FIR Filter with Variable Fractional Delay and ... - Semantic Scholar

FIR Filter With Variable Fractional Delay and Phase Shift: Efficient Realization and Design Using Reweighted 𝑙1-Norm Minimization Håkan Johansson and Amir Eghbali Division of Electronics Systems, Department of Electrical Engineering, Linköping University, Sweden Email: [email protected], [email protected]

Abstract— This paper introduces a finite-length impulse response (FIR) digital filter having both a variable fractional delay (VFD) and a variable phase shift (VPS). The realization is reconfigurable online without redesign and without transients. It can be viewed as a generalization of the VFD Farrow structure that offers a VPS in addition to the regular VFD. The overall filter is composed of a number of fixed subfilters and a few variable multipliers whose values are determined by the desired FD and PS values. It is designed offline in an iterative manner, utilizing reweighted 𝑙1 -norm minimization. This design procedure generates fixed subfilters with many zero-valued coefficients, typically located in the impulse response tails.

I. I NTRODUCTION Variable (reconfigurable) digital signal processing algorithms are gaining an increasing interest with the increasing demand for reconfigurable systems. Variable building blocks are required in, e.g., emerging communication systems supporting several different standards and operation modes [1], [2]. This paper deals with finite-length impulse response (FIR) digital filters having both a variable fractional delay (VFD), say 𝑑, and a variable phase shift (VPS), say 𝛼. Such filters are needed in, e.g., reconstruction of sub-Nyquist sampled sparse multi-band signals [3]. The objective of this paper is to derive a realization that is efficient and reconfigurable (adaptable) online without redesign. Furthermore, it should not exhibit transients when it is adapted, i.e., when 𝑑 or 𝛼 is altered (or both). For a VFD filter alone, the well known Farrow structure, or modifications of it, can be used [4]–[8]. The Farrow structure makes use of 𝐿 + 1 fixed subfilters weighted with 𝑑𝑘 , 𝑘 = 1, 2, . . . , 𝐿, and is adapted by simply changing the value of 𝑑. The VPS filter can be realized similarly utilizing a fixed Hilbert transform filter and variable multipliers for approximating its real and imaginary parts. A simple way to achieve the desired variable FD and PS (VFDPS) filter is to cascade the VFD filter with the VPS filter. This offers an efficient online-adaptable filter, but it will have transients whenever 𝑑 or 𝛼 is changed. As an alternative, to avoid transients, this paper introduces an FIR filter whose realization can be viewed as a generalization of the VFD Farrow structure that offers a VPS in addition to the regular VFD. In terms of arithmetic complexity, the proposed structure is more expensive than that of the cascade structure. The main advantage is that it has no transients and

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it is therefore preferred in applications where 𝑑 and/or 𝛼 changes frequently and where transients cannot be allowed. In addition, the proposed structure has a shorter integer delay and thereby requires fewer delay elements (less memory) in the implementation. The proposed filter is composed of a number of fixed subfilters and a few variable multipliers whose values are determined by the desired FD and PS values. An iterative design procedure is proposed, which utilizes reweighted 𝑙1 norm minimization. This procedure generates fixed subfilters with many zero-valued coefficients, typically located in the tails of the impulse responses. This corresponds to subfilters of different effective orders. Following this introduction, Section II states the problem under consideration and gives some prerequisites. Section III introduces the proposed realization. Section IV considers the design whereas Section V provides a design example. Finally, Section VI concludes the paper. II. P ROBLEM F ORMULATION AND P REREQUISITES The desired frequency function 𝐷𝐻 (𝑗𝜔, 𝑑, 𝛼) is 𝐷𝐻 (𝑗𝜔, 𝑑, 𝛼) = 𝑒−𝑗𝜔(𝑁𝐻 /2) 𝑒−𝑗(𝜔𝑑+𝛼×sgn(𝜔)) (1) ∪ for 𝜔 ∈ [−𝜔2 , −𝜔1 ] [𝜔1 , 𝜔2 ], 𝑑 ∈ [−1/2, 1/2], and 𝛼 ∈ [−𝜋/2, 𝜋/2]. For each pair of values 𝑑 and 𝛼, 𝐷𝐻 (𝑗𝜔, 𝑑, 𝛼) corresponds to an allpass filter with an FD of 𝑑 and a PS of 𝛼. Furthermore, the filter has an additional fixed delay of 𝑁𝐻 /2, where 𝑁𝐻 denotes the filter order. The problem is now to form an FIR filter transfer function 𝐻(𝑧, 𝑑, 𝛼) so that: 1) the corresponding filter frequency response 𝐻(𝑒𝑗𝜔 , 𝑑, 𝛼) approximates 𝐷𝐻 (𝑗𝜔, 𝑑, 𝛼), 2) the corresponding realization (structure) is adaptable online without redesign, and 3) the corresponding realization does not exhibit transients when it is adapted, i.e., when 𝑑 or 𝛼 is altered (or both). Before proceeding, it is noted that we assume the filter to be real which means that 𝐻(𝑒𝑗𝜔 , 𝑑, 𝛼) is conjugate symmetric. This implies that it suffices to consider the frequency interval 𝜔 ∈ [𝜔1 , 𝜔2 ] in the design and in the sequel. A straightforward way to achieve the first two goals stated above is to realize 𝐻(𝑧, 𝑑, 𝛼) of order 𝑁𝐻 = 𝑁𝐹 + 𝑁𝐺 in a cascade form as 𝐻(𝑧, 𝑑, 𝛼) = 𝐹 (𝑧, 𝑑)𝐺(𝑧, 𝛼)

(2)

where 𝐹 (𝑧, 𝑑) =

𝐿 ∑

d

𝑑𝑘 𝐹𝑘 (𝑧)

FL(z)

𝐺(𝑧, 𝛼) = 𝑧 −𝑁𝐺 /2 cos(𝛼) + 𝐸(𝑧) sin(𝛼).

𝐷𝐹 (𝑗𝜔, 𝑑) = 𝑒−𝑗𝜔(𝑁𝐹 /2+𝑑) , 𝜔 ∈ [𝜔1 , 𝜔2 ]

(5)

(6)

for 𝛼 ∈ [−𝜋/2, 𝜋/2]. The subfilters 𝐹𝑘 (𝑧) and 𝐸(𝑧) are fixed, with 𝐹𝑘 (𝑒𝑗𝜔 ) approximating the weighted differentiators 𝑒−𝑗𝜔𝑁𝐹 /2 (−𝑗𝜔)𝑘 /𝑘!, and 𝐸(𝑒𝑗𝜔 ) approximating −𝑒−𝑗𝜔𝑁𝐺 /2 𝑗sgn(𝜔). It is possible to let 𝐹𝑘 (𝑧) and 𝐸(𝑧) be linear-phase FIR filters which roughly halves the multiplication complexity due to their coefficient symmetries. Since 𝐹𝑘 (𝑧) and 𝐸(𝑧) are fixed, the overall filter is online-adaptable without redesign as it suffices to alter 𝑑 and/or 𝛼 when a new mode is desired. However, a drawback of using this cascade realization is that transients will distort the signal whenever 𝑑 or 𝛼 is altered (depending on whether 𝐹𝑘 (𝑧) or 𝐸(𝑧) is placed first in the cascade realization). This is because either 𝑑 or 𝛼 will be realized and altered in front of the next subfilter, i.e., in front of delay elements (memory). To avoid transients, all variable parameters must be located after the delay elements, thus never at the input of a subfilter. In the next section, we introduce such a structure. III. P ROPOSED T RANSFER F UNCTION AND S TRUCTURE To avoid transients, we expand the product of 𝐹 (𝑧, 𝑑) and 𝐺(𝑧, 𝛼) above so that all variable parameters can be placed at the output of the resulting subfilters. We then obtain the transfer function form 𝐿 ∑ 𝑘=0

𝑑𝑘 𝐹𝑘 (𝑧) + sin(𝛼)

𝐿 ∑

F1(z)

F0(z) y(n)

x(n)

for 𝑑 ∈ [−1/2, 1/2]. Further, 𝐺(𝑧, 𝛼) is a VPS filter with 𝐺(𝑒𝑗𝜔 , 𝛼) approximating 𝐷𝐺 (𝑗𝜔, 𝛼) given by 𝐷𝐺 (𝑗𝜔, 𝛼) = 𝑒−𝑗(𝜔𝑁𝐺 /2+𝛼) , 𝜔 ∈ [𝜔1 , 𝜔2 ]

F2(z)

(4)

Here, the filter 𝐹 (𝑧, 𝑑) is a regular Farrow-structure based VFD filter with 𝐹 (𝑒𝑗𝜔 , 𝑑) approximating the filter response 𝐷𝐹 (𝑗𝜔, 𝑑) given by

𝐻(𝑧, 𝑑, 𝛼) = cos(𝛼)

cos(a)

d

(3)

𝑘=0

and

d

𝑑𝑘 𝐺𝑘 (𝑧), (7)

𝑘=0

with 𝑁𝐻 = 𝑁𝐹 = 𝑁𝐺 as the filter is now in parallel form instead of cascade form. In this short paper, we assume that the orders of the subfilters are even. Odd-order subfilters can alternatively be used, but this option will not be treated here. With even-order filters, 𝐹𝑘 (𝑧) are linear-phase filters of Type I (Type III) for even (odd) values of 𝑘, thus 𝑓𝑘 (𝑛) = (−1)𝑘 𝑓𝑘 (𝑁𝐻 − 𝑛). For 𝐺𝑘 (𝑧), it is the other way around, i.e., they are linear-phase filters of Type III (Type I) for even (odd) values of 𝑘, thus 𝑔𝑘 (𝑛) = −(−1)𝑘 𝑔𝑘 (𝑁𝐻 − 𝑛). This is explained as follows. First, it is well known that 𝐹𝑘 (𝑧) are Type I (Type III) filters for even (odd) values of 𝑘 in the regular Farrow filter transfer function 𝐹 (𝑧, 𝑑) in (3) when even-order subfilters are used. Second, the first filter component of 𝐺(𝑧, 𝛼) in (4) is a pure integer delay which

82

GL(z)

G2(z) d

G1(z) d

Fig. 1.

G0(z) d

sin(a)

Proposed realization.

corresponds to a Type I filter. The second component, 𝐸(𝑧), which is used for approximating −𝑒−𝑗𝜔𝑁𝐺 /2 𝑗sgn(𝜔), is a Type III filter when it is of even order. Based on these facts, the product 𝐹 (𝑧, 𝑑)𝐺(𝑧, 𝛼) then results in the proposed transfer function in (7) with the subfilter types stated above. The corresponding proposed structure is seen in Fig. 1. As all variable multipliers are placed after the subfilter outputs, there are no transients when altering 𝑑 and 𝛼. It is noted that cos(𝛼) and sin(𝛼) need to be computed whenever 𝛼 changes. These computations can, e.g., be efficiently done via low-order polynomials in 𝛼. In the filter design considered in this paper (see below), we assume that we have a sufficiently accurate representation of cos(𝛼) and sin(𝛼), as well as 𝑑. This is a normally adopted approach when finding the optimum subfilters in polynomial impulse response based filters. However, in a full-length paper under way, we also address the additional problem of finding the optimum polynomial representations of cos(𝛼) and sin(𝛼) for a given specification. IV. F ILTER D ESIGN Our objective is to minimize the number of nonzero impulse response values 𝑓𝑘 (𝑛) and 𝑔𝑘 (𝑛) subject to the constraint that a filter specification is satisfied. In this way, the sparsity for the given specification is maximized and thus the number of multiplications required in the implementation is minimized. This corresponds to an integer programming problem where the 𝑙0 -norm is minimized. This becomes impractical to solve for larger problems. To alleviate this problem, we adopt 𝑙1 norm minimization which is known to generate sparse impulse responses whenever this is feasible [7], [9]. Although 𝑙1 norm minimization may not yield the global optimum, it gives good suboptimal results and it can be used for much larger problems. More precisely, we will use iteratively reweighted 𝑙1 -norm minimization [10], [11], which tends to give more sparse solutions than the unweighted 𝑙1 -norm minimization utilized in [7], [9]. The main difference from [11] is that we consider variable FIR filters, not regular fixed FIR filters, and that we ensure that a specification is satisfied, namely that the complex error modulus is below a prescribed value. In [11], in contrast, the weighted 𝑙1 -norm and an approximation error were minimized simultaneously. We design the overall filter 𝐻(𝑧, 𝑑, 𝛼) by solving the following approximation problem: Approximation Problem: Find the unknowns 𝑓𝑘 (𝑛) = (−1)𝑘 𝑓𝑘 (𝑁𝐻 − 𝑛) for 𝑛 = 0, 1, . . . , 𝑁𝐻 /2 and 𝑘 =

1, 2, . . . , 𝐿1 , 𝑔𝑘 (𝑛) = −(−1)𝑘 𝑔𝑘 (𝑁𝐻 − 𝑛) for 𝑛 = 0, 1, . . . , 𝑁𝐻 /2 and 𝑘 = 0, 1, . . . , 𝐿, to minimize the weighted 𝑙1 -norm 𝑓=

𝑃𝑘 𝐿 ∑ ∑ 𝑘=1 𝑛=0

𝑤1𝑘 (𝑛)∣𝑓𝑘 (𝑛)∣ +

𝑄𝑘 𝐿 ∑ ∑

𝑤2𝑘 (𝑛)∣𝑔𝑘 (𝑛)∣

(8)

𝑘=0 𝑛=0

with 𝑤1𝑘 (𝑛), 𝑤1𝑘 (𝑛) > 0, subject to ∣𝐸(𝑗𝜔, 𝑑, 𝛼)∣ ≤ 𝛿,

(9)

𝐸(𝑗𝜔, 𝑑, 𝛼) = 𝐻(𝑒𝑗𝜔 , 𝑑, 𝛼) − 𝐷𝐻 (𝑗𝜔, 𝑑, 𝛼),

(10)

where

for 𝜔 ∈ [𝜔1 , 𝜔2 ], 𝑑 ∈ [−1/2, 1/2], and 𝛼 ∈ [−𝜋/2, 𝜋/2]2 . Further, 𝐻(𝑒𝑗𝜔 , 𝑑, 𝛼) is given by (7) with 𝑧 = 𝑒𝑗𝜔 whereas 𝐷𝐻 (𝑗𝜔, 𝑑, 𝛼) is the desired function in (1). Finally, 𝑃𝑘 = 𝑁𝐻 /2 for even 𝑘 and 𝑃𝑘 = 𝑁𝐻 /2 − 1 for odd 𝑘 due to the zero-valued center taps of Type III filters. For the same reasons, 𝑄𝑘 = 𝑁𝐻 /2 for odd 𝑘 and 𝑄𝑘 = 𝑁𝐻 /2 − 1 for even 𝑘. To obtain the best feasible solution, we utilize the the weighted 𝑙1 -norm minimization above in the following procedure where 𝑀𝑞 denotes the total number of non-zero impulse response values of 𝑓𝑘,𝑞 (𝑛) and 𝑔𝑘,𝑞 (𝑛) obtained in iteration 𝑞. 1) For a given 𝐿, find the required order 𝑁𝐻 = 𝑁𝐺 = 𝑁𝐹 by designing 𝐻(𝑧, 𝑑, 𝛼) so that 𝐸(𝑗𝜔, 𝑑, 𝛼) in (10) satisfies ∣𝐸(𝑗𝜔, 𝑑, 𝛼)∣ ≤ 𝜀𝛿 with 𝜀 ≈ 0.53 . Denote the corresponding impulse responses as 𝑓𝑘,0 (𝑛) and 𝑔𝑘,0 (𝑛). 2) Set 𝑞 = 1. 3) Minimize 𝑓 in (8) with 𝑤1𝑘 (𝑛) = 1/∣𝑓𝑘,𝑞−1 (𝑛)∣ and 𝑤2𝑘 (𝑛) = 1/∣𝑔𝑘,𝑞−1 (𝑛)∣ subject to (9). This gives 𝑓𝑘,𝑞 (𝑛) and 𝑔𝑘,𝑞 (𝑛). 4) If 𝑀𝑞 < 𝑀𝑞−1 , set 𝑞 = 𝑞 + 1 and go to Step 3. Otherwise, go to Step 5 5) Set 𝑓𝑘 (𝑛) = 𝑓𝑘,𝑞 (𝑛) and 𝑔𝑘 (𝑛) = 𝑔𝑘,𝑞 (𝑛). In Step 1, the order 𝑁𝐻 can be found through a few designs of a VFDPS filter with different orders around an estimated ˆ𝐻 given by minimum order 𝑁 2 ˆ𝐻 = − 2𝜋 log10 (10𝛿 ) , Δ𝜔 = min{𝜔1 , 𝜋 − 𝜔2 }. (11) 𝑁 3 Δ𝜔 This formula emanates from order estimations of regular frequency-selective FIR filters with a passband and stopband ripple of 𝛿 and a transition band of Δ𝜔 [12]. The same formula is applicable for the Hilbert transform filter 𝐸(𝑧), and thus the subfilters 𝐺𝑘 (𝑧), because it has such a transition band and such ripples (albeit not being frequency selective). It is also applicable for the subfilters 𝐹𝑘 (𝑧), which realize the VFD filter part. This can be shown based on the fact that the polyphase components of a regular frequency selective 𝑀 th-band filter approximate FD filters with different FDs.

filter 𝐹0 (𝑧) is a pure delay and thus 𝑓0 (𝑛) = 𝛿(𝑛 − 𝑁𝐻 /2). the filter design, it suffices to consider only half of the interval of either 𝑑 or 𝛼, due to linear-phase subfilters and conjugate symmetry. 3 The VFDPS filter in Step 1 can be designed in essentially the same way as for the regular Farrow structure (including the determination of 𝐿), but with two sets of subfilters and two variable parameters. 1 The 2 In

83

Steps 2–4 correspond to the iterative reweighted 𝑙1 -norm design discussed earlier. In each new iteration, the weights are simply the reciprocals of the corresponding impulse response values from the previous iteration. In practice, the weights are set to a maximum value, whenever the impulse response values are small, to avoid division with small values. Typically, only a few iterations are needed in practice (three in the example to be presented in Section IV). The above optimization problem is convex and therefore has a unique global optimum. The initial solution is therefore not crucial. However, the speed of the optimization can be improved by first designing 𝐹𝑘 (𝑒𝑗𝜔 ) to approximate (−𝑗𝜔)𝑘 /𝑘! and 𝐺𝑘 (𝑒𝑗𝜔 ) to approximate 𝑗(−𝑗𝜔)𝑘 /𝑘!, and then using the corresponding 𝑓𝑘 (𝑛) and 𝑔𝑘 (𝑛) as the initial solution. The above problems can be solved using any regular solver for convex problems. In the example of this paper, we have used the general-purpose routine fminimax4 in MATLAB together with the real-rotation theorem [13], which states that minimizing ∣𝑓 ∣ is equivalent to minimizing ℜ{𝑓 𝑒𝑗Θ }, ∀Θ ∈ [0, 2𝜋]. The optimization problem is then solved with 𝜔, 𝑑, 𝛼, and Θ discretized to dense enough grids. V. D ESIGN E XAMPLE The following specification is considered: 𝜔1 = 0.2𝜋, 𝜔2 = 0.8𝜋, and 𝛿 = 0.01. With the structure in Fig. 1, designed as outlined in Section IV, the specification is met with 𝐿 = 4 and 𝑁𝐻 = 𝑁𝐺 = 𝑁𝐹 = 14. Figure 2 plots the complex error modulus for a set of pairs 𝑑 and 𝛼. Tables I and II give the impulse response values of the subfilters 𝐹𝑘 (𝑧) and 𝐺𝑘 (𝑧), respectively. As seen, only 34 impulse response values are nonzero, out of the total number of 67 values used in the design. It is also seen that the zeros are typically located in the tails of the impulse responses, which corresponds to subfilters of different effective orders. In addition, the subfilters 𝐺2 (𝑧) and 𝐺4 (𝑧) have an extra zero-valued impulse response value, whereas every second impulse response value of the subfilter 𝐺0 (𝑧) is zero. The latter feature is due to the symmetric specification (𝜔1 + 𝜔2 = 𝜋). The overall filter requires 34 fixed multipliers, 54 fixed adders, and 14 delay elements provided they are shared between the subfilters. The additional number of general multiplications and additions required for the variable parts are 10 and 9, respectively. As a reference, we have also designed a cascade realization according to (2). It is designed in a similar manner as the proposed filter, and the specification is met by cascading 𝐹 (𝑧, 𝑑) in (3) with 𝐺(𝑧, 𝛼) in (4), both with subfilters of order 14. The overall cascaded filter requires 22 fixed multipliers, 37 fixed adders, and 28 delay elements provided they are shared between all the subfilters. The additional number of general multiplications and additions, required for the variable parts, are 6 and 5, respectively. Hence, in terms of multiplier/adder complexity, the proposed structure is more expensive. As to the delay and the number of delay elements, which translates 4 The function fminimax is an optimization program for nonconvex problems, but it works well also for convex problems.

TABLE I I MPULSE RESPONSE VALUES OF THE SUBFILTERS 𝐹𝑘 (𝑧). 𝑘

0

1

2

3

4

𝑓𝑘 (0) = (−1)𝑘 𝑓𝑘 (14) 𝑓𝑘 (1) = (−1)𝑘 𝑓𝑘 (13) 𝑓𝑘 (2) = (−1)𝑘 𝑓𝑘 (12) 𝑓𝑘 (3) = (−1)𝑘 𝑓𝑘 (11) 𝑓𝑘 (4) = (−1)𝑘 𝑓𝑘 (10) 𝑓𝑘 (5) = (−1)𝑘 𝑓𝑘 (9) 𝑓𝑘 (6) = (−1)𝑘 𝑓𝑘 (8) 𝑓𝑘 (7) = (−1)𝑘 𝑓𝑘 (7)

0 0 0 0 0 0 0 1

-0.018195203769563 0.022051137613484 -0.065667080603246 0.101423864605232 -0.243253068919431 0.424110646826419 -0.959395164496708 0

0 0 0 0 0.020832736911578 -0.131154013539234 0.925615594926796 -1.465144776484994

0 0 0 -0.043334394615083 0.294552964377848 -0.509375374193068 0.496383590077041 0

0 0 0 0 0 0 -0.559419982234582 0.223616465710477

|E(jω,d,α)|

TABLE II I MPULSE RESPONSE VALUES OF THE SUBFILTERS 𝐺𝑘 (𝑧). 𝑘

0

1

2

3

4

𝑔𝑘 (0) = −(−1)𝑘 𝑔𝑘 (14) 𝑔𝑘 (1) = −(−1)𝑘 𝑔𝑘 (13) 𝑔𝑘 (2) = −(−1)𝑘 𝑔𝑘 (12) 𝑔𝑘 (3) = −(−1)𝑘 𝑔𝑘 (11) 𝑔𝑘 (4) = −(−1)𝑘 𝑔𝑘 (10) 𝑔𝑘 (5) = −(−1)𝑘 𝑔𝑘 (9) 𝑔𝑘 (6) = −(−1)𝑘 𝑔𝑘 (8) 𝑔𝑘 (7) = −(−1)𝑘 𝑔𝑘 (7)

-0.009426550401105 0 -0.048665137856736 0 -0.152253144909209 0 -0.611358204703674 0

0 0 0 0 0.012958384592719 0 0.567515157386206 -1.559708816964933

0 -0.034978693477021 0.077193322216295 -0.169704660493031 0.279259934116362 -0.670280747940336 0.788136808577663 0

0 0 0 0 0 0.241254097171053 -0.671873030886289 1.035993980903543

0 0 0 0 0 0.354995315481949 0 0

0.01

structure is somewhat more expensive, but at the same time, its memory cost and delay are lower.

0.005 0 0.2π

0.4π

ωT [rad]

0.6π

0.8π

Fig. 2. Modulus of the complex error for 66 pairs of FD and PS values: 11 values of 𝛼 evenly distributed between −𝜋/2 and 𝜋/2 and 6 values of 𝑑 evenly distributed between 0 and 1/2 (see Footnote 1).

to memory requirements in an implementation, the proposed structure is more efficient. Depending on the implementation cost of multipliers/adders versus memory, the proposed structure may have a somewhat higher total implementation complexity. However, the main advantage of the proposed structure is that it is transient free, whereas the cascade realization exhibits transients when 𝑑 or 𝛼 is changed. It is also noted that an additional advantage of the proposed filter, over the cascaded filter, is that it has a considerably shorter delay (7 compared to 14). VI. C ONCLUSION This paper introduced an FIR filter with a VFD and a VPS. The realization is reconfigurable online without redesign and without transients. The overall filter is composed of a number of fixed subfilters and a few variable multipliers whose values are determined by the desired FD and PS values. The filter is designed offline and iteratively via reweighted 𝑙1 -norm minimization. This design procedure generates fixed subfilters with many zero-valued coefficients, typically located in the impulse response tails. The proposed structure is transient free, which is the main advantage over the straightforward cascade realization. In terms of arithmetic complexity, the proposed

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