We have shown earlier that the 1st-order lowpass transfer function

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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 We have shown earlier that the st-order lowpass transfer function + H LP and the st-order highpass transfer function + H HP are doubly-complementary pair Moreover, they can be expressed as H ) LP [ + A ] H ) HP [ A where + A is a st-order allpass transfer function )] 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 4single-multiplier allpass structures described earlier 4 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 Figure below shows the composite magnitude responses of the two filters for two different values of Magnitude ω/π

2 Tunable Bandpass and The nd-order bandpass transfer function H BP β + ) + and the nd-order bandstop transfer function + β + H BS β + ) + also form a doubly-complementary pair Tunable Bandpass and Thus, they can be expressed in the form [ A )] H BP H ) BS [ + A ] where β + ) + A β + ) + is a nd-order allpass transfer function 7 8 Tunable Bandpass and A realiation of H BP and H BS based on the allpass-based decomposition is shown below Tunable Bandpass and The final structure is as shown below 9 The nd-order allpass filter is realied using a cascaded single-multiplier lattice structure 0 In the above structure, the multiplier β controls the center frequency and the multiplier controls the -db bandwidth Tunable Bandpass and Figure below illustrates the parametric tuning property of the overall structure Magnitude β Magnitude β 0.8 β 0. Realiation of an All-pole IIR Transfer Function Consider the cascaded lattice structure derived earlier for the realiation of an allpass transfer function ω/π ω/π

3 A typical lattice two -pair here is as shown below W m+ ) W m S m+ ) S m Its input-output relations are given by Wm Wm + km Sm S k W S ) m+ m m + m 4 From the input-output relations we derive the chain matrix description of the two -pair: W + ) i ki Wi S + ) i k S i i The chain matrix description of the cascaded lattice structure is therefore X ) k k k W Y ) k S k k 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, The transfer function W / X is thus an all-pole function with the same denominator as that of the rd-orderallpass function A : W X + d + d + d Gray-Markel Method A two -step method to realie an Mth-order arbitrary IIR transfer function H P / D M Step : An intermediate allpass transfer M function AM DM ) / DM is realied in the form of a cascaded lattice structure M 8 Step : A set of independent variables are summed with appropriate weights to yield the desired numerator P M To illustrate the method, consider the realiation of a rd-order transfer function P p0 + H D + d p + + d p + + d p

4 In the first step, we form a rd-order allpass transfer function Y / X D )/ D A Realiation of A has been illustrated earlier resulting in the structure shown below Objective: Sum the independent signal variables Y, S, S, and S with weights { i } as shown below to realie the desired numerator P 9 0 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 From the figure it follows that S k + ) S d" + ) S and hence ) " S d + X D In a similar manner it can be shown that + S ) d + d ) S Thus, ) S d + d + X D ote: The numerator of S i / X is precisely the numerator of the allpass transfer function A S / W i i i 4 We now form Y o Y S S S X + X + X + X 4 X 4

5 Substituting the expressions for the various transfer functions in the above equation we arrive at d + d + d + ) " Y o + d + d + ) + d + ) + X D 4 Comparing the numerator of Y o / X with the desired numerator P and equating like powers of we obtain d + d " + d + 4 p0 d + d + p d + p p 5 6 Solving the above equations we arrive at p p d p d d " 4 p0 d d d Example - Consider P H D The corresponding intermediate allpass transfer function is given by A D D ) The allpass transfer function A was realied earlier in the cascaded lattice form as shown below X Y In the figure, k d 0., k d k " d Other pertinent coefficients are: d 0.4, d 0.8, d 0., d p, p 0.44, p 0.6, p 0.0, 0 0 Substituting these coefficients in p p d p d d " 4 p0 d d d 5

6 Tapped Cascaded Lattice Realiation Using MATLAB 0.0, , The final realiation is as shown below 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 k , k 0.708, k 0. 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 An arbitrary th-order FIR transfer function of the form n H + n pn can be realied as a cascaded lattice structure as shown below 4 5 From figure, it follows that X X k m m + m Ym ) Ym km Xm + Ym In matrix form the above equations can be written as X ) m km Xm Y ) m k Ym m where m,,..., 6 Denote X H m m, Gm X Ym X ) 0 0 Then it follows from the input-output relations of the m-th two-pair that Hm Hm + km Gm G k H G ) m m m + m 6

7 7 From the previous equation we observe H + k, G k + where we have used 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 8 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 9 Substituting G ) H ) in the two previous equations we get H H + k H ) G k H + H ow we can write G k H + H ) [ k ) )] H + H H G ) is the mirror-image of H ) ) 40 In the general case, from the input-output relations of the m-thtwo-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 m Gm Hm ), m,,...,, G m is the mirror-image of H m 4 To develop the synthesis algorithm, we express H and G m m ) in terms of H m and G m for m,,...,, arriving at H G { H k G } k ) { k H G + k ) )} Substituting the expressions for n H + n pn and n G H ) n 0 pn + in the first equation we get + { k ) p n pn knp k H ) n n + p ) k } 4 7

8 4 If we choose k p, then H reduces to an FIR transfer function of order and can be written in the form + n H n pn where pn k p n pn, n k Continuing the above recursion algorithm, all multiplier coefficients of the cascaded lattice structure can be computed 44 Example - Consider 4 H From the above, we observe k4 p Using pn 4 4 k p n p, n n k4 we determine the coefficients of H : p 0.79, p.79 p As a result, H Thus, k p 0.79 Using p " n k p n p, n n k we determine the coefficients of H : p".0, p.0 " 46 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 k 0.5, k, k 0.79, k Realiation Using MATLAB The M-filetflatc 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 8