View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by Directory of Open Access Journals
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
RESEARCH ARTICLE OPEN ACCESS
Filter Design: Analysis and Review
Anirudh Singhal Thapar University, Patiala (Punjab), India
Abstract This paper represents the review analysis of various types of filters design. In this paper we discuss the designing of filters. For discussing the designing of filters we consider the some standard paper which is based on filter design. First we will discuss about filter then we will discuss about types of filter and give the review on different ways of designing of filter. Butterworth Filter I. Introduction A filter is generally a frequency-selective Chebyshev Filter device. Some frequencies are passed through the Eliptical Filter filter and some frequencies are blocked by filter. The Linkwitz-Riley Filter frequencies of signals that are pass through the filter Eletronic Filter are called pass band frequencies and those Biquad Filter frequencies that are stop by the filter are call stop Notch Filter band frequencies. For pass band frequencies, the Comb Filter system function magnitude is very large and ideally Infinite and Finite Impulse Response Filter is a constant while for a stop band frequencies, the (IIR and FIR) system function magnitude is very small and ideally High Pass and Low pass Filter (HPF and is zero. The filter is also work as a remove the noise LPF) component from the signal and passes the remaining Bessel Filter signal. Ideally a good filter removes the noise Bilinear Filter component completely from the signal without any Adaptive Filter loss of meaningful part of signal. Basically, filters are Kalman Filter two types:- Wiener Filter ARMA Filter In this paper we will take some filters which are mention above and comparison the various type of designing on different filters
II. BUTTERWORTH FILTER:- Fig 1.1 Description of filters The magnitude- frequency response of low- pass Butterworth filters are characterized by [1] Analog filter is simply to implement and 11 requires few electrical components like resistors, |H ( ) |2 capacitors and inductors while for implementation of 1 ( / c)2NN 1 2 ( / p ) 2 digital filter, first we convert the ananlog signal into (2.1) digital signal by taking a samples value of analog Where A is a filter gain, N is a order of a filter, c is signal then we implement the digital filter with the help of adders, subtracts, delays etc. which are a -3db frequency, p is the pass band edge classified under digital logic components. Analog 1 frequency and is the band edge value of filter’s characteristics are fixed by circuit design and 12 component values. If we want to change the filter 2 characteristics than we have to make major the|H ( ) | . In Butterworth filter the magnitude modification in circuit while in digital filters do not response decreases monotonically as the frequency require the major modification. But in digital filter increases. When the number of order of a filter we remove the noise easily from the signal as increases, the Butterworth filter magnitude response compare to analog filter that’s why generally we use more closely approximates the ideal response. The digital filters. Butterworth filter is also said to have maximally flat Another classification of filters is:-
www.ijera.com 236 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259 response. The phase response of a butterworth filter Butterworth low-pass filter. So the damping factor δ becomes nonlinear as N increase. is close to 1 and 0. The conjugate complex poles 2.1 Designing Method distribute closer with image axis, the corresponding Paper 1 damping factor δ is smaller, the overshoot is larger, Butterworth low pass filter is also used for a and the system has poor stability. remove some noise from the signals like electrical For improving the stability of nth-order noise, drift rates etc. in navigation system. The Butterworth low-pass filter they chosen the poles navigation system plays an important role to from (n+2)th order of Butterworth low-pass filter. determine the position or attitude in the dynamics or For example, for designing of 4th order Butterworth stationary condition. So, Ping et al [2] designed an low-pass filer they choose the poles from 6th order active low pass Butterworth filter using symbolic Butterworth low-pass filter. They choose four poles codes with the help of Sallen key topology structure. far from image axis as poles of improved Butterworth For designing a Butterworth low pass filter they used low-pass filter and the angle θ of 4th order improved an operational amplifier as the active component like Butterworth low-pass filter will equal 45o . Similarly, some resistors and capacitors which provide an RLC for designing of 5th order Butterworth low-pass filter like filter performance. They take a standard and they choose the poles from 7th order Butterworth low- discrete value of resistors and capacitors then select o the cut-off frequency. They used a Sallen-key pass filter and the angle of θ will equal to 38.57 . topology to design a transfer function. Sallen-key They observed that if the order n of the circuit is look like a voltage control voltage source. improved filter is even, then the transfer function of Sallen- key topology is better if the improved filter is given as Gain accuracy is important 1
Unity gain filter is required G(s) n Pole-pair Q (quality factor) is low 2 ss2 nd Ping et al make 2 order Butterworth low 2 2(cosk ) 1 pass filter from Sallen-key topology then for 4th order k1 wwcc Butterworth low pass filter they used a cascade Where connection. After that for every stage MATLAB code (2k 1) used and simulated into PSPICE. This Butterworth k filter is the combination of resistors, capacitors and 2(n 2) op-amp. It is very reliable and functional filter but If the order n of the improved Butterworth Filter is this process of designing is very slow. This filter odd, then the transfer function of the improved filter provides better analysis and verification. is given as 1 Paper 2 G(s) n1 Another method of designing a Butterworth s 2 s2 s low-pass filter is given by Li Zhongshen [3]. They 1 2 2(cosk ) 1 present the nth-order improved Butterworth low-pass wc k1 w c w c filter, whose n poles are chosen from the poles of Where (n+2)th-order Butterworth low-pass filter. The k conventional Butterworth low-pass filter exist the k contradiction problems between test precision, n 2 stability and response time. For improvement this Now, the improved Butterworth low-pass type of problem they designed the improved filter is analyzed. They compared with the Butterworth low-pass filter. conventional Butterworth low-pass filter f the same Butterworth low-pass filter is a type of order, the improved Butterworth low-pass filter has whole-pole filter, each pole of the Butterworth low- rapid response, smaller overshoot, quality of unit step pass filter is distributed on the complex plane. response is more ideal and good stability. This kind Phase-angle difference of two bordering poles is of filter can be used to enhance the test precision. 1800 n , the angle θ between the pole which is Paper 3 When we increase the order of Butterworth 1800 nearest to image axis and image axis is 2n . For low pass filter then the filter will approximate the ideal filter characteristics. But it will increase the cost 4th order Butterworth low-pass filter, the angle of θ = and circuit complexity of the filter. So Md Idro and o o 22.5 and for 5th order θ =18 . They notice that the Abu Hasan [4] discovered the filter’s order on the conjugate complex poles distribute far from and close ideal filter approximation due to this cost and circuit with image axis with increasing the order n of the complexity reduced. For this they used the Sallen-key
www.ijera.com 237 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259 topology, because Sallen-key topology requires fewer components and they assumed that resistors and capacitors are assumed to be equal. They used an operational amplifier to reduce the problem of Start uniformly and balancing due to sensitive of analog signal circuit variation in electrical parameter. For increasing the order of filter they used cascading of lower order filter. The process is show in the flow chart 2.1. First, they design the basic two stages op- amp. The op-amp is design based on the following Design basic op-amp based on specification specification:- 1. Phase Margin > 600 2. Unity Gain Frequency > 5MHz 3. DC Gain > 100 (40 dB) 4. Voltage Swings = 1.0V to 4.0V Design first order Low Pass Filter (LPF) 5. Slew rate > 5V/μs Some equations are related to the above specifications. By implement both specifications parameters and equations they design a basics two stages op-amp and determine the transistor size Design second order LPF (W/L). After that they designed a first order LPF. (the Sallen key topology)
Design higher order LPF by cascading lower order filters
Analyze data Fig 2.1:- First-order non-inverting LPF
Layout design
End
Fig 2.1:- Flow chart of Process of the Experiment
The transfer function for the above circuit is
www.ijera.com 238 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
R2 b 1 R 1 R3 2 fC A(s) c 1 wc R11 C s a1 1 A0 33 R2 Where DC gain A 1 and b1 Q 0 R 3 The value of depends only on quality factor Q and a 1 vice versa. After that the final step is design the R1 2 fCc 1 layout for first order Butterworth LPF. So Sallen-key topology is good choice to reducing the circuit RRA2 3( 0 1) complexity.
Where value of coefficient a1 is taken from Butterworth coefficient tables and calculate the value Paper 4 Another design is proposed by Gilbert and of resistor R1 and R2 . From these values they design Fleming [5]. They design a high order of Butterworth the first-order LPF. filter with using either series connection or parallel After the designing of first-order LPF they connection or combination of both connections. They designed the second-order LPF using the Sallen-key design a filter by using double precision method to tpology. The general Sallen-key LPF is given below overcome the numerical accuracy rather than a direct method. Due to this method the simplicity and economy of programming are easy but the programming become tough compare to direct form. For designing a Butterworth filter using a direct expansion method they obtained a transfer function which a digital filter may be realized is given in the form of z- transform N p Ezp p0 H (z) N (2.2) Fz p p Fig 2.2:- Generall Sallen-key LPF p0 Where EEE01, ,... N and FFF01, ,... N are the filter The transfer function of the above figure 2.2 coefficients and N is the order of filter. is: For the digital Butterworth low-pass filter they
A0 applied bilinear transformation in equation (2.2) A(s) 22 1wcc [(R C1 1 R)(1A)R 2 0 1 C 2 ] s w R 1 R 2 C 1 C 2 s 1 H(z) p (2.3) N 2(1 z1 ) Since the Sallen-key equal component is used, the ap 1 transfer function of figure 2.2 would be different p0 T(1 z ) from above equation Where A A(s) 0 1w RC (3 A)s(wRC) 22 s (m 1) cc0 p cos Where p 2N apc p 1,2,..., N a w RC(3 A ) m1 m 10c sin 2N R4 A10 a0 1 R3
2 b1 (wc RC) T is the sampling period, and c is the cut off
Set the value of C and calculate the R and A0 . frequency in rad/sec. They expand powers of (1 z1 ) and (1 z1 ) using Binomial expansion
www.ijera.com 239 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
N filter. They applied the double precision method only pN zCp in denominator of transfer function and calculated the H (z) p0 denominator coefficients, since the numerator Nr r N r coefficients are integer. They observed that due to a2 [( 1)p z p C r ] [ z q C N r ] r T p q this method the simplicity and economy of r0 p 0 q 0 programming are easy but the programming become (2.4) tough compare to direct form. Where N! 2.2 Discussion:- 0 pN Table 1 provides an analysis of designing of N (N p)!!p Cp Butterworth filter. Sallen-key topology provides 0 otherwise functional and reliable filter. It is also provide the fast method to analysis and verification of filter. And in It should be noted that the summation another designing of Butterworth filter Sallen-key variables in the denominator of (2.4) are rearranged topology is used to reduce the cost and complexity at this stage in order to arrive at the final form of the filter. But Sallen-key topology has a slow H(z) with the desired summation variable p. They designation process. In double precision method, the defined simplicity and economy of programming are easy but pr programming is tough compare to direct form. ACpp(r) ( 1)
B (r) CNr 2.3 Comparison Table:- qq Strength They took a denominator of equation (2.4) and Paper Approach Weakness rearranged it in the following way:- Paper Filter design Reliable and Design 1 using functional filter. process is r N r N symbolic Fast method to slow r p r q N r p ( 1)z Cp z C q z D p (r) codes with analysis and p0 q 0 p 0 help of verification Where sallen-key p topology DAp(r) q (r) B p q (r) Paper Take a nth Better p0 2 order pole frequency In the above expansion, terms are included for from (n+2)th performance. order filter to Smaller convenience that may be zero, e.g., BN (r) 0 if r > improve the overshoot and 0. filter more stability From the denominator of equation (2.4)
Paper Increase the Reduce the cost NNNN r r 3 number of and complexity a2 T zpp D (r) z a2 D (r) r p r T p order with r0 p 0 p 0 r 0 the help of Hence, equation (2.4) can be written as a sallen-key N topology due zC pN p to this filter p0 H (z) behaves like NNr z p a2 D (r) a ideal filter. rp T pr00Paper Design a Simplicity and Applied the This is the direct form of equation (2.2) with 4 filter using economy of double the filter coefficients given by double programming precision N precision are easy method only ECpp method to in N r overcome the denominator F a2 D (r) numerical of transfer p r T p r0 accuracy function and rather than calculated the Using double precision method they direct form denominator designed the direct form of Butterworth low-pass coefficients,
www.ijera.com 240 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
since thenearest to origin maximum of frequencies but not numerator modifying in the normalized edge frequency. coefficients Chebyshev filters phase variation depends are integer. upon the chebyshev polynomial order, that is, if the Table 2.1:- Comparison Table of butterworth chebyshev filter has greatest polynomial order then filter desing filter gives the worst phase response. Using this technique Lopez et al give the improvement in the delay response of chebyshev filter. For this technique III. 3.Chebyshev Filter Lopez et al using equation (3.2) because they use
There are two types of chebyshev filters [1]. normalized edge frequency c =1rad/sec. For Type 1 chebyshev filters are all pole filters and exhibit equiripple behaviour in the pass band and a finding the roots of TN (x) , they have to calculate monotonic characteristic in stop band. Type 2 the value of X such that TN (x) =0 and find the value chebyshev filters contain both poles and zeros and of x. Now this value put into the chebyshev transfer exhibit a monotonic behaviour in pass band and function and find that no zeros at x=0. Now they equiripple behaviour in stop band. The zeros of this modified the chebyshev polynomial such that type of filter lie on the imaginary axis in the s-plane. The magnitude squared of the frequency response TN (x) has a pair of zeros at the origin and edge characteristics of a type 1 chebyshev filter is given as frequency does not change. This change of variable
2 1 consists in the shifting of the nearest root to origin at |H ( ) | (3.1) the same origin as two roots equal to zero. 22 1 TN ( ) In short we conclude that for low orders p filters the delay response will get considerably better Where is a parameter of the filter related to the after they made the proposed mapping. Additionally to this improvement, the gain response gives us the ripple in passband and T (x) is the Nth order N possibility to make an LC ladder realization t be chebyshev polynomial defined as equally terminated for even order function. 1 TN (x) cos( N cos x ) |x| 1 (3.2) 1 Paper 2 = cosh(Nx cosh ) |x| > 1 (3.3) Another method to designing chebyshev filter is given by Hisham L. Swady [7]. Swady The chebyshev polynomials can be introduced a new generalized chebyshev filter- like represented by a recursive equation approximation for analog filters. This analyse only for odd order filter. This work presents a general TNNN11(x) 2 xT (X) T (x) N =1,2..... (3.4) approach for analog filter designing for chebyshev-
like approximation with odd order.In this approach The type2 chebyshev filter magnitude chebyshev-like filter differ from classical chebyshev squared of frequency response is given by filter in the ripple factor which is not equal amplitude 1 |H ( ) |2 (3.5) with classical one. In this method consider the general magnitude T 2 s squared function of a low pass filter of order N N p (N=2n+1):- 1 2 2 1 2 s |F (jw) | (3.6) TN 22 1 P (w ) Where 22 1 Where is the Nth order chebyshev polynomial p Where is the passband gain and is the minimum and is the stopband frequency. p value of a |H( )|.This approach is equal to classical 3.1 Designing Method chebyshev filter authr satisfy the following condition: Paper 1 22P(w ) 1 In designing and analysis of chebeyshev 0<
www.ijera.com 241 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
Swady approach depends on choosing the chebyshev polynomial in cylindrical coordinates 22 p |H( )| 1 in passband and obtained a formula system. Where xy. Now chebyshev after server manipulations:- polynomial can be represented in the form of new max[P(w2 )] 1 (3.7) 1 variable ρ a TN ( ) cos(Ncos ) - 22 For a special case with choosing P(w ) TN (w) 1<|ρ|<1 (3.10) 1 equation (3.5) becomes identical to magnitude TN ( ) cos(Ncosh ) |ρ|>1 (3.11) squared function of the chebyshev filter. The value of By the use of special transformation ρ can be n is determined from the stopband requirement that represented as satisfy: w |F(jw)| , w w (3.8) cos( ) -π ≤ w ≤ π (3.12) s s 0 2 Where s and ws are the stopband tolerance and Where 0 is the maximum value of ρ. From the stopband edge frequency. By an analytical equation (3.12) it is notice that as w varies from 0 to manipulation Equ(3.8) can be transformed into the following result: π, ρ will vary from to 0 and vice versa. 2 1 Therefore, they conclude that this transform converts 2 s (3.9) the polynomial to low pass filter. if we wish to design P(w ) 2 p 1 a high pass filter from the polynomial than ρ can be From the equ (3.7) and (3.9) calculate the represented as: filter parameters and pole location of the filter. w sin( ) -π ≤ w ≤ π (3.13) From the observation it is clear that the new 0 2 proposed approximation has non-equal ripple in Now from equation (3.12) for low pass filter passband and more flatness than chebyshev filter (as polynomial can be defined as: shown in fig) and improvement in time domain response can be observed. uv22 T ( ) cos{Ncos1 ( )} -1<|ρ|<1 (3.14) N 0 2 uv22 T ( ) cos{Ncosh1 ( )} |ρ|>1 (3.15) N 0 2 Where w= uv22 is mapped onto a 2D plane of frequencies. Similarly, we can represent the polynomial for high pass filter. To realize this filter
0 can be represented in term of order of filter and attenuation require for side bands. 1 0 cosh(coshb / N) (3.16) Where N is the order of the filter and b is the absolute value of the attenuation and is defined by (attenuation in dB /20) b 10 (3.17) Fig.3.1: Passband performance of the proposed They also defined the value of stop band design filter (“A”) & the Chebyshev filters (“B”) frequency ws and pass band frequency wp as
11 Paper 3 wbs 2cos [1/{cosh(1/ Ncosh )}] (3.18) Another method of design a chebyshev filter is given by Sunil Bhooshan and vinay kumar [8]. 1 They proposed a designing a 2-dimensional 1 cosh{(1/ N)cosh (b / 2)} wp 2cos chebyshev non-recursive filter with linear phase for cosh(1/ N cosh1 b ) processing image. In this designing of filter pass band to stop band ratio is critical because they design for (3.19) the side bands to be a certain amount lower than the Now from this approach we can design a 2D pass band. linear filter with the help of chebyshev polynomial. To convert the 1D Chebyshev polynomial, They conclude that if the order of the filter increases defined in equation (3.2) and (3.3), to 2D they change than the transition band becomes sharper and sharper. the variable X into a new variable ρ, which represent In this approach no approximation involved,
www.ijera.com 242 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
computation time is less and it is easier to the power line frequency 60 Hz and its harmonics be understand. eliminated. For creating a null in the frequency response of a filter at a frequency w , we simply 3.2 Discussion 0 introduce a pair of complex-conjugate zeros on the In the first paper Lopez and Fernandez give the desigining of modified chebyshev filter by unit circle at an angle w0 . That is, shifting the frequency near to origin. Due to this jw ze 0 improvement in the delay response of the chebyshev 1,2 filter but it is only for even order chebyshev filter. (4.1) After that we discuss the paper of Hisham L. Swady Thus the system function for an FIR notch filter is where they give the designing of chebyshev filter by iw0011 iw H(z) b0 (1 e z )(1 e z ) using analog filter. They used the magnitude 12 response of low pass filter of order N and convert b00(1 2cos w z z ) into chebyshev filter. Due to this technique we get (4.2) the more flat response in the pass band frequency But in the FIR notch filter notch has a relatively large range but this technique is valid only for odd order bandwidth, which means that other frequency chebyshev filters. After that we discuss the paper of components around the desired null are severely Sunil Bhooshan and Vinay Kumar where they give attenuated. To reduce the bandwidth of the null we the designing of a 2D chebyshev filter using the 1D introduce poles in the system function. Suppose that chebyshev polynomials. It is very easy to design, no we replace a pair of complex-conjugate poles at approximation involved and complexity is less. jw0 p1,2 re 3.3 Comparison Table (4.3) Paper Approach Strength Weakness The effect of poles is to introduce a resonance in the Paper Shifting the Improvement Only for vicinity of the null and thus to reduce the bandwidth 1 frequencies in delay even order of the notch. The system function for the resulting nearest to response chebyshev filter is origin and filter not change 1 2cos w z12 z (4.4) in the cut- H (z) b 0 off 0 1 2r cos w z1 r 2 z 2 frequency 0 In addition to reducing the bandwidth of the Paper use Mangnitude Only for notch, the introducing of a pole in the vicinity of the 2 magnitude response is odd order null may result in a small ripple in a pass band of the response of more flat in chebyshev filter due to the resonance created by poles. The low pass pass band filter effect of ripple can be reduced by introducing poles filter and frequencies and/or zeros in the system function of notch filter. convert into
chebyshev 4.1Designing Method filter Paper 1 magnitude In designing and analysis of Notch filter Lee response and Tseng [9] give the design of 2D notch filter Paper Convert 1D Complexity is using band pass filter and fractional delay filter. For 3 chebyshev less and no designing of 2D notch filter decompose the filter into polynomial approximation the 2D parallel-line filter and straight-line filter. int 2D involved Then, the parallel-line filter is designed by band pass polynomial filter and the straight-line filter is designed by Table 3.1:- Comparison Table of chebyshev filter fractional delay filter. The purpose of this paper is to desing establish the relation between 2D notch filter and the fractional delay filter such that 2D notch filter can be IV. Notch Filter designed by using well-documented design method Notch filter [1] contains one or more deep of fractional delay filter. notches. It has perfect nulls in its frequency response The ideal frequency response of 2D notch filter is characteristics. Notch filters are very useful in many given by applications where the specific frequency D(w , w ) 0 , (w , w ) (w w ) components must be eliminated. For example, 12 1 2 1NN , 2 instrumentation and recording systems require that =1,others (4.5)
www.ijera.com 243 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
Where ( ww1NN , 2 ) is the notch frequencies. Where H B1 is the 1D band pass filter. Moreover, the straight-line filter is given by Now design a 2D filter H (z12 ,z ) which is H (z ,z ) H (z) | sB1 2 2 z z z approximate the D(w12 , w ) . They choose the 12 simple algebraic method to make the IIR notch filter. Where H B2 is another 1D band pass filter and In the simple algebraic method, the design of 2D w2N notch filter is decompose into 2D . Now put the value in the equation w1N parallel-line filter H p (z12 , z ) and 2D straight-line (4.6), the design notch filter is given by
filter H s (z12 , z ) . The transfer function of 2D notch 11az1 z 2 z z 1 filter is given by 2 2 1 2 H(z12 ,z ) 1 1 1 2 2 1 1 11raz2 r z 2 rz 1 z 2 HHH(z,z)1 2 1ps (z,z) 1 2 (z,z) 1 2 (4.9) (4.6) Where r is the small positive number and Where the parallel-line filter is 12 a 2cos(w2N ) . 1 a2 a 1 z 2 z 2 H p (z12 ,z ) 1 Case 2: When | w | | w | , then parallel-line 21a z12 a z 12NN 1 2 2 2 filter is design by the 1D band pass filter. (4.7) H (z ,z ) H (z) | With the coefficients p1 2 B 1 z z1 BW Where is the 1D band pass filter. Moreover, the 1 tan straight-line filter is given by 2cos(w2N ) 2 a , a 1 2 H (z ,z ) H (z) | BW BW sB1 2 2 z z z 1 tan 1 tan 1 2 2 2 Where is another 1D band pass filter BW is a small positive number. Now for designing w straight-line filter they used the analog inductance- and 1N . Now put the value in the equation resistance network and bilinear transformed. w2N (4.6), the design notch filter is given by 1 1 1 1 1z1 z 2 z 1 z 2 Hs (z12 ,z ) RLLRLLRLLRLL 1 2 1 1 2 1 2z1 1 2 z 1 1 2 z 1 z 1 11az z z z RRRR1 2 1 2 1 1 1 2 H(z12 ,z ) 1 1 1 2 2 1 1 (4.8) 11raz1 r z 1 rz 1 z 2 Where the parameters are (4.10)
1 1 Where a cos(w2N ) . L1 , L2 w1N w2N Case 3: if ww , then 2D notch filter is tan tan 12NN 2 2 and R is a small positive number. For the bounded 11az1 z 2 z z 1 1 1 1 2 input/bounded output (BIBO) stability of the 2D H(z12 ,z ) 1 1 1 2 2 1 1 11raz r z rz z notch filter, the parameters need to satisfy the 1 1 1 2 (4.11) constraints L1 > 0 and L2 > 0. This implies that notch And if ww , then 2D notch filter is frequency needs to satisfy the following condition:- 12NN
w 0 and w 0 1N 2N 1 2 1 1 11az1 z 1 z 1 z 2 The above constraints limit the applicability H(z12 ,z ) 1 1 1 11raz1 r 2 z 2 rz 1 z 1 of notch filter. so they used the fractional delay filter 1 1 1 2 to design the straight-line filter such that constraint (4.12)
can be removed. Where a cos(w2N ) . This is designing of the 2D Now the proposed design method divides into 3 notch filter using band pass filter and fractional delay following case: filter. This is the close form design, so it is easy to Case-1: when |w12NN | | w | , then parallel-line use. filter is design by the 1D band pass filter. H (z ,z ) H (z) | Paper 2 p1 2 B 1 z z2
www.ijera.com 244 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
Another method of designing of notch filter Where the positive definite symmetric matrix Q and is given by Tseng and Pei [10]. In this paper they vector P are given by represent the designing of 2D IIR Notch filter and 2D Q C(w,w ) Ct (w,w )dw dw FIR Notch filter. They explain the 2D IIR Notch 1 2 1 2 1 2
filter by simple algebraic method which we have discussed in the above paper and 2D FIR Notch filter P C(w , w )dw dw 1 2 1 2 explain by Lagrange method. FIR filter requires more
arithmetic operation than IIR filter during Therefore, the optimal solution is given by implementation. AQCCCP[11 (w ,w ){Ctt (w ,w )Q 1 (w ,w )} 1 (w ,w )]Q 1 A 2D linear phase FIR digital filter with transfer opt1 NN 2 1 NN 2 1 NN 2 1 NN 2 (4.19) function H (z ,z ) has a frequency response 12 Based on this discussion, they design a 2D
FIR Notch filter with notch frequency (w12NN , w ) . NN22 H(eiw1 ,e jw 2 ) e jN 1 w 1 e jN 2 w 2 h (m,n)e j(mw 1 nw 2 ) They used this filter to remove the single sinusoidal m N11 n N interference superimposed on an image. For real-time (4.13) processing purpose, the 2D notch filter is preferred. Where the impulse response h(m,n) satisfy the half- In 2D notch filter case, the notch frequency can be plane symmetrical condition, i.e. chosen exactly the same as the sinusoidal frequency, h(m,n) = h(-m,-n) (4.14) so there is nearly no information loss and notch filter After some manipulation it is easily shown that the technique is much more efficient than conventional magnitude response G(w , w ) of H (z ,z ) is FFT method in computational complexity. 12 12 given by Paper 3 R Another method of designing of Notch filter Ga(w,w)1 2 (i)g(w,w) 1 2 (4.15) is given by Shen et al [11]. In this paper based upon i1 transfer function, they design a digital notch filter to Where the dimension eliminate the 50 Hz noise. The filter was realized by R (N1 1) N 2 (2 N 1 1) and the software implementation in VC++ environment and simulated by MATLAB. After this process g (m,n) cos(mw nw ) (4.16) i 12 power-line interference removes effectively. Its ai(i) h(0,0) 1 operation was simple and applicable. According to given sampling frequency, the transfer (4.17) 2hi (m,n) 1 function could be constructed as: They rewrite the in the following vector 1 z fs form H (z) f (4.20) s t 50 GC(w,w)1 2 A (w,w) 1 2 (4.18) 1 z jw Where Substituting ze in equation (4.20)
wfs t ffsssin A [ a (1) a (2)....a(R)] jw He(jw ) e 2 100 2 (4.21) t wfs C(w1 ,w 2 ) [ g 1 (w 1 ,w 2 ) g 2 (w 1 ,w 2 )....gR (w 1 ,w 2 )] sin 100 They design to approximate the In the digital domain the range of the frequencies is and in analogue domain , ideal notch filter frequency response in least square 02w 0 ffs error sense. In this sense, the filter design problem where f s is the sampling frequency. formulated as follow: jw If f= f s then w 2 , and |H (e ) |could simply express as: Minimize| G (w , w ) 1|2 dw dw 1 2 1 2 t 2 jw sin( f) | A C(w1 , w 2 ) 1| dw 1 dw 2 |H (e ) | f tt sin A QA 21 P A 50 t Subjected to C (w12NN , w ) A 0
www.ijera.com 245 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
Another design is presented by the ffss And the phase shift Srisangngam et al [12]. This paper represents the 2 100 design of symmetrical IIR Notch filter using pole 0f 50. k ( k 0,1,2...) position displacement. It has modified the transfer |H (jw) | function equation with the optimum pole positions 50f 50. k ( k 0,1,2...) for the symmetry of pass-band gain and transition- The magnitude frequency characteristics shown in band gain. First, they present the IIR Notch filter figure below, which was drawn in MATLAB. design then they present IIR Notch filter proposed design and finally simulates the result. Now the transfer of the IIR Notch filter is given by 1 2cos w z12 z 0 (4.22) H(z) b0 1 2 2 1 2r cos w0 z r z From this equation it is show that the angle of pole is the same as angle of zero. The simulation result of this method reveals the uncontrollable gain at frequency DC and π. The constant for controlling Fig 4.1:- Magnitude Frequency Characteristics of the pass-band gain rate b0 is unknown because of filter this unknown constant asymmetric and
uncontrollable pass-band gain at DC and π While passing through the transfer function, frequencies. Then they modified the pole angle and the series of signals could be demodulated with the the result will be a transfer function of the IIR Notch frequency component. Notch filter could be obtained filter as shown by taking the demodulated series from original series. 12 According to transfer function, the differential 1 2cos w z z H(z) b 0 (4.23) equation would be calculated. 0 1 2r cos( w ) z1 r 2 z 2 Assuming that the original signal series was x(n) and 0 demodulated signal series was y(n) 1 2r cos(w ) r2 b 0 (4.24) f 0 s 2 2cos w0 y(n) y n x (n) x(n fs ) 50 2 1 1 r Then the band-stopped series R (n) could be ww00cos cos (4.25) e 2r calculated as follow: After changing the pole position of an IIR Notch ffss filter, it could control the magnitude response at DC Res(n) x n x (n) x(n f ) 2 100 frequency and π frequency but it could not control transition-band gain. Therefore, the magnitude ff ss response of an IIR Notch filter is not symmetry. 2 100 Now they add pole and zero to compensate magnitude of transition-band as shown in equation The power-line frequency interference has below ˆ ˆ been removed in series Re (n) . HH(z) b0 (z) HADD (z) According to the principle of transfer 12 1 2cos w0 z z function, the FIR notch filter was designed via H(z) 1 2 2 software procedure in VC++. Through the specified 1 2r1 cos(w 0 1 ) z r 3 z x(n) and the differential equation f the notch filter, 1 2r cos(w ) z1 r 2 z 2 the demodulated series y(n) and R (n) could be H (z) 2 0 2 2 e ADD 1 2r cos(w ) z1 r 2 z 2 calculated.In this paper, the transfer function H(z) of 3 0 3 3 digital filter can demodulated the frequency Because of the zero position has no effect on the components that was integral multiple of 50 Hz. The transition-band gain and pass-band gain. Thus, 2 experiment showed that the 50 Hz notch filter should be assigned as zero and r r r r . worked properly with stable and reliable 1 2 3 performance. The 13 is an angle of pole that will symmetrical with notch frequency in both positive Paper 4 and negative sides.
www.ijera.com 246 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
Where is easy to 22 use. ˆ 1 2rr cos(w00 ) r 1 2 cos(w ) r b0 2 Paper 2D FIR For real- FIR filter 2 2cosw00 1 2 r cos w r 2 Notch time requires filter processing more The magnitude response of propose IIR explain by purpose, arithmetic Notch filter could control the pass-band gain and Lagrange the 2D operation transition-band gain. Because of distance from zero method. notch filter than IIR position of each side are equal. Then, the magnitude is preferred filter during response of an IIR Notch filter is symmetry. In and no implementati conclusion, the proposed method for the design of information on and only IIR Notch filter by modifying the pole position can loss and fixed 2D ensure the symmetry of pass-band and transition- notch filter notch filter band gain at the target level. technique is not discuss much more the 3D notch 4.2 Discussion:- efficient filter design In paper 1 Lee and Tseng give the design of than using same 2D notch filter using band pass filter and fractional convention method delay filter and using simple algebraic method they al FFT design a IIR Notch filter. Due to this technique they method in remove the sinusoidal interferences corrupted on a computatio desired signal and it is closed-form design, so it is nal easy to use but they explain only for 2D Notch filter complexity. not discuss the 3D notch filter design using same Paper Based Power-line method. In paper 2 Tseng and Pei presents the 3 upon interference designing of 2D IIR Notch filter and 2D FIR Notch transfer removes filter. They explain the 2D IIR Notch filter by simple function effectively. algebraic method 2D FIR Notch filter explains by design a Its Lagrange method. We observe that FIR filter requires digital operation more arithmetic operation than IIR filter during notch filter was simple implementation and they also could not explain the to and 3D notch filter design using same method. In paper 3 eliminate applicable. Shen et al based upon transfer function design a the 50 Hz digital notch filter to eliminate the 50 Hz noise. The noise with filter was realized by the software implementation in the help of VC++ environment and simulated by MATLAB. software After this process power-line interference removes implement effectively. Its operation was simple and applicable. ation in In paper 4 Srisangngam et al represents the design VC++ of symmetrical IIR Notch filter using pole position environme displacement. Magnitude response of an IIR Notch nt and filter is symmetry and by modifying the pole position simulated can ensure the symmetry of pass-band and transition- by band gain at the target level. MATLAB Paper Design of Magnitude 4.3 Comparison Table:- 4 symmetric response of Paper Approach Strength Weakness al IIR an IIR Paper Using Remove the only fixed Notch Notch filter 1 simple sinusoidal 2D notch filter using is algebraic interference filter not pole symmetry method s corrupted discuss the position and by design a on a 3D notch displacem modifying IIR Notch desired filter design ent the pole filter signal and using same position it is closed- method can ensure form the design, so it symmetry
www.ijera.com 247 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
of pass- Where, x is the discrete sequence obtained from the band and input signal sampled at time (nT), T is the sampling transition- interval, y is the output sequence and (NT) is the time band gain interval of the delay. Taking the z-transform of at the target equation (5.1), then we get level. Y (z) H(z)1 1 2 zNNN z 22 (1 z ) 1 X (z) Table 4.1: Comparison table of Notch Filter (5.2) Design This transfer function has (2N) zeros. Each double zeros are equally spaced around the unit circle in the V. COMB Filter z-plane at location given by: Comb filter [1] is a type of notch filter in j2/ k N which nulls occurs periodically across the frequency zk e, k 0,1,2....(N 1) band, hence comb has periodically space teeth. Comb The frequency response of the filter is: filters are use in a wide range of practical systems H (ejwT ) (1 e jwNT )2 such as in the rejection of power-line harmonics, in 1 separation of solar and lunar components from Thus, jwT ionospheric measurements of electron concentration. |H1 (e ) | 2(1 cos(wNT)) To illustrate a simple form of a comb filter, consider Which means that zeros occur at the frequencies that a moving average (FIR) filter describe by the are integral multiple of Hz and is the difference equation (fs / N) f s 1 M sampling frequency. yx(n) (n k) Now, if in equation (5.2), the subtraction is replaced M 1 k 0 by addition, the transfer function becomes The system function of the FIR filter is Y (z) NNN 22 1 M H(z)1 1 2 z z (1 z ) Hz(z) k 2 X (z) M 1 k 0 (5.3) And its frequency response is Thus, there are double zeros equally spaced around M 1 jwM sin w unit circle at e 2 2 H (w) j2 ( k 0.5)/ N M 1 w zk e, k 0,1,2....(N 1) sin 2 And the frequency response becomes From the above equation we observe that the jwT |H 2 (e ) | 2(1 cos(wNT)) filter has zeros on the unit circle at In this case zeros occur at the frequencies j2k M 1 z e, k 1,2,3....., M that are integral multiple of (fs / 2 N) Hz. Note that the pole at z=1 is actually cancelled by the Now, they represent the conventional zero at z=1, so that in effect the FIR filter does not COMB filter through cascade both transfer function contain poles outside z=0 [1]. and together with DM are cascade with conventional filter. This is the suggested structure given by them. 5.1 Designing Method DM is cascaded with conventional filter’s structure in Paper 1 order to convert the analog input signal into uni-bit For designing of Comb filter Makarov and digital waveform. Odda [13] present a paper. In this paper the second Now, the simulated model for second order order COMB filter is consist of two conventional COMB filter computer program is operated by COMB filter in cascade. The purpose of this paper is reading the parameters of the filter and DM. The to simulate this type of filter by using delta Mean Squared Error (MSE) is calculated to modulation (DM) technique which gives the investigate the performance of suggested structure realization low cost, simplicity and an efficient result according to different values of hysteresis width of in real time processing. After that the suggested DM and timing factor of program processing. MSE is structure has been simulated through an appropriate defined as follow: computer program to achieve the desired frequency fs /2 response. This structure is based on DM as analog to 2 2 MSE( Hds (f) H (f)) digital converter (ADC) for continuous input signal. fs f 1 The second order COMB filter is: Where H (f) and H (f) are the desired and y (nT) x(nT) 2 x(nT NT) x(nT 2 NT) d s 1 simulated frequency response respectively. (5.1)
www.ijera.com 248 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
2 In this paper, they constructed and designed j N of second order filter using computer simulation. Where WeN This design simplifies the structure of this filter using th DM as analog to digital converter for input signal. Zeroing every M value of the result of The simulation results are obtained that MSE multiplying the FFT matrix times the input, x, is decrease by increasing clock-frequency and equivalent to multiplying the input vector by a decreasing hysteresis width of DM. multiplication of the FFT matrix with every M th row zeroed. Let A be the FFT matrix with every M th Paper 2 r Another method of designing of COMB row zeroed. Thus, Ar is given by filter is given by J. L. Rasmussen [14]. One 0 0 ... 0 application of the COMB filter is to remove of the 11 ... (N 1) fundamental sinusoidal signal and its harmonics from WWNN 12 ... 2(N 1) a signal of interest (SOI). In this condition, it is not WWNN feasible to use simple IIR or FIR filter to remove 1 these interfering signals without significant N 0 0 ... 0 degradation of SOI. In this case, we can use the FFT COMB filter to remove these interfering signals but the FFT method has not often used because of its 1(N 1) ... (N 1)(N 1) WWNN computational complexity. In this paper they present a new formulation of the FFT COMB filter that Let Cr be the result of multiplication, eliminates all limitation of the FFT method and CAA 1 (5.4) provides advantages over the IIR or FIR COMB rr filter. Ar can also be represented as the matrix (A- Ae ), In earlier method of FFT COMB filter where A is the matrix A with all its rows zeroed out collect the N samples of data, performing an FFT on e th this data, zeroing every M th value of the resulting except the first row and every M row. Then from data, and then performing IFFT to obtain the time equation (5.4) 1 1 domain filtered result. The FFT and IFFT both take CAre(A A ) IAAe (5.5) order NNlog multiplications or 2NN log th 2 2 Since only every M element of the multiplications for whole process. Due to this process columns of A is non-zero, the summation is only computational complexity increases. There is another e 1 problem in FFT COMB filter if we want to extend over 0 to (N/M-1). The elements of matrix AAe are this method to larger bandwidth. First, we assume sums of N/M products and can be expressed as that the original FFT frequency resolution must be 1 N maintained. Under this constraint, if the SOI n mod(i j, ) 0 AA1 bandwidth is increased, both sampling frequency e MM 0 otherwise f s and N need to be increased. Thus, the FFT size th has increased. The FFT/IFFT processing time is also Then, the ij element of increased. Cr is For solving the above problem of FFT COMB filter they give the new design of FFT COMB filter. They use the FFT and IFFT matrix 1 ij 1 multiplication process to simplify the above M problems. It observe that if N and M are choose such that N/M is an integer, the product of the IFFT matrix 1 N Cr (ij) n mod(i j, ) 0,and i j with the FFT matrix with every M th rows zeroed MM produces a sparse, multi-diagonal structure. Let A 0 otherwise and A1 be the size N FFT and IFFT matrices. A is given by (5.6) 1 1 ... 1 Thus, a multi-diagonal matrix results with 2M-1 diagonal, include the main diagonal. If the input 1WW1 ... (N 1) 1 NN vector, x, is multiplied by this matrix, then in general N the result for the k th output is (N 1) (N 1)(N 1) 1WWNN ...
www.ijera.com 249 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
k M k M 1 1 1 2 1 HH(z) [H12 (z)] [ (z )] xˆk x k x N (5.7) j0 kj M M (5.9) For k equal 0 to N/M-1. Because of the cyclic nature Where 11 z M1 of the matrix Cr , only the first N/M outputs need to H1(z) 1 ; be calculated as in equation (5.7). The rest of the (N- Mz1 1 N/M) outputs can be calculated using the first N/M 11 z MM12 outputs as M1 H 2 (z ) M xˆˆ x x x (5.8) Mz1 1 kmod(k, N ) k mod(k, N ) 2 MM (5.10) For k equal N/M to N. From this method the total number of The choice of M1 is a matter of the compromise multiplications reduces to N/M. Thus, this method of between having less complex polyphase components implementation COMB filtering is a factor of in the first stage, and making the filter in the second stage working at, as much as possible lower rate. For 2MN log2 more efficient than the FFT method. This simplified method solves some of problems that this condition they proposed that the factors M1 and were identified. First, computation time has been M 2 are close in values as much as possible to each reduced. Second, there is no restriction that a radix 2 value for the size of the FFT must be used. The only other, with MM12 . k1 and k2 are the number of restriction is that N/M must be an integer. There is cascade comb filters at first and second sections still requirement that all N samples must be collected respectively. The filter in the first section is realized before the output can be obtained. The time involved in the non recursive form, might be too large for some Time Division Multiple k1 1 M1 1 Access or network systems. It can be implemented as Hzk1 (z) i (5.11) an FIR filter. Now, its computational complexity is 1 M1 i0 reduce so we can use this filter is to remove of the fundamental sinusoidal signal and its harmonics from Using ployphase decomposition, the filters of the first a SOI. section can be moved t operate at a lower rate.Two- stage structure is shown in figure
Paper 3
Another method of designing a COMB filter k is given by Gordana Jovanovic Dolecek [15]. This 2 H paper presents the design of the 2 stage COMB based M1 2 decimation filter with a very low wideband pass band droop and a high stop band attenuation of the overall filter. In this paper they used polyphase decomposition method to stage 1 to avoid the filtering at the high input rate. Then, they applied the compensation filters for both stages and sharpening M 2 technique is applied in the 2 stage to the cascade of comb filter and the compensation filter. The comb filter must have a high alias Fig 5.1:- Two-Stage Structure rejection around the zeros of comb filter and a low Now, they try to improve the pass band pass band drop in the pass band in order to avoid the distortion of the decimated signal. However the comb characteristics of the overall structure. To improve filter has a high pass band droop and low folding the overall pass band characteristics, they propose to use compensation filters: band attenuation. So the main goal of this paper is to M1 M 12 M 1 k 11 decrease the pass band droop and keeping good stop Gz1(z ) [B(1 Az )] (5.12) band attenuation. In this paper first, they describe two MMM2 k22 (5.13) stage structures, then introduce the compensation Gz2 (z ) [B(1 Az )] filter and finally give the proposed filter. Where the parameters k11 and k22 are related with For two stage structure they consider the case where the decimation factor M can be presented the parameters k1 and k2 , and 4 as MMM 12. The corresponding comb-based B 2 decimation filter is given as A [24 2]
www.ijera.com 250 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
The compensation filters (5.12) and (5.13) not depend on the values of M and M . In the have many favourable characteristics such as 1 2 parameters B and A equal in both compensators, and proposed design of the 2 stage COMB based decimation filter wideband pass band droop is very do not depend on M and M1 , filters are multiplier low and stop band attenuation is high of the overall less, and can be moved to a lower rate but the number filter. of cascade filters k and k depend on k and k . 11 22 1 2 5.2 Discussion:- The two-stage structure with the compensators is In paper 1 Makarov and Odda give the shown in figure below designing of second order COMB filter using delta modulation which gives the realization low cost, k Mk H 1 (z) G (z)G (z22 ) H (z) simplicity and an efficient result in real time 1 M1 1 2 2 processing. After that the suggested structure has been simulated through an appropriate computer program to achieve the desired frequency response. This structure is based on DM as analog to digital M converter (ADC) for continuous input signal. In 2 paper 2 J. L. Rasmussen gives the new method of designing a FIR COMB filter. They present a new formulation of the FFT COMB filter that eliminates Fig 5.2:- Compensated two-stage structure all limitation of the FFT method and provides advantages over the IIR or FIR COMB filter. They For proposed structure they apply the design a COMB filter using FFT and IFFT matrix k2 sharpening technique to the cascaded filters H 2 and multiplication process. Due to this technique computational complexity is reduced and there is no the compensator G (zM 2 ) , at the second stage. 2 restriction that a radix 2 value for the size of the FFT kM22 must be used but there is some restriction. N/M must HHG2c (z) 2 (z) 2 (z ) (5.14) be an integer, there is still requirement that all N In the sharpening technique they used the simplest samples must be collected before the output can be 2 sharpening polynomial in the form of 2HH obtained and the time involved might be too large for 2 some Time Division Multiple Access or network Sh{ H2c (z)} 2H 2 c (z) H 2 c (z) systems. In paper 3 Gordana Jovanovic Dolecek HH22cc(z)[2z (z)] (5.15) gives the designing of cascade comb filter. Using Where, Sh{x} means sharpening of x. The delay τ polyphase technique, compensation filter and sharpening technique they design a cascade comb is introduced to keep the linear phase of the filter. filter. Due to this technique there are some The overall proposed transfer function is advantages. We need amultiplier less structure and k1 M 1 k 2 M 1 M H p (z) H1 (z)G 1 (z )Sh{H 2 (z )G 2 (z )} the design parameters are practically do not depend (5.16) on the values of and but there is a one
disadvantage that is number of cascade filters and The corresponding structure is shown in figure below depend on and . k M 1 1 1 1 k1 i G (z) Hz1 (z) 1 5.3 Comparison Table:- M i0 1 Pape Approach Strength Weakness
r
Paper simulate low cost, kM21 M 1 second order simplicity and Sh{H 22 (z )G (z )} COMB filter an efficient
by using result in real
delta time Fig 5.3:- Proposed Filter modulation processing
(DM) This is the two-stage compensated technique sharpened COMB based decimator. The important Paper Using FFT Computationa N/M must features of the proposed filter are multiplier less 2 and IFFT l complexity be an structure and the design parameters are practically do matrix is reduced integer,
www.ijera.com 251 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
multiplicatio and there is there is still With B0 (s) 1and B1(s) s 1 as initial n process to no restriction requiremen conditions. design a that a radix 2 t that all N An important characteristic of Bessel filters COMB filter value for the samples is the linear-phase response over the pass band of the size of the must be filter. Bessel filter has a larger transition bandwidth, FFT must be collected but its phase is linear within the pass band. used. before the
output can 6.1 Designing Method be obtained Paper 1 and the Johnson et al present [16] the new time designing method of Bessel filter. They used the involved continued fraction to obtain the transfer function of might be Bessel filter is modified by adding a parameter a > -1 too large to the odd integers in the expansion. The resulting for some transfer function is realizable and has a time delay at Time w=0 which is independent of the order of the filter, Division and which may be varied from 0 to ∞ by changing α. Multiple So first we will discuss about the modified transfer Access or function and then discuss about the time delay. network The transfer function of generalized Bessel filter is systems. a Paper Using Multiplier Number of B (0) Haa (s)n , 1 (6.5) 3 polyphase less structure cascade n Ba (s) technique, and the filters n compensatio design Where k11 and a a a n filter and parameters Bn(s) M n (s) N n (s) sharpening are practically k22 depen Ma (s) and Na (s) are the even and odd part technique to do not depend d on n n obtained from the continued fraction expansion design a on the values k and k . cascade 1 2 a of M1 and a M n (s) a 11 comb filter Fn (s) Nsa (s) a 31 M 2 . n s 1 Table 5.1:- Comparison table of COMB filter an21 desing s (6.6) VI. Bessel Filter 0 When a=0, Bn (s) is obtained i.e. Bnn(s) B (s) . Bessel filters [1] are a class of all-pole filters a that are characterized by the system function Bn (s) is Hurwitz if and only if a≥-1. 1 To obtain the expressions for a , from H (s) (6.1) Bn BN (s) equation (6.6) note that
a a 11 Where BN (s) is the Nth-order Bessel polynomial. Fn (s) a2 These polynomials can be expressed in the form sFn1 (s) N From which we obtain B(s) a sk (6.2) Nk Ma(a 1)M a22 sN a k 0 n n11 n Naa sM 2 Where the coefficients { ak } are given as nn1 (2 N k)! or a k=0,1,.....,N (6.3) k 2Nk k!(N k)! a a22 a M n(s) (a 1)M n11 (s) sN n (s) Alternatively, the Bessel polynomial may be aa2 generated recursively from the relation Nnn (s) sM1 (s) BB(s) (2 N 1)B (s) s2 (s) (6.4) Therefore we may write NNN12 a a22 a Bn (s) sB n11 (s) (a 1)M n (s) (6.7)
www.ijera.com 252 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
a2 a2 Or, since Mn1 is the even part of Bn1 , we have aad THn(0) n (s) | s0 1 ds a2 a 2 a 2 M n1(s) B n 1 (s) B n 1 ( s) 2 From equations (6.5) and (6.9) is given by And therefore a A1 a(2s a 1) a22 (a 1) a Tn (0) (6.10) Bn (s) B n11 (s) B n ( s) A 22 0 a2 (6.8) By equation (6.7) and (6.9), A1 is equal to Bn1 (0) . From equation (6.8) a a2 B (0) n But from (6.8), B (0) is equal to n , ak n1 (a 1) Bnk (s) As , n=1,2,3..... (6.9) k0 A which is turn is 0 . Substituting this A list of the Ak for various value of n is given in (a 1) Table 6.1, where they define expression for into equation (6.10)
i a i, i 1,2,3... (6.10) a 1 Tn (0) (6.11) N a 1 An An1 An2 An3 An4 ThusA nthe5 time delay at w=0 is independent of n, as in the case of Bessel filter. Also they note that if -1 < a 1 1 a 1 < 0, then 1 < Tn (0) < ∞, and if a > 0 then 0 < 2 1 a 3 13 Tn (0) < 1. The dividing line is the case of Bessel 3 1 a 2 35 1 3 5 3 filter a = 0, in which case Tn (0) =1. 4 1 The modified Bessel filter transfer function is a 2 5 335 3 5 7 1 3 5 7 realizable function which reduces to that of the 5 1 3 5 357 43 5 7 3 5 7 9 Bessel1 3 5 function 7 9 when the parameter ‘a’ is zero. The time delay of the modified Bessel filter at w=0 is a independent of the order of the filter and can be Table 6.1:- Coefficient of Bn varied from 0 to ∞ by changing the parameter ‘a’.
The modified Bessel filter having transfer Paper 2 function (6.5) does not have maximally flat time Another method of designing a Bessel filter delay, except in the case a=0 of the Bessel filter. is given by Johnson et al [17]. They present an However, by keeping a near 0, one may retain extension of the Bessel filter where transfer function approximately the desirable phase properties of the is a rational function with finite zeros. They combine Bessel filter while obtaining a variety of amplitude the constant magnitude response of the all-pass filter response. with the linear phase response of the Bessel filter. By varies ‘a’ from 0 to in one direction or the other, This paper presents an extension to the Bessel filter observe the time delay of Bessel filter. Let us based on interesting approximation suggested be consider the time delay T a (w) of the modified Budak [18]. n Budak’s approximation is obtained by considering Bessel filter at w=0. So, s ' the ideal normalized transfer function e in the a dH(w) (jw) form Tn (w) Re dw H(jw) eks eks ,0 (6.12) Where H ' (jw) represent differentiation with e(k 1)s When the Bessel filter transfer function is used to respect to (jw) and since both H ' (jw) and independently approximate eks and e(k 1)s , the H(jw) are real at w=0, then Bessel rational filter approximant results, given by ' a H (0) ' THn (0) (0) B (0) B [(k 1)s] H (0) esk H(s)nm , m , n 1,2,..., m n mn B (0) B (ks) Because Ha (0) 1. Therefore, mn n (6.13)
www.ijera.com 253 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
When k=1 equation (6.13) becomes equal to equation n (2n i)!si (6.1). For 0 < k < 1, equation (6.13) becomes B (s) (6.17) n ni nonminimum phase function and for k1 , it is a i0 2 (n i )!i! minimum phase function. By using (6.17) it is interesting to note that Now, they obtained the time delay from that of the s (2n)! denormalized Bessel filter BPnn (s) 2 2n n! d Where argBnn (j w) [1 f ( w)] dw n n!(2n i)!si Where Pn (s) 2 i0 (2n)!(n i)!i ! fn (w) 22 Thus for the special case, the Bessel rational filter wj[nn11 (w) j (w)] can be written as 22 1 P ( s) And j are the spherical Bessel function. For 2 n n1 H nn (s) 2 Pn (s) the normalized Bessel filter, the time delay is given Which is the (n,n) Pade [19] approximate of by the function es . T (w) 1 f (w) (6.14) nn So, in this paper they used to Budak’s and Pade And for the normalized Bessel rational filter, the time approximate to design a transfer function of Bessel delay is given by rational filter. They also combine the constant k Tmn(w) 1 kf n (kw) (1 k)f m [(1 k ) w ] magnitude response of the all-pass filter with the (6.15) linear phase response of the Bessel filter. Due to this they note that the time delay of the all-pass filter is From (6.15) and the MacLaurin series for fn (u) , flat for approximately twice the frequency range as given by that of the Bessel filter of same order. 2 n 2 n 2 2 n 4 2nu! 2n 2(n 2)u Paper 3 fun (u) 2 ... (2n)! 2n 1 (2n 1) (2n 3) Another method of designing a Bessel filter is given by Susan and Jayalalitha [20]. In this They notice that the Bessel rational filter has paper, they simulate the passive component L using maximally flat time delay in the case where a) k=1 Generalized Impedance Converter (GIC). This (normalized Bessel Filter), b) m=0 (denormalized simulated inductor is applied for the realization of Bessel Filter), and c) m=n. There is an another Bessel filter. The circuit simulation is done using PSPICE. The use of passive component such as important case of (6.13) where m=n and k=0.5, which is an all-pass filter with maximally flat time delay inductor in analog circuit is very difficult at low given by frequencies because the size of the inductor becomes 1 very large. Due to this reason they simulate the 2 w inductor and then this inductor used for realization of Tnn(w) 1 f n ( ) (6.16) 2 Bessel filter. From (6.14) it is note that the all-pass time delay is For simulating the inductor they used the GIC. They 1 related to the Bessel filter byTT2 (w) w . obtained the simulated inductor circuit from the GIC nn n 2 which consists of the active component the Therefore, the time delay of the all-pass filter is flat operational amplifier along with the capacitor. The for approximately twice the frequency range as that GIC invented by Antoniou is given in figure (6.1). of the Bessel filter of same order. From the equation (6.2) Bessel polynomial can be defined by
www.ijera.com 254 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
Fig 6.1:- Antoniou
Circuit The input impedance with respect to ground is ZZZ Z 1 3 5 (6.18) ZZ24 Where
ZRZXZRZRZR1 1,,,, 2 C 2 3 3 4 4 5 5 then, sR R R C Z 1 3 5 2 (6.19) R Fig 6.3:- Frequency Response of Low Pass 4 Filter This is equivalent to an inductor. The value of L is Similarly to design a Bessel filter they replaced the RRRC inductor L in the basic circuit of the high pass and L 1 3 5 2 R all-pass filters by the simulated inductor as shown in 4 fig (6.5) and (6.6). The frequency response obtained If RRRRR1 3 5 4 and CC2 then using PSPICE is given in fig (6.7) and (6.8). L CR2 (6.20)
From this procedure they obtained the value of simulated inductor. Now this value of simulated inductor they used to design of Bessel filter. For simulation of Bessel filter they replaced the inductor L in the low pass filter by the simulated inductor as shown in fig (6.2) and the frequency response obtained using PSPICE is given in fig (6.3). Fig 6.5:- High Pass Filter using Simulated L
Fig 6.2:- Circuit Diagram of LPF with Simulated L Fig 6.6:- All-Pass Filter using Simulated
www.ijera.com 255 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
transformation method named as bilinear transformation. To get the Bessel polynomials they used the following MATLAB function [z,p,k]=besselap(N) % N-th order normalized prototype Bessel analog low pass filter [sos,g]=zp2sos(z,p,k) %Zero-pole-gain to second order section model conversion Using this MATLAB code they obtained polynomials factors as shown in table 6.1
Filter '''' Order D '(p) B1 (p) B 2 (p) B 3 (p) B 4 (p) Polynomials Factors
1 (p+1)
2 2 ( p +1.7321p +1)
Fig 6.7:- Frequency Response of High Pass Filter 3 2 ( p + 1.4913p + 1.0620)(p + 0.9416)
4 2 2 ( p + 1.3144p + 1.1211)( p + 1.8095p + 0.8920)
5 2 2 ( p + 1.1812p + 1.1718)( p + 1.7031p + 0.9211)(p + 0.9264) Table 6.1:- N-order normalized polynomials of Bessel
In designing of Bessel digital filter, first they design the Bessel digital low pass filter. For this they obtained the N-th order normalized Bessel analog low-pass filter transfer function from table 6.1 1 G(p) ' (6.21) DN (p) Then they obtained the N-th order Bessel order analog low-pass filter transform function by Fig 6.8:- Frequency Response of All-Pass Filter removing normalization of equation (6.21) HG(s) (p) (6.22) a p s In this paper first they obtained the c simulated inductor using GIC then design of Bessel Where is pass band cut-off frequency in rad/sec. filter from the low pass, high pass and all-pass filter c with the help of simulated inductor. Now, they transform H a (s) into H(z) using the bilinear transformation. They obtained Paper 4 M 12 Another method for designing Bessel filter b0k b 1 k z b 2 k z HH(z) (s) 21z1 is given by Chengyun et al [21]. In this paper a a s 12 T 1z1 a a z a z method for the design of digital Bessel filter is k1 0k 1 k 2 k (6.23) presented. It is based on MATLAB function besselap and zp2sos to obtain the Bessel polynomials. They Any multi-order Bessel filter can be obtain the analog filter transform function H a (s) by designed by using normalized Bessel polynomials search N-th order normalized Bessel polynomials. presented in Table 6.1. The general form of the first order polynomial is: Then, transform H a (s) into H(z) using the bilinear transformation and obtain the general coefficients D'(p) p d (6.24) function of digital Bessel filters from H(z). MATLAB provides a function called bilinear to implement this mapping. This mapping is the best
www.ijera.com 256 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
From equations (6.21), (6.22) and (6.23) they obtain determine the frequency response for 5th order Bessel relevant digital filter coefficients B, A digital filter as shown in figure below. (where B[ b 0, b 1, b 2], A [ a 0, a 1, a 2]):
cc B , ,0 (6.25) 11dd cc
dc 1 A 1, ,0 (6.26) 1d c Now, they consider the general form of second order polynomial is; D'(p) p2 kp c (6.27) Again from equations (6.21), (6.22) and (6.23) they obtain relevant digital filter coefficients B, A: Fig 6.9:- Bessel digital low-pass filter frequency response 2 2 2 c2 c c B 222,, 111kc c c k c c c k c c c (6.28)
2c22 2 1 k c c c c (6.29) A 1,22 , 11k c c c k c c c
From this method they obtain the general polynomials coefficients of Bessel filter. After designing of Bessel low-pass digital filter, they design the Bessel high-pass filter by similar way. The difference is removing normalization. For removing Fig 6.10:- Bessel digital low-pass filter frequency normalization they used the following equation: response
HG(s) (p) (6.30) a p c s The results of figure (6.9) and (6.10) show They obtained relevant digital filter coefficients B, A that the designed filter meet the requirements. of order 2 (where B[ b 0, b 1, b 2], A [ a 0, a 1, a 2]) 6.2 Discussion:- In the paper 1, Johnson et al present the 1 2 1 new designing method of Bessel filter. They used the B 222,, c k c k c k continued fraction to obtain the transfer function of c c c c c c Bessel filter .The resulting transfer function is (6.31) realizable and has a time delay at w=0 which is 22independent of the order of the filter, and which may 22c c c k c c A 1,22 , be varied from 0 to ∞ by changing α. But adding a c k c c c k c c parameter, in a Bessel filter transfer function, a > -1 (6.32) to the only for odd integers in the expansion not for And the relevant digital filter coefficients B, A of even integer. In paper 2, Johnson et al present an order 1 is: extension of the Bessel filter where transfer function 11 is a rational function with finite zeros. They combine B , ,0 the constant magnitude response of the all-pass filter dd cc with the linear phase response of the Bessel filter. In this paper they used the Budak and Pade c d approximation to design Bessel filter. In paper 3, A 1, ,0 d Susan and Jayalalitha simulate the passive c component L using GIC. This simulated inductor is To ensure the methods mentioned above applied for the realization of Bessel filter. The circuit correct they validate them by MATLAB program and simulation is done using PSPICE. After that this
www.ijera.com 257 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259 simulated L use in low-pass, high-pass and all-pass Paper Using the Linear phase filter and design Bessel filter and get a maximally flat 4 bilinear along with response being equivalent to a time delay and linear transformat good phase response. This simulated L can be used for the ion obtain magnitude design low frequency analog circuits which make use the general flatness is of the high value of unrealized L. In paper 4, coefficients obtained and designing of Bessel filter is given by Chengyun et function of method of al. In this paper a method for the design of digital digital determine Bessel filter is presented. It is based on MATLAB Bessel the general function besselap and zp2sos to obtain the Bessel filters from coefficients polynomials. Using the bilinear transformation they H(z) and function of obtain the general coefficients function of digital verified by digital Bessel filters from H(z) and verified by MATLAB. MATLAB. Bessel filter Linear phase along with good magnitude flatness is can be obtained and method of determine the general deduced. coefficients function of digital Bessel filter can be Table 6.2:- Comparison Table of Bessel Filter deduced. VII. Conclusion 6.3 Comparison Table:- Filter can be designed in many ways. In this Paper Approach Strength Weakness paper we discussed the Butterworth filter, Chebyshev Paper Using the At w=o time Adding a filter, Notch filter, Comb filter and Bessel filter and 1 continued delay is parameter, gave the designing of filters on the basis of some fraction to independent in a Bessel standard paper which is based on filter design. In obtain the of order of filter Butterworth filter, we discussed the four papers transfer Bessel filter transfer which is based upon the Sallen-key topology ,double function of and the function, a precision method and take a nth order pole from Bessel modified > -1 to the (n+2)th order filter to improve the filter methods. In filter. Bessel filter only for Chebyshev filter, we discussed the three papers transfer odd which is based upon shifting the frequencies nearest function is a integers in to origin and not change in the cut-off frequency, use realizable. the magnitude response of low pass filter and convert expansion. into Chebyshev filter magnitude response and Paper Combine Time delay Step Convert 1D Chebyshev polynomial into 2D 2 the of the all- response polynomial. In Notch filter, we discussed the four constant pass filter is has a papers which is based upon simple algebraic method, magnitude flat for nonzero largen method, MATLAB and pole-position method. response of approximate initial In Comb filter, we discussed the three papers which the all-pass ly twice the value. This is based upon delta modulation, matrix multiplication filter with frequency could be and polyphase technique. In Bessel filter, we the linear range as that undesirable discussed four papers which is based upon continue phase of the Bessel for some fraction, combine the all pass filter, GIC and response of filter of application MATLAB methods. After discussing the methods of the Bessel same order. s. each filter we discussed the strength and weakness of filter each method in comparison table and give the review Paper Obtained Due to the of filters. 3 the simulated simulated inductor the References:- inductor use of [1] Proakis, John G. and Manolakis, Dimitris circuit passive G. Digital Signal Processing. s.l. : Dorling from the component Kindersley (India) Pvt. Ltd, 2007. GIC which such as [2] A new method for effcient design of is use for inductor in Butterworth filter based on symbolic realization analog calculus. Guo-Ping, Huo, Ling-jaun, Miao of Bessel circuit is and Hui, Ding. s.l. : IEEE, 2010. filter. very easy at [3] Design and Analysis of Improved low Butterworth Low Pass Filter. Zhongshen, frequencies. Li. s.l. : IEEE, 2007.
www.ijera.com 258 | P a g e
Anirudh Singh Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 1( Version 3), January 2014, pp.236-259
[4] A design of butterworth low pass filter's layout basideal filter approximation on the ideal filter approximation. Idros, Bin Md, et al. s.l. : IEEE, 2009. [5] Direct form expansion of the transfer function for a digital Butterworth low-pass filter. Gilbert, N. and Fleming, J. s.l. : IEEE, 1982. [6] Modified Chebyshev filter design. Baez- Lopez, D and Jimenez-Fernanadez, V. s.l. : IEEE, 2000. [7] Generalized Chebyshev-like Approximation for Low-pass Filter. Swady, L. Hisham. s.l. : IEEE, 2010. [8] Design of two dimensional linear phase Chebyshev FIR filters. Bhooshan, S. and Kumar, V. s.l. : IEEE, 2008. [9] Design of two-dimensional notch filter using bandpass filter and fractional delay filter. Tseng, Chien-Cheng and Lee, Su-Ling. s.l. : IEEE, 2013. [10] Two dimensional IIR and FIR digital notch filter design. Pei, Soo-Chang and Tseng, Chien-Cheng. s.l. : IEEE, 1993. [11] New Method of Designing Digital Notch Filter of Mains Frequency. Shen, Guanghao, et al. s.l. : IEEE, 2009. [12] Symmetric IIR notch filter design using pole position displacement. Srisangngam, P., et al. s.l. : IEEE, 2009. [13] Simulation of second order comb filter based on delta modulation. Makarov, Sergey B. s.l. : IEEE. [14] New method of FIR comb filtering. Rasmussen, J.L. s.l. : IEEE, 2012. [15] Low Power Sharpened Comb Decimation Filter. Jovanovic Dolecek, G. s.l. : IEEE, 2010. [16] A Modification of the Bessel Filter. Johnson, D., Johnson, J. and Eskandar, A. s.l. : IEEE, 1975. [17] A Bessel Rational Filter. Marshak, A., Johnson, D. and Johnson, J. s.l. : IEEE, 1974. [18] A maximally flat phase and controllable magnitude approximation. Budak, A. s.l. : IEEE, 1965, Vols. CT-12. [19] Time-Domain Synthesis of Linear Networks. Su, K. L. s.l. : IEEE, 1971. [20] Bessel Filters Using Simulated Inductor. Susan, D. and Jayalalitha, S. s.l. : IEEE, 2011. [21] Design and Implementation of Real Time Crossover Based on Bessel Digital Filter. Chengyun, Zhang, Minyi, Chen and Jie, Wang. s.l. : IEEE, 2010.
www.ijera.com 259 | P a g e