Analog Lowpass Filter Analog Lowpass Filter Specifications Specifications • Typical magnitude response H a ( j Ω ) of an •In the passband, defined by 0 ≤ Ω ≤ Ω p , we analog lowpass filter may be given as require indicated below 1− δ p ≤ Ha ( jΩ) ≤1+ δ p , Ω ≤ Ω p i.e., H a ( j Ω ) approximates unity within an error of ± δ p •In the stopband, defined by Ω s ≤ Ω ≤ ∞ , we require Ha ( jΩ) ≤ δs , Ωs ≤ Ω ≤ ∞ i.e., H a ( j Ω ) approximates zero within an 1 2 error of δs Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Analog Lowpass Filter Analog Lowpass Filter Specifications Specifications • Ω p - passband edge frequency • Magnitude specifications may alternately be •-Ωs stopband edge frequency given in a normalized form as indicated •-δ p peak ripple value in the passband below •-δs peak ripple value in the stopband • Peak passband ripple α p = −20log10 (1− δ p ) dB • Minimum stopband attenuation αs = −20log10(δs ) dB 3 4 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Analog Lowpass Filter Analog Lowpass Filter Design Specifications • Two additional parameters are defined - • Here, the maximum value of the magnitude in the passband assumed to be unity Ω (1) Transition ratio k = p Ωs •-1/ 1+ ε 2 Maximum passband deviation, given by the minimum value of the For a lowpass filter k <1 magnitude in the passband ε (2) Discrimination parameter k = 1 A2 −1 •-1 Maximum stopband magnitude Usually k <<1 A 1 5 6 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 1 Butterworth Approximation Butterworth Approximation • The magnitude-square response of an N-th order analog lowpass Butterworth filter 2 is given by • Gain in dB is G(Ω) =10log10 Ha ( jΩ) 2 1 Ha ( jΩ) = 2N 1+ (Ω/ Ωc ) • As G ( 0 ) = 0 and 2 • First 2 N − 1 derivatives of H a ( j Ω ) at Ω = 0 G(Ωc ) =10log10 (0.5) = −3.0103 ≅ −3dB are equal to zero Ωc is called the 3-dB cutoff frequency • The Butterworth lowpass filter thus is said to have a maximally-flat magnitude at Ω = 0 7 8 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Butterworth Approximation Butterworth Approximation • Typical magnitude responses with Ωc =1 • Two parameters completely characterizing a Butterworth Filter Butterworth lowpass filter are Ω c and N N = 2 • These are determined from the specified 1 N = 4 0.8 N = 10 bandedgesΩ p and Ω s , and minimum 2 nitude 0.6 g passband magnitude 1 / 1 + ε , and Ma 0.4 maximum stopband ripple 1/ A 0.2 0 0 1 2 3 Ω 9 10 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Butterworth Approximation Butterworth Approximation • Since order N must be an integer, value • Ω and N are thus determined from c obtained is rounded up to the next highest 2 1 1 Ha ( jΩ p ) = = integer 1+ (Ω / Ω )2N 1+ ε 2 p c • This value of N is used next to determine Ω 2 1 1 c H ( jΩ ) = = by satisfying either the stopband edge or the a s 1+ (Ω / Ω )2N A2 s c passband edge specification exactly • Solving the above we get • If the stopband edge specification is log [(A2 −1)/ε 2 ] log (1/ k ) satisfied, then the passband edge N = 1⋅ 10 = 10 1 specification is exceeded providing a safety 2 log (Ω / Ω ) log (1/ k) 10 s p 10 margin 11 12 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 2 Butterworth Approximation Butterworth Approximation •Example- Determine the lowest order of a • Transfer function of an analog Butterworth Butterworth lowpass filter with a 1-dB cutoff lowpass filter is given by frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz ΩN ΩN H (s) = C = c = c •Now a D (s) N N −1 l N ⎛ 1 ⎞ N s + ∑ =0 d s ∏ =1(s − p ) 10log ⎜ ⎟ = −1 l l l l 10⎝1+ ε 2 ⎠ where which yields ε 2 = 0.25895 j[π (N +2l−1) / 2N ] p = Ωce , 1≤ l ≤ N l and ⎛ 1 ⎞ • Denominator D ( s ) is known as the 10log ⎜ ⎟ = −40 N 10 A2 Butterworth polynomial of order N ⎝ ⎠ which yields A2 =10,000 13 14 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Butterworth Approximation Chebyshev Approximation 2 • The magnitude-square response of an N-th • Therefore 1 = A −1 =196.51334 k ε order analog lowpass Type 1 Chebyshev filter 1 is given by 1 Ωs and = = 5 2 1 k H (s) = Ω p a 2 2 1+ ε TN (Ω / Ω p ) • Hence where T N ( Ω ) is the Chebyshev polynomial log (1/ k ) of order N: N = 10 1 = 3.2811 log10 (1/ k) ⎧ cos(N cos−1 Ω), Ω ≤1 TN (Ω) = ⎨ −1 • We choose N = 4 ⎩cosh(N cosh Ω), Ω >1 15 16 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Chebyshev Approximation Chebyshev Approximation • If at Ω = Ω s the magnitude is equal to 1/A, • Typical magnitude response plots of the then analog lowpass Type 1 Chebyshev filter are 2 1 1 shown below Ha ( jΩs ) = 2 2 = 2 Type 1 Chebyshev Filter 1+ ε TN (Ωs / Ω p ) A N = 2 1 N = 3 • Solving the above we get 0.8 N = 8 −1 2 −1 cosh ( A −1/ε) cosh (1/ k1) nitude 0.6 g N = −1 = −1 Ma 0.4 cosh (Ωs / Ω p ) cosh (1/ k) 0.2 •Order N is chosen as the nearest integer 0 0 1 2 3 greater than or equal to the above value Ω 17 18 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 3 Chebyshev Approximation Chebyshev Approximation • The magnitude-square response of an N-th • Typical magnitude response plots of the order analog lowpass Type 2 Chebyshev analog lowpass Type 2 Chebyshev filter are (also called inverse Chebyshev) filter is shown below given by Type 2 Chebyshev Filter 2 1 N = 3 H ( jΩ) = 1 N = 5 a 2 0.8 N = 7 2 ⎡TN (Ωs / Ω p )⎤ nitude 0.6 1+ ε ⎢ ⎥ g ⎣ TN (Ωs / Ω) ⎦ Ma 0.4 0.2 where T N ( Ω ) is the Chebyshev polynomial 0 0 1 2 3 of order N Ω 19 20 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Chebyshev Approximation Elliptic Approximation • The order N of the Type 2 Chebyshev filter • The square-magnitude response of an is determined from given ε , , and A elliptic lowpass filter is given by Ωs using 2 1 −1 2 −1 cosh ( A −1/ε) cosh (1/ k ) Ha ( jΩ) = 2 2 1 1+ ε R (Ω/ Ω ) N = −1 = −1 N p cosh (Ωs / Ω p ) cosh (1/ k) where R N ( Ω ) is a rational function of order •Example- Determine the lowest order of a N satisfying R ( 1 / Ω ) = 1 / R ( Ω ) , with the Chebyshev lowpass filter with a 1-dB cutoff N N frequency at 1 kHz and a minimum attenuation of roots of its numerator lying in the interval 40 dB at 5 kHz - 0 < Ω <1 and the roots of its denominator −1 lying in the interval 1< Ω < ∞ cosh (1/ k1) N = −1 = 2.6059 21 cosh (1/ k) 22 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Elliptic Approximation Elliptic Approximation • For given Ω p , Ω s , ε , and A, the filter order • Example - Determine the lowest order of a elliptic can be estimated using lowpass filter with a 1-dB cutoff frequency at 1 2 log (4/ k ) kHz and a minimum attenuation of 40 dB at 5 kHz N 10 1 ≅ 1/ k =196.5134 log10 (1/ ρ) Note: k = 0.2 and 1 where k'= 1− k 2 • Substituting these values we get k'= 0.979796, ρ0 = 0.00255135, ρ = 1− k' 0 2(1+ k') ρ = 0.0025513525 5 9 13 ρ = ρ0 + 2(ρ0 ) +15(ρ0 ) +150(ρ0 ) • and hence N = 2.23308 • Choose N = 3 23 24 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 4 Analog Lowpass Filter Design Elliptic Approximation •Example- Design an elliptic lowpass filter of lowest order with a 1-dB cutoff • Typical magnitude response plots with Ω p = 1 frequency at 1 kHz and a minimum are shown below attenuation of 40 dB at 5 kHz Elliptic Filter • Code fragments used N = 3 1 N = 4 [N, Wn] = ellipord(Wp, Ws, Rp, Rs, ‘s’); 0.8 [b, a] = ellip(N, Rp, Rs, Wn, ‘s’); nitude 0.6 g Ma 0.4 with Wp = 2*pi*1000; 0.2 Ws = 2*pi*5000; 0 0 1 2 3 Rp = 1; Ω 25 26 Rs = 40; Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Design of Analog Highpass, Analog Lowpass Filter Design Bandpass and Bandstop Filters • Steps involved in the design process: • Gain plot Step 1 - Develop of specifications of a Lowpass Elliptic Filter prototype analog lowpass filter H L P ( s ) 0 from specifications of desired analog filter -20 HD (s) using a frequency transformation Step 2 - Design the prototype analog Gain, dB -40 lowpass filter -60 Step 3 - Determine the transfer function H (s) 0 2000 4000 6000 D Frequency, Hz of desired analog filter by applying the inverse frequency transformation to H (s) 27 28 LP Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Design of Analog Highpass, Analog Highpass Filter Design Bandpass and Bandstop Filters •Let s denote the Laplace transform variable • Spectral Transformation: of prototype analog lowpass filter H (s) Ω Ωˆ LP s = p p and sˆ denote the Laplace transform sˆ variable of desired analog filter H (sˆ) D where Ω is the passband edge frequency of • The mapping from s-domain to -domain is p sˆ H (s) and Ωˆ is the passband edge given by the invertible transformation LP p frequency of H (sˆ) s = F(sˆ) HP • On the imaginary axis the transformation is •Then H (sˆ) = H (s) D LP s=F (sˆ) ˆ Ω pΩ p H (s) = H (sˆ) −1 Ω = − LP D sˆ=F (s) Ωˆ 29 30 Copyright © 2005, S.