Lecture Notes on Elliptic Filter Design

Lecture Notes on Elliptic Filter Design Sophocles J. Orfanidis Department of Electrical & Computer Engineering Rutgers University, 94 Brett Road, Piscataway, NJ 08854-8058 Tel: 732-445-5017, e-mail: [email protected] November 20, 2006 Contents 1 Introduction 2 2 Jacobian Elliptic Functions 4 3 Elliptic Rational Function and the Degree Equation 11 4 Landen Transformations 14 5 Analog Elliptic Filter Design 16 6 Design Example 17 7 Butterworth and Chebyshev Designs 19 8 Highpass, Bandpass, and Bandstop Analog Filters 22 9 Digital Filter Design 26 10 Pole and Zero Transformations 26 11 Transformation of the Frequency Specifications 30 12 MATLAB Implementation and Examples 31 13 Frequency-Shifted Realizations 34 14 High-Order Parametric Equalizer Design 40 These notes and related MATLAB functions are available from the web page: www.ece.rutgers.edu/~orfanidi/ece521 1 1. Introduction Elliptic filters [1–11], also known as Cauer or Zolotarev filters, achieve the smallest filter order for the same specifications, or, the narrowest transition width for the same filter order, as compared to other filter types. On the negative side, they have the most nonlinear phase response over their passband. The following table compares the basic filter types with regard to filter order and phase response: In these notes, we are primarily concerned with elliptic filters. But we will also discuss briefly the design of Butterworth, Chebyshev-1, and Chebyshev-2 filters and present a unified method of designing all cases. We also discuss the design of digital IIR filters using the bilinear transformation method. The typical “brick wall” specifications for an analog lowpass filter are shown in Fig. 1 for the case of a monotonically decreasing Butterworth filter, normalized to unity gain at DC. Fig. 1 Brick wall specifications for a Butterworth filter. The passband and stopband gains Gp,Gs and the corresponding attenuations in dB are defined in terms of the “ripple” parameters εp,εs as follows: 1 −Ap/20 1 −As/20 Gp = = 10 ,Gs = = 10 (1) 2 2 1 + εp 1 + εs which can be inverted to give: − A =− G = ( + ε2 ) ε = G 2 − = Ap/10 − p 20 log10 p 10 log10 1 p p p 1 10 1 ⇒ (2) 2 As =−20 log Gs = 10 log (1 + ε ) −2 A / 10 10 s εs = Gs − 1 = 10 s 10 − 1 Associated with these specifications, we define the following design parameters k, k1: Ωp εp k = ,k1 = (3) Ωs εs where k, k1 are known as the selectivity and discrimination parameters, respectively. Both are less than unity. A narrow transition width would imply that k 1, whereas a deep stopband 2 or a flat passband would imply that k1 1. Thus, for most practical desired specifications, we will have k1 k 1. The magnitude responses of the analog lowpass Butterworth, Chebyshev, and elliptic filters are given as functions of the analog frequency Ω by:† 1 Ω |H(Ω)|2 = ,w= 2 2 (4) 1 + εpFN(w) Ωp where N is the filter order and FN(w) is a function of the normalized frequency w given by: ⎧ ⎪ N ⎪w , Butterworth ⎪ ⎪ ⎨CN(w), Chebyshev, type-1 F (w)= N − (5) ⎪ −1 −1 1 ⎪ k1CN(k w ) , Chebyshev, type-2 ⎪ ⎩⎪ cd(NuK1,k1), w = cd(uK, k), Elliptic − where CN(x) is the order-N Chebyshev polynomial, that is, CN(x)= cos(N cos 1 x),andcd(x, k) denotes the Jacobian elliptic function cd with modulus k and real quarter-period K. The Chebyshev-2 definition looks a little peculiar, but it is equivalent to that given in [12]. k−1w−1 = (Ω /Ω )(Ω /Ω)= Ω /Ω ε = ε k−1 Indeed, noting that s p p s and that s p 1 , we have: 1 1 |H(Ω)|2 = = (6) + ε2 k−2/C2 (k−1w−1) + ε2/C2 (Ω /Ω) 1 p 1 N 1 s N s The normalized frequency w = 1 corresponds to the passband edge frequency Ω = Ωp, whereas the value w = Ωs/Ωp = 1/k corresponds to the stopband edge Ω = Ωs. The require- ment that the passband and stopband specifications are met at the corners Ω = Ωp or w = 1 − and Ω = Ωs or w = k 1 gives rise to the following conditions: 1 1 |H(Ω )|2 = = ⇒ F ( )= p 2 2 2 N 1 1 1 + εpFN(1) 1 + εp (7) ε 2 1 1 −1 s 1 |H(Ωs)| = = ⇒ FN(k )= = 2 2 −1 2 1 + εpFN(k ) 1 + εs εp k1 Thus, in all four cases, the function FN(w) is normalized such that FN(1)= 1 and must † satisfy the following “degree equation” that relates the three design parameters N, k, k1: F (k−1)= k−1 N 1 (degree equation) (8) In particular, we find that the degree equation takes the following forms in the Butterworth and both Chebyshev cases: −1 ln(k ) ln(εs/εp) k−N = k−1 ⇒ N = 1 = 1 − ln(k 1) ln(Ωs/Ωp) (9) −1 acosh(k ) acosh(εs/εp) C (k−1) = k−1 ⇒ N = 1 = N 1 − acosh(k 1) acosh(Ωs/Ωp) These equations may be solved for any one of the three parameters N, k, k1 in terms of the other two. Often, in practice, one specifies Ωp,Ωs and εp,εs, which fix the values of k, k1. Then, † Ω is in units of radians per second and is related to the frequency f in Hz by Ω = 2πf. For digital filter design, Ω is related to the physical digital frequency ω = 2πf/fs via the appropriate bilinear transformation, e.g., for a lowpass design, Ω = tan(ω/2). † For Chebyshev-2, it is FN(1)= 1 that provides the desired relationship among N, k, k1. 3 Eqs. (9) may be solved for N, which must be rounded up to the next integer value. Since N is slightly increased, Eqs. (9) may be used to recompute either k in terms of N, k1, or alternatively, k1 in terms of N, k. Because k is an increasing function of N,andk1, a decreasing one, it follows that in either case, the final design will have slightly improved specifications, either by making the transition width narrower, or by increasing the stopband or decreasing the passband attenuations. Fig. 2 shows an example designed with Butterworth, Chebyshev types-1&2, and elliptic filters. Fig. 3 shows the corresponding phase responses (their piece-wise nature arises because they are always wrapped modulo 2π to lie within [−π, π].) The specifications were as follows: f = ,G= . ,A=− G = . p 4 p 0 95 p 20 log10 p 0 4455 dB (10) f = . ,G= . ,A=− G = . s 4 5 s 0 05 s 20 log10 s 26 0206 dB where the radian frequencies were computed as Ωp = 2πfp, Ωs = 2πfs. The design parameters k, k1 were computed to be: √ Ω f ε Ap/10 − 0.04455 − k = p = p = . ,k= p = √10 1 = √10 1 = . 0 8889 1 A / . 0 0165 (11) Ωs fs εs 10 s 10 − 1 102 60206 − 1 We note that the elliptic design has the smallest filter order N, and the Butterworth, the largest. The difference between the Chebyshev designs is that type-1 is equiripple in the passband, whereas type-2 is equiripple in the stopband. It follows from Eq. (5) that the replacement 1 CN(w)−→ −1 −1 k1CN(k w ) causes the type-1 case to be replaced by the type-2 case, and the equal ripples in the passband to become equal ripples in the stopband. In the elliptic case, we want a filter that is equiripple in both the passband and the stopband, as shown in Fig. 2. This will be accomplished if we can find a filter function FN(w) that is equiripple in the passband and satisfies the identity: 1 FN(w)= (12) −1 −1 k1FN(k w ) which is equivalent to εpFN(Ω/Ωp)= εs/FN(Ωs/Ω), so that in this case the magnitude response can be written as follows and will have ripples in both the passband and stopband: 1 1 |H(Ω)|2 = = 2 2 2 2 (13) 1 + εpFN(Ω/Ωp) 1 + εs /FN(Ωs/Ω) We note that the Butterworth filter also satisfies Eq. (12), because of the degree equation N k1 = k , but in this case FN(w) is monotonic in both the passband and the stopband. 2. Jacobian Elliptic Functions Jacobian elliptic functions are a fascinating subject with many applications [13–20]. Here, we give some definitions and discuss some of the properties that are relevant in filter design [8]. The elliptic function w = sn(z, k) is defined indirectly through the elliptic integral: φ dθ w dt z = = ,w= sin φ (14) 0 1 − k2 sin2 θ 0 (1 − t2)(1 − k2t2) 4 Butterworth, N = 35 Chebyshev− 1, N = 10 1 1 0.9 0.9 0.8 0.8 0.7 0.7 )| 0.6 )| 0.6 f f ( ( H 0.5 H 0.5 | | 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 f f Chebyshev− 2, N = 10 Elliptic, N = 5 1 1 0.9 0.9 0.8 0.8 0.7 0.7 )| 0.6 )| 0.6 f f ( ( H 0.5 H 0.5 | | 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 f f Fig.

Load more