Empirical evidence of ideal filter approximation: pheriperal and selected EU countries application

ONLINE FIRST PRAGUE ECONOMIC PAPERS / ONLINE FIRST 1 © UNIVERSITY OF ECONOMICS, PRAGUE EMPIRICAL EVIDENCE OF IDEAL FILTER APPROXIMATION: PERIPHERAL AND SELECTED EU COUNTRIES APPLICATION Jitka Poměnková, 1Roman Maršálek* Abstract: We compare three fi lters commonly used for business cycle analysis: the Baxter-King, the Christiano-Fitzgerald and the Hamming window fi lter. Empirical contribution of the paper is numerical evaluation of the approximation of the ideal band-pass fi lters in the discussion of the fi lters’ theoretical properties (gain and attenuation within the business cycle frequencies, as well as the leakage in the remaining frequencies). We consider the truncation factor for the BaxterKing fi lter and the sample size for the latter two. We show that the leakage and attenuation of the Christiano-Fitzgerald and the Hamming window fi lter perform similarly across the range of chosen sample sizes and better than the Baxter-King fi lter. Moreover, we apply the fi lters to data of selected EU countries and point out diff erences in their estimation of growth business cycles. Our fi ndings indicate that Christiano-Fitzgerald fi lter and the Hamming window both are appropriate for the identifi cation of abusiness cycle. The Hamming window fi lter introduces smaller attenuation near the edges but in case of small samples its approximation of ideal fi lter is very rough. Keywords: band-pass fi lters, business cycle, frequency transfer function, gain, attenuation, leakage. JEL Classifi cation: C15, C02, E32, E37 1. Introduction Economic literature presents several defi nitions and methods on how to measure business cycles. Burns and Mitchell (1946) defi ne a classical business cycle concept referring to the cycle in the levels of the output series. Harding and Pagan (2005) develop a business cycle concept distinguishing classical and growth (deviation) cycles. Growth cycles are cycles obtained from an input time series by removing the permanent component (Canova, 1998). The sensitivity of results to the method used for isolating the business cycle from the input data is also discussed (Harding and Pagan, 2005; Canova, 1998). The identifi cation of growth business cycle is in the front of empirical work especially in the context of analysis small samples, such as transition or peripheral economies. The focus is on the process of economic integration, on analysis of business cycle comovement, synchronization of business cycles during the crisis, analysis of international trade linkages and many others. * 1 Jitka Poměnková, Department of Radio Electronics (DREL), Brno, University of Technology, Czech Republic ([email protected].cz); Roman Maršálek, Department of Radio Electronics (DREL), Brno, University of Technology, Czech Republic ([email protected].cz). The research described in the paper was fi nanced by Czech Ministery of Education in frame of National Sustainability Program under grant LO1401. For research, infrastructure of the SIX Center was used. We appreciate helpful comments from Jarko Fidrmuc and Svatopluk Kapounek. Prague Economic Papers DOI: 10.18267/j.pep.512 Accepted: 25. 6. 2014 Published: 24. 6. 2015 ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 2 Our motivation for evaluating the quality of ideal fi lter approximation by various band-pass fi lters derives from the very frequent use of these fi lters for identifying economic cycles. The aim of this paper follows consequent steps. The fi rst aim/step is to numerically evaluate the quality of the approximation of selected band-pass fi lters to the ideal fi lter frequency transfer function. For this we select the Baxter-King (Baxter and King, 1999), the Christiano-Fitzgerald (Christiano and Fitzgerald, 2003) and the Hamming Window fi lter (Iacobucci and Noulez, 2005). As approved by many empirical studies, these three fi lters represent common widely use band-pass fi lters suitable for business cycle analysis. There also exist other methods such as a high-pass Hodrick-Prescott fi lter, ARMA processes, and fi rst order difference (FOD) or regression functions (Bonenkamp et al., 2001; Poměnková, 2012) for business cycle identifi cation. But those are criticised in the literature as having limited ability to let only the business cycle frequencies pass (Harvey and Jager, 1993; Guy and St-Amant, 2005). Therefore, we do not consider them. The quality of the approximation is assessed by using three metrics: undesired gain (further denoted as gain) in the business cycle region, undesired leakage outside the business cycle region (further denoted as leakage) and attenuation in the business cycle region (further denoted as attenuation). In the second step we provide an empirical evidence of ideal fi lter approximation. Here we use application on selected European countries, namely Greece, Ireland, Portugal, Spain, Italy and Austria. The aim is an empirical contribution of obtained knowledge from the step one. Because the transfer function of the Baxter-King fi lter is infl uenced by the cut-off parameter K while the Christiano-Fitzgerald’s and the Hamming Window fi lter’s approximations both depend on the sample size, we also consider the percentage of data loss for several selected sample sizes. The last third step is focused on the comparison of different fi ltering techniques appropriate to empirical analysis focussing on time periods affected by global crisis shocks. Therefore, we also conduct a correlation analysis between the business cycles identifi ed for the selected countries and the business cycles identifi ed for Germany. We include Germany as the heart of the euro area and its economically most signifi cant country. The empirical analysis reveals additional problems such as edge effects of the Hamming window. Note that edge (boundary) effects will be referred to a situation at the end of the sample size where estimated values of fi ltered time series are biased. Our fi ndings suggest that the Christiano-Fitzgerald and Hamming window fi lters both are appropriate for identifying business cycles. The Christiano-Fitzgerald fi lter is suitable even for small sample sizes, while the Baxter-King and the Hamming window fi lter require a comparably larger sample size. The Hamming window fi lter also introduces smaller attenuation near the edges but in case of small samples its approximation of ideal fi lter is very rough. The paper is organized as follows: in the next section we outline the literature review and consequently methodological background of the three selected fi lters. The third section contains a description of the chosen data set and evaluation of the fi lters. In Section 4 we present our results and their practical and theoretical implications. Section 5 is focused on comparison of the results via correlation analysis. Section 6 concludes and summarizes the paper. ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 3 2. Literature Review The literature paying attention to the fi ltering method useful for the business cycle identifi cation is extensive. Generally, we can see two streams which are nicely presented by Canova (1998). He mentioned statistical procedures (polynomial functions of time, fi rst order difference, Beveridge and Nelson’s procedure, frequency domain methods, unobserved components model) and economic procedures (a model of common deterministic or stochastic trends, Hodrick-Prescott fi lter). Even this categorisation application of any method cannot be done without satisfaction of assumption supplemented about economic background. As we mentioned in the introduction band-pass fi lters are commonly used for fi ltering. The suitability of band-pass fi lters is given by their property of wholly selecting only the data component belonging to a specifi c frequency band (called pass-band) while eliminating all other components of outside this specifi c band (Baxter and King 1999). Originally, the use of band-pass fi lters for business cycle frequency analysis was proposed by Burns and Mitchel (1946). Modern empirical macroeconomics uses a variety of techniques to perform the decomposition of time series into trend components and cyclical components such as deterministic model, stochastic model or fi lters. Many of those suitable for business cycle analysis are based on the application of a two-sided moving average. A precise (perfect) band-pass fi lter is an infi nite order moving average fi lter that lets only the components in a given frequency range pass. This is only a theoretical concept and infeasible in practice; such a fi lter is thus called “an ideal fi lter”. For practical applications it is therefore necessary to use an approximation of this ideal fi lter. The main problem then becomes constructing the closest possible approximation. As stated by Baxter and King (1999) it is suitable for such an approximation that should let as much as possible of the data of the predefi ned band of frequency pass, while affecting the other frequencies as little as possible. Then it is the optimal approximation of the ideal fi lter. Three band-pass fi lters commonly used for the business cycle analysis in recent literature are the Baxter-King fi lter (Baxter and King, 1999), the Christiano-Fitzgerald fi lter (Christiano and Fitzgerald, 2003) and the Hamming Window fi lter (Iacoboucci and Noulez, 2005). Guaya and St-Amant (2005) assess the ability of the Baxter-King fi lter and the Hodrick-Prescott fi lter to extract the business-cycle component of macroeconomic time series by using two different defi nitions of the business-cycle component. They show that both fi lters do relatively well when applied to series that have a peak in their spectra at business-cycle frequencies. But they do poorly with series whose spectra decrease sharply and monotonically at higher frequencies. Therefore, as wrote Harvey and Jager (1993) or Haug and Devald (2004) the Christiano-Fitzgerald fi lter can be taken as improvement of Baxter-King fi lter, because it chose the weights of the fi lter in frequency domain, i.e. it uses spectra estimations as weighted function. Haug and Devald (2012) also use band-pass fi lters to extract cycles (not even business cycles) in pre-defi ne range (2-8 and 8-40 years) from time series. They use fl uctuations for correlation analysis and for assessment of comovement between series. Because the band-pass fi lters provide possibility to fi lter predefi ned frequency range, they distinguish the long-term component and short-term component by specifi cation of fi lters bands. This approach allows analysis of parts of time series separately. From the group of band-pass fi lters Haug and Devald (2012) chose the Christiano-Fitzgerald fi lter. For robustness they ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 4 checked their results with the Baxter-King fi lter. They found that the fi ltered components were almost identical and that the phase shift (which can occur in the Christiano-Fitzgerald fi lter case due to its non-symmetry) in the fi ltered series is likely negligible. According to their fi ndings the Christiano-Fitzgerald fi lter provides the closest approximation to the ideal fi lter. Our fi ndings in this article support this statement. Identifi cation of cyclical fl uctuation in the context of convergence analysis is used in Drake and Mills (2011). They focused on examination of properties of GDP in the euro area with the stress to the adoption euro in 1999. Drake and Mills (2011) have particular interest in the time series decomposition into trend and cyclical components using Christiano-Fitzgerald fi lter, and the Baxter-King fi lter. They take the Christiano-Fitzgerald fi lter as superior to the traditional Hodrick-Prescot fi lter. They support their decision by the fact that the asymmetric version is better at estimating cycle in real time and near at the end of the sample. As they also mentioned, there are two approaches for convergence analysis, comparison of cycles between themselves or comparison of cycles with the benchmark countries such as Germany. We are going to use this idea in different way. On the basis of known empirical results we can evaluate the suitability of selected band-pass fi lters fi rst according to the level of the measurement for ideal fi lter approximation and in the context of the results for comovement analysis with Germany. Croux et al. (2001) focused on theory and empirics of comovement of economic variables asking whether it can be explained by large aggregate shocks or if the answer should be found in non-linear propagation mechanism. They propose dynamic correlation and cohesion as the relevant measurement for comovement analysis. Macroeconomic literature often presents standard approach of correlation pre-fi ltered (high-pass or band-pass fi lter application) data. Croux et al. (2001) discuss the difference between correlation of pre-fi ltered data and application of dynamic measure. They prefer two-sided fi lter which eliminates all the inappropriate waves. As inappropriate waves they denote all the waves whose frequency is out of the relevant interval and leaves unchanged the amplitude of the waves within the interval. From a methodical point of view in the last decade the time domain and the frequency domain (Iacobucci, 2003; Iacobucci and Noullez, 2005) analysis has been extended to an integrated view of the time-frequency domain (Croux et al., 2001; Hallett and Richter, 2007; Rua, 2010; Maršálek et al., 2013). In all these fi elds the importance of appropriately identifying business cycle phases of fl uctuations in economic activity arises. Therefore the ability to precisely identify business cycles can increase the effi ciency of economic policy instruments and the assessment of the comovement of economies. 3. Selected Band Pass Filters We can analyse a time series, yn, n = 1, ..., N either in the time or the frequency domain. The approach proposed by Baxter and King (1999) or Christiano and Fitzgerald (2003) is to perform the fi ltering in the time domain, while the requirements are specifi ed in the frequency domain. In the time domain representation of the ideal, though infeasible, two-sided linear fi lter is given by the infi nite moving average producing fi ltered time series un: , njnj j uby     (1) ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 5 where yn is the input time series and bj are fi lter weights (Christiano and Fitzgerald, 2003). This linear transform selects only the data components in the specifi ed band of angular frequencies [ω1, ω2], called pass-band. The components outside of this band are eliminated. The adjective “ideal” corresponds to the requirement of an infi nite amount of data. The frequencies specifying the pass-band (the so-called cut-off frequencies) are 112 2 2/ , 2/ ,qq    where q1 and q2 denote the longest and shortest period of cycles passed through the fi lter. In the frequency domain, the ideal band-pass fi lter is defi ned by the frequency transfer function G(ω) equal to 1 for frequencies in the range [ω1, ω2], and zero for all other frequencies. The power spectrum SU , (ω), of the fi ltered time series can be computed as    2, Uy SGS   (2) where Sy , (ω) is the spectrum of the input time series. The fi lter frequency transfer function can be decomposed as G(ω) = |G (ω)|ei  (ω) where the absolute value of G(ω) denoted as |G(ω)| is a module characteristic representing how the amplitude of frequency components are altered by the fi lter,  (ω) is the phase shift caused by the fi lter and describing how different frequency components are delayed and i is the complex unit. The squared module |G(ω)|2 thus determines the weights corresponding to the components of the power spectrum Sy (ω) at the angular frequencies ω. For more details regarding the module and phase characteristics, we refer readers e.g. to Pollock (2009). Note that the symmetric fi lters have linear phase characteristics as a function of frequency. Consequently, a group delay for all frequency components is fl at (constant). This is advantageous, as in such a case all components at the fi lter input, regardless their frequency are delayed by the same amount. Thus the phase distortion is avoided. The main representatives of fi lters based on a feasible approximation of the infi nite moving average are the Baxter-King, the Christiano-Fitzgerald and the Hamming window fi lters. 3.1 The Baxter-King (BK) fi lter The BK fi lter is a two-sided linear moving average band-pass fi lter. In the case of business cycle frequencies its pass-band corresponds to cycle periods between six quarters and eight years. The components outside this range of frequencies are removed. Baxter and King (1999) propose the approximation of an ideal fi lter by the fi nite symmetric linear moving average fi lter of the odd order M = 2K+1 such that ˆ ˆ. K njnj jK uby    (3) The weights of the fi lter ˆj b are computed in the frequency domain by minimizing the loss function Q (Baxter and King, 1999; Christiano and Fitzgerald, 2003) of the differences between the ideal fi lter G(ω) and the feasible fi lter H(ω): 2 1() () , 2 QGHd        (4) where () . Kij j jK Hbe      According to Kowal (2005), this fi lter has a number of desirable properties. First, since it is real and symmetric, it does not introduce a phase shift ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 6 and leaves the extracted components unaffected except for their amplitudes. Second, being of constant fi nite length and time-invariant, it is stationary. 3.2 The Christiano-Fitzgerald (CF) fi lter Another fi lter which well approximates the ideal fi lter is the CF fi lter (Christiano and Fitzgerald, 2003). A fi ltered estimate ˆn u of the N-observations long data set yn can be written as , , ˆ ˆ p pf njnj jf uby    (5) where f = N – n and p = n – 1 for n = 1, ..., N and , ˆpf j bare the time-varying fi lter weights. Similarly to the BK fi lter, the CF fi lter weights are designed with the aim of minimizing the mean square error between the output of the ideal fi lter and its approximation. In the frequency domain this problem can be written in the form 2 , 1() () () , 2 pf y QGBSd        (6) where ,, () p pf pf i j j jf Bbe     . The fi lters weights are set with respect to the importance of the spectrum in the given frequency and therefore depend on the property of the analysed time series. If possible, the trend component should be removed from the original time series prior to the CF fi lter application (Christiano and Fitzgerald, 2003; Haug and Devald, 2004; Iacobucci and Noullez, 2005). We have followed this recommendation in our analysis. 3.3 The Hamming-Window (HW) fi lter In the case of fi nite-length sample sizes it is impossible to design an ideal band-pass fi lter. An interesting fi lter that provides very simple (easily applicable) and effi cient solution to an ideal fi lter approximation developed with respect to application on a short time series was proposed by Iacoboucci and Noulez (2005). In contrast to CF fi lter, the HW fi lter is symmetric and stationary, in contrast to BK fi lter its application results in no data loss. As verifi ed further (see Figure 1) its frequency transfer function is much fl atter than in the case of CF and BK fi lters. It consists in smearing the ideal fi lter response with a selected window and it leads to good attenuation of the spectral power outside the passband, allowing almost complete removal of undesired frequency component. The use of a window is advantageous for suppression of a Gibbs phenomenon, Haykin and van Veen (2003), that can result in the appearance of spurious oscillations at the end of time series (edge effects). In contrast to the two above-mentioned (BK, CF) fi lters, the HW fi lter is applied in the frequency domain. Thus, although the other two fi lters can be defi ned easily in the time domain, the HW fi lter has to be defi ned in the frequency domain. In the following we adopted the original HW fi lter defi nition provided in Iacoboucci and Noulez (2005). First a time series syn is converted to the frequency domain by the calculation of its discrete Fourier transform yn , 1 2/ 0 , 0,1... / 2 N iknN kn n Yye k N       . (7) ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 7 After that, the Hamming windowed fi lter is applied according to the formula  * kkkk UWGY, (8) where Gk represents the frequency transfer function of the ideal fi lter, i.e. the fi lter that fi lters out all the frequency components outside the business cycle band, Wk corresponds to the selected window function, e.g. Hamming window (Iacobucci, 2003) and the operator * denotes a linear convolution. Finally, the fi ltered time series is computed coming back from the frequency domain to the time domain using the Inverse Discrete Fourier Transform, Haykin and van Veen (2003) of Uk  /2 2/ *2/ 1 0 0 ˆ, 0,1... 1 N iknN iknN nkk N k uU Ue Ue n N             , (9) where * in the superscript means complex conjugation. 4. Evaluation of the Filters 4.1 Data The evaluation is done with respect to the infl uencing factor, which is the truncation factor for the BK fi lter and the sample size for the CF and HW fi lter. For the application we use GDP data in quarterly values, denoted in millions of national currency, transformed into chain-linked volumes and based on the reference year 2000 (including ‘euro fi xed’ series for euro area countries). The data are seasonally adjusted and transformed by natural logarithm for business cycle identifi cation. The source of our data is the statistical offi ce of the European Union (Eurostat, 2012). We choose Austria as a benchmark country, representing a stable and developed economy in the EU core, and peripheral euro area countries in available sample sizes. For comparing the approximation results of the fi lters we also use Germany as another representative of a core EU country. Available sample sizes are: Portugal (N =68) from 1995/Q1–2011/Q4, Ireland (N=59) from 1997/Q1–2011/Q3, Italy (N=84) from 1991/Q1–2011/Q4, Greece (N=45) from 2000/Q1–2011/Q11, Spain (N=68) from 1995/Q1–2011/Q4, Austria (N=96) from 1988/Q1–2011/Q4 and Germany (N=84) from 1991/Q1–2011/Q4. These countries were selected due to topicality of the economic situation during the debt crisis. Another important factor is the different sample size of the selected countries. 4.2 Quality of approximation As stated in Christiano and Fitzgerald (2003) if the raw data before application of bandpass fi lters have a non-zero mean or are covariance stationary about a trend, then the trend has to be removed prior to analysis of optimal fi lter design. This statement is followed by Iacobucci and Noullez (2005) in work focused on a frequency selective fi lter used for short-length time series. They also recommend as one solution of optimal fi lter design to subtract deterministic trend before fi ltering. Therefore, we use for this the high-pass Hodrick-Prescott (HP) fi lter (Hodrick and Prescott, 1997). The use of the BK fi lter does not require this step. ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 8 The frequency transfer function plots (Figure 1, Figure 3 in Appendix A) are presented in the normalized frequency range (0, 0.5). In the range (0.5, 1) the frequency transfer function of the band-pass fi lter for business cycle frequencies is zero. Note that the normalized frequency 1 stands for half of the sampling frequency and that there is a straightforward relationship between the normalized frequency and the cyclic component period T: fnormalized = 2/T. Normalized frequencies of 0.0625 (T1 = 32 quarters) and 0.33 (T2 = 6 quarters) correspond to the periods delimitating the business cycle frequency band (Burnsch and Mitchell, 1946). Figure 1 | Example of the Ideal Filter Approximation Using the Band-Pass Filters, K=10, N=45 (Solid Line Area: The Gain, Dashed and Dotted Line Area: The Attenuation, Dotted Line Area: The Leakage) To determine the quality of approximation, we quantify the (undesired) gain of the fi lter approximation in the business cycle frequency range, the (undesired) attenuation in the business cycle frequency range and the (undesired) leakage of the fi lter approximation outside the business cycle frequency range. In all cases the metric value is calculated as the area under/above the fi lter frequency transfer function with respect to the ideal fi lter. The motivation to select these three metrics is given by the shape of the rectangle of ideal fi lter. Looking at the Figure 1 we decided to evaluate how big the area of leakage is, because in case of big leakage the fi lter gives the pass of such cyclical movements close to ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 9 the established bands which does not belong to defi ned range. We also decided to evaluate attenuation and the gain of the cyclical component belonging to the defi ned range (in our case business cycle range). In case of big gain (attenuation) the fi ltered time series can excaudate (inhibit) importance of cyclical component. Therefore, in all three cases of approximation of ideal fi lter rectangle consecutive analysis (comovement analysis etc.) using time series fi ltered by such fi lter can indicate bias results. As the transfer function of the CF fi lter is time variant, these metrics were computed in the time instant n = N/2 (the best ideal fi lter approximation). For the description and illustration of the selected metrics see Figure 1 above Among these three observed quantities, the leakage and the attenuation are more critical for business cycle isolation than the undesired gain in the business cycle frequency range. The gain magnifi es the identifi ed components so in some sense it can be benefi cial for this application. 4.2.1 Baxter-King fi lter approximation The description of the BK fi lter in Equation 3 shows that both the length of its impulse response and its frequency transfer function as well depend on the truncation factor K only and not on the input sample size. Therefore, the measures of the approximation to the ideal fi lter using the BK fi lter will be the same for an arbitrary sample size. The variation of sample size results only in a relative loss of data change. This loss is caused by shortening the analysed time series by K observations from both sides of the data set. Therefore, we defi ne the indicator of relative data loss (RDL) as a ratio between the numbers of reduced data from both sides (2K) and the sample size (N): 2100 K N RDL  . (10) We varied the truncation factor in a range of K = 5, …, 19. The results are presented in Table 1. A detailed graphical representation of the results in Table 1 can be found in Appendix A (Figure 3). As we mentioned above, the frequency transfer function does not depend on the analysed data or on its sample size, but only on the truncation factor K. Thus, the optimum value of the parameter K is the value for which the approximation to the ideal fi lter results in minimum values of (undesired) gain, attenuation and leakage with respect to minimized data loss. We can state that this condition is fulfi lled for the value K=10. For K greater than 10, the value of leakage has a descending tendency, but on the contrary the relative loss of data is ascending. For practical reasons during the application on real data we require a loss of data as small as possible. We can observe that undesired gain is also rising for higher K values and the difference between the leakage of approximation for parameter K=10 (0.0125) and K=11 (0.0107) is not signifi cant. It is thus preferable to accept a marginally higher value of leakage in order to achieve a smaller data loss. Note that this result is in good accordance with the original recommendation for an optimum selection of the K value (Baxter and King, 1999) without considering data loss. ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 16 loss close to a minimum. This result is in good accordance with the original recommendation for optimum K value selection (Baxter and King, 1999). Our theoretical fi ndings show that regarding leakage and attenuation the Christiano-Fitzgerald and the Hamming window fi lter perform similarly across the range of chosen sample sizes, while yielding better results than the Baxter-King fi lter. Secondly, we apply the fi lters to GDP data of selected EU countries. The empirical analysis reveals additional problems such as edge effects and shape of frequency transfer function of fi lters. We suggest that the Christiano-Fitzgerald fi lter might be the most appropriate for the identifi cation of business cycles even for small sample sizes, while the Hamming window fi lter or Baxter-King especially require a comparably large sample size. Our fi ndings are supported by the results of a correlation analysis between Germany and Ireland, Greece, Spain, Portugal, Italy and Austria. References Baxter R., King R. G. 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(2001), “Cyclical Aspects of Business Cycle Turning Points.” International Journal of Forecasting, Vol. 17, No. 3, pp. 369–382. ONLINE FIRST ONLINE FIRST PRAGUE ECONOMIC PAPERS 18 Appendix A Figure 3 | Filter Approximation Using the Baxter-King Filter with Truncation Factor K= 5, …, 19 (Solid Line Area: The Gain, Dashed and Dotted Line Area: The Attenuation, Dotted Line Area: The Leakage), (X-Label: Normalized Frequency, Y-Label: The Frequency Transfer Function)