Part One Ef Cient Digital Filters Copyrighted Material-Books Pdf

Part One Ef cient Digital Filters COPYRIGHTED MATERIAL
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Lost Knowledge Refound,Sharpened FIR Filters,Matthew Donadio. Night Kitchen Interactive, What would you do in the following situation Let s say you are diagnosing. a DSP system problem in the field You have your trusty laptop with your. development system and an emulator You figure out that there was a problem. with the system specifications and a symmetric FIR filter in the software won t do. the job it needs reduced passband ripple or maybe more stopband attenuation. You then realize you don t have any filter design software on the laptop and the. customer is getting angry The answer is easy You can take the existing filter and. sharpen it Simply stated filter sharpening is a technique for creating a new filter. from an old one 1 3 While the technique is almost 30 years old it is not. generally known by DSP engineers nor is it mentioned in most DSP textbooks. 1 1 IMPROVING A DIGITAL FILTER, Before we look at filter sharpening let s consider the first solution that comes to. mind filtering the data twice with the existing filter If the original filter s transfer. function is H z then the new transfer function of the H z filter cascaded with. itself is H z 2 For example let s assume the original lowpass N tap FIR filter. designed using the Parks McClellan algorithm 4 has the following. characteristics,Number of coefficients N 17,Sample rate Fs 1. Streamlining Digital Signal Processing A Tricks of the Trade Guidebook Edited by Richard G Lyons. Copyright 2007 Institute of Electrical and Electronics Engineers. 4 Chapter 1 Lost Knowledge Refound Sharpened FIR Filters. 0 0 1 0 2 0 3 0 4 0 5,0 0 05 0 1 0 15 0 2 0 25, Figure 1 1 H z and H z 2 performance a full frequency response b passband response.
Passband width fpass 0 2,Passband deviation pass 0 05 0 42 dB peak ripple. Stopband frequency fstop 0 3,Stopband deviation stop 0 005 46 dB attenuation. Figure 1 1 a shows the performance of the H z and cascaded H z 2 filters. Everything looks okay The new filter has the same band edges and the stopband. attenuation is increased But what about the passband Let s zoom in and take a look. at Figure 1 1 b The squared filter H z 2 has larger deviations in the passband than. the original filter In general the squaring process will. 1 Approximately double the error response ripple in the passband. 2 Square the errors in the stopband i e double the attenuation in dB in the. 3 Leave the passband and stopband edges unchanged, 4 Approximately double the impulse response length of the original filter. 5 Maintain filter phase linearity,1 1 Improving a Digital Filter 5. 0 0 2 0 4 0 6 0 8 1,Stopband Passband, Figure 1 2 Various F H z functions operating on H z.
It is fairly easy to examine this operation to see the observed behavior if we. view the relationship between H z and H z 2 in a slightly unconventional way We. can think of filter squaring as a function F H z operating on the H z transfer func. tion We can then plot the output amplitude of this function H z 2 versus the ampli. tude of the input H z to visualize the amplitude change function. The plot for F H z H z is simple the output is the input so the result is the. straight line as shown in Figure 1 2 The function F H z H z 2 is a quadratic. curve When the H z input amplitude is near zero the H z 2 output amplitude is. closer to zero which means the stopband attenuation is increased with H z 2 When. the H z input amplitude is near one the H z 2 output band is approximately twice. as far away from one which means the passband ripple is increased. The squaring process improved the stopband but degraded the passband The. improvement was a result of the amplitude change function being horizontal at 0. So to improve H z in both the passband and stopband we want the F H z ampli. tude function to be horizontal at both H z 0 and H z 1 in other words have a. first derivative of zero at these points This results in the output amplitude changing. more slowly than the input amplitude as we move away from 0 and 1 which lowers. the ripple in these areas The simplest function that meets this will be a cubic of the. F x c0 c1 x c2 x 2 c3 x 3 1 1,Its derivative with respect to x is. F x c1 2c2 x 3c3 x 2 1 2, Specifying F x and F x for the two values of x 0 and x 1 allows us to solve 1 1. and 1 2 for the cn coefficients as,F x x 0 0 c0 0 1 3. 6 Chapter 1 Lost Knowledge Refound Sharpened FIR Filters. F x x 0 0 c1 0 1 4,F x x 1 1 c2 c3 1 1 5,F x x 1 0 2c2 3c3 0 1 6. Solving 1 5 and 1 6 simultaneously yields c2 3 and c3 2 giving us the. F x 3 x 2 2 x 3 3 2 x x 2 1 7, Stating this function as the sharpened filter Hs z in terms of H z we have.
Hs z 3 H z 2 2 H z 3 3 2 H z H z 2 1 8, The function Hs z is the dotted curve in Figure 1 2. 1 2 FIR FILTER SHARPENING, Hs z is called the sharpened version of H z If we have a function whose z. transform is H z then we can outline the filter sharpening procedure with the aid. of Figure 1 3 as the following,1 Filter the input signal x n once with H z. 2 Double the filter output sequence to obtain w n,3 Subtract w n from 3x n to obtain u n. 4 Filter u n twice by H z to obtain the output y n. Using the sharpening process results in the improved Hs z filter performance. shown in Figure 1 4 where we see the increased stopband attenuation and reduced. passband ripple beyond that afforded by the original H z filter. It s interesting to notice that Hs z has the same half power frequency 6 dB. point as H z This condition is not peculiar to the specific filter sharpening example. used here it s true for all Hs z s implemented as in Figure 1 3 This characteristic. useful if we re sharpening a halfband FIR filter makes sense if we substitute 0 5 for. H z in 1 8 yielding Hs z 0 5,3x n u n y n,Figure 1 3 Filter sharpening process.
1 3 Implementation Issues 7,0 0 1 0 2 0 3 0 4 0 5,0 0 05 0 1 0 15 0 2 0 25. Figure 1 4 H z and Hs z performance a full frequency response b passband response. 1 3 IMPLEMENTATION ISSUES, The filter sharpening procedure is very easy to perform and is applicable to a broad. class of FIR filters including lowpass bandpass and highpass FIR filters having. symmetrical coefficients and even order an odd number of taps Even multi. passband FIR filters under the restriction that all passband gains are equal can be. From an implementation standpoint to correctly implement the sharpening. process in Figure 1 3 we must delay the 3x n sequence by the group delay N 1 2. samples inherent in H z In other words we must time align 3x n and w n This. is analogous to the need to delay the real path in a practical Hilbert transformer. Because of this time alignment constraint filter sharpening is not applicable to filters. having nonconstant group delay such as minimum phase FIR filters or infinite. impulse response IIR filters In addition filter sharpening is inappropriate for. Hilbert transformer differentiating FIR filters and filters with shaped bands such as. 8 Chapter 1 Lost Knowledge Refound Sharpened FIR Filters. sinc compensated filters and raised cosine filters because cascading such filters cor. rupts their fundamental properties, If the original H z FIR filter has a nonunity passband gain the derivation of. 1 8 can be modified to account for a passband gain G leading to a sharpening. polynomial of,3 H z 2 2 H z 3 3 2 H z,Hs gain 1 z H z 2 1 9. G G2 G G 2, Notice when G 1 Hs gain 1 z in 1 9 is equal to our Hs z in 1 8.
1 4 CONCLUSIONS, We ve presented a simple method for transforming a FIR filter into one with better. passband and stopband characteristics while maintaining phase linearity While. filter sharpening may not be often used it does have its place in an engineer s. toolbox An optimal Parks McClellan designed filter will have a shorter impulse. response than a sharpened filter with the same passband and stopband ripple and. thus be more computationally efficient However filter sharpening can be used. whenever a given filter response cannot be modified such as software code that. makes use of an unchangeable filter subroutine The scenario we described was. hypothetical but all practicing engineers have been in situations in the field where. a problem needs to be solved without the full arsenal of normal design tools Filter. sharpening could be used when improved filtering is needed but insufficient ROM. space is available to store more filter coefficients or as a way to reduce ROM. requirements In addition in some hardware filter applications using application. specific integrated circuits ASICs it may be easier to add additional chips to a. filter design than it is to design a new ASIC,1 5 REFERENCES. 1 J Kaiser and R Hamming Sharpening the Response of a Symmetric Nonrecursive Filter by. Multiple Use of the Same Filter IEEE Trans Acoustics Speech Signal Proc vol ASSP 25. no 5 1977 pp 415 422, 2 R Hamming Digital Filters Prentice Hall Englewood Cliffs 1977 pp 112 117. 3 R Hamming Digital Filters 3rd ed Dover Mineola New York 1998 pp 140 145. 4 T Parks and J McClellan A Program for the Design of Linear Phase Finite Impulse Response. Digital Filters IEEE Trans Audio Electroacoust vol AU 20 August 1972 pp 195 199. EDITOR COMMENTS, When H z is a unity gain filter we can eliminate the multipliers shown in Figure. 1 3 The multiply by two operation can be implemented with an arithmetic left shift. by one binary bit The multiply by three operation can be implemented by adding. a binary signal sample to a shifted left by one bit version of itself. Editor Comments 9,Hs gain 1 z,Figure 1 5 Nonunity gain filter sharpening.
To further explain the significance of 1 9 the derivation of 1 8 was based. on the assumption that the original H z filter to be sharpened had a passband gain. of one If the original filter has a nonunity passband gain of G then 1 8 will not. provide proper sharpening in that case 1 9 must be used as shown in Figure 1 5. In that figure we ve included a Delay element whose length in samples is equal to. the group delay of H z needed for real time signal synchronization. It is important to realize that the 3 G and 2 G2 scaling factors in Figure 1 5. provide optimum filter sharpening However those scaling factors can be modified. to some extent if doing so simplifies the filter implementation For example if. 2 G2 1 8 for ease of implementation the practitioner should try using a scaling. factor of 2 in place of 1 8 because multiplication by 2 can be implemented by a. simple binary left shift by one bit Using a scaling factor of 2 will not be optimum. but it may well be acceptable depending on the characteristics of the filter to be. sharpened Software modeling will resolve this issue. As a historical aside filter sharpening is a process refined and expanded by the. accomplished R Hamming of Hamming window fame based on an idea originally. proposed by the great American mathematician John Tukey the inventor of the. 1 1 Improving a Digital Filter 5 It is fairly easy to examine this operation to see the observed behavior if we view the relationship between H z and H z 2 in a slightly unconventional way We

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