frequency domain.
  2. From the available hardware and software, choose an allowed class of filters (e.g. a length-N FIR digital filter).
  3. From the application, set a measure or criterion of “goodness" for the response of an allowed filter compared to thedesired response.
  4. Develop a method to find (or directly generate) the best member of the allowed class of linear phase FIR filters as measured by the criterionof goodness.

This approach is often used iteratively. After the best filter is designed and evaluated, the desired response and/or the allowedclass and/or the measure of quality might be changed; then the filter would be redesigned and reevaluated.

The ideal response of a lowpass filter is given in [link] .

This figure contains three graphs. For all of these graphs the x axis is labeled Normalized Frequency, f and the y axis is labeled Ideaal Amplitude, A_d. The top graph is labeled a. Ideal Lowpass Filter Amplitude Response. The graph consist of a right angle formed by a line extending from the y axis at 1 and then a line extending from the x axis at .4. The area contained with in this right angle is labeled with two arrows pointing up from the word passband and then the area to the right of the line extending from the x axis was labeled the same way with the word stopband. The second graph is labeled b. Ideal Lowpass Filter with Transition Function and consist of a line extending from the y axis at 1 and then at about .3 there is a diagonal line extending down and to the right until it the line reached 0 on the x axis. Below the x axis parallel to the the line extending from the y axis are to arrows designation the area of the passband and then another two arrow under the diagonal line designation the area of the transitionband. To the right of where the diagonal line stops is a similar designation for the stopband. Another arrow points to the diagonal line labeling it transition function. The third graph is similar to the last graph except that the diagonal line and the area of the x axis that below it have been omitted as has the arrow labeling the diagonal line.
Ideal Amplitude Responses of Linear Phase FIR Filters

[link] a is the basic lowpass response that exactly passes frequencies from zero up to a certain frequency, then rejects(multiplies those frequency components by zero)the frequencies above that. [link] b introduces a “transitionband" between the pass and stopband to make the design easier and more efficient. [link] c introducess a transitionband which is not used in the approximation of the actual to the ideal responses. Each ofthese ideal responses (or other similar ones) will fit a particular application best.

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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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