Analog Lowpass Filter Specifications

Transcription

1 Analog Lowpass Filter Specifications Typical magnitude response analog lowpass filter may be given as indicated below H a ( j of an Copyright 005, S. K. Mitra

2 Analog Lowpass Filter Specifications In the passband, defined by require p a 0 " " p % # " H ( j " $ #, ", we i.e., H ( j a approximates unity within an error of ( # p In the stopband, defined by s " ' &, we require H ( j " #, " ' & a i.e., H ( j a error of # s s s p approximates zero within an p Copyright 005, S. K. Mitra

3 Analog Lowpass Filter Specifications 3 p - passband edge frequency - stopband edge frequency s # p - peak ripple value in the passband # s - peak ripple value in the stopband Peak passband ripple * p % 0log0( % # p db Minimum stopband attenuation * s % 0log0( # s db Copyright 005, S. K. Mitra

4 Analog Lowpass Filter Specifications Magnitude specifications may alternately be given in a normalized form as indicated below 4 Copyright 005, S. K. Mitra

5 Analog Lowpass Filter Specifications Here, the maximum value of the magnitude in the passband assumed to be unity / $ + - Maximum passband deviation, given by the minimum value of the magnitude in the passband 5 - Maximum stopband magnitude A Copyright 005, S. K. Mitra

6 Analog Lowpass Filter Design Two additional parameters are defined - ( Transition ratio k p s For a lowpass filter k ' ( Discrimination parameter Usually k '' k A + % 6 Copyright 005, S. K. Mitra

7 Butterworth Approximation 7 The magnitude-square response of an -th order analog lowpass Butterworth filter is given by H a j ( $ ( / c First % derivatives of ( j at are equal to zero H a The Butterworth lowpass filter thus is said to have a maximally-flat magnitude at 0 0 Copyright 005, S. K. Mitra

8 Butterworth Approximation Gain in db is G ( 0log H ( j 0 a As c G( 0 0 and G( 0log (0.5 % 3.003, % 3 0 db c is called the 3-dB cutoff frequency 8 Copyright 005, S. K. Mitra

9 Butterworth Approximation Typical magnitude responses with c Magnitude Butterworth Filter = = 4 = Copyright 005, S. K. Mitra

10 Butterworth Approximation Two parameters completely characterizing a Butterworth lowpass filter are and These are determined from the specified bandedges and, and minimum p s passband magnitude / $ +, and maximum stopband ripple / A c 0 Copyright 005, S. K. Mitra

11 Butterworth Approximation c and are thus determined from H a ( j p $ ( / p c ( / A s c H ( j a s $ $ + Solving the above we get - log 0 log [( A 0 ( % / s / + p ] log log 0 0 (/ k (/ k Copyright 005, S. K. Mitra

12 Butterworth Approximation Since order must be an integer, value obtained is rounded up to the next highest integer This value of is used next to determine by satisfying either the stopband edge or the passband edge specification exactly If the stopband edge specification is satisfied, then the passband edge specification is exceeded providing a safety margin c Copyright 005, S. K. Mitra

13 Butterworth Approximation Transfer function of an analog Butterworth lowpass filter is given by H a ( s where C D ( s Denominator s $ / c % d s. ( s % p 0 is known as the Butterworth polynomial of order j[0 ( $ % / ] p e, " " c D (s c 3 Copyright 005, S. K. Mitra

14 Chebyshev Approximation The magnitude-square response of an -th order analog lowpass Type Chebyshev filter is given by H a ( j $ + T ( / p 6 where of order : T T ( is the Chebyshev polynomial % 3 cos( cos, % cosh( cosh, ( " 4 Copyright 005, S. K. Mitra

15 Chebyshev Approximation Typical magnitude response plots of the analog lowpass Type Chebyshev filter are shown below Type Chebyshev Filter 7 Magnitude = = 3 = Copyright 005, S. K. Mitra

16 8 Chebyshev Approximation If at then H a s ( j s Solving the above we get cosh the magnitude is equal to /A, $ + T ( / A s p % cosh ( % A ( s % / / Order is chosen as the nearest integer greater than or equal to the above value p + cosh cosh % % (/ k (/ k Copyright 005, S. K. Mitra

17 9 Chebyshev Approximation The magnitude-square response of an -th order analog lowpass Type Chebyshev (also called inverse Chebyshev filter is given by H where a ( of order j T ( $ + : T 8 9 T ( s ( s / is the Chebyshev polynomial / p Copyright 005, S. K. Mitra

18 Chebyshev Approximation Typical magnitude response plots of the analog lowpass Type Chebyshev filter are shown below Type Chebyshev Filter 0 Magnitude = 3 = 5 = Copyright 005, S. K. Mitra

19 Chebyshev Approximation The order of the Type Chebyshev filter is determined from given +, s, and A using cosh % cosh ( % A ( s % / / p + cosh cosh % % (/ k (/ k Copyright 005, S. K. Mitra

20 Elliptic Approximation The square-magnitude response of an elliptic lowpass filter is given by where H a satisfying ( j $ + R ( / R ( is a rational function of order roots of its numerator lying in the interval 0 ' ' and the roots of its denominator lying in the interval ' ' & R ( / / (, with the R p 3 Copyright 005, S. K. Mitra

21 Elliptic Approximation For given,,, and A, the filter order p s + can be estimated using where ;, k ' % k 0 % ( $ log log k' 0 0 k' (4/ (/ ; k (; 5 5( ; 9 50( ; ; ; $ $ $ 4 Copyright 005, S. K. Mitra

22 Elliptic Approximation Typical magnitude response plots with are shown below Elliptic Filter p 6 Magnitude = 3 = Copyright 005, S. K. Mitra

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