Tunable IIR Digital Filters

Transcription

1 Tunable IIR Digital Filters We have described earlier two st-order and two nd-order IIR digital transfer functions with tunable frequency response characteristics We shall show now that these transfer functions can be realied easily using allpass structures providing independent tuning of the filter parameters Copyright 00, S. K. Mitra

2 Tunable Lowpass and Highpass Digital Filters We have shown earlier that the st-order lowpass transfer function α H LP α and the st-order highpass transfer function α H HP α are doubly-complementary pair Copyright 00, S. K. Mitra

3 Tunable Lowpass and Highpass Digital Filters Moreover, they can be expressed as H LP [ A ] H HP [ A where α A α is a st-order allpass transfer function ] Copyright 00, S. K. Mitra

4 Tunable Lowpass and Highpass Digital Filters A realiation of H LP and H HP based on the allpass-based decomposition is shown below H HP The st-order allpass filter can be realied using any one of the 4 single-multiplier allpass structures described earlier 4 Copyright 00, S. K. Mitra

5 Tunable Lowpass and Highpass Digital Filters One such realiation is shown below in which the -db cutoff frequency of both lowpass and highpass filters can be varied simultaneously by changing the multiplier coefficient α 5 Copyright 00, S. K. Mitra

6 Tunable Lowpass and Highpass Digital Filters Figure below shows the composite magnitude responses of the two filters for two different values of α 0.8 α 0.4 α 0.05 Magnitude ω/π Copyright 00, S. K. Mitra

7 Tunable Bandpass and Bandstop Digital Filters 7 The nd-order bandpass transfer function α H BP β α α and the nd-order bandstop transfer function α β H BS β α α also form a doubly-complementary pair Copyright 00, S. K. Mitra

8 Tunable Bandpass and Bandstop Digital Filters Thus, they can be expressed in the form where A H BP [ A H BS [ A ] α β α β α is a nd-order allpass transfer function ] α 8 Copyright 00, S. K. Mitra

9 Tunable Bandpass and Bandstop Digital Filters A realiation of H BP and H BS based on the allpass-based decomposition is shown below 9 The nd-order allpass filter is realied using a cascaded single-multiplier lattice structure Copyright 00, S. K. Mitra

10 Tunable Bandpass and Bandstop Digital Filters The final structure is as shown below In the above structure, the multiplier β controls the center frequency and the multiplier α controls the -db bandwidth 0 Copyright 00, S. K. Mitra

11 Tunable Bandpass and Bandstop Digital Filters Figure below illustrates the parametric tuning property of the overall structure 0.8 β 0.5 α 0.8 α 0.4 α β 0.8 β 0. Magnitude Magnitude ω/π ω/π Copyright 00, S. K. Mitra

12 IIR Tapped Cascaded Lattice Realiation of an All-pole IIR Transfer Function Consider the cascaded lattice structure derived earlier for the realiation of an allpass transfer function Copyright 00, S. K. Mitra

13 IIR Tapped Cascaded Lattice A typical lattice two-pair here is as shown below W m W m Its input-output relations are given by S S m S m Wm Wm m km m m km Sm W S Copyright 00, S. K. Mitra

14 Copyright 00, S. K. Mitra 4 IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice From the input-output relations we derive the chain matrix description of the two-pair: The chain matrix description of the cascaded lattice structure is therefore S W k k S W i i i i i i S W k k k k k k Y X

15 IIR Tapped Cascaded Lattice From the above equation we arrive at X { [ k k kk] [ k ] kk k k } W d d d W using the relation S W and the relations k d", k d ' k d, 5 Copyright 00, S. K. Mitra

16 IIR Tapped Cascaded Lattice The transfer function W / X is thus an all-pole function with the same denominator as that of the rd-order allpass function A : W X d d d 6 Copyright 00, S. K. Mitra

17 IIR Tapped Cascaded Lattice Gray-Markel Method A two-step method to realie an Mth-order arbitrary IIR transfer function H PM / DM Step : An intermediate allpass transfer M function AM DM / DM is realied in the form of a cascaded lattice structure 7 Copyright 00, S. K. Mitra

18 Copyright 00, S. K. Mitra 8 IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice Step : A set of independent variables are summed with appropriate weights to yield the desired numerator To illustrate the method, consider the realiation of a rd-order transfer function P M 0 d d d p p p p D P H

19 IIR Tapped Cascaded Lattice In the first step, we form a rd-order allpass transfer function A Y / X D / D Realiation of A has been illustrated earlier resulting in the structure shown below 9 Copyright 00, S. K. Mitra

20 IIR Tapped Cascaded Lattice Objective: Sum the independent signal variables Y, S, S, and S with weights { α i } as shown below to realie the desired numerator P 0 Copyright 00, S. K. Mitra

21 IIR Tapped Cascaded Lattice To this end, we first analye the cascaded lattice structure realiing and determine the transfer functions S / X, S / X, and / X S X Y We have already shown S X D Copyright 00, S. K. Mitra

22 Copyright 00, S. K. Mitra IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice From the figure it follows that and hence " S d S k S " D d X S

23 IIR Tapped Cascaded Lattice In a similar manner it can be shown that Thus, ' ' S d d S S X d ' ' d D Note: The numerator of S i / X is precisely the numerator of the allpass transfer function A S / W i i i Copyright 00, S. K. Mitra

24 Copyright 00, S. K. Mitra 4 IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice We now form X S X S X S X Y X Y o 4 α α α α

25 Copyright 00, S. K. Mitra 5 IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice Substituting the expressions for the various transfer functions in the above equation we arrive at D d d d X Y o α 4 α α α " ' ' d d d

26 IIR Tapped Cascaded Lattice Comparing the numerator of Y o / X with the desired numerator P and equating like powers of we obtain α d αd ' αd" α4 p0 α d αd ' α p αd α p α p 6 Copyright 00, S. K. Mitra

27 Copyright 00, S. K. Mitra 7 IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice Solving the above equations we arrive at p α " ' 0 4 d d d p α α α α ' d d p α α α d p α α

28 A IIR Tapped Cascaded Lattice Example - Consider P H D The corresponding intermediate allpass transfer function is given by D D Copyright 00, S. K. Mitra

29 IIR Tapped Cascaded Lattice The allpass transfer function A was realied earlier in the cascaded lattice form as shown below X 9 Y In the figure, k d., k d 0 ' k " d Copyright 00, S. K. Mitra

30 IIR Tapped Cascaded Lattice 0 Other pertinent coefficients are: d 0.4, d 0.8, 0., ' d d p, p 0.44, p 0.6, p 0 0 Substituting these coefficients in α p α p αd α p α d α ' α d ' " 4 p0 αd αd αd , Copyright 00, S. K. Mitra

31 IIR Tapped Cascaded Lattice α 0.0, α α.765, 0.5 α The final realiation is as shown below k , k 0.708, k 0. Copyright 00, S. K. Mitra

32 Tapped Cascaded Lattice Realiation Using MATLAB Both the pole-ero and the all-pole IIR cascaded lattice structures can be developed from their prescribed transfer functions using the M-file tflatc To this end, Program 6_4 can be employed Copyright 00, S. K. Mitra

33 Tapped Cascaded Lattice Realiation Using MATLAB The M-file latctf implements the reverse process and can be used to verify the structure developed using tflatc To this end, Program 6_5 can be employed Copyright 00, S. K. Mitra

34 FIR Cascaded Lattice An arbitrary Nth-order FIR transfer function of the form N H n N n pn can be realied as a cascaded lattice structure as shown below 4 Copyright 00, S. K. Mitra

35 FIR Cascaded Lattice 5 From figure, it follows that Xm X k m m Ym Y k X Y m m m m In matrix form the above equations can be written as X m k m Xm Y m k Ym m where m,,..., N Copyright 00, S. K. Mitra

36 Copyright 00, S. K. Mitra 6 FIR Cascaded Lattice FIR Cascaded Lattice Denote Then it follows from the input-output relations of the m-th two-pair that, 0 0 X Y G X X H m m m m G k H H m m m m G H k G m m m m

37 FIR Cascaded Lattice 7 From the previous equation we observe H k, G k where we have used the facts H X / X G Y0 / X 0 X 0 / X0 It follows from the above that G k H G is the mirror-image of H Copyright 00, S. K. Mitra

38 FIR Cascaded Lattice From the input-output relations of the m-th two-pair we obtain for m : H H k G G k H G Since H and G are st-order polynomials, it follows from the above that H and G are nd-order polynomials 8 Copyright 00, S. K. Mitra

39 Copyright 00, S. K. Mitra 9 FIR Cascaded Lattice FIR Cascaded Lattice Substituting in the two previous equations we get Now we can write is the mirror-image of H H k G ] [ H H H k H G H k H H H H k G G H

40 FIR Cascaded Lattice In the general case, from the input-output relations of the m-th two-pair we obtain Hm H k m m Gm G k H G m m m m It can be easily shown by induction that G m m Hm, m,,..., N, N G m is the mirror-image of H m 40 Copyright 00, S. K. Mitra

41 FIR Cascaded Lattice To develop the synthesis algorithm, we express H m and G m in terms of H m and G m for m N, N,...,, arriving at H G N { } H k G k N N N N N { k H G k N N N N } 4 Copyright 00, S. K. Mitra

42 Copyright 00, S. K. Mitra 4 FIR Cascaded Lattice FIR Cascaded Lattice Substituting the expressions for and in the first equation we get N n n n N p H 0 N n N n n N N N p H G { N n n n N n n N N N N p k p p k k H } N N N k p

43 FIR Cascaded Lattice 4 If we choose k N p N, then H N reduces to an FIR transfer function of order N and can be written in the form N H ' n N n pn where ' pn kn pn n pn, n N k N Continuing the above recursion algorithm, all multiplier coefficients of the cascaded lattice structure can be computed Copyright 00, S. K. Mitra

44 FIR Cascaded Lattice 44 Example - Consider H From the above, we observe k p 0.08 Using 4 p ' n p n k 4 4 n, n k p we determine the coefficients of H : p' 0.79, ' p.79 p'.79 Copyright 00, S. K. Mitra

45 FIR Cascaded Lattice 45 As a result, Thus, k Using H 79 ' p p ' 0.79 k p' n k " n n p, n we determine the coefficients of H : p".0, p.0 " Copyright 00, S. K. Mitra

46 FIR Cascaded Lattice As a result, H From the above, we get k " p The final recursion yields the last multiplier coefficient k " p / k 0.5 The complete realiation is shown below 46 k 0.5, k, k 0.79, k Copyright 00, S. K. Mitra

47 FIR Cascaded Lattice Realiation Using MATLAB The M-file tflatc can be used to compute the multiplier coefficients of the FIR cascaded lattice structure To this end Program 6_6 can be employed The multiplier coefficients can also be determined using the M-file polyrc 47 Copyright 00, S. K. Mitra