Signal Processing and Linear Systems-B.P.Lathi copy

O ur e arlier tmaximally flat amplitude response

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Unformatted text preview: us types of filters. b andstop filters. I n t his c hapter we s hall discuss t he design procedures for t hese four types of filters. Fortunately, t he h ighpass, bandpass, a nd b andstop filters can be o btained from a basic lowpass filter by simple frequency transformations. For example, replacing s w ith wei s i n t he low pass filter t ransfer f unction results in a highpass filter. Similarly, o ther frequency transformations yield t he b andpass a nd b andstop filters. Hence, i t is n ecessary t o develop a design procedure only for a basic lowpass filter. Using a ppropriate t ransformations, we c an t hen d esign o ther t ypes of filters. We shall consider here two well known families of filters: t he B utterworth a nd t he C hebyshev filters. 7.5 Butterworth Filters T he a mplitude r esponse IHUw)1 o f a n n th o rder B utterworth lowpass filter is given by (7.31) Observe t hat a t w = 0, t he g ain IHUO)I is u nity a nd a t w = W e, t he g ain IH(jwc)1 = 1 /\1'2 o r - 3 dB. T he g ain drops by a factor \1'2 a t w = W e. B ecause t he power is p roportional t o t he a mplitude s quared, t he power r atio ( output power t o i nput 506 7 Frequency Response a nd Analog Filters 7.5 507 B utterworth F ilters t I:H(joo)1 neven n odd 0.707 . -1 " o F ig. 7.20 (b) (a) Amplitude response of a normalized lowpass Butterworth filter. power) drops by a factor 2 a t W = W e' For this reason W e is called t he h alf-power f requency or t he 3 d B-cutoff f requency ( amplitude ratio of v'2 is 3 d B). Fig. 7.21 Poles of a normalized even-order lowpass Butterworth Filter transfer function and its conjugate. Substituting s = jw in this equation, we o btain Normalized Filter In t he d esign procedure it proves most convenient to consider a normalized filter h (s), w hose half-power frequency is 1 r ad/s (we = 1). For such a filter, t he a mplitude characteristic in Eq. (7.31) reduces t o 1 h (s ) h( - s) = 1 + ( s/j)2n T he poles of h (s ) h( - s) a re given by s2n = _ (j)2n (7.32) In this result we use t he fact t hat - 1 = We c an prepare a table of normalized transfer functions h (s) which yield t he frequency response i n Eq. (7.32) for various values of n . Once t he normalized transfer function is o btained, we can obtain t he desired transfer function H (s) for any value of W e by simple frequency scaling, where we replace s by s /w e in t he normalized h (s ). T he a mplitude response Ih(jw)1 of t he normalized lowpass B utterworth filters is d epicted in Fig. 7.20 for various values of n . From Fig. 7.20 we observe t he following: 1. T he B utterworth a mplitude response decreases monotonically. Moreover, t he first 2 n - 1 derivatives of t he a mplitude response are zero a t W = O. For this reason t his c haracteristic is called m aximally f lat a t W = O. Observe t hat a c onstant c haracteristic (ideal) is maximally flat for all W < 1. In t he B utterworth filter we t ry t o retain this property a t least a t t he origin. t 2. T he filter g ain is 1 (0 dB) a t W = 0 a nd 0.707 ( -3 dB) a t w = 1 for all...
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