VORAPOJ PATANAVIJIT: COMPUTER MODELLING AND SIMULATION OF BILINEAR TRANSFORMATION ON .

Computer Modelling and Simulation of Bilinear Transformation on the Digital Infinite Impulse Response

Vorapoj Patanavijit

Assumption University of Thailand Bangkok, Thailand. e-mail: [email protected]

Abstract – During the past two and half decades digital telecommunication technology and digital models have seen accelerated development and implementation to support the electronic infrastructure of the communication industry. The digital IIR (Infinite Impulse Response) filter has been the focus as the primary component within this area with dramatic progress in term of design, precision, fast operation and easy implementation. Our paper here systematically orders the significance of the digital IIR filter design approach based on Butterworth filtering and bilinear transformation for both statistical and simulation models computation. First, the magnitude response and the phase response of the of the analog filter are computed by Butterworth filtering approach. Next, the magnitude response and the phase response of the transfer function of the are computed by bilinear transformation. The simulation results of the analog filter and the digital filter are computationally analyzed in term of performance.

Keywords - digital IIR (Infinite Impulse Response) filter, Butterworth Filter Desiring Technique, Bilinear Transform, Digital Signal Processing (DSP)

I. INTRODUCTION AND LITERATURE REVIEW II. THE BUTTERWORTH FILTER MODEL

Traditionally, a system [1-5] which sends some fragment For the statistical property of Butterworth low-pass filter frequency collection but suppresses all other frequency [11,17], the magnitude response in both stopband and collection, is designated as band-pass filter or frequency- passband is smooth attribute (or monotonically declining). 2 selective filter, which is one of the most well-known and The magnitude squared response ( Hj ) of this applicable set of LTI (Linear Time-Invariant) systems [6,7]. c However, a system which alters fragment frequency Butterworth low-pass filter can be algebraic written as collection and sends all other frequency collection, can be afterward. 1 designated as the frequency-selective filter in broader sense. 2N 2  (1) Because of the required transfer function characteristics, the Hjc 1   digital filter based on IIR (Infinite Impulse Response) [8, 9, c 10] is one of the primary components in digital where N is the order of the Butterworth low-pass filter telecommunication and digital signal processing [13, 14, 15, and c is the low-pass cutoff frequency (rad/sec) 16] where many design approaches of a digital IIR filter have The arrangement of the Butterworth low-pass filter can developed for over the last two and half decades in term of be expressed as following steps: design, precision, fast operation, easy implementation, etc. From algebraic point of view, the Butterworth filter approach -1. Calculate the order of Butterworth low-pass filter N [11, 17] has been initially used to design the analog filter from the specification: R (passband ripple parameter) and with the required transfer function because this approach is p (stopband attenuation parameter) simple and widely understood and accepted. Subsequently, As the transfer function of analog filter is converted to the A R  s p  transfer function of digital filter by using the bilinear 10 10 s (2) N log 10 1 10 1 2log  transform [12]. Thus, this paper undertake to analyze the     p  digital IIR filter design approach based on Butterworth where: filtering model and bilinear transformation for both statistical computation and simulation computation.   is the round up operator.  R is the passband ripple parameter (dB). p  is the stopband attenuation parameter (dB). As

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-2. Calculate the filter parameter (or the cutoff c Td (9) =2arctan  frequency of the Butterworth CT filter) from the 2 specification: , , , and . N Rp As  p s The transformation between CT complex plain and DT complex plain, which can be illustrated as following figure, can be algebraic written as afterward. R 10 2N 10p  1 (3.1) cp  The Res  0 is mapped into Rez  1 (or inside or the unit circle). 2N  As 10 (3.2)  The is mapped into (or inside cs10  1 Res  0 Rez  1 the unit circle). -3. Calculate the poles of the transfer function of the  The Res  0 is mapped into Re z  1 (or Butterworth CT filter (from the filter parameter and ) N c outside the unit circle).

jk for N is odd. pkNkcexp ,  0,1,2, , 2  1 One-to-One N Transformation Imaginary Part Imaginary Part 12 sT (4.1) z  d 12 sTd k for N is even. pjkcexp  , k  0,1,2, , 2 N  1 Unit circle 2NN (4.2) Real Part Real Part

The stable and causal filter can be defined by Hc s limiting poles in the left half-plain.

-4. Calculate the transfer function ( ) of the s-plane z-plane Hc s Figure 1. The bilinear transformation mapping of the from Butterworth CT filter the s-plane to z-plane. N c (5)  Hsc  s  pk LHP  III. BILINEAR TRANSFORMATION MODELS Td =2arctan  The bilinear transformation concept is a nonlinear 2 mapping that converts the continuous–time variable ( s ) of  the s-plane to the discrete-time variable ( z ) of the z-plane. Therefore, the mathematical relationship between the continuous-time variable s and the discrete-time variable z  can be algebraic written as follows: 21 z1 where T is a sample period. (6) Figure 2. The bilinear transformation mapping of the continuous–time s  1 d Tzd 1 frequency ( ) to the discrete-time frequency (  ). Therefore, the DT transfer function ( H z ) can be mathematically expressed as: The design of the DT IIR low-pass filter by using the 21 z1 bilinear transformation can be algebraically written as steps Hz H (7)  c 1 as follows: Tzd 1 -1. Calculate the continuous frequency of passband ( ) The mathematical relationship between the continuous–  p and stopband ( ) from the specification: , and time frequency (      ) of the s-plane to the discrete- s  p s Td time frequency (     ) of the z-plane can be 2  p (10) algebraic written as:  p =tan Td 2 2  where  is the analog frequency (8) and = tan  2 s Td 2 and  is the digital frequency (11) s =tan or Td 2

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-2. Calculate the transfer function of continuous-time transformation concept (the detail of the continuous-time Hc s low-pass filter (Butterworth filter) from the specification: filter design is expressed in the preceding section):

 ,  , R and A (the detail of the continuous-time filter p s p s 0.1370 Hs (13.3) design is expressed in the preceding section). c sss321.0310 0.5315 +0.1370 -3. Calculate the transfer function H z of discrete-time low-pass IIR filter from the transfer function H s of c   Step 3: Calculate the transfer function ( H z ) of the DT continuous-time low-pass filter by using bilinear, which can filter (from the transfer function ( H s ) of the Butterworth be algebraic written as: c  CT filter) for by using bilinear transformation: Td 1 1 21 z (12) Hz Hc  1 Tz1 1 21 z d  Hz H (12)  c 1 The later part presents numerous experimental cases of Tzd 1 the designing of CT Butterworth lowpass filter by using 21z1 bilinear transform for determining the transfer function and, Hz H  c 1 next, the magnitude response and the phase response of the 11z 1 transfer function is illustrated for examining the performance 1 z Hz Hc 2 of this filter. 1 z1  0.1370 IV. SIMULATION RESULTS Hz  32 11  11zz In this section, all computer results are simulated by 211 1.0310 2  11zz  MATLAB software and executed by PC at specification:  CPU Intel i7-6700HQ and RAM Memory: 16 GB. 1 z1 0.5315 2 +0.1370 1 1 z A. Computer Results for Case 1 

0.0103 0.0308zzz123 0.0308 0.0103 By using the bilinear transformation concept, design the Hz (13.4) Butterworth IIR digital filter where the passband gain 1 2.0002zzz12 1.4428 0.3604  3 ( 00.2  ) between 0 dB and -7 dB, and stopband First, the magnitude in decibels (dB), the magnitude and ( 0.3 ) has attenuation of -16 dB where T 1 . d the phase of this of the analog filter Sketch the magnitude in decibels (dB), the magnitude, the H s [17] can be illustrated as figure 3. Later, the phase and the group delay of this frequency response of this c magnitude and the phase of this frequency response of the Butterworth IIR digital filter j The design of the DT IIR low-pass filter by using the digital filter H e  can be illustrated as figure 4. From bilinear transformation can be expressed in the following these computer results, the bilinear transformation concept steps: can perfectly converse from the analog filter to the digital filter for magnitude perspective as shown in Fig. 3(a) and Step 1: Calculate the continuous frequency of passband Fig. 4(a). Moreover, the phase of the frequency response of ( ) and stopband ( ) from the specification: , and  p s  p s the digital filter, which is converted from analog filter, is T slightly distorted from original analog filter as shown in Fig. d 3(c) and Fig. 4(c).  2 p (10)  20.2 (13.1)  p =tan p = tan 0.6498 Td 2 12 B. Computer Results for Case 2 and By using the bilinear transformation concept, design the 2 s (11)  20.3 (13.2) s =tan s tan  1.0191 Butterworth IIR digital filter where the passband gain Td 2 12 ( 00.2  ) between 0 dB and -1 dB, and stopband

( ) has attenuation of -15 dB where . Step 2: Calculate the transfer function of 0.3   Td 1 Hc s continuous-time low-pass filter (Butterworth filter) [17] Sketch the magnitude in decibels (dB), the magnitude, the phase and the group delay of this frequency response of this from the specification:  ,  , R and A for bilinear p s p s Butterworth IIR digital filter

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The design of the DT IIR low-pass filter by using the Step 2: Calculate the transfer function of Hc s bilinear transformation can be expressed as following step. continuous-time low-pass filter (Butterworth filter) [17]

from the specification:  ,  , R and A for bilinear Step 1: Determine the continuous frequency of passband p s p s (  ) and stopband (  ) from the specification:  ,  and transformation concept (the detail of the continuous-time p s p s filter design is expressed in the preceding section): T d 0.1480 (14.3) Hsc  2  p 20.2 6543  =tan (10)  = tan 0.6498 (14.1) s 2.8100sss 3.9482 3.5168 p  p   Td 2 12 2 2.0884ss 0.7862 0.1480 and

2  s (11)  20.3 (14.2) Step 3: Calculate the transfer function ( ) of the DT s =tan s tan  1.0191 H z Td 2 12 filter (from the transfer function ( H s ) of the Butterworth c  CT filter) for by using bilinear transformation Td 1 concept.

Figure 3 (a) The relationship between the magnitude in decibels (dB) of the

frequency response of the analog filter, j , and  . Figure 4 (a) The relationship between the magnitude in decibels (dB) of the 20log10 H e  frequency response of the digital filter, j , and  . 20log10 H e 

Figure 3 (b) The relationship between the magnitude of the frequency

response of the analog filter, , and analog frequency  . Figure 4 (b) The relationship between the magnitude of the frequency H c j response of the digital filter, H e j , and digital frequency  .

Figure 3 (c) The relationship between the phase of the frequency response

of the analog filter, , and analog frequency . Hc  j   Figure 4 (c) The relationship between the phase of the frequency response of the digital filter, H e j  , and digital frequency  .

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1 21 z (12) Hz Hc  REFERENCES Tz1 1 d  21z1 [1] A. V. Oppenhiem, A. S. Willsky, and S. H. Nawab, “Signals and Hz H Systems”, Prentice-Hall, 2nd ed., 1997.  c 1 11z [2] A. V. Oppenheim, and R.W. Schafer, “Discrete-Time Signal Processing”, Prentice Hall, 2nd ed., 1998. 1 1 z [3] A. V. Oppenheim, and R.W. Schafer, “Discrete-Time Signal Hz Hc 2 1 z1 Processing”, Prentice Hall, 3nd ed., 2010.  [4] C. Ray Wylie & Louis C. Barrett, “Advanced Engineering Mathematics”, McGraw-Hill Companies, Inc., 6th ed., 1995 [5] Erwin Kreyszig, “Advanced Engineering Mathematics”, John Wiley 0.1480 Hz & Sons, Inc., 10th ed., 2011. 654[6] Hwei P. Hsu, “SCHAUM'S OUTLINES OF Theory and Problems of 111zzz111     Signals and Systems” SCHAUM's Outlines, McGraw-Hill 2111 2.8100 2  3.9482 2  111zzz   Companies, Inc., 1995.  32 [7] L. Phillips, J. M. Parr, and E. A. Riskin, “Signals, Systems, and 11zz11      Transforms”, Prentice-Hall, 4th ed., 2007. 3.5168 211 2.0884 2  [8] Monson H. Hayes, "Schaum's Outline of Theory and Problems of 11zz   Digital Signal Processing" SCHAUM's Outlines , McGraw-Hill  1 z1 Companies, Inc., 1999 [9] Rafael C. Gonzalez & Richard E. Woods, “Digital Image 0.7862 21 0.1480 1 z Processing”, Prentice-Hall, 3th ed., 2010.   [10] S. Haykin and B. V. Veen, “Signals and Systems”, John Wiley & (14.4) Sons, Inc., 2nd ed., 2003. [11] Shibendu Mahata, Suman Kumar Saha, Rajib Kar, Durbadal Mandal, First, the magnitude in decibels (dB), the magnitude and Accurate integer-order rational approximation of fractional-order low-pass Butterworth filter using a metaheuristic optimisation the phase of this frequency response of the analog filter approach, IET Signal Processing, 2018. [17] can be illustrated as shown in figure 5, see next [12] Soo-Chang Pei and Hong-Jie Hsu, Fractional Bilinear Transform for Hc s Analog-to-Digital Conversion, IEEE Transactions on Signal page after the references. Then, the magnitude and the phase Processing, Volume: 56 , Issue: 5, 2008 of this frequency response of the digital filter H e j  can [13] Vinay K. Ingle and John G. Proakis, “Digital Signal Processing using Matlab”, Brooks/Cole Thomson Learning, 2000. be illustrated as in figure 6. From these experimental [14] Vorapoj Patanavijit and Kornkamol Thakulsukanant, “Mathematical simulation results, the is perfectly Tutorial of Discrete-Time Analysis of and Non-Aliasing transferred from the analog filter to the digital filter for Periodic Sampling Concept with Fourier Analysis for Digital Signal Processing and Digital Communication Prospective”, SDU Research magnitude perspective as shown in Fig. 5(a) and Fig. 6(a). Journal Sciences and Technology, Suan Dusit Rajabhat University, Moreover, the phase of the frequency response of the digital Vol. 9, No. 3, Sep.-Dec. 2016. (indexed by TCI Group 1 and ACI) filter, which is converted from analog filter, is different very [15] Vorapoj Patanavijit, “Mathematical Tutorial of Discrete-Time slightly from the original analog filter as shown in Fig. 5(c) Analysis of Sampling Rate Changing Concept for Digital Signal Processing and Digital Communication Prospective”, RMUTT and Fig. 6(c). Journal Sciences and Technology, RMUTT, Vol. 6, No. 2 , July.- Dec. 2017. (indexed by TCI Group 1 and ACI) V. DISCUSSION OF RESULTS AND CONCLUSION [16] Vorapoj Patanavijit, “Conceptual Framework of Super Resolution Reconstruction Based on Frequency Domain From Aliased Multi- Low Resolution Images: Theory Part”, Panyapiwat Journal, We considered the analytical models of the digital IIR Panyapiwat Institute of Management (PIM), Thailand, Vol. 8, No. 2, filter design approach based on Butterworth filtering and May. – Aug. 2016. (indexed by TCI Group 1 and ACI) bilinear transformation for both statistical computation and [17] Vorapoj Patanavijit, Experimental Simulation of Digital IIR Filter simulation computation. The computer results illustrate the Design Technique Based on Butterworth Concept and Impulse Invariance Concept, The IJSSST (International Journal of filter design procedure in both mathematical modelling and Simulation: Systems, Science and Technology), United Kingdom, computer simulation. The simulation results of the analog Vol. 19, No. 6 , Dec. 2018. (indexed by Scopus). filter and the digital filter were computationally analyzed to enhance the design quality. The computer results confirm RESULTS CONTINUE ON THE NEXT PAGE. this design technique has high accuracy for magnitude response and phase response requirements.

ACKNOWLEDGMENT

The research project was funded by Assumption University.

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Figure 5 (a) The relationship between the magnitude in decibels (dB) of the Figure 6 (a) The relationship between the magnitude in decibels (dB) of the frequency response of the analog filter, j , and  . frequency response of the digital filter, j , and  . 20log10 H e  20log10 H e 

Figure 5 (b) The relationship between the magnitude of the frequency Figure 6 (b) The relationship between the magnitude of the frequency response of the analog filter, , and analog frequency  . response of the digital filter, j , and digital frequency  . H c j H e

Figure 6 (c) The relationship between the phase of the frequency response Figure 5 (c) The relationship between the phase of the frequency response of the digital filter, j , and digital frequency  . of the analog filter, , and analog frequency . H e  Hc  j  

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