Chapter 18. The Bilinear Transformation And IIR Filter . - RecordingBlogs
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsChapter 18. The bilinear transformation and IIR filter transformationsIn chapter 17, we computed the Butterworth filter using the inverse Laplace transform, whichwas cumbersome. The bilinear transformation is the standard approach used to convert aLaplace transform transfer function into a Z transform transfer function.18.1. The bilinear transformationOne of the first results of the previous chapter was that the Laplace transform of a discretesignal evaluates to the Z transform with z esT. The bilinear transformation is anapproximation of this result.The bilinear transformation is the substitution ݖ ൌ ݁ ௦ ் ൎʹ ܶ ݏ ʹ െ ܶ ݏ (18.1)or alternatively ݏ ൎʹ ݖ െͳܶ ݖ ͳ(18.2)where T is the sampling time, or 1 / fs for the sampling frequency fs. Usually, when thebilinear substitution is performed, T is assumed to be to 1.The result above follows from the Taylor series expansion of the function esT/2. ݖ ൌ݁௦ ்ൌ݁݁௦ ்ଶି௦ ்ଶ ܶ ݏ ଵ ܶ ݏ ଶ ܶ ݏ ଷቀʹቁቀʹቁቀʹቁ ܶ ݏ ͳ ͳǨ ʹǨ Ǩ ڮ ͳ ʹൌൎ ܶ ݏ ܶ ݏ ଶ ܶ ݏ ଷ ܶ ݏ ଵͳെቀെ ʹ ቁቀെ ʹ ቁቀെ ʹ ቁʹ ڮ ͳ ͳǨʹǨ Ǩ(18.3)Setting T 1 for simplicity, we note that if Re(s) σ 0, then187
DSP for Audio Applications: Formulaeȁ ݖ ȁ ൌ ฬChapter 18. The bilinear transformation and IIR filtertransformationsʹ ߪ െ ݆߱Ͷ െ ሺߪ ଶ ߱ଶ ሻ െ Ͷ݆߱ʹ ݏ Ͷ െ ሺߪ ଶ ߱ଶ ሻቤ ͳ ฬൌฬฬൌቤͶ ሺߪ ଶ ߱ ଶ ሻ െ ͶߪͶ ሺߪ ଶ ߱ ଶ ሻ െ Ͷߪʹ െ ߪ ݆߱ʹെ ݏ (18.4)Thus, if we have the Laplace transform transfer function of a stable filter with roots of thedenominator in the left part of the s- complex plane, the transfer function that we will obtainwith the bilinear transformation would have roots that are inside the unit circle and the filter willstill be stable. The bilinear transformation preserves stability.18.2. Butterworth filters with the bilinear transformationIf we use the bilinear transformation on the Laplace transform transfer function of the secondorder Butterworth filter in equation 17.31, we will obtain the following. ߨ ܪ ሺ ݖ ሻ ൌ ሺ߱ ଶ ʹ ߱ ଶ ି ݖ ଵ ߱ ଶ ି ݖ ଶ ሻ Ȁ ሺͶ െ Ͷ ߱ ܿ ݏ ൬ ൰ ߱ ଶ ሺെͺ ʹ߱ ଶ ሻ ି ݖ ଵͶ ߨଶିଶ ൬Ͷ Ͷ ߱ ܿ ݏ ൬ ൰ ߱ ൰ ݖ ሻ Ͷ(18.5)Suppose again that the cutoff frequency is Zc 0.6. After scaling to obtain 1 at the beginning ofthe denominator, we getͲǤͲͷͻͶ ͷ ͲǤͳͳͺͺ Ͳ ି ݖ ଵ ͲǤͲͷͻͶ ͷ ି ݖ ଶ ܪ ሺ ݖ ሻ ൌ ͳ െ ͳǤʹͲͳͻͲͶ ି ݖ ଵ ͲǤͶ ͻ Ͷ ି ݖ ଶ(18.6)This filter is different than the impulse invariant filter described in the previous chapter, becausethe bilinear transformation is an approximation. Figure 112 compares the magnitude responseof the two filters in the pass band and in the transition band.188
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsFigure 112. Bilinear and impulse invariant Butterworth filtersdB010-1-2-3-4-5-6-7-8-9Frequency (Hz)110210Impulse invarianceBilinearAlthough in the discrete digital case the bilinear transformation filter performs better, in theanalog case this same filter warps the frequency spectrum.18.3. High pass and other Butterworth filtersWe could attempt the filter transformations that we used to derive FIR high pass, band pass, andband stop filters when designing IIR filters as well. We would have to choose an easy anappropriate all pass filter and that could be difficult, considering that we would have to matchthe phase response of the all pass filter to the phase response of the low pass filter. It could alsobe fruitless, given that IIR filters do not behave as nicely in the stop band as do FIR filters.The general prototype for an all pass filter was shown in equation 16.38. An all pass filter hasthe same, but inverted coefficients in the numerator and the denominator. If, for example, ourlow pass filter is the same as in the one in equation 18.6 then an easy all pass filter would be ܪ ሺ ݖ ሻ ൌͲǤͶ ͻ Ͷ െ ͳǤʹͲͳͻͲͶ ି ݖ ଵ ି ݖ ଶ ͳ െ ͳǤʹͲͳͻͲͶ ି ݖ ଵ ͲǤͶ ͻ Ͷ ି ݖ ଶ(18.7)The high pass filter would be the difference between the twoͲǤ ͺͲʹͲͺ െ ͳǤ ʹͲ Ͳ ି ݖ ଵ ͲǤͻͶͲͷͷ ି ݖ ଶ ܪ ሺ ݖ ሻ ൌ ͳ െ ͳǤʹͲͳͻͲͶ ି ݖ ଵ ͲǤͶ ͻ Ͷ ି ݖ ଶ(18.8)The magnitude response of this high pass filter, however, is not as well behaved as we mighthave hoped. It has a normalized magnitude response that is greater than 1 in the pass band andthat decreases slowly with higher frequencies, although that can be corrected to some extent bysimply applying some gain to the coefficients in the numerator of the transfer function.189
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsAn easier filter transformation would be to realize that if the following is the magnituderesponse of a low pass filterȁ ܪ ሺ݆Zሻȁ ൌ ܩ Z ଶ ටͳ ቀ ቁZ (18.9) (18.10)then a good magnitude response of a high pass filter isȁ ܪ ሺ݆Zሻȁ ൌ ܩ ଶ ටͳ ቀZ ቁZAs Z increases up to and past Zc, this magnitude response would increase and settle atapproximately 1.Given a low pass transfer function H(s / Zc), we can typically create a high pass transferfunction by substituting s / Zc with Zc / s. Substituting s / Zc with (s2 Zc2) / (B s)produces a band pass filter, where Zc is the midpoint of the pass band and B is the width ofthe band. Substituting s / Zc with B s / (s2 Zc2) produces a band stop filter.We can then define the Butterworth filter again as the filter with the transfer function ܪ ሺ ݏ ሻ ൌ ܩ ܤ ሺܵሻ(18.11)where n is the order of the filter and Bn(S) are the normalized Butterworth polynomials given by Ȁଶ ܤ ሺܵሻ ൌ ෑʹ݇ ݊ െ ͳ൬ܵ ଶ െ ʹ ܵ ܿ ݏ ൬ߨ൰ ͳ൰ ǡ ݊ ݁ ݊݁ݒ ʹ݊ ୀଵ ିଵଶʹ݇ ݊ െ ͳ൬ܵ ଶ െ ʹ ܵ ܿ ݏ ൬ ܤ ሺܵሻ ൌ ሺܵ ͳሻ ෑߨ൰ ͳ൰ ǡ ݊ ݀݀ ʹ݊ ୀଵ(18.12)Substituting S s / Zc produces the Butterworth low pass filter, where Zc is the cutofffrequency of the filter. Substituting S Zc / s produces the Butterworth high pass filter.Substituting S (s2 Zc2) / (B s) produces the Butterworth band pass filter, where Zc is the190
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsmidpoint of the pass band and B is the width of the band. Substituting S B s / (s2 Zc2)produces the Butterworth band stop filter. Since the transfer function and the Butterworthpolynomials above are written irrespective of the cutoff frequency Zc (i.e., they use the cutofffrequency Zc 1) and the various forms of the Butterworth filter require substitutions, thetransfer function and the polynomials are called normalized.Take the transfer function for the second order Butterworth high pass filter with G 1. ܪ ሺ ݏ ሻ ൌ ݏ ଶ ߨ߱ ଶ െ ʹ ߱ ݏ ܿ ݏ ቀ Ͷ ቁ ݏ ଶ (18.13)Use the bilinear transformation s 2 (z – 1) / (z 1) to rewrite this transfer function as follows. ܪ ሺ ݖ ሻ ൌ ሺͶ െ ͺ ି ݖ ଵ Ͷ ି ݖ ଶ ሻȀ ሺ൫Z ଶ െ ʹξʹZ Ͷ൯ ൫ʹZ ଶ െ ͺ൯ ି ݖ ଵ ൫Z ଶ ʹξʹZ Ͷ൯ ି ݖ ଶ ሻ (18.14)When Zc 0.6 for example, then the transfer function is ܪ ሺ ݖ ሻ ൌ ͲǤ Ͳ െ ͳǤ ʹͲͺ ି ݖ ଵ ͲǤ ͲͶ ି ݖ ଶ ͳ െ ͳǤʹͲͳͻ ି ݖ ଵ ͲǤͶ ͻ ି ݖ ଶ(18.15)Take the sampling frequency of fs 2000 Hz. The cutoff frequency translates to Zc 191 Hzand the transfer function above produces a filter with the magnitude response shown on figure113.Figure 113. Magnitude response of an example high pass Butterworth filterdB010-2-4-6-8-10-12-14-16-18-20Frequency (Hz)210410610810191
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsMagnitude response of an example high pass second order Butterworth filter with Zc 0.6.The following is the formula for the second order Butterworth band pass filter after the bilineartransformation. ܪ ሺ ݖ ሻ ൌ ܽ ܽଶ ି ݖ ଶ ܽସ ି ݖ ସ ܾ ܾଵ ି ݖ ଵ ܾଶ ି ݖ ଶ ܾଷ ି ݖ ଷ ܾସ ି ݖ ସܽ ൌ Ͷ ܤ ଶ ܽଶ ൌ െͺ ܤ ଶ ܽସ ൌ Ͷ ܤ ଶ ܾ ൌ ͳ ͺξʹ ܤ ͶሺʹZ ଶ ܤ ଶ ሻ ʹξʹ ܤ Z ଶ Z ସ (18.16)ܾଵ ൌ െ Ͷ െ ͳ ξʹ ܤ Ͷξʹ ܤ Z ଶ ͶZ ସ ܾଶ ൌ ͻ െ ͺሺʹZ ଶ ܤ ଶ ሻ Z ସ ܾଷ ൌ െ Ͷ ͳ ξʹ ܤ െ Ͷξʹ ܤ Z ଶ ͶZ ସ ܾସ ൌ ͳ െ ͺξʹ ܤ ͶሺʹZ ଶ ܤ ଶ ሻ െ ʹξʹ ܤ Z ଶ Z ସ For example, using Zc 0.6 and B 1 and scaling to obtain b0 1 produces the filter ܪ ሺ ݖ ሻ ൌ ͲǤͳͳ ʹ െ ͲǤʹʹ Ͷ ି ݖ ଶ ͲǤͳͳ ʹ ି ݖ ସ ͳ െ ʹǤ ͺͻ ି ݖ ଵ ʹǤ ͶͻͲ ି ݖ ଶ െ ͳǤʹͳ ି ݖ ଷ ͲǤ Ͳʹͳ ି ݖ ସ(18.17)Given a sampling frequency of, say, fs 2000 Hz, the midpoint frequency translates to Zc 191Hz and the band width translates to B 318 Hz. The magnitude response of this band passfilter is shown on figure 114.192
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsFigure 114. Magnitude response of an example band pass Butterworth filterdB010-2-4-6-8-10-12-14-16-18-20Frequency (Hz)210410610810Magnitude response of a second order band pass Butterworth filter with Zc 0.6 and B 1.18.4. Higher order Butterworth filtersWhen designing higher order Butterworth filters we have two choices. We can use the methodsdescribe above. Those require many computations and may be tedious. Alternatively, we candesign higher order filters by stacking lower order filters. Note, for example, that in a low passButterworth filter of order four, the roots of the denominator form two complex conjugatepairs. s1 and s4 are a pair and s2 and s3 are a pair.ହగ ݏ ଵ ൌ Z ݁ ଼ గ ݏ ଶ ൌ Z ݁ ଼ ݏ ଷ ൌ Z ݁ ݏ ସ ൌ Z ݁ ଽగ଼ଵଵగ଼ൌ Z ݁ି గ଼ (18.18)ହగൌ Z ݁ ି ଼ We can rewrite the transfer function of the fourth order low pass Butterworth filter as follows. ܪ ሺ ݏ ሻ ൌ ቆZ ଶZ ଶቇቆቇ ሺ ݏ െ ݏ ଵ ሻሺ ݏ െ ݏ ସ ሻ ሺ ݏ െ ݏ ଶ ሻሺ ݏ െ ݏ ଷ ሻ(18.19)This is a multiplication of two transfer functions of two filters of order two. These two filtersare not Butterworth filters – their roots are not the roots of the Butterworth filter of order two.Notwithstanding, we can then treat these two transfer functions as two separate filters.193
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsThe product of two Laplace transforms is the Laplace transform of the convolution of the tworespective functions. In this case specifically, the multiplication of two transfer functions is theconvolution of two impulse responses. This means that the filters are to be stacked one afterthe other. The output of the first filter is the input to the second filter.Figure 115 compares the magnitude responses of the second and fourth order low passButterworth filters, where the fourth order filter is implemented as two second order filtersstacked one after the other. The two filters themselves are computed after using bilineartransformations on the two separate multiples of the transfer function. Note that the two filterscross at around -3 dB, which, with Butterworth filters, happens at the cutoff frequency. Thenormalized cutoff frequency in this example is Zc 0.6 radians per second, which is equivalentto approximately 191 Hz, given the sampling frequency of fs 2000 Hz.Figure 115. Magnitude response of Butterworth filters of different orderdBFrequency (Hz)0-510110210310-10-15-20-25-302nd order4th orderComparison of the magnitude response of a fourth order and a second order Butterworth filterswith Zc 0.6 and fs 2000 Hz.18.5. Phase response of the Butterworth filterSince we know the transfer function H(z) of the Butterworth filter, we can compute its phaseresponse with equation 16.13.ߔሺ߱ሻ ൌ ܽ ʹ݊ܽݐ ቆܴ݁൫ ܪ ሺ ݖ ሻ൯ቇ ݉ܫ ൫ ܪ ሺ ݖ ሻ൯For example, if the transfer function of some second order low pass Butterworth filter is194(18.20)
DSP for Audio Applications: Formulae ܪ ሺ ݖ ሻ ൌChapter 18. The bilinear transformation and IIR filtertransformationsܽ െ ܽଵ ି ݖ ଵ ܽଶ ି ݖ ଶ ܽ ݖ ଶ െ ܽଵ ݖ ܽଶൌ ଶ ͳ െ ܾଵ ି ݖ ଵ ܾଶ ି ݖ ଶ ݖ െ ܾଵ ݖ ܾଶ(18.21)Then its phase response at z e-j Z cos(Z) – j sin(Z) is ܥ ܣ ܦ ܤ ଶଶߔሺ߱ሻ ൌ ܽ ʹ݊ܽݐ ቌ ܤ ܦ ቍ ܥ ܤ െ ܦ ܣ ܤ ଶ ܦ ଶ ܣ ൌ ܽ ܿ ݏ ሺʹ߱ሻ ܽଵ ܿ ݏ ሺ߱ሻ ܽଶ (18.22) ܤ ൌ െܽ ݊݅ݏ ሺʹ߱ሻ ܽଵ ݊݅ݏ ሺ߱ሻ ܥ ൌ ܿ ݏ ሺʹ߱ሻ ܾଵ ܿ ݏ ሺ߱ሻ ܾଶ ܦ ൌ െ ݊݅ݏ ሺʹ߱ሻ ܾଵ ݊݅ݏ ሺ߱ሻ The phase response of our example second order low pass Butterworth filter is shown on figure116 for the normalized Z between 0 and π. Unlike the phase response of symmetric FIR filters,this phase response is not linear.Figure 116. Phase response of a Butterworth 02.53.0Phase response of a second order Butterworth filter with Zc 0.6.18.6. Equivalent noise bandwidth of the Butterworth filterWhile chapter 14 discusses the term equivalent noise bandwidth only as it applies to windows,the term is very appropriate for filters.195
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformationsThe simplest (first order) low pass Butterworth filter, for example, is defined by the transferfunction ܪ ሺ ݏ ሻ ൌͳͳ ݏ ൌ߱ ͳ ͳ ݆߱ ߱ (18.23)The magnitude response of this filter isȁ ܪ ሺ݆߱ሻȁ ൌ߱ ඥ߱ ଶ ߱ ଶ (18.24)Since the magnitude response of the Butterworth filter is monotonically decreasing, Hmax H(0) 1. The equivalent noise bandwidth is ܤ ே ൌͳ ஶ ߱ ଶߨ ߱ ߱ න݀߱ ൌൎ ͳǤͷ ଶଶʹߨʹߨ ߱ ߱ʹ ʹߨ(18.25)Thus, the equivalent noise bandwidth is proportional to the normalized cutoff frequency. Notethat the magnitude response at the cutoff frequency isȁ ܪ ሺ݆߱ሻȁ ൌ߱ ඥ߱ ଶ ߱ ଶൌͳξʹൎ െ ݀ ܤ The ENBW relative to the -3 dB frequency is 1.57.With filters of higher orders, the equivalent noise bandwidth decreases.Order 1 ENBW 1.57Order 2 ENBW 1.11Order 3 ENBW 1.05Order 4 ENBW 1.03Order 5 ENBW 1.02Order 6 ENBW 1.01196(18.26)
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformations18.7. Warping of the frequency domain and the biquad transformationThe bilinear transformation filter performs better and is easier to derive, but warps themagnitude response of the filter. The properties of the resulting filter are evaluated at s -jω orz e-jω. However,(18.27) ܪ ሺ ݏ ሻ ൌ ܪ ሺെ݆߱ሻ whereas ఠ ఠ ఠ߱݁ ି ଶ ൬݁ ି ଶ െ ݁ ଶ ൰െʹ ݆ ݊݅ݏ ቀ ʹ ቁʹ ሺ ݖ െ ͳሻʹ ൫݁ ି ఠ െ ͳ൯ൌ ܪ ቌʹ ܪ ൬൰ ൌ ܪ ቆ ି ఠቇ ൌ ܪ ൮ʹ ఠ ఠ ఠ ൲߱ ቍ ݖ ͳ݁ ͳିିʹ ܿ ݏ ቀ(18.28)݁ ଶ ൬݁ ଶ ݁ ଶ ൰ʹቁ߱ൌ ܪ ቀെʹ ݆ ݊ܽݐ ቀ ቁቁ ʹThe discrete filter H(z), obtained after the bilinear transformation of H(s), will behave at thediscrete frequency ωd the same way the continuous filter H(s) behaves at the frequency ωa 2tan(ωd/2) (alternatively, ωd 2 arctan(ωa/2)).For example, the squared magnitude response of the second order low pass Butterworth filter,computed from H(s), isȁ ܪ ሺ݆߱ሻ ܪ ሺെ݆߱ሻȁ ൌ߱ ସ ߱ ସ ߱ ସ(18.29)The squared magnitude response of the filter after bilinear transformation can be computedfrom ܪ ሺ ݖ ሻ ൌ߱ ଶ ݖ ʹ߱ ଶ ߱ ଶ ି ݖ ଵ൫Ͷ ʹξʹ߱ ߱ ଶ ൯ ݖ ሺെͺ ʹ߱ ଶ ሻ ൫Ͷ െ ʹξʹ߱ ߱ ଶ ൯ ି ݖ ଵ (18.30)and is197
DSP for Audio Applications: FormulaeChapter 18. The bilinear transformation and IIR filtertransformations߱ ସ ሺ ሺ߱ሻ ͳሻଶ߱ ସ ሺ ሺ߱ሻ ͳሻଶ ͳ ሺ ሺ߱ሻ ͳሻଶ െ Ͷ ሺ߱ሻ߱ ସ߱ ସൌൌ ߱ ସሺ ሺ߱ሻ െ ͳሻଶସସ߱ ቀʹ ቀ ʹ ቁቁ߱ ͳ ሺ ሺ߱ሻ ͳሻଶห ܪ ൫݁ ఠ ൯ ܪ ൫݁ ି ఠ ൯ห ൌ(18.31)This warping of the frequency domain is small for small ω and increases as ω increases. At ωa 1, for example, ωd 0.927.When designing digital filters, this frequency warping can be remedied in one of two ways. First,a discrete filter with the cutoff frequency ωd can be designed with the bilinear transformation onthe continuous filter with the cutoff frequency ωa 2 tan(ωd/2). Alternatively, instead of thebilinear transformation, one can use the biquad transformation ݏ ൌ ͳ ݖ െͳǡ ݖܭ ͳ ܭ ൌ ቀ߱ ቁ ʹAn example filter designed with the biquad transformation is presented in chapter 19.198(18.32)
DSP for Audio Applications: Formulae Chapter 18. The bilinear transformation and IIR filter transformations 189 Figure 112. Bilinear and impulse invariant Butterworth filters Although in the discrete digital case the bilinear transformation filter performs better, in the analog case this same filter warps the frequency spectrum. 18.3.
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The Generalized Bilinear Model Bilinear model: Infer b RNb and c RNc from bilinear measurements y m b TΦ mc w m, m 1.M, where {Φ m} are known matrices and {w m} are independent noise samples. Generalized bilinear model: Infer band cfrom y m f bTΦ mc w m, m 1.M, where f(·)is possibly non-linear (e.g., quantization, loss of phase).
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