Tunable Lowpass and Highpass Digital Filters. Tunable IIR Digital Filters. We have shown earlier that the 1st-order lowpass transfer function

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1 Tuable IIR Digital Filters Tuable Lowpass a We have escribe earlier two st-orer a two -orer IIR igital trasfer fuctios with tuable frequecy respose characteristics We shall show ow that these trasfer fuctios ca be realie easily usig allpass structures proviig iepeet tuig of the filter parameters We have show earlier that the st-orer lowpass trasfer fuctio α H LP α a the st-orer highpass trasfer fuctio α H HP α are oubly-complemetary pair Tuable Lowpass a Moreover, they ca be expresse as H ) LP [ A ] H ) HP [ A )] Tuable Lowpass a A realiatio of H LP a H HP base o the allpass-base ecompositio is show below H HP where α A α is a st-orer allpass trasfer fuctio The st-orer allpass filter ca be realie usig ay oe of the 4 sigle-multiplier allpass structures escribe earlier 4 Tuable Lowpass a Oe such realiatio is show below i which the -B cutoff frequecy of both lowpass a highpass filters ca be varie simultaeously by chagig the multiplier coefficiet α Tuable Lowpass a Figure below shows the composite magitue resposes of the two filters for two ifferet values of α 0.8 α 0.4 α 0.05 Magitue ω/π

2 Tuable Bapass a Bastop Digital Filters The -orer bapass trasfer fuctio α H BP β α) α a the -orer bastop trasfer fuctio α β H BS β α) α also form a oubly-complemetary pair Tuable Bapass a Bastop Digital Filters Thus, they ca be expresse i the form H ) BP [ A H ) BS [ A ] where α β α) A β α) α is a -orer allpass trasfer fuctio )] 7 8 Tuable Bapass a Bastop Digital Filters A realiatio of H BP a H BS base o the allpass-base ecompositio is show below Tuable Bapass a Bastop Digital Filters The fial structure is as show below 9 The -orer allpass filter is realie usig a cascae sigle-multiplier lattice structure 0 I the above structure, the multiplier β cotrols the ceter frequecy a the multiplier α cotrols the -B bawith Tuable Bapass a Bastop Digital Filters Figure below illustrates the parametric tuig property of the overall structure Magitue β 0.5 α 0.8 α 0.4 α Magitue β 0.8 β 0. Realiatio of a All-pole IIR Trasfer Fuctio Cosier the cascae lattice structure erive earlier for the realiatio of a allpass trasfer fuctio X ω/π ω/π Y

3 A typical lattice two-pair here is as show below W m ) W m S m ) S m Its iput-output relatios are give by Wm Wm km Sm S k W S ) m m m m From the iput-output relatios we erive the chai matrix escriptio of the two-pair: W m k m Wm S ) m k S m m The chai matrix escriptio of the cascae lattice structure is therefore X ) k k k W Y ) k S k k 4 From the above equatio we arrive at X { [ k k) kk] [ k kk k)] k } W ) ) W usig the relatio S W a the relatios k ", k k, The trasfer fuctio W / X is thus a all-pole fuctio with the same eomiator as that of the r-orer allpass fuctio A : W X Gray-Markel Metho A two-step metho to realie a Mth-orer arbitrary IIR trasfer fuctio H PM / DM Step : A itermeiate allpass trasfer M fuctio AM DM )/ DM is realie i the form of a cascae lattice structure 8 Step : A set of iepeet variables are summe with appropriate weights to yiel the esire umerator P M To illustrate the metho, cosier the realiatio of a r-orer trasfer fuctio P p0 H D p p p

4 I the first step, we form a r-orer allpass trasfer fuctio Y / X D )/ D A Realiatio of A has bee illustrate earlier resultig i the structure show below Objective: Sum the iepeet sigal variables Y, S, S, a S with weights { α i } as show below to realie the esire umerator P X Y 9 0 To this e, we first aalye the cascae lattice structure realiig a etermie the trasfer fuctios S / X, S / X, a / X S X Y We have alreay show S X D From the figure it follows that S k ) S " ) S a hece ) " S X D From Slie o. 0, we have S S S " ) S ) ) W S ) ) ) " W S ) From the last equatio we get " W ) S Substitutig W " ) S ) " S ) S i ) S W S we arrive at ) a S ) " ) ) " S ) S [ " ) ] S 4 4

5 From Thus, " ) " ) ) ) S S we observe 5 ) Thus, S X D ote:the umerator of S i / X is precisely the umerator of the allpass trasfer fuctio S A i i W 6 i We ow form Y o X Y S S S α X α X α X 4 X α Substitutig the expressios for the various trasfer fuctios i the above equatio we arrive at α ) Y o α ) α " ) X D α4 7 8 Comparig the umerator of Y o / X with the esire umerator P a equatig like powers of we obtai α α α" α4 p0 α α α p α α p α p Solvig the above equatios we arrive at α α α α p p α p α α " 4 p0 α α α 9 0 5

6 Example- Cosier P H D The correspoig itermeiate allpass trasfer fuctio is give by A D D ) The allpass trasfer fuctio A was realie earlier i the cascae lattice form as show below X Y I the figure, k., k 0 k " Other pertiet coefficiets are: 0.4, 0.8, 0., p, p 0.44, p 0.6, p 0.0, 0 0 Substitutig these coefficiets i α p α p α α p α α α " 4 p0 α α α α 0.0, α 0.5 α 0.765, α The fial realiatio is as show below k , k 0.708, k 0. 4 Tappe Cascae Lattice Realiatio Usig MATLAB Both the pole-ero a the all-pole IIR cascae lattice structures ca be evelope from their prescribe trasfer fuctios usig the M-file tflatc To this e, Program 6_4 ca be employe Tappe Cascae Lattice Realiatio Usig MATLAB The M-file latctf implemets the reverse process a ca be use to verify the structure evelope usig tflatc To this e, Program 8_5 ca be employe 5 6 6

7 7 A arbitrary th-orer FIR trasfer fuctio of the form H p ca be realie as a cascae lattice structure as show below 8 From figure, it follows that Xm X k m m Ym Y k X Y ) m m m m I matrix form the above equatios ca be writte as X ) m k m Xm Y ) m k Ym m where m,,..., 9 Deote X Y H m m, Gm X X ) m 0 0 The it follows from the iput-output relatios of the m-th two-pair that Hm Hm km Gm G k H G ) m m m m From the previous equatio we observe H k, G k where we have use the facts H X / X G Y0 / X0 X0 / X0 It follows from the above that G k ) H G ) is the mirror-image of H 40 ) From the iput-output relatios of the m-th two-pair we obtai for m : H ) ) H k G G k H G ) Sice H a G are st-orer polyomials, it follows from the above that H ) a G are -orer polyomials 4 4 Substitutig G H ) i the two previous equatios we get H H k H ) G k H H ow we ca write G k H H ) [ k ) )] H H H G ) is the mirror-image of H ) ) 7

8 I the geeral case, from the iput-output relatios of the m-th two-pair we obtai Hm H k m m Gm G k H G ) m m m m It ca be easily show by iuctio that G m H ), m,,..., m m, G m is the mirror-image of H m To evelop the sythesis algorithm, we express H m a G m i terms of H m a G m for m,,...,, arrivig at H G { ) )} H k G k ) { k H G k ) )} 4 44 Substitutig the expressios for a G H p ) H 0 p i the first equatio we get { k ) p k H p k p ) p ) k } If we choose k p, the H reuces to a FIR trasfer fuctio of orer a ca be writte i the form H ) p where p k p p, k Cotiuig the above recursio algorithm, all multiplier coefficiets of the cascae lattice structure ca be compute 47 Example- Cosier 4 H From the above, we observe k4 p Usig p 4 4 k p p, k 4 we etermie the coefficiets of H : p 0.79, p.79 p As a result, H 79 Thus, k p Usig p 0.79 p " k p, k we etermie the coefficiets of H : p".0, p.0 " 8

9 As a result, H From the above, we get k " p The fial recursio yiels the last multiplier coefficiet k " p / k) 0.5 The complete realiatio is show below Realiatio Usig MATLAB The M-file tflatc ca be use to compute the multiplier coefficiets of the FIR cascae lattice structure To this e Program 8_7 ca be employe The multiplier coefficiets ca also be etermie usig the M-file polyrc 49 k 0.5, k, k 0.79, k