Minimum Phase Property of Chebyshev-Sharpened Cosine Filters

We prove that the Chebyshev sharpening technique, recently introduced in literature, provides filters with a Minimum Phase (MP) characteristic when it is applied to cosine filters. Additionally, we demonstrate that cascaded expanded Chebyshev-Sharpened Cosine Filters (CSCFs) are also MP filters, and we show that they achieve a lower group delay for similar magnitude characteristics in comparisonwith traditional cascaded expanded cosine filters.The importance of the characteristics of cascaded expandedCSCFs is also elaborated. The developed examples show improvements in the group delay ranged from 23% to 47% at the cost of a slight increase of usage of hardware resources. For an application of a low-delay decimation filter, the proposed scheme exhibits a 24% lower group delay, with 35% less computational complexity (estimated in Additions per Output Sample) and slightly less usage of hardware elements.


Introduction
A Minimum Phase (MP) digital filter has all zeros on or inside the unit circle [1].We consider in this paper MP Finite Impulse Response (FIR) filters, which find applications in cases where a long delay, usually introduced by Linear Phase (LP) FIR filters, is not allowed.Examples of these cases include data communication systems or speech and audio processing systems [2][3][4].
The basic building block analyzed in this paper, the cosine filter, is a simple FIR filter whose transfer function and frequency response are, respectively, given by This filter is of special interest because of the following main reasons: (a) It has MP property because its zero lies on the unit circle.(b) It has a low computational complexity because it does not require multipliers, which are the most costly and power-consuming elements in a digital filter [5].(c) It has a low usage of hardware elements, which can be translated into a low demand of chip area for implementation.
The fact that a cosine filter has the Minimum Phase characteristic becomes significant because these basic building blocks can be used to design filters with a low delay and simultaneously a low computational complexity and a low usage of hardware resources.Due to the aforementioned characteristics, cascaded expanded cosine filters were investigated in [6].A cascaded 2 Mathematical Problems in Engineering expanded cosine filter is defined as a filter with transfer function and frequency response, respectively, given by with () being the cosine filter given in (1), whereas  and  are arbitrary integers.In method [6], Rouche's theorem was employed to demonstrate that when cascaded expanded cosine filters are sharpened with a modified version of the technique from [7] (originally devised for LP FIR filters), the result is an overall FIR filter that has all its zeros on or inside the unit circle; that is, it satisfies the MP characteristic.Nevertheless, the resulting filter still has a high group delay because a large number of cascaded expanded cosine filters are needed to meet a given attenuation specification, as we see in the examples from [6].
On the other hand, the Chebyshev sharpening technique was recently introduced in [8] to design LP FIR filters based on comb subfilters for decimation applications.In that method the sharpening is performed with an th degree Chebyshev polynomial of first kind, defined as where   are integers [9].When applied to comb filters, Chebyshev sharpening provides solutions with advantages like a simple and elegant design method, a low-complexity resulting LP FIR filter, and improved attenuation characteristics in the resulting filter.However, filters from [8] are not guaranteed to have MP characteristic.
From the aforementioned literature we can extract the following observations: (a) In MP FIR filters the reduction of the group delay is a priority.(b) The use of cosine filters results in low-complexity multiplierless MP FIR filters.(c) The recent Chebyshev sharpening method from [8] can improve the attenuation of cosine filters and is a potentially useful approach to preserve a simple multiplierless solution with a lower group delay in comparison with simple cascaded expanded cosine filters.
Motivated by the remarks listed above, we present in this paper the following contributions: (1) The mathematical proof that the use of Chebyshev sharpening in cosine filters, which produces Chebyshev-Sharpened Cosine Filters (CSCFs), guarantees resulting multiplierless filters with all of their zeros placed on the unit circle, that is, with MP property: this demonstration hinges upon the factorization of the transfer function of the CSCF into secondorder sections, taking advantage of the antisymmetry of the roots of the Chebyshev polynomial.
(2) The mathematical proof that cascaded expanded CSCFs have also all of their zeros placed on the unit circle: this demonstration is a simple extension of aforementioned proof for CSCFs.
(3) The explanation of how cascaded expanded CSCFs can be efficiently employed in the design of MP FIR filters. (

Definition of Chebyshev-Sharpened Cosine Filter (CSCF) and Cascaded Expanded CSCF
We define the transfer function and the frequency response of an th order Chebyshev-Sharpened Cosine Filter (CSCF), respectively, as with where   are the coefficients of the Chebyshev polynomial of first kind, represented in (5), and () is given in (1).To obtain a low-complexity multiplierless implementation, the constant  must be expressible as a Sum of Powers of Two (SOPOT).To this end, we set where (, ) denotes "the closest value less than or equal to  that can be realized with at most  adders" and ⌊⌋ denotes rounding  to the closest integer less than or equal to .
To provide an improved attenuation around the zero of the cosine filter,  must be as close as possible to its upper limit [8].This is achieved by increasing the integer .The value  in ( 8)-( 9) is usually set as an integer equal to or greater than 2 for applications in decimation processes [8].However, we will allow for more flexibility to the parameter  in this paper, as will be explained in the next section.
The transfer function and frequency response of a cascaded expanded CSCF are, respectively, defined as where the integer  indicates the number of cascaded CSCF blocks, each of them repeated   times, with  = 1, 2, . . ., .Every value of  is a distinct factor that expands a different CSCF whose corresponding order is   .These CSCFs have different factors   , which can be obtained using (9), just replacing  by   and  by   , where   and   are integer parameters that correspond to the th CSCF in the cascade.Figure 1(a) shows the structure of the CSCF, where we have that   =  2+V , with  = 0, 1, 2, . . .,  = ( − V)/2 and with V = 1 if  is odd or V = 0 if  is even.Dashed blocks in Figure 1(a) appear only if  is odd. Figure 1(b) presents the structure of the cascaded expanded CSCF whose transfer function is given in (10).

Proof of Minimum Phase Property in CSCFs
The proof starts with the expression of the Chebyshev polynomial from (5) in the form of a product of first-order terms as [9] On the other hand, we rewrite the transfer function of the CSCF from (6) as Using (12), and after simple rearrangement of terms, we express  , (, ) as follows: which can be rewritten as where ⌈⌉ denotes rounding  to the closest integer greater than or equal to .At this point, it is worth highlighting that the antisymmetry relations ⌈/2⌉ = 0 for  odd (17) hold [9].Thus, replacing (17) in (16), and after simple manipulation of terms, we have From (18) we have that  , (, ) consists of a product of either several terms   () if  is even or several terms   () and a term () if  is odd, with  = 1, 2, . . ., .Thus, to prove the MP property of the CSCF, it is only necessary to ensure that   () and () have MP characteristic for all values .Using (1), it is easy to see that the term () has a root on the unit circle and thus it corresponds to a MP filter.On the other hand, replacing (1) into (19) and after simple rearrangement of terms, we get From (20) it is easy to show that the roots of   () are placed on the unit circle; that is, if the argument   ⋅  −1 in ( 22) is preserved into the range [−1, 1].From (13) we have that −1 ≤   ≤ 1 holds.Additionally, by setting  ≥ 0.5 (23) in ( 8)-( 9), we ensure  ≥ 1.Under this condition for , we have that −1 ≤  −1 ≤ 1 holds.In this case,   () has its roots on the unit circle for all the valid values  and, as a consequence, the filter  , (, ) has a MP characteristic.

Proof of Minimum Phase Property in Cascaded Expanded CSCFs
The proof starts with the expression of every CSCF of the cascaded expanded CSCFs from (10) in the form of a product of second-order expanded transfer functions using ( 18) and (20); that is, where  = By replacing (1) in the term   (  ) and then making the resulting expression equal to zero, we can find the  roots of   (  ).These roots turn out to be the  complex roots of −1, which have unitary magnitude.Thus,   (  ) has MP characteristic, since its roots are placed on the unit circle.On the other hand, using (21) we can express (25) as follows: To preserve the argument Under this condition for   , we have that −1 ≤  −1  ≤ 1 holds.In this case, the respective  roots of factors (1− 2   − ) and (1− −2   − ) in (25) are the  roots of the complex numbers  2  and  −2  , which have unitary magnitude for all the valid values .Therefore,   (  ) has MP characteristic, since its roots are placed on the unit circle.Finally, since   (  ) and   (  ) have MP characteristic, the overall cascaded expanded CSCF from (9), (), also has MP characteristic.

Characteristics and Applications of Cascaded Expanded CSCFs
A cascaded expanded CSCF has both MP and LP characteristics.The former was proven in Section 4, whereas the latter is easily seen from the frequency response (  ) given in (11).A consequence of this is that the cascaded expanded CSCF has a passband droop in its magnitude response.Due to this passband droop, the cascaded expanded CSCF should be employed only to provide a given attenuation requirement of an overall MP FIR filter over a prescribed stopband region (depending on the application).Thus, the resulting structure to design an overall MP FIR filter can be associated with the prefilter-equalizer scheme of [10], shown in Figure 4, where the prefilter provides the required attenuation whereas the equalizer corrects the passband droop of the prefilter.
The cascaded expanded CSCF, with transfer function () defined in (10), can be used as prefilter.Note that since a cascaded expanded cosine filter (whose transfer function   () is defined in ( 3)) also has both LP and MP properties, it is used as prefilter in [6].
Even though this paper is not focused on the design of the equalizer, it is worthwhile to spend some words on how this filter could be designed.An Infinite Impulse Response (IIR) filter with optimally located poles based, for example, in the Least Squares criterion as shown in [11] can form a proper equalizer.However, FIR filters are usually preferred over their IIR counterparts because they have guaranteed stability, they are free of limit-cycle oscillations, and their polyphase decomposition in multirate schemes allows them to reduce the computational load, among other characteristics [12].Thus, we are more concerned here with FIR equalizers.Since a FIR equalizer with LP characteristic has its zeros placed in quadruplets around the unit circle [1], it does not accomplish the MP characteristic.Therefore, a MP FIR equalizer (i.e., that filter whose zeros appear inside the unit circle) does not have a Linear Phase.
When a LP FIR filter is designed by sharpening cascaded expanded cosine filters with the traditional sharpening polynomial 3 2 − 2 3 from [7], the resulting filter has a prefilter given by [  ()] 2 and a LP equalizer given by [3 − − 2  ()], where  is the group delay of   () used to preserve the Linear Phase characteristic.In method [6] the delay  has been removed to obtain a MP FIR equalizer.Thus, a first option would be to use the same approach of [6] to design a FIR equalizer.However, it is worth highlighting that the recent LP droop compensators proposed in literature (see, e.g., [13,14]) are novel low-complexity alternatives to the aforementioned LP equalizer based on the traditional sharpening.Inspired by these alternatives, a more convenient approach would be to design MP droop compensators as counterparts of the MP equalizer based on sharpening, proposed in [6].
Besides method [6], other design methods for MP FIR filters have been introduced, for example, in [15][16][17][18][19][20][21][22].They can be classified in general terms as methods based on cepstrum [15][16][17] and methods based on the design of a LP FIR filter [18][19][20][21][22].However, in general, these methods have the inconvenience of producing filtering solutions that require multipliers, which are the most costly elements in a digital filter [5].To solve this problem, the cascaded expanded CSCF can be used as a prefilter to implement an overall MP FIR filter using several multiplierless CSCFs.A similar approach can be followed with the use of a cascaded expanded cosine filter from [6].Nevertheless, as we mentioned earlier, the problem with method [6] is that the resulting filter requires a large cascade of expanded cosine filters, increasing the group delay of the resulting filter.Finally, it is worth highlighting that, in comparison to the filter   (), the filter () has many more parameters, namely,   ,   ,   , and   , with  = 1, 2, . . ., , to be tuned in order to find a desired attenuation.This characteristic provides more flexibility for the design of MP FIR filters in comparison to   ().Moreover, by setting  = ,   = 1,   = 0.5, and   =  for all  in (10), we obtain the same expression as (3).Thus, the cascaded expanded CSCF from ( 10) is a generalized case of (3).

Examples and Discussion
This section presents a couple of examples (Examples 1 and 2) that compare the cascaded expanded CSCFs () with cascaded expanded cosine filters   () from [6].This comparison is made in terms of (a) group delay, measured in samples and defined as follows [1]: where (  ) is the frequency response of the corresponding filter; (b) implementation complexity, measured in the required number of adders and delays for a given attenuation over a prescribed stopband region.
Additionally, an engineering application is provided in Example 3, namely, the antialiasing filtering process used in the first stage of a two-stage decimation structure applied in a low-delay Sigma-Delta ADC for audio systems, detailed in [3].In this case, comparisons are made in terms of group delay referred to high rate, computational complexity counted in Additions per Output Sample (APOS), and number of hardware elements assuming that both filters, the one used in method [3] and the proposed filter, are implemented in recursive form.
Example 1. Design a MP FIR filter with minimum attenuation equal to 60 dB over the range from  = 0.17 to  =  (see Figure 1 of [6]).
In [6], the filter employed to accomplish such characteristic is obtained from (3) using  = 5 and  = 3.The group delay is obtained by replacing these values in (4) and then using (4) in (29).This filter requires 15 adders and 45 delays, but it has a group delay of 22.5 samples.
Example 2. Design a MP FIR filter with minimum attenuation equal to 100 dB over the range from  = 0.15 to  =  (see Example 1 of [6]).
In [6], the filter employed to accomplish such characteristic is obtained from (3) using  = 7 and  = 4.The group delay is obtained by replacing these values in (4) and then using (4) in (29).This filter requires 24 adders and 120 delays.However, its group delay is 56 samples.
Figure 6 shows the magnitude responses and group delays of both filters.Tables 3 and 4, respectively, show the first half of the symmetric impulse response of the filter of method [6] and the proposed filter in Example 2. Table 5 summarizes the results from the previous examples.From them we observe that the cascaded expanded CSCFs achieve a lower group delay in comparison to the cascaded expanded cosine filters from [6].This characteristic, desirable for MP filters, occurs because CSCFs take advantage of the zero-rotation effect (see Figures 2 and 3), provided by Chebyshev sharpening, to achieve a given attenuation using less cascaded filters.Nevertheless, since every CSCF needs in general a few more hardware resources (adders and delays) than their cascaded cosine counterparts, the price to pay is an increase in the implementation complexity.Note, however, that the resulting filters still are low-complexity multiplierless solutions.
In the following example we show the design of the antialiasing filter used in the first stage of a two-stage decimation structure applied in a low-delay Sigma-Delta ADC for audio systems, where the second stage is a FIR filter with droop compensation characteristic followed by a downsampler by 2 [3].

𝜋;
for  = 16. (30) In [3], the filter employed to accomplish such characteristic is obtained from method [23].Its transfer function  comb () and frequency response  comb (  ) are, respectively, given as Mathematical Problems in Engineering Figure 7: CIC structure for the first-stage decimation filter of Example 3, used in method [3].This filter has a group delay of (10 × 31)/2 = 155 samples.Moreover, its Cascaded Integrator-Comb (CIC) structure, shown in Figure 7, performs 10 × 32 + 10 = 330 Additions per Output Sample (APOS) and uses 20 adders and 20 delays.Now consider the proposed filter, whose transfer function  prop () is given by where () is an expanded CSCF given by The frequency response  prop (  ) is given by This filter has a group delay of [(6 × 31)/2 + 8 × 3] = 117 samples, which is approximately 24% less delay than that of [3].The filter () can be moved after a downsampling by 16 because it is actually a CSCF expanded by 16.Thus, the resulting CIC-based structure, shown in Figure 8, performs (6 × 32 + 6) + (3 × 2 + 5 + 6) = 215 Additions per Output Sample (APOS), that is, nearly 35% less computational complexity with regard to the filter used in [3].Moreover, this filter uses 20 adders and 16 delays, which represents 10% lower usage of hardware resources compared with method [3].
Figure 9 shows the magnitude responses and group delays referred to high rate of both the filter used in method [3] and the proposed filter.Tables 6 and 7 show the first half of the symmetric impulse response referred to high rate of the filters obtained with method [3] and the proposed method, respectively, whereas the summary of results is given in Table 8.
It is worth highlighting that the implementation of the comb decimation filter in a CIC structure has been employed in method [3] due to its regularity and simplicity, which has a low usage of hardware resources (see Figure 7).However, the price to pay for such desirable characteristics is a high computational complexity.Our proposed solution has taken advantage of the possibility of factorize the decimation factor  = 32 as  = 16 × 2. With this we have used an expanded-by-16 CSCF as an additional filter that contributes to improving the attenuation in the first stopband, where the comb filter has the worst attenuation.The first advantage of doing so is observed in the reduction of the number of Integrator-Comb stages from 10 to 6.Moreover, since the CSCF can operate at a sampling rate reduced by 16, the computational complexity of the decimation process is reduced and the number of hardware elements is also reduced.Of course one can resort to other types of architectures, such as the nonrecursive form of the comb filter and its subsequent polyphase decomposition.However, this decreases the computational complexity at the cost of a considerable increase of the number of hardware elements.

Conclusion
In this paper we have presented the mathematical demonstration that the application of Chebyshev sharpening to cosine filters results in filters with zeros on the unit circle, that is, with Minimum Phase (MP) characteristic.From this, we have proven that filters composed by a cascade of Chebyshev-Sharpened Cosine Filters (CSCFs) expanded by different factors, called cascaded expanded CSCFs, also have MP property.The cascaded expanded CSCFs are useful prefilters that provide the attenuation in an overall MP FIR filter.Moreover, these filters are a general case where the cascaded expanded cosine filters are a subset.The CSCFs are low-complexity filters, since they do not need multipliers.It has been shown with three examples that, for a desired attenuation in the magnitude response, cascaded expanded CSCFs achieve a lower group delay in comparison to cascaded expanded cosine filters.This lower group delay is desirable in the design of MP FIR filters.Since the purpose of this paper is to prove the suitability of cascaded expanded

Example 3 .
Design a MP FIR filter for a decimation factor  = 32 and residual decimation factor  = 2, with minimum

Figure 8 :
Figure 8: Proposed CIC-based structure for the first-stage decimation filter of Example 3.

Table 1 :
[6]st half of the symmetric impulse response of the filter designed with method[6]in Example 1.

Table 2 :
First half of the symmetric impulse response of the proposed filter in Example 1.

Table 3 :
[6]st half of the symmetric impulse response of the filter of method[6]in Example 2.

Table 4 :
First half of the symmetric impulse response of the proposed filter in Example 2.

Table 5 :
Comparison of results in Examples 1 and 2.

Table 6 :
[3]st half of the symmetric impulse response of the filter of method[3], referred to high rate, in Example 3.

Table 7 :
First half of the symmetric impulse response of the proposed filter, referred to high rate, in Example 3.