NOVEMBER 13, 1996 EE 4773/6773: LECTURE NO. 37 PAGE 1 of 5

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1 NOVEMBER 3, 996 EE 4773/6773: LECTURE NO. 37 PAGE of 5 Characteristics of Commonly Used Analog Filters - Butterworth Butterworth filters are maimally flat in the passband and stopband, giving monotonicity priority over transition-band attenuation. These are all-pole filters (in the analog domain) characterized by the frequency response: H( Ω) ( Ω Ω c ) 2N ε 2 ( Ω Ω p ) 2N where N is the order, Ω c is the cutoff (3 db) frequency, Ω p is the passband edge frequency, and ( + ε 2 ) is the gain at the edge of the passband (at Ω Ω p ). Where are the poles of this filter? s k Ω c e j2π 2 e j ( 2k + )π ( 2N ) k 0,,, N H(s) H(-s) We normally design a Butterworth filter by specifying a required attenuation, δ 2, at a given frequency, Ω s : ε 2 ( Ω Ω p ) 2N 2 δ 2 A relationship between the order and the attenuation is: N 2 log[ ( δ 2 ) ] log( Ω s Ω c ) Thus, a Butterworth filter is completely specified by N, δ 2, ε, and the ratio Ω s Ω c.

2 NOVEMBER 3, 996 EE 4773/6773: LECTURE NO. 37 PAGE 2 of 5 Type I: Type II: Chebyshev Type I and II Filters Equiripple in the passband, monotonic in the stopband Monotonic behavior in the passband, equiripple in the stopband Maimize attenuation at the epense of ripple The transfer function for a Type I filter is: H( Ω) ε T N ( Ω Ωp ) where ε is a parameter controlling passband ripple, and T N () is the N -th order Chebyshev polynomial defined as: T N () cos( Ncos ), cosh ( Ncosh ), > The Chebyshev polynomials can be generated by the recursive equation: T N + () 2T N () T N (), N 2,, where T 0 () and T (). A Chebyshev filter has poles distributed about an ellipse that lies within the unit circle (see Figure 8.38 on page 637). The transfer function for a Type II filter contains poles and zeroes (located on the imaginary ais): H( Ω) ε [ T N ( Ω Ωp ) T N ( Ωs Ω) ] A Chebyshev filter is completely specified by N, δ 2, ε, and the ratio Ω s Ω p. We can derive a design equation for the order in terms of these parameters: N log δ 2 + δ 2 ( + ε ) ( εδ ) log ( Ω s Ω p ) + ( Ω s Ω p ) 2 cosh ( δ ε) cosh ( Ω s Ω p ) where δ 2 + δ 2.

3 NOVEMBER 3, 996 EE 4773/6773: LECTURE NO. 37 PAGE 3 of 5 Elliptic (Cauer) Filters Equiripple in both the stopband and passband. The transfer function is given by: H( Ω) ε 2 U N ( Ω Ω p ) where U N () is the Jacobian elliptic function of order N, and ε is a parameter related to the passband ripple. The zeros of this filter lie on the imaginary ais. These filters are the most efficient, because they distribute ripple equally in the stopband and passband. This implies we get the maimum attenuation for a given order. What about phase? A design equation for the filter order is given by: K K( Ω p Ω s )K( ( ε 2 δ 2 )) K( ε δ)k ( Ω p Ω s ) 2 where K () is the complete elliptic integral of the first kind, defined as: K () π 2 dθ sin 2 θ and δ 2 + δ 2. The passband ripple is 0log + ε 2. 0 ( ) These filters are very popular for consumer applications because of their efficiency. There are many third-party software packages that support the automated design of these filters (very hard to design by hand!).

4 NOVEMBER 3, 996 EE 4773/6773: LECTURE NO. 37 PAGE 4 of 5 Bessel Filters Bessel filters are all-pole filters with a system function given by: Hs () where B N () s is the N -th order Bessel polynomial. These polynomials are of the form: where the coefficients { a k } are given as ( 2N k)! a k N k k 0,,, N k! ( N k)! These can also be computed recursively: B N () s with B 0 () s and B () s s + as initial conditions. N B N () s a k s k k 0 B N () s ( 2N )B N () s + s 2 B N 2 () s Bessel filters have a linear phase response in the passband in the analog domain why might this be useful. However, when the filter is transformed into the discrete domain, the linear phase characteristic is lost. Finally, consider Picone filters: Hs () P N () s what would it take to convince you that such a filter was worthwhile?

5 NOVEMBER 3, 996 EE 4773/6773: LECTURE NO. 37 PAGE 5 of 5 A Comparison of Digital Filter Designs Specification: ma. ripple of 2 db in the passband 60 db atten. in the stopband passband edge freq.: ω p 0.25π stopband edge freq.: ω s 0.30π Butterworth N39 Chebyshev Type N3 Chebyshev Type II N3 Elliptic N7

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